ADAMS'S NEW ARITIBIETIC- REVISED EDITION ARITHMIETIC, IN WHICH THE PRINCIPLES OF OPERATING BY NUMBERS ANALYTICALLY EXPLAINED AtD SYNTHETICALLY APPLIED. ILLJUSTRATED BY COPIOUS EXAMPLES DESIGNED FOR THE USE OF SCHOOLS AND ACADEIIIES BY DANIEL ADAMS, Mv.'I.. AUTHOR OF THE SCHOLAR'S ARITHMETIC; SCHOOL *EOGRAFHY, EKT BOSTON: PHILLIPS, SAMPSON, AND COMPANY. NEW YORK: COLLINS AND BROTHER KEENE, N. H.: J. H. SPALTER. Eltterd according to Act of Congr~ss, in the year 1848, by DANIEL ADAMS, M. D. mn the Clerk's Office of the District Court of the District of New Hampshire. PREFACE. THn' Scho ar's Arithinetic," by the author of the present work, was first published in IS01. The great favor with which it was received is an evidence tiat it was adapted to the wants of schools at the time. At a subsequent rperiod the analytic method of instruction was applied to arithmetic, with much iLtenuity and success, by our late lamented countryman, WARREN COLBURN. This was the great improvement in the modern method of teaching arithmetic. The author then yielded to the solicitations of numerous friends of ed-ucation, and prepared a work combining the analytic with the synthetic method, which was published -n 1827, with the title of " Adams' New Arithlnetic." Few works ever issued from the American press have acquired so great popularity as the "( New Arithmetic." It is almost the only work on arithmetic used in extensive sections of New England. It has been re-published in Canada, and adapted to the currency of that province. It has been translated into the language of Greece, and published in that country. It has found its way into every part of the United States. In the state of New York, for example, it is the text-book in ninetythree of the one hundred and fifty-five academies, which reported to the regents of the University in 1847. And, let it be remarked, it has secured this extensive circulation solely by its merits. Teachers, superintendents, and committees have adopted it because they have found it fitted to its purpose, not because hired agents have made unfair representations of its merits, and, of the detects of other works, seconding their arguments by liberal pecuniary offers -a course of dealing recently introduced, as unfair as it is injurious to the cause of education. The merits of the "New Arithmetic" have sustained it very successfully against such exertions. Instances are indeed known, in which it has been thrown out of schools on account of the " liberal offers" of those interested in other works, but has subsequently been readopted without any efforts from its publishers or author. The "New.Arithmetic" was the pioneer in the field which it has occupied. It is not strange, then, that teachers should find defects and deficiencies in it which they would desire to see removed, though they might not think that they would be profitedrby exchanging it for any oth.r work. The repeate&: calls of such have induced the author to undertake a revision, in which labor he would present acknowledgments to numerous friends for inportant and valuable suggestions. Mr. J. HOMER FRENCH, oif Phelps, N. Y., well known as a teacher, has been engaged with the author in this revision, and has rendered important aid. Mr. W. B. BUNNELL, alSp, principal of Yates Academy, N. Y., formerly principa of an academy in Vermont, has assisted throughout the work, having prepared many of the articles. The revision after Percentage is mostly his work. in PREFACE. The characteristics of the "New Arithmetic," which have given the work so great popularity, are too well known to require any notice here These, it is believedt, will be found in the new work in an improved form. One of the peculiar characteristics cf the new work is a more natura~ an' lrhilosophical arrangement. After the consideration of simple whole numbers, that of simple fractional nambers should evidently be introduced, since a part of a thing needs to be considered quite as frequently as a whole thing. Again; since the money unit of the federal currency is divided decimally, Federal Money certainly ought not to precede Decimal Fractions. It has been thought best to consider it in connection with decimals. Then follow Compound Numbers, both integral and fractional, the reductions preceding the other operations, as they necessarily must. Percentage is made a general subject, under which are embraced many particulars. The articles on Proportion, Alligation, and the Progressions will be found well calculated to make pupils thor oughly acquainted with these interesting but difficult subjects. Care has been taken to avoid an arbitrary arrangement, whereby the processes will be pureiy mechanical to the learner. If, for instance, all the reductions in common fractions precede the other operations, the pupil will have occasion to divide one fraction by another long before he shall have learned the method of doing it, and must proceed by a rule, to himself perfectly unintelligible. The studied aim has been throughout the entire work to enable the ordinary pupil to understand every thing as he advances. The author is yet to be convinced that mental discipline will be promoted, or any desirable end be subserved, by conducting the pupil through blind, mechanical processes. Just so far as he can understand, and no farther, is there prospect of benefit. No good results from presenting things, however excellent in themselves, if they are beyond the comprehension of the learner. Those teachers who prefer to examine their classes by questions, will find that little will escape the pupil's attention, who shall correctly answer all those In the present work, while teachers who practise the fax superior meth d of recitation by analysis, will find the work admirably adapted to their purpose. The examples, it is hoped, will require very full applications of the principles. Many antiquated things, which it has been fashionable to copy in arithmetics, from time immemorial, have been omitted or improved, while new and practical matter has been introduced. A Key to this revision is in progress. With these remarks, the work is submitted to the candid examination of the public, by THiE AUTHOR. Keene, N, H., February; 1848. SUGGESTIONS TO TEACHERS. TaEI writer complies with the request of the venerable author of " Adaxis' Arithmetic,s" to preface the new work with a few suggestions to his associates in the work of instruction. Though he has been engaged for sometime past in assisting to make the work better fitted to accmplish its design, he is perfectly satisfied that improvement in school educatioa is rather to be sought in improved use of the books which we now have, than in making better books. Better arithmeticians would be made by the book as it was before the present revision, using it as it might be used, than will probably be. made in most cases with the new wvork, even though the former were very defective, the latter perfect. Exertion, then, to bring teachers to a higher standard, will be more effective in improving school education, than any efforts at improving school books can possibly be. It is here where the great improvement must be soughL. Without the cooperation of competent teachers, the greatest excellences in an.g book -will remain unnoticed, and unimproved. Pupils will frequently complain that they have never found one that could explain some particular thing, of which a full explanation is given in the. book which. they have ever used, and their attention only needed to have been called to the explanation. Then let teachers. make themselves, in the first place, thoroughly acquainted, with arithmetic. The idea that they can'" study and keep ahead of their. classes," is an absurd one. They must have surveyed the whole field in. order to. conduct inquirers over any part, or there will be liability to ruinous misdirection. Young teachers are little aware of their deficiencies in knowledge, and still less aware of the injurious effects which these deficiencies exert upon pupils, who are often disgusted with school education, because they are made to see in it so little that is meaning. In the next place, let no previous familiarity with the subject excuse teachers from carefully preparing each lesson before meeting their classes. Thereby alone will they feel that freshness of interest, which will awaken a kindred interest among their pupils; and if on any occa aion they are compelled to omit such preparation, they will discover a declining of interest with their classes. Teachers who are obliged to have their books open, and watch the page while their classes recite, are unfit for their work. Pupils should be taught how to study. That, after all, is the great object of edu,'ating. The facilities for merely acquiring knowledge are abundant, if persons know how to improve them.' The members of classes will often fail in recitation, not because they have not tried, but have not known how to get their lesson. They neglect trying, because,hey can do so little to advantage. They may read over a statement n their book a dozen times they say, but cannot remember it, —bec ase thl d, n"at understand it. An hour spent with each pupil individr 11v 1* VI SUGGESTIONS TO TEACHERS. in questioning him on the meaning of each sentence, which ne may be required to read, will be of incalculable advantage. When pupils shall have been taught how to study, let them be re. quired to get their lessons, and rec4te them. If the present book is nol thought by teachers to contain a sufficient description, and a sufficient explanation of everything, let them try to-find one that does, for if pupils present themselves before the blackboard at the time of recitation, with the expectation that the teacher is to explain to the class, and help them through with what they cannot go through themselves, they wi3l not feel that they must have studied themselves; and the paltry oralizing of the teacher will not be listened to, or if heard, will not be understoodi or at best, not retained in memory. Pupils may be made to see things for the moment, while no abiding impression will remain on their minels. They will often proceed,. in such a manner, through a book, and, with the mistaken idea that they understand its contents, the evil of superficialism may be perpetuated by them, perhaps, as teachers. Pupils will never have a sufficient understanding of a subject' till they shall have studied it carefully themselves, and mastered each part by severe personal application. Recitation by analysis will be found more conducive to thorouga scholarship than adherence'to any written questions. Let the class, or any member of the class, be able to commence at the beginning and go through with the entire lesson without any suggestion from the teacher, — a thing that is perfectly practicable and easily attainable. Let pupils le called on, at the pleasure of the teacher, in any part of the class, to io on with the recitation, even to proceed with it in the midst of a sub-,ect, the topic in no case ever being named by the teacher. They will thereby become accustomed to give their attention to the recitation, and they will be profited from it, besides securing habits of attention, which will be of incalculable value. In fine, let arithmetic be studied properly, and more valuable mental discipline will be acquired from it, than is often attained from the whole course in mathematics usually ass gned by cdtlege faculties. It is not the extent, bit the value of acqiisition3 in mathematics, which is desireable. W. B. B. SILtPLE NUMBERS.. Nation anh N umerati)n,...... 9 Contractions in Division,..... Addition,.... 16 Review of Division,......... 63 Review of Numeration and Addition,.22 Miscellaneous Exercises,....... 65 Subtraction,..........23 Problems in the Measurement of RecReview of Subtraction,... 29 tangles and Solids,...69 Multiplication,........31, illustration by Diagram,. 71 C-ontractn, illustration by Diagram, 34 Definitions,.............73 Contractions in Multiplication,....40 General Principles of Division,.74 Review of Multiplication,......45 Cancelation,............ 75 Division....47 Common Divisor, 78 - lustration by Diagram,.. 50 Greatest Common Divisor,..... 78 COMMON FRACTIONS. Notation of Common Fractions,....80 Multiplication of whole numbers by a Pr6per, Improper, &c.,.........82 fraction, 95 Reduction of Fractions,....... 83 Multiplication of one fraction by To reduce a fraction to its lowest another,.. 96 terms,...85 General Rule,........ 97 Addition and Subtraction of Fractions,. 87 Examples in Cancelation,...... Common Denominator,........87 Division of Fractions,........ 99, st method,. 88 by a whole num2d method,.. 89 ber, two ways,....... 100 Least Common Denominator, or Least Division of whole numbers by a fraction, 101 Common Multiple,......... 90 Division of one fraction by another,. 103 New Numerators,'..........91 General Rule,.... 103 General Rule,............91 Reduction of Complex to Simple F'racMultiplication of Fractions,.... 93 tions,............. 104 by a whole Promiscuous Examples,....... 106 number, two ways,........ 94 Review of Common Fractions,.... 106 DECIMAL FRACTIONS AND FEDERAL MONEY. Decimal Fractions,......... 108 Addition and Subtraction of Decimal Notation of Decimal Fractions,.. 110 Fractions,... Table,............ 111 Addition and Subtraction of Federal To read Decimals,... 112 Money, 120 To write Decimals,.112 Multiplication of Decimal Fractions, 121 Reduction of Decimal Fractions,.. 113, illustration by Diagram, 122 ------ of Common to Decimal Frac- of Federal Money,. 123 tions.......114 Division of Decimal Fractions,....124 Federal Money............ 116 - - of Federal Money,..... 126 Reduction of Federal Money,... 118 Review of )scimal Fractions,. 127 Bills,......... 129 COMPOUND NUMBERS. Definition,..... 131 MBASURE OF BXIENSION. Reduction of Compound Numbers,. 132 Reduction of, 1. Linear Measure,. 138 - English Money,....132 Cloth Measure,.. 139 WIOGHT. II. Land, or Square L AvoirdupoiseWeight, 135 Measure,... 140.. Troy Weight,....136 I.- Cubic Measura,.. 141 m. Apthe.ary's weiht,; 137 I U1i INDEX. MRASRE F CAPACrrY. Subtraction of Frattional Compouri! Iuctlon of, I. Wine Measure,. 143 Numbers,............ 164 - - - - II. Beer Measure,... 144 Multiplication and Division of Corn-- - lI. Dry Measure,... 144 pound Numbers,......... 165... Time,.........145 Difference in longitude and time be-— t Circular easre,... 146 tween different places. Diagram,.. 171 iscelaneous Table,..... 147 Review of Compound Nuembers,.. 172 edlluction of Fractional Compound Analysis.............. 174 Nunlmers..147 Given, price of unity, the quantity, to i ) reduce a fraction of a higher de- find the price of quantity,..... 175 nomination to one of a lower,... 148 Given, quantity, price of quantity, to t ) reduce a fraction of a lower to a find the price olf unity...... 175 rni,,her denomination.. 148 Given, price of unity, price of quantity, CI reduce a fraction of a higher to in- to find the quantity......... 175.e4,crs of a lower derWmination,.. 149 Practice. Aliquot Parts,.178 to reluce integers of a lower to frac- Articles sold by 100, 18C tions of a higher denomination 149 --- by the ton of 2000 lbs.,. 1.eduction of Decimal Compound iNum- price, aliquot part of a pound, ers,.151 &c... 183,4eview of Reduction of Compound Articles, quantity less than unity,. 184 Nulmbers..... 153 To reduce shillings, pence, &c., to the.Iddition of Cotmpound Numbers,. 156 decimal of a pound,......... 185 Fractional Compound Num- - To reduce the decimal of a pound to bers,..160 shillings, pence and farthings,... 187 Subtraction of Compound Numbers,. 160 PERCENTAGE. Definiti3n, 187. Rule,........18 Discount, 21:. Commission, 216 Insurance,.............190 Timle, rate, interest, to find the princiMutual Insurance,.........191 pal.217 Stocks,.............. 193 Principal, interest, titne, to find the rate, 218 Brokerage..... 194 Principal, rate, interest; to find the Profit anti Loss...... 194 tie,... 18 Interest, 195. General Rule..... 199 Percentage to find the rate,. 219 Easy way of castillg interest when tle Bankruptcy,.....220 rate is 6 per cent.2..200 Glneral Average,..........221 To compute interest on pounds, shil — Partnlership,. 222 lings. pence., &c.........205 on Tine,.223 To colnlute interest when partial pay- Banlking,.224 ments tave been tttade,...... 205 Taxes, method of assessing,..... 225 Compound Interest.......... 209 Duties,............... 227 - Table...... 211 - Specific,........... 228 Annual Interest,. 212 -- Ad Valorem,......... 229 Time. rate, and amlount given, to find Review of Percent.ae,'.......230 the principal,..........214 Eluation of Payments,.233 Ratio,..............'235 Extraction of the Square Root,.... 262 Inverted and Direct Ratios,.235 Practical Exercises,. 266 Compound Ratio,..........236 Extraction of the Cube Root,.269 Proportion,............236 Practical Exercises,. 273 Rule of Three,........ 237 Review,... 274'o invert bothl Ratios,. 238 Arithmletical Progression,. 275 ----- one Rati,......... 239 Simple Interest y P'rogression,.277 To find the fourth terln of a proportion Annuities by A rithmetical Progression, 21 9 when three are given,. 239 Geotmetrical Proression,... 281 Cancelation App)lied,. 240 Compound lIterest by Progression,.283 Compouind Proportion,.. 242 Comipound Discount.-Table,.. 285 Rule,...244 Annuities at Complound Interest,. 288 Review of Proportion,........ 245 Present worth of Annuities at ComAlligation Medial,......; 246 pound Interest,..299 -*- Alternate,........ 247 Present worth of Annif.ties. Table,. 290 Exchange,..251 in Rever-- * with England, -......253 sion,..............291 - Frarnce,........ 254 Perpetual Annuities, 292 Value of Gold Cois,......... 255 Permutation,...293 Duodecimals,.256 Miscellaneous Examples,..294 - -, scale for taking Dimen- Measurement of Surfaces,... 298 sions in feet and Derimals of a foot, 259 - Solids,....00 Involution,......260 Guagi ng..........3801 E.volutlon,2... 62 I Fornas of Notes. &c.,....... 303 1 - Bills,.., - 304 ARITHMETIC NOTATION AND NUMERATION. ~ 1. A single thing, as a dollar, a horse, a man, &c., is called a unit, or one. One and one more are called two, two and one more are called three, and so on. Words expressing how many (as one, two, three, &c.) are called numbers. This way of expressing numbers by words would be very slow and tedious in doing business. Hence two shorter methods have been devised. Of these, one is called the Roman* method, by letters; thus, I represents one; V, five; X, ten, &c., as shown in the note at the bottom of the page. The other is called the Arabic method, by certain characters, called figures. This is that in general use. * In the Roman method, by letters, I represents one; V,.five; X, ten; L, fifty; C, one hundred; D,.five hundred; and M, one thousand. As often as any letter is repeated, so many times its value is repeated, unless it be a letter representing a less number placed before one representing a greater; then the less number is taken from the greater; thus, IV represents four; IX, nine, &c., as will be seen in the following TABLE. One I. Ninety LXXXX. or XC Two II. One hundred C. Three III. Two hundred CC. Four IIII. or IV. Three hundred CCC. Five V. Four hundred CCCC. Six VI. Five hundred D. or I3.* Seven VII. Six hundred DC. Eight VIII. Seven hundred DCC. Nine VIIII. or IX. Eight hundred D(CCC. Ten X. Nine hundred DCCCC. Twenty XX. One thousand M. or CID.t Thirty XXX. Five thousand IO3. or V.: Forty XXXX. or XL. Ten thousand CCI33. or 3. Fifty L. Fifty thousand IOD3. Sixty LX. Hundred thousand CCCI303. or C Seventy LXX. One million M. Eighty LXXX. Two million MM. * 13 is used instead of D to represent five hundred, and for every additional 3 an nexted at the right hand, the number is tnareased ten times. t CO1 is used to represent ohe thousand, and for every C and D put at each end, thi number is increased ten times. I 4 line over any number increases its value one thioisand times. 10 NOTATION AND NUIMERAT'ION. s 2, 3. In the Arabic method the first nine numbers have each a separate character to represent it; thus. ~W 2. A unit, or single h Note 1. These nine charthing, is represented by this = acters are called significant Character.... 1 fiures, because they each Two units, by this character, 2. represent some number. Three units, by this character, 3. Sometimes, also, they are Four units, by this character, 4. called digits. Five units, by this character, 5. Note 2. The value of Six units, by this character, 6. these figures, as here shown, Seven units, by this character, 7 is called their simple value. Seven units, by this character, 7. It is their value always when Eight units, by this character, 8. single. Nine units, by this character, 9. Nine is the largest numoer wnici can be expressed by a single figure. There is another character, 0; it is called a cipher, naught, or nothing, because it denotes the absence of a thing. Still it is of frequent use in expressing numbers. By these ten characters, variously combined, any number may be expressed. The unit I is but a single one, and in this sense it is called a unit of the first order. All numbers expressed by one figure are units of the first order. ~[ 3. Ten has no appropriate character to represent it, but It is considered as forming a unit of a second or higher order, consisting of tens. It is represented by the same unit figure 1 as is a single thing, but it is written at the left hand of a cipher; thus, 10, ten. The 0 fills the first place, at the right hand, which is the place of units, -and the 1 the second place from the right hand, which is the place of tens. Being put in a new place, it has a new value, which is ten times its simple value, and this is what is called a local value. Questions. -~ 1. What is a single thing called? What is a numoer? Give some examples. How many ways of expressing numoers shorter than writing them out in words? What are they called? Which is the method in general use? In the Arabic method, how many numbers have each a separate character? t~ 2. Hovow is one represented? Make the characters to nine. What are these nine characters called? Why? What is the simple value of figures? What is the largest number which can be represcnted by a single figure? What other character is frequently used?'Why is it zalled naught? How many are the Arabic charicotersI "N lat are urabeiz Expressing single things called? T 4. NOTATION AND NUMERATION. if There maj be one, two,.. or more tens, just as there D are one, two, or more units, One ten is... 10 ten. or single things; it takes Two tens are.. 20 twenty ten cents to make one ten- Three tens". 30 thirty cent piece; just so it takes Four tens ".. 40 forty. ten single things to make Five tens ".. 50 fifty. one ten. All figures in the Six tens ".. 60 sixty. second place express units Seven tens".. 70 seventy. of the 2d order, that is, Eight tens ".. SO0 eighty. units of tens. Nine tens ".. 90 ninety. One ten and one unit, 11, One ten, one unit, 11 e even. are called eleven; one ten One ten, two units, 12 twelve. and two units, 12, twelve, &c. In this w way the units Note. Twenty, thirty, &c., are &cthe In thi r wayre unitsd.contractions for two tens, three tens, of the 1st order are united with the tens, that is, with the units of the 2d order, to form the numbers from 10 to 20, from 20 to 30, to 40, and so on to 99, which is the largest number that can be represented by two figures. The weeks in a year are 5 tens and 2 units, (5 of the second order and 2 of the first order n6w described,) and are expressed thus, 52, (fifty-two.) In'the same manner express on your slate, or on the blackboard, the two orders united, so as to form'all the numbers from 10 to 99. I 4. Ten tens are called one hundred, which forms a unit of a still higher, or 3d order, and is ex- L I pressed by writing two ciphers at the: right hand of the unit 1,.. thus, 100 one hundred. Note. When there are no units or tens, we 200 two hundred write ciphers in their places, which denote the 300 three hundred, absence of a thing, (~] 2.) &c. Questions.- ~ 3. How is ten represented? What is it considered as forming? Consisting of what?.VWhat place does the cipher fill? The one? Where is unit's place, and where ten's place, counting from the right? How much is the value of a figure increased by being removed from a lower to a higher place? In which place does it retain its simple value? In ten's place, what is its value called? NWhat is 1 ten and 1 unit called? 1 ten and 2 units? How are the numbers irom 10 to 99 expressed? Of what is the number forty made up? Ans. 4 tens and io units. Sixty? What do you'nite, to form the number twenty, hrre? thirty-seven? seventy-five? &c. Of what are twenty, thirty, Vc., contractions? What is the largest, and what the least, number you Zan express by one figur e? by trwo figures? 12 NOTATION AND NUMERATION. ~ 6, 6. Three hundred sixty-five, the days in a year, are expressed thus, 365; 3.being in the place of hundreds, 6 in the place of tens, and 5 in the place of units. After the same manner, the pupil may be required to unite the three orders, and express any number from 99 to 999. T 5. Wre have seen that figures have two values, viz., simnple and local. The simple value of a figure is its value when standing zione; thus, the simple value of 7 is seven. The local value of a figure is its value according to its distance from the place of units; thus, the local value of 7, in the number 75, is 7 tens, or seventy, while its szmple value is seven; in the number 756, its local value is seven hundred. Note. From the fact that 10 is I more than 9, it follows, as may be found by trial, that the local value of every figure at the left of units, except 9, exceeds a certain number of nines by the simple value of the figure. Take the number 623; 2 (tens) is 2 more than 2 nines, and 6, (hundreds,) 6 more than a certain number of nines. On this principle is founded a method of proof in the subsequent rules, by casting out the nines. ~r 6. Ten hundred make one thousand, which is called a unit of the next higher, or 4th order, consisting of thousands, anld is expressed by writing three ciphers at the right hand of the unit 1, giving it a new local value; thus, 1000, one thousand. To thousands succeed tens and hundreds of thousands, forming units of the 5th and 6th orders. Questions. - ~ 4. What are 10 tens called? What do they form 1 [low many places are required to express hundreds? How much does l cipher, placed at the right hand of 1, increase it? 2 ciphers? How dlo you express two hundred? &c. What are 4 hundreds, 9 tens, and F units called? How is one hundred ninety-three expressed? What place does the 3 occupy? the 9? the 1? How do you express the ab. sence of an order? How is the number of days in a year expressed? ~r 5. How many values have figures? What are they? What 1%,he simple value? local value? What is the value of 5 in 59? Is it its simple, or a local value? Is the value of 8, in 874, simple or local f the 7? of the 4? 1~ 6. How do you express one thousand? seven thousand? A thousand is a unit of what order? How many thousands are 30 hundreds? What after thousands, and of what crder? The 6th order is what? In writing nine hundrei and two thousand and nine, where do yol place tiphers? Why? ~ 7, 8. NOTATION A-ND NUMERATION. 1 ~[ 7. In this table of the six orders now TABLw Jescribed, you see the unit 1 moving from right to left, and at each removal forming the unit of a higher order. There are other or- E ders yet undescribed, to form which the unit E moves onward still towards the left, its value H eing increased ten times by each removal, a Note 1. The Ordinal numbers, 1st, 2d, 3d, &c., " C may be called indices of their respective orders. Note 2. Various Readings. In the number l 546873, the left hand figure 5 expresses 5 units of 1 0 the 6th order, or it may be rendered in the next 1 0'ower order with the 4, and together they may be read 54 units of the 5th order, (ten thousands,) and 1 0 0 0 connecting with the 6, they may be read, 546 units 1 0 0 0 0 of the 4th order, or 546000. Hence, units of any I 0 0 0 0 0 higher order' may be rendered in units of any lower,,,,, order. 99 ol~der. 9 9 9 9 9 9 To hundreds of thousands succeed units, tens, and hun-. dreds of millions. ~fr S To millions succeed billions, trillions, quadrillions, iuintillions, sextillions, septillions, octillions, nonillions, decilions, undecillions, duodecillions, tredecillions, &c., to each of which, as to units, to thousands, and t6 millions, are assigned three places, viz., units, tens, hundreds, as in the following examples: Questions, - ~ 7. How is the unit 1 of the'st order made a unit of the 2d order? of the 3d order, &c., to the 6th order? What may the irdinal numbers, 1st, 2d, 3d, &c., be called? 7 units of the 6th order %re how many units of the 4th order? The teacher will multiply such 7uestions. What is the least, and what the largest, number which can be expressed by 2 places? 3 places? &c. What after hundreds of thousands? Of what order will millions be? tens of millions? hundreds of mnillions? ~ 8. What after millions? How many places are allotted to hit lions? to trillions? &c. Give the names of the orders after trilliors In reading large numbers, Nxhat is frequently done? Why? The lst period a t-che right is the period of what? the 2d? the 3d? the 4 h & c 14 NOTATION AND NUMERATION. ~ 9.,3 O O o o o o o — O o o Q>- 2 2 (1 2X2 clq 3 0 8 2 7 1 5, 2 0 30 1 7 4 5 9 2 8 3 7 4 6 3 5 1 2 To facilitate the reading of large numbers, we may point them of into periods of threefigures each, as in the 2d example. The names and the order of the periods being known, this division enables us to read numbers consisting of many figures as easily as we can read those of only three figures. Thus, in looking at the above examples, we find the first period at the left hand to contain one fi(ure only, viz., 3. By io:kincg Mder it, we see that it stands in the 9th period from unlits, which is the period of septillions; therefore we read it 3 8 2 7 15 3s, and so on, 2 sextillions, 715 quintillions, 6 3 5 03 quadr, 592 billions, 3 7 millions, 4,, 83, 463 thou sands, 5122. T 9. From the foregoing we dedue u the following pinciples': Nufnbers incase cn r toef n rease from right to left and decrease fom eft to riglit, in a ten-fold ratio; and it is A FUNDAMENTAL LAW OF THEf ARABiC NOTATION; that, Questions. a n 9. How do numbers increase? how detrease, a20 im what proportion? To what is 1 ten equal? 1 hundred? 1 thoulsand? &c. To what are 10 units equal? 10 h'limdreds? &c. What is a fundanental law of the Arabic notation? What is notation? numera tion? How do you write numbers? read nunbers? If you were tc write a lnumber containin units, ens hur Areds, and millions, but no tbousands, how would youexpress it? ~ 10. NOTA'IrlON AND NUMERATION. 16 I. Removing any figure one place towards the left, increases its value ten timnes, and Il. Removing any figure one place towards the right, decreases its value ten times. The expressing of numbers as now shown is called Notatcon. The reading of any number set down in figures is called Numeration. To write numbers.-Begin at the left hand, and write in their respective places the units of each order mentioned in the number. If any of the intermediate orders of units be omitted in the number mentioned, supply their respective places with ciphers. To read numbers.:-Point them off into periods of three figures each, beginning at the right hand; then, beginning at the left hand, read each period separately. Let the pupil write down and read the following numbers: Two million, eighty thousand, seven hundred and five. One hundred million, one hundred thousand and one. Fifty-two million, sixty thousand, seven hundred and three. One hundred thirty-two billion, twenty-seven million. Five trillion, sixty billion, twenty-seven million. Seven hundred trillion, eighty-six billion, arid nine. Twenty-six thousand, five hundred and fifty men. Two million, four hundred thousand dollars. Ninety-four billion, eighty thousand minutes. Sixty trillion, nine hundred thousand miles. Eighty-four quintillion, seven quadrillion, one hundred million grains of sand. 4~f 10. Numbers are employed to express quantity. Quantity is anything which can be measured. Thus, Time is quantity, as we can measure a portion of it by days, hours, &c. Distance is quantity, as it can be measured by miles, rods, &c. By the aid of numbers quantities may either be added together, or one q iantity may be taken from another. Arithmetic is the art Qf making calculations upon quantities by means of numbers. Questions. -~ 10. Numbers are employed to express what? What is quantity? By what is a quantity of grain measured? a quantity of cloth? What is arithmetic? Wa t is an abstract number? a denominate number? What is the unit of a number! What is the unit value of 8 bushels? of 16 yards? of 20 pounds of sugar? of 3 quarts of milk? of 9 dozen of buttons? of 18 tons of hay? f 16 hogs. eads of molasves? 16 ADDITION OF SIMPLE NUMBERS. 1B 1 A number applied to no kind of thing, as 5, 10, 18, 36, is nalled an abstract number. A number applied to some kind of thing, as 7 horses, 25 Aollars, 250 men, is called a denominate number. The unit, or unit value of a number, is one of the kind which the number expresses; thus, the unit of 99 days is 1 day; the unit of 7 dollars is I dollar; the unit of 15 acres is 1 acre. In like manner the unit of 9 tens may be said to be 1 ten; the unit of 8 hundred to be 1 hundred; the unit of 6 thousand to be 1 thousand, &c. ADDITION OF SIMPLE NUMBERS. ~ 11. lo1. James had 5 peaches, his mother gave him 3 more; how many had he then? Ans. 8. Why? Ans. Because 5 and 3 are 8. 2. Henry, in one week, got 17 merit marks for perfect lessons, and 6 for good behavior; how many merit marks did he get? Ans --—. Why? 3. Peter bought a wagon for 36 cents, and sold it so as to gain 9 ceai's; how muany cents did he get for it? 4. Frank gave 15 walnuts to one boy, 8 to another, and had 7 left; how many walnuts had he at first? 5. A man bought a cha;se for 54 dollars; he expended 8 dollars in repairs, and then sold it so as to gain 5 dollars; how many dollars did he get for the chaise? The putting together of two or more numbers, (as in the foregoing examples,) so as to make one whole number, is called Addition, and the whole number is called the Sum, or Amount. 6. One man owes me 5 dollars, another owes me 6 dollars, another 8 dollars, another 14 dollars, and another 3 dollars what is the sum due to me? 7 What is the amount of 4, 3, 7, 2, 8, and 9 dollars? 8. In. a certain school, 9 study grammar, 15 study arith mretic, 20 attend to writing, and 12 study geography; what is the whole numnber bf scholars? Questions. - ~ 11. What is addition? What is the answer, oz number sought, called? What is the sign of addition? What does it show? How is it sometimes read? Whence the word pilus, and what's its signification? What is the sign of equality, and( what does it shod. P ~ 11. ADDITION OF SIMPLE NUMBERS. 17 SIGNS. — A cross, +-, one line horizontal and the other rers pendicular, is the sign of Adcdition. It shows that numbers writh this sign between them are to be added together; thus, 4 + 7 + 14 + 16 denote that 4, 7, 14, and 16 are to be added together. It is sometimes read plus, which is a Latin word signifying more. Two parallel, horizontal lines, =, are the sign of Equality. It signifies that the number before it is equal to the number after it; thus, 5 + 3 = —8 is read 5 and 3 are 8; or, 5 plus 3 are equal to 8. In this manner let the pupil be instructed to commit the following ADDITION TABLE. 2+0= 2 3+0= 3 4+0= 4 5+0= 2+ — 1 3 - 1= 4 4 1 5 5+ 6 2+2= 4 3 -2= 5 54~2 6 5 2t 7 2+3= 5 3+3= 6 4+= 7 5+3 8 2 4 —4= 6 3+4- 7 4+4= 8 4= 9'2+5- 7 3-+-5= 8 4-+5= 9 5 --- 10 2+6 - 8 3+-6= 9 4+6 10 5+ 611 2 -7 9 33+7=10 4- 7=11 5 +7=12 2 — S 10 3 -+8=11 4_ 8 12- 5+8= 13 2 + — 9 11 3 + 9 12 4 9 13 5 -+ 9 14 6+0= 6 7+0 7+0 - - 8+0= 8 9-4-0= 9 6+1= 7 7+1= 8 8~1= 9 9~1=10 6 + 2 8 7+2= 9 8 2 =10 9 2= 11 6 3= 9 7 3=- 3 10 8- 3-11 9 -- 12 6 +4- 10 7- 4 11 8 4 12 9 4 4=13 6 + 5 11 7 +-5 12 8 + — 5 13 9 +5= 14 6 ~- 12 7+6=13 8+6=14 9+6=15 6- 7=13 7+7.=14 8+7=15 9+7=16 6- -814 7+8=15 8+8=16 9 8 17 6 +9=15 7 9=16 8+9=17 9 9 18 5 + 9 how many? 8 + 7 how many? 4 + 3 + 2 how many? 6 + 4 + 5 - how many? 2 - O + 4 + 6 how many? 7 1 + O + 8 how many? 3- - O + 9+ 6 how, many 18 ADDITION OF SIMPLE NUMBERSo ~ 12 9 + 2 + 64-4+ 5-howv many? 1 +3 + ++ 7 8+ how many? 1 +2+3+ 4 + 5 + 6 how many? 8+ — q 0 — 2 +4+ 95 —- how many? 6 +! + 5 + 0 +s + 3. how many? 11 12. When the numbers to be added are small, the ad dition is readily performed in the mind, and this is called:nenlaJl arithmetic; but it will frequently be more convenient and even necessary, when the numbers are large, to write theim down before adding them, and this is called written arithmetic. 1. Harry had 43 cents, his father gave him 25 cents more' how many cents had he then? SOLUTION. —One of these numbers contains 4 tens and 3 units. The other number contains 2 tens and 5 units. To unite these two nuIlmbers together into one, write them down one un43 cents. der the other, placing the unilts of one number directly 25 cents. under units of the other, and the tens of one number directly under tens of the other, and draw a line underneath. 43 cents. 25 ce7nts. Beginning at the column of units, we add each column separately; thus, 5 units and 3 units are 8 units, which we set down in units' place. 43 cents. We then proceed to the column of tens, and say, 25 cents. 2 tens and 4 tens are 6 tens, or 60, which we set - down directly under the column in tens' place, and Ans. 68 cents. the work is done. It now appears that Harry's whole number of cents is 6 tells and 8 units, or 68 cents; that is, 43 t 25 = 6S. Units are written under units, tens under tens, &c.; because none but figures of the same unit value can be added to each other; for 5 units and 3 tens will make neither S tens nor S units, just as 5 cows and 3 sheep will make neither 8 cows nor 8 sheep. Questions. - T 12. What distinction do you-make between mental and writtezn arithmetic? How do you write numbers for addition I W'here do you begin to add? and where do you set the amount? How do ycu proceed Why do you write units under units, tens under tens, &e1.. ~ 13. ADDITION OF SIMPILE NUMBERS. 19 2. A farmer bought a chaise for 210 dollars, a horse for 70 dollars, and a sadde for 9 dollars; what was the whole amount? Write the numbers as before directpd, with units under units, tens under tens, &c. OPERATION,. Chaise, 210 dollars. Horse, 70 dollars. Add as before. The units will be 9, the Saddle, 7 dollars. tens 8, and the hundreds 2; that is, 210 + Saddle, 9 dollars. 70+ 9_ 89. 70 -- 09 = 289. Answer, 289 dollars. After the same manner are performed the following examples, in which the amount of no column exceeds nine. 3. A man had 15 sheep in one pasture, 20 in another pasture, and 143 in another; how many sheep had he in the three pastures? 15 + 20 + 143 - how many? 4. A man has three farms, one containing 500 acres, another 213 acres, and another 76 acres; how many acres in the three farms? 500 + 213- +76 = how many? 5. Bought a farm for 2316 dollars, and afterwards sold it so as to gain 550 dollars; what did I sell the farm for? 2316 + 550 = how many? 6. A chair-maker sold, in one week, 30 Windsor chairs, 36 cottage, 102 fancy, and 21 Grecian chairs; how many chairs did he sell? 30 + 36 + 102 - 21 — how many? 7. A farmer, after selling 500 bushels of wheat to a commission merchant, 320 to a miller, and sowing 117 bushels, found he had 62 bushels left; how many bushels had he at first? 500 +- 320 +- 117 + 62 = how nmany? 8. A dairyman carried to market at one time 231 pounds of butter, at another time 124, at another 302, at another 20, and at another 12; how many pounds did he carry in all? Ans. 689 pounds. 9. A box contains 115 arithmetics, 240 grammars, 311 geographies, 200 reading books, and 133 spelling books; how many books are there in the box? Ans. 999. ~ 13. Hitherto the amount of any one column, when added up, has not exceeded 9, al d Consequently has been expressed by a single figure. But it will frequently happen that the' amount of a single column will emceed 9, requiring two or more figures to express it. 1. There are three bags of money. The first contains S76 W0 ADDITION OF SIMPLE NUMBERS. ~ 13. dollars, the second 653 dollars, the third 426 dollars; what is the amount contained in all the bags? OPERATION. SOLUTION.- Writing the numbers as First bag, 876 a:}llars. already described, we add the units, and Second " 653 " find them to be 15, equal to 5 units, which Third 6( 426 we write in units' place, adding the 1 ten with the tens; which being added together are 15 tens, equal to 5 tens, to 1955 " be written in tens' place, and 1 hundred, to be added to the hundreds. The hundreds being added are 19, equal to 9 hundreds, to be written in hundreds' place, and 1 thousand, to be written in thousands' place. Ans. 1955 dollars. PROOF. -We may reverse the order, and, beginning at the top, add the figures downwards. If the two results are alike, the work may be supposed to be right, for it is not likely that the same mistake will be made twice, when the figures are added in a different order. NOTE.- Proof by the excess of nines. If the work be right, there will be just as many of any small number, as 9, with the same remainder, in the amount, as in the several numbers taken together. Hence, In the upper number, 8 (hundreds) is 8 more than a OPERATION. certain number of nines, (~ 5) 7 (tens) is 7 more. 876 5 Adding the 8 and 7, and the 6 units together, the sum 426 3 is 21-2 nines and 3 remainder, which we set down at the right hand, as the excess of nines in this number. 1955 2 In the same manner, 5 is found to be the excess of nines in the second number, and 3 in the third number. These several excesses being added together, make 1 nine and an excess of 2. which is the same as the excess of nines in the general amount, found in the same manner. This method will detect every mistake, except it be 9, or an exact number of nines. To find what will be the excess after casting the nines out of any number, begin at the left hand, and add together the figures which express the number; thus, to cast the nines out of 892, we say 8 (passing over 9) +2 (dropping 9 from the sum) = 1. From the examples and illustrations now given, we derive the following RULE. I. Write the numbers to be added, one under another, placeQuestions. - ~ 13. If the amount of the column does not exceed 9, wha do you do? What whery it exceeds 9? How do you add each column? What do you do with the amount of the left column? For what number do you carry? If the amount (f a column be 36, what would you set down, and how man? would you carry? On what printiple do you do this? How is addition proved? Why? Repeat the ruie for addition. ~ 13. ADDITION OF SIMPLE NUMBERS. 2 ir:g units uinder units, tens under tens, &c., and draw a imne underneath. II. Begin at the unit column, and add together all the fig ures contained in it; if the amount does not exceed 9, write it under the column; but if it exceed 9, write the units in units' place, and carry the tens to the column of tens. III. Add each succeeding column in the same manner, and vet down the whole amount of the last column. EXAMPLES FOR PRACTICE, 2. 3. 2863705421061 4367583021463 3107429315638 1752349713620 6253034792 6081275306217 247135 5652174630128 8673 8703263472013 4. Add together 587, 9658, 67, 431, 2S670, 85, 100000, 6300, and 1. Amount, 145799. 5. What is the amount of 8635, 7, 2194, 16, 7421. 93, 5063, 135, 2196, 89, and 1225? Ans. 27074. 6. A man being asked his age, answered that he left England when he was 12 years old, and that he had afterwards spent 5 years in Holland, 17 years in Germany, 9 years in France, whence he sailed for the United States in the year 1827, where he had lived 22 years; what was his age? Ans. 65 years. 7. A company contract to build six warehouses; for the first they receive 36850 dolls.; for the second, 43476 dolls.; for the third, 18964 dolls.; for the fourth, 62S40 dolls.; for the fifth, 71500 dolls.; for the sixth, as much as for the first three; to what do these contracts amount? Ans. 332920 dollars. 8. James had 7 marbles, Peter had 4 marbles more than James, and John had 5 more than Peter; how many marbles in all? Ans. 34. 9. There are seven men; the first man is worth 67850 dollars; the second man is worth 0500 dolls. more than the first man; the third, 3168 dolls. more than the second; the fourth, 16973 dolls. more than the third; the fifth, 40600 dolls. more than the fourth; the sixth, 19SS dolls. more than the fifth; and the seventh, 49676 dolls. more than the sixth; how many dollars are they all worth? Ans. 784934 dollars. 10. What is the interval in years between a transaction ADDITION (F SIMPLE NUMIBERS. ~1 14 that happened 275 years ago, and one that will happen 125 years hence? Ans. 400 years. I1. What is the amount of 46723, 674-2, and 986 dollars? 12. A man has three orchards; in the first there are 140 trees that bear'apples, and 64 trees that bear cherries; in the second, 234 trees bear apples, and 73 bear cherries; in the third, 47 trees bear plums,'t6 bear pears, and 25 bear cherries - how many trees in all the orchards, and how many of each kind? _ns. 619 trees; 374 bear apples; 162 cherries; 47 plums and 36 pears. 13. A gentleman purchased a farm for 7854 dollars; he paid 194 dollars for having it drained and fenced, and 300 dollars for having a barn built upon it; how much did it cost him, and for how much must he sell it, to gain 273 dollars? A.ns It cost him S348 dollars. He must sell it for 8621 dollars. IV 14. Review of Numeration and Addition Questions. — What are numbers? What are the methods of expressing numbers? What is numeration? notation? fundamental law in the Arabic notation? How does the Arabic differ from the Roman method? What is understood b, units of different orders? What is quantity? Arithmetic? What is utderstood by the simple value of figures? the local value? the unit value of a number? Explain the difference between an abstract and a denominate number. What is addition? the rule? proof? For what number do yoll carry, and why? EXElRCISE.S. 1. Washington was born in the year of our Lord 1732; he was 67 years old when he died-; in what year did he die? Ans. 1799. 2. The invasion of Greece by Xerxes took place 4S1 years before Christ; how long ago is that this current year? 3. There are two numbers; the less is 8671, the difference between the numbers is 597; what is the greater number? Ans. 9268. 4. A man borrowed a sum of money, and paid in part 684 dollars; the sum left unpaid was S76 dollars; what was the sum borrowed? 5. There are Four numbers; the fit-t 317, the second 812 the third 1350, and the fourth as nmach as the other three; wh5at is the sunm of them all? Ans. 4958. 6. A gentleman left his daughter 16 thousand 16 hundred I 15. SUBTRACTION OF SXIPLE NUMBERS. ~ and 16 dollars; he left his son 1800 more than his daughter what was his son's portion, ard what was the amount of the whole estate? Son's portion, 19416. Whole estate, 37032. 7. A man, at his death, left his estate to his four children who, after paying debts to the amount of 1476 dollars, received 4768 dollars each; how much was the whole estate? Ans. 20548. 8. A man bought four hogs, each weighing 375 pounds; how much did they all weigh? Ans. 1500. 9. The fore quarters of an ox weigh one hundred and eight pounds each, the hind quarters weigh one hundred and twenty-four pounds each, the hide seventy-six pounds, and the tallow sixty pounds; what is the whole weight of the ox? Ans. 600. 10. The imports into the several States in 1842 were as follows: Me. 606864 dollars, N. H. 60481, Vt. 209868, Mass. 179S6433, R. 1. 323692, Ct. 335707, N. Y. 57875604, N. J. 145, Pa. 73S5S58, Del. 3557, Md. 4417078, D. C. 29056, Va. 316705, N. C. 187404, S. C. 1359465, Ga. 341764, Al. 363871, La. 8033590, 0. 13051, Ky. 17306, Tenn. 5687, Mlich. 80784, Mo. 31137, Fa. 176980 dollars; what was the entire amourt? Ans. 100162087. SUBTRACTION OF SIMPLE NUMBERS. ~T 15. 1. Charles, having 18 cents, bought a book, for which he gave 6 cents; how many cents had he left? 2. John had 12 apples; he gave 5 of them to his brother' how many had he left? 3. Peter played at marbles; he had 23 when he began, but when, he had done he had only 12; how many did he lose? 4. A man bought a cow for 17 dollars, and sokl her again for 22 dollars; how many dollars did he gain? 5. Charles is 9 years old, and Andrew is 13; what is the difference in their ages? 6. A man borrowed 50 dollars, and paid all but 18; how many dollars did he pay? that is, take 18 from 50, and how many would there be left? The taking of a less number from a greater (as in the fore. going examples) is called Subtraction. The greater numbei 24 SUBTRACTION OF SIMPLE NUMBERS. ~ 16 is called the MIinuend, the less number the Subtrahend, and what is left after subtraction is called the Difference, or Re. mainder. 7. If the minuend be 8, and the subtrahend 3, what is the difference or remainder? Ans. 5. 8. If the subtrahend be 4, and the minuend 16, what is the re mainder? SIGN. -A short horizontal line, -, is the sign of subtraction. It is usually read minus, which is a Latin w9rd signifying less. It shows that the number after it is to be-taken from the number before it. Thus, 8 -3 = 5 is read 8 minus or less 3 is equal to 5; or, 3 from 8 leaves 5. The latter expression is to be used by the pupil in committing the following SUBTRACTION TABLE. 2 -2 -- 0 6 -3 37 3-2=-1 7- 5- 5 6=0 I7- 870 4 - 22=2 87-4-3 7-6- 51 8- 9 5 2=3 9- 36 8-6= 3 10 -7=3 4 — 2 102 - 3 76 ~6-2=4 10-37 — 964 7- 2 4-= 10Z-3 -- -860 78-268 - - hw9-8 3 9 - 2 6-4=2 67- 6_ 10 9 —8=2 90-2 7-4 3 7- = 61 - -9 0 3 —3=0 8-4=4 8-6=2 10- 9=1 I-3=1 9 4 =5 9 -6=3 5-3 2 10 - 4 6 10-6=48-5=h10owmany? 28- 7=how any? 9 - 4 ==how many? 22 - 13-how-many? 12-3 -how many? 33 - -- how many? 1 - 4 -how many? 41 - 15 - how many ~ 16. When the numbers are small, as in the foregoing examples, the taking of a less number from a greater is ~eadilv done in the mind; but wien the numbers are large, Questions. - ~ 15. What is subtraction? What is the greatel number called? the less number? that which is left after subtraction? Wlhat is the sign of subtraction? How is it usually read? What does mninus mean? What does the sign of subtraction show? ~ 116 SUBTRACTION OF SIMPLE NUMIBERS. 2 the operation is most easily performed part at a time, and therefore it is necessary to write the numbers down before performing the operation 1. A farmer, having a flock of 237 sheep, lost 114 of then) by disease; how many had he left? Here we have 4 units to be taken from 7 units, I ten to in taken from 3 tens, and 1 hundred to be taken from 2 hundreds. It will therefore be most convenient to write the less number under the greater, observing, as in addition, to place units under units, tens under tens, &c., thus: OPERATIDN. SOLUTION. - We begin with the From 237 the mzinuend, units, saying, 4 (units) from7, (units,) Takel 114 the subtrahend and there remain 3, (units,), which we set down directly under the column in units' place. Then proceeding to 123 the remainder. the next column, we say, 1 (ten) from 3, (tens,) and there remain 2, (tens,) which we set down in tens' place. Proceeding to the next column we say, 1 (hundred) from 2, (hundreds,) and there remains 1, (hundred,) which we set down in hundreds' place, and the work is done. It now appears that the number of sheep left was 123; that is, 237 - 114 = 123, Ans. NOTE. — We write units under units, tens under tens, &c., that those of the same unit value may be subtracted from each other; for we can no more take 3 tens from 7 units than we can take 3 cows from 7 sheep. Examznples in which each figure in the subtrahend is less than the figure above it. 2. There are two farms; one is valued at 3750, and the other at 1500 dollars; what is the difference in the value of the two farms? Ans. 2250. 3. A man's property is worth 8560 dollars, but he has debts o the amount of 3500 dollars; what will remain after paying his debts? Ans. 5060. 4. From 746 subtract 435. Rem. 311. 5. From 4983 subtract 2351. Rem. 2632. 6. From 658495 subtract 336244. Rem. 322251. 7. From 8764292 subtract 7653181. Rem. 1111111. Questions. —X 16. When the numbers are small, how may the.cperation be performed? When they are large, what is more convee.ent? How are the two numbers to be written? Whk.rc do you begin Ate subtraction? 26 SUBTRACTION OF SMPLE NUS3ERS. I 1. 9T 7.o 1. James, having 15 certs, bought a pen-knife for which he gave 7 cents, how many cents had he left? OPERATION. 15 cents. A difficulty presents itself here; for we cannot 7 cents. take 7 from 5; but we can take 7 from 15, anti there will remain 8. S cents left. 2. A man bought a horse for 85 dollars, and a cow for 27 dollars; what did the horse cost him more than the cow? OPERATIO[. SOLUTION.- The same difficulty presents itself here 85 as in the last example, that is, the unit figure in the 27 subtrahend is greater than the unit figure in the minuend. To obviate this difficulty, we may take I (ten) from the 8 (tens) in tuhe nminuend, which will leave 7 (tens,) and add it to the 5 units, making 15 units, (7 tens + 15 units = 85,) thus, TENS. UNITS, 7 15 We now take 7 units from 15 units, and!2tens from 2 7 7 tens, and have 5 tens and 8 units, or 58 remainder; that is 85 -27 = 58 dollars more for the horse than 5 8 for the cow. The operation may be shortened as follows. OPERATION. We have 8 tens and 5 units in the minuend, 1-orse, 85 dollars. and 2 tens and 7 units in the subtrahend. We Cow, 217 16 can now, in the 7mind, suppose I ten taken from Cow, 27 " the 8 tens, which would leave 7 tens, anu this 1 ten we can suppose joined to the 5 units, Diff. 5S " making 15. We can now take 7 from 15, as before, and there will remain 8, which we set down. The talking of 1 ten out of 8 tens, and joining it with the 5 units, is called borrowing ten. Proceeding to the next higher order, or tens, wae must consider the upper figure, 8, from which we borrowed,.1 less, calling it 7; then, taking 2 (tens) from 7, (tens,) there will remain 5, (tens,) which we set down, making the difference 58 dollars, Ans. Questions. - ~ 17. In subtracting 7 from 15, what difficulty pre. sents itself'? How do you obviate it? In taking 27 from 85, instead of taxing 7 from 5 what do you take it from? Whence the 15? From what do you subtract the 2 tens? Why not from 8 tens instead of 7.ens? Whmat is this operation called? Explain how the operation is nerformed in example 3. There is another method, often practised, erroneously called borroling ten, - explain the principle on vwhich it is don@.,When we subtra<,t units fiom units, of what ns.nm:- will the femamillder be? tens fi'on tenS, what? hundreds from hundrekdi,, a'hat; 18. SUBTRACTION OF SIMPLE NUMBERS. St NOTE.- It has been usual to perform subtraction, where the figutre in the subtrahend is larger than the figure above it, on another principle. If to two unequal numbers the samne number be added, the difierence between them will remain the same. Thus, the diff&rence between 17 and 8 is 9, and the difference between 27 and 18, each being increased by 10, is also 9. Take the last example.'ENS. UNITS. 8ENS. UNITS. Adding 10 units to 5 units in the minuend, and 1 3 15 ten to 2 tens in the subtrahend, we have increased both by the same number, and the remainder is not 5 8 altered, being 58. This method, which has been erroneously called borrowving ten, may be practised by those who prefer, though the former is moro dimple and equally convenient. 3. From 10000 subtract 9. OPERATION. SOLUTION.- In this example we have 0 units from 10000 which to subtract 9 units, and going to tens of tht 9 minuend, we have 0 tens, nor hundreds; nor thousands, but we have 1 ten thousand from which, borrowingr 10 units, we have 9990, that is, 9 thousands, 9 hundreds 9991 and 9 tens left. Taking 9 units from 10 units, we have 1 unit, then no tens in the subtrahend from 9 tens m the minuend leave 9 tens, no hundreds from 9 hundreds leave 9 nundreds, no thousands from 9 thousands leave 9 thousands. 4. A man borrowed 713 dollars and paid 475 dollars; how much did he then owe? Ans. 23S dollars. 5. From 1402003 take 681404. Rem. 720599. 6. What is the difference between 36070324301 and 280. 40373315? Ans. S029950986. 7. From 81324036521 take 2546057867. Rem. 78777978654. ~ST 1 S. To PROVE ADDITION AND SUBTRACTION. - Addition and subtraction are the reverse of each other. Addition is putting together; subtraction is taking asunder. 1. A man bought 40 sheep 2. A man sold 18 sheep and sold 18 of them; how and had 22 left; how many many had he left? had he at first? 40 - 18 22 sheep left. 18 +-t- 22 - 40 sheep at first. Ans. Ans. Hence, subtraction may be proved by addition, and addition by subtraction. To prove subtraction, add the remainder to the subtrahend, and, if the work is right, the amount will be equal to the ninuend. '2~ kSUBTRACTION OF SIMPLE NUMBERS. "I 18 To prove addition, subtract, successively, from the amount the several numbers which were added to produce it, and, if the work is right, there will be ao remainder. Thus 7 + 8 +- 6=-21; proof, 21-6 =- 15, and 15 - 8 — 7, and 7 - 7 — 0. NOTE. -Proof b6 excess of nines. We may cast out the nines in the remainder and subtrahend; if the excess equals the excess found by casting out the nines from the minuend, the work is presumed to be right. From the remarks and illustrations now given, we deduce the following I. Write down the numbers, the less under the greater, placing units under units, tens under tens, &c., and draw a line under them. II. Beginning with units, take successively each figure in the lower numbei from the figure over it, and write the remainder directly below. III. When a figure of the subtrahend exceeds the figure of the minuend over it, borrow 1 from the next left hand figure of the minuend; and add it to this upper figure as 10, in which case the left hand figure of the minuend must be considered one less. NOTE. - Or when the lower figure is greater than the one above it we may add 10 to the upper figure, and 1 to the next lower figure EXAMPLES FOR PRACTICE. 1. If a farm and the buildings on it be valued at 16000, and the buildings alone be valued at 4567 dollars, what is the value of the land? Ans. 5433 dollars. 2. The population of New York in 1830 was 1,918,608; in 1840 it was 2,428,921; what was the increase in ten years? Ans. 510,313. 3. George Washington was born in the year 1732, and died in the year 1799; to what age did he live? Ans. 67 years, 4. The Declaration of Independence was published July 4th, 1776; how many years to July 4th the present year? Questions, —~ 18. Addition is the reverse of what? Subtrac tion, of what? How will you show that they are so? How do yon prove subtraction? How can you prove addition by subtraction? ARe peat the rule for subtraction, ~ 19. SUBTRACTION OF SIMPLE NUMBERS. 29 5. The Rocky 1M[ountains, in N. A., are 12,500 feet above the level of the ocean, and tile Anes, in S. A., are 21,440 feet; how raany feet higher are the Andes than the Rocky Mountains? Ans. 8,940 feet. NOTE. - Let the pupil be required to prove the following examples. 6. What is the difference between 7,64S,203 and 92S,671? Ans. 6,719,532. 7. How much must you add to 358,642 to make 1,4S7,945? Ans. 1,129,303. 8. A man bought an estate for 13,6S2 dollars, and scld it again for 15,293 dollars; did he gain or lose by it? and how much? Ans. 1,611 dollars 9. From 364,710,82f5,193 take 27,940,386,574. 10. From 831,025,403,270 take 651,30S,604,782. 11, From 127,36S,047,216,843 take 978,654,827,352. ~ 19. Review of Subtraction. Questions. —Vhat is subtraction? What is the rule? What is understood by borrowing ten? Of what is subtraction the reverse I How is subtraction proved? How is addition proved by subtraction EXERCISES. 1. How long from the discovery of America by Columbus, in 1492, to this present year? 2. Supposing a man to have been born in the year 1773, how old was he in 1847? Ans. 74. 3. Supposing a man to have been 80 years old in the year 1846, in what year was he born? Ans. 1766. 4. There are two numbers, whose difference is 8764; the greater number is 156S7; I demand the less. A.ns. 6923. 5. What number is that which, taken frora 3794, leaves 865? Ans. 2929. 6. What number is that to which if you add 789, it will become 6350? Ans. 5561. 7. A man possessing an estate of twelve thousand dollars, gave two thousand five hundred dollars to each of his two daughters, and the remainder to his son; what was his son's share? Ans. 7000 dollars. 8. From seventeen million'lake fifty-six thousand, and what will remain? A s. 16,944,000. s0 SUBTRACT ON OF SIMPLE NUMBERS. ~i 19 9. What number, together with these three, viz., 1301 2561, and 3120, will make ten thousand? Ans. 3018. 10. A man bought a horse for one hundred and fourteen dollars, and a chaise for one hundred and eighty-seven dol. larq; how much more did he give for the chaise than for the norse? 11. A man borrows 7 ten dollar bills and 3 one dollar bills, and pays at one time 4 ten dollar bills and 5 one dollar bills; how many ten dollar bills and one dollar bills must he afterwards pay to cancel the debt? Ans. 2 ten doll. bills and 8 one doll. 12. The greater of two numbers is 24, and the less is 16; what is their difference? 13. The greater of two numbers is 24, and their difference 8; what is the less number? 14. The sum of two numbers is 40, the less is 16; what is the greater? EXERCISES IN ADDITION AND SUBTRACTION. 15. A man carried his produce to market; he sold his pork for 45 dollars, his cheese for 38 dollars, and his butter for 29 dollars; he received, in pay, salt to the value of 17 dollars, 10 dollars' worth of sugar, 5 dollars' worth of molasses, and the rest in money; how much money did he receive? Ans. 80 dollars. 16. A boy bought a sled for 28 cents, and gave 14 cents to have it repaired; he sold it for 40 cents; did he gain or lose by the bargain? and how much? Ans. He lost 2 cents. 17. One man travels 67 miles in a day, another man fol lows at the rate of 42 miles a day; if they both start from the same place at the same time, how far will they be apart at the close of the first day? -- of the second? of the third? - of the fourth? Ans. To the last, 100 miles. 18. One man starts from Boston Monday morning, and travels at the rate of 40 miles a day; another starts from the same place Tuesday morning, and follows on at the rate of 70 miles a day; how far are they apart Tuesday night? Ans. 10 miles. 19. A man, owing 379 dollars, paid at one time 47 dollars at another time 84 dol'ars, at another time 23 dollars, and as another time 143 dollars; how much did he then owe? Ans. 82 dollars. 20. Four men bought a lot of land for 482 dollars; the first NW in paid 274 dollars, the second man 194 dollars less than E 20. MULTIPL CATION OF SIMPLE NUMBERS..31 the firsts and the third man 20 dollars less than the second, how much did the second, the third, and the fourth man pay - The second paid 80. Amn. The third paid 60. TZhe fourth paid 68. 21 Four men bought a horse; the first man paid 21 dol. lars, the second 18 dollars, the third 13 dollars, and the fourth as much as the other three, wanting 16 dollars; how much did the fourth man pay? and what did the horse cost? Ans. Fourth man paid - dolls.; horse cost 88 dolls. 22. From 1,000,000 take 1, and what remains? (See ~T 17 Ex. 3.) MULTIPLICATION OF SIMPLE NUVIDERS. IT2[. 1. If one orange costs 5 cents, how many cents must I give for 2 oranges? -— how many cents for 3 oranges? - for 4 oranges? 2. One bushel of apples cost 20 cents; how many cents must I give for 2 bushels? --- for 3 bushels? 3. One gallon contains 4 quarts; how many quarts in 2 gallons? - in 3 gallons? - in 4 gallons? 4. Three men bought a horse; each man paid 23 dollars for his share; how many dollars did the horse cost them? 5. In one dollar there are one hundred cents; how many cents in 5 dollars? 6. How much will 4 pairs of shoes cost at 2 dollars a pair? 7. How much will two pounds of tea cost at 43 cents a pound? 8. There are 24 hours in one day; how many hours in 2 days -- in 3 days - in 4days? - in days? 9. Six boys met a beggar, and gave him 15 cents each; how many cents did the beggar receive? In this example we have 15 cents (the number which each boy gave the beggar) to be repeated 6 times, (as many times as there were boys.) When questions occur where the same number is to be repeated several times, the operation may be shortened by a rule called Multiplication. In multiplication the number to be repeated is called the Mulijjicand. The number which shows how mazny times the multiplicana s4 to be repeated, is called the Multiplier. 52 MULTIPLICAl3~'ON OF SIMPLE NUMBERS. 11 20 The result, or answer, is called the Product. The multiplicand and nru-ltiplier taken together are called Factors, or producers, becaise when multiplied together they prioduce the product. 10. There is an orchard in which are 5 rows of trees, and 27 trees in each row; how many trees in the orchard? bzn the first row,... 27 trees. SOLUTION.-The whole numit second,.. 27 " ber of trees will be equal to the'" third, e 27 it amount of five 27's added toit fourth,.. 27 " gether. 6 fourifthA,)... 27 t' In adding, we find that 7 ":fifth,.....27 " ntaken five times amounts to 35. We write down the five units, In the whole orchard, 135 trees. and reserve the three tens; the amount of 2 taken five times is 10, and the 3, which we reserved, makes 13, which, written at the left of units, makes the whole number of trees 135. If we have learned that 7 taken 5 times amounts to 35, and that 2 taken 5 times amounts to 10, it is plain we need write the number 27 but once, and then, setting the multiplier under it, we may say, 5 times 7 are 35, writing Multiplicand, 27 trees in each row. down the 5, and reserving Multiplier,. 5 rows. the 3 (tens) as in addition. Again, 5 times 2 (tens) are ~Product,. 135 trees,i An~s. 10, (tens,) and 3, (tens,) which we reserved, make 13, (tens,) as before. From the above example, it appears that multiplication is a short way of performing many additions, and it may be defined, The method of repeating one of two numbers Vs many times as there are units in the other. SIGN. -Two short lines, crossing each other in the form of the letter X, are the sign of multiplication. When placed between numbers it shows that they are to be multiplied together; thus, 3 X 4- 12, signifies that 3 times 4 are equal to 12, or 4 times 3 are equal to 12;. and thus, 4 X 2 > 7 = 56, signifies that 4 multiplied by 2, and this product by 7, equals 56. Questions. -- 20. When questions occur in which the same number is to be repeated several times, how may thie operation be shortened? In multiplication, what is the multiplicand? the multiplieir? the product? What are factors? Why? What is multiplication? Illustrate by the two methods of performing the 10th example. How do rou define multiplicatin? What is the sign? Repeat the table. " 20. MULTIPLICATION OF SIMPLE NUMBERS. 3. 3 NOTE. - Before any progress can be made in this rule, the follow ing table must be committed perfectly to memory. MU1LTIPLICATION TABLE. 2 times O are 0 4X 10 —-40 7X 6= 42 10 X 3- 30 2X 1= 2 4 X 11 44 7X 7= 49 10X 4= 40 2X 2=- 4 4X12=48 7X 8= 56 10X 5 — 50 2X 3= 6 7X 9- 63 10 X 6 60 2X 4- 8 5X O== 0 7X 10= 70 10X 7= 70 2X 5 =10 5X 1= 5 7X11- 77 10X 8= 80 2X 6=12 5X 2=10 7 X 12 84 10X 9= 90 2 X 7 —14 5X 3 —-15 10 X 10 —-100 2X 8=16 5 X 4=20 8X 0= 0 10X 11110 2X 9=18 56X 5 —25 8X 1= 8 10X 12 120 2X10= —20 5X 6-30 8x 2 16 2X11=22 5X 7-35 X 3= 2 llX 0= 0 2 X 12=24 5X 8-40 8X 4= 32 11 X 1 11 -- SX 9-45 8 X = 40 111X 2= 22 3X 0= 0 5X1050 SX 6= 48 lix 3= 33 3X 1= 3 5X11-55 8X 7= 56 liX 4= 44 3X 2= 6 5X12=60 8X 8= 64 11X 5- 55 3 X 3= 9 - 8X 9= 72 11x 6 66 a X 4=12 6X 0= 0 8X10= 80 11 X Y= 77 3X 5 151 6X 1- 6 SXl2= 88 11X 8= 88 3X 6= -18 6X 212 8X12= 96 lX 9- 99 3X 7=-21 6X 3=18 9X O- O llX 10 —1 7 3x 8=24 6X 4= 24 9X 1- 9 11Xll= 3 X 9=27 6 X 5-30 9X 2= 18 11X12=132 3 X 10 30 6 X 6=36 9 x 3= 27 3X 11=33 6X 7=42 9 X 4=- 36 12X 0= O 3x12=36 6x 8=48 9X 5 45 12x 1= 12 6>X 9_54 9X 6= 54 12X 2- 24 4 X 0 0 6 X 10=60 9 X 7- 63 12 X 3=36 4X 1= 4 6Xll=66 9X 8= 72 12X 4= 48 4 x 2= 8 6X12=72 9X 9= 81 12 X 5= 60 4X 3-12_ 9 10= 90 12X 6- 72 4X 4-16 7X 0= 0 9X11= 99 12x 7= 84 4X 5=20 7X 1= 7 9X12=108 12X 8= 96 4 X 6 24 7 X 2 — 14 _12X 9 10 4 X 7 283 7 X 3-21 10 l O_ 01) X 10x — 0-)0 - 8 —327 X 4=28 OX 1= 10 12 x 11 132 4X 9 90=36 7 X 5=3 7-5 f0 X - P0 12X' 2= 1441 34 MULTIPLICATION OF SIMPLE NUMBERS. 91 21 9 X 2 how many? 4 >< 3 X 2 - 24. 4 X 6- how many? 3 X 2 X 5=how many 8 X 9 - how many? 7 X 1 X 2 how many 3 X 7 how many? 8 X 3 X 2 =how many? 5 X 5 =how many? 3 X 2 X 4 X 5 how many? ~2'1t. 1. There are on a board, 3 rows of stars, and 4 stars in a row; how many stars on the board? DIAGRAM OF STARS. A slight inspection of the diagram will * * * * show that the number of stars may be found by considering that there are either * * * * 3 rows of 4 stars each, (3 times,4 are 12,),* * * or 4 rows of 3 stars each, (4 times 3 are 12;) therefore, we may use either of the given numbers for a multiplier, as best suits our convenience. We generally write the numbers as in subtraction, the larger uppermost, with units under units, tens under tens, &c. Thus, lMultiplicand, 4 stars. Multiplier, 3 rows. NOTE. —4 and 3 are the factors, which produce the product 12. Product, 12 stars, Ans. This diagram of stars is commended to the particular attention of the pupil, as it is intended to make use of it hereafter in illustrating operations in multiplication and also in division. First, you will notice the terms of the diagram, and theii application. TERMS OF THE DIAGRAM. ( Using this term as a representation or Stars in a row. 1 symbol of the multiplicand and one factor ( of the product. Number of rows. Using this term as a symbol of the multiplier and the other factor of the product. e Using this term as a symbol of the product, for when the stars in a row are taken Number of stars as many times as there are rows of stars, then the product will be the whole number of stars contained in the diagram. As the stars in a row symbolize the multiplicand, it follows that the nmultiplier (number of rows) in reality simply expresses the number of times the multiplicand (stars in a row) is to ~ 21. MULTIPLICATION OF SIMPLE NUMBERS. 36 ne taken. Hence, to multiply by 1, (1 row of stars,) is to take the multiplicand (stars in a row) 1 time; to multiply by 2 (2 rows,) is to take the multiplicand (stars in a row) 2 times; to multiply by 3, (3 rows,) is to take the multiplicand (stars in a row) 3 times, and so on. ILLUSTRATION. - What cost 7 yards of cloth at 3 dollars Ea yard? (7 rows, 3 stars in a row.) The two numbe-s as given in the question are both denominate; but how are they to be considered in the operation? The price of 7 yards will evidently be 7 times the price of 1 yard, that is, 7 times 3 dollars; dollars (number of stars) is the thing sought by ti question; and hence, 3 dollars being of the same name as tht thing or answer sought, is the true multiplicand. That number which was yards in putting the question, being taken for thqe multiplier, in this relation is not to be considered yards, but times of talcing the multiplicand;, and hence, in the operation, it must always be considered an abstract number For multiplication is a short way of performing many ad ditions, and to talk of adding 3 dollars to itself 7 yards times is nonsense. But we can repeat 3 dollars as many times aa 1 yard is repeated to make 7 yards. There is then a true multiplicand and a true multiplier. The true multiplicand is that number which is of the same name as the answer sought; the true multiplier,.is that number which indicates the times the true multiplicand is to be repeated, or taken; but as it respects the operation, it has been shown above that we may use either of the given numbers as the multiplier; that is, the multiplicand and multiplier may change places; still, the product will always be of the same name as the true multiplicand.'rhis application of the terms of the diagram to the terms of the question we shall call symbolizing the question. Questions. - ~ 21. You have in your book a diagram of stars; what is the first use made of it? What is the difference between 4 times 7, and 7 times 4? Which of the given numbers may be used fcr the multiplier? What are the terms of the diagram? What do these terms symbolize? 6 times 7 are 42, — which of these numbers is the multiplicand? the multiplier? the product? and what, in the diagram, is a symbol of each? What does the multiplier express? Show by the llagram what it is to multiply by 1, by 2, by 3, &c. What must the multiplier always be considered? What do you undle: tandlk y the true multiplicand? by the true multiplier? What will the product a.ways be? Give an example to show that you understand these principles What lo roat utderstand by symbolizing a question? 36 MULTIPLICATION OF SIMPLE N UMBERS. ~ 22 NOTE. - Let the teacher see to it that these principles are well understood by the pupil before he proceeds. As the pupil advances, the teacher should, from time to time, refer him back to a review of these principles. ~U 2.o 1. What will 84 barrels of flour cost, at 7 dollars a barrel? SOLUTION. - The price of 84 barrels will evidently be 84 times the price of 1 barrel, 7 stars in one low X by 84, number of rows, 7 dollars is the true multiplicand, &c.; but as it will be more convenient, the multiplicand and multiplier may change places, and we may consider it 7 rows of 84 stars in a row, and multiply the number of barrels, 84, by the price of 1 barrel, thus -'Writing the larger number uppermost, OPERATION. as in subtraction, (~ 16,) and the multiMuid~tiplicand, 84 plier under units of the multiplicand, we Zlultiplier, 7 begin at the right hand and say, 7 X 4 (units) = 28 (units) = 2 tens and 8 units; we set down the 8 units in units' place, as Product, 5SS dolls. in addition, and reserving the 2 tens, we say, 7 X 8 (tens) = 56 (tens,) and 2 (tens) which we reserved, make 58 (tens,) or five hundred and 8 tens, which we set down at the left of the 8 units, and the whole make 588 dollars, the cost of 84 barrels of flour, at 7 dollars a barrel, Ans. 2. A merchant bought 273 hats, (stars in a row,) at 8 dol lars each, (number of rows;) what did they cost (number of stars)? Ans. 21S4 dollars. 3. How many inches are there in 253 feet, (stars in a row,) every foot being 12 inches (number of rows)? OPERATION. SOLUTION. —The product of 12, with each of the 253 significant figures or digits, having been committed to 12 memory from the multiplication table, it is just as easy --- to multiply by 12 as by a single figure. Thus, 12 Ans. 3036 times 3 are 36, &c. 4.- What will 476 barrels of fish cost, at 11 dollars a barrel? Ans. 5236 dollars. From these examples we deduce the following Questions. - ~ 22. How will you explain the first example? When you multiply units by units, what is your product? Wrhen tens ny units, what? How can you multiply by 12? How do you write down numbers for multiplication? How do you perform multiplication when the mnultiplit does irot exceed 12? 11 23. MULTIPLICATION OF SIMPLE NUMBERS. 37 RULE. I. To set down numbers for multiplication. Write down the multiplicand, under which write the multiplier, setting units under units, tens under tens, &c. II. To pefform multiplication when the mnultiplier does not exceed 12. Begin at the right hand, and multiply each figure in the multiplicand by the multiplier, setting down and carrying as in addition. EXAMPLES FOR PRACTICE. 5. A farmer sold 29 bags of wheat, each bag containing 3 oushels; how many bushels did he sell? 29 X 3 = how many? 6. A farmer, who had two farms, raised 361 bushels of wheat on one, and 5 times as much on the other; how many bushels did he raise on both? Ans. 2166 bushels. 7. A miller sold 42 loads of flour, each load containing 9 barrels, at 7 dollars a barrel; how many barrels of flour did he sell, and what did the whole cost? Ans. He sold - barrels; cost, 2646 dollars. 9f 23. 1. A piece of valuable land, containing 33 acres, (number of rows,) was sold for'246 dollars an acre, (stars in a row;) what did the whole cost? NOTE 1. - When the multiplier exceeds 12, it is more convenient to multiply by each figure separately: — FIRST OPERATION. SOLUTION.- In sljultiplicand, 246 dollars, price of 1 acre. this example, the.Multiplier, 33 number of acres. multiplier consists of 3 tens and 3 st product, 738 dollars, price of 3 acres. tiplts. Fying by t, heul3 units, gives us 738 dollars, the price of 3 acres. Having found the price of 3 acres, our next step is to get the price of 30 acres. SECOND OPERATION. To do this, we multiply by the 246 dollars, price of 1 acre. 3 tens, (thirty,) and write the 33 numbelr of acres. first figure of the product (8) in tens' place, that is, directly under 738 dollars, price of 3 acres. the figure by which we multiply. 738 (tens) price of 30 acres. For the price of 30 acres being 10 times the price of 3 acres, it 8118 dollars, price of 33 acres. will consist of the same figures each being removed 1 place to wards the left, by which its value is increased 10 times. Then addA "3 IMULTIPLICATION OF SIMPLE NUMBERS. ] 23 ing together the price of 3 acres, and the price of 30 acres, we have the price of 33 acres. NOTE.- The correctness of the above operation results from the fact that when units (1st order) are multiplied by units, (1st order,) the product is units, 1st order. Tens (2d order) X units, (lst cr(er,) the product is units of the 2d order. Hundreds (3d order) X tent, (2d order,) the product is units of the 4th order. And universally, — If a figure of any order be multiplied by some figure of another order, the product will be units of that order indicated by the szum of their indices, minus 1. Thus, 7 of the 5th order, (70000,) multiplied by-4 of the 3d order, (400,) their indices being 5 + 3 - S, and 8 - 1 = 7, their product will be 28 units of the 7th order, that is, 28 millions. 2. How many yards in'3 pieces of broadcloth, each piece containing 67 yards? OPERATION. SOLUTION. - Multiplying 67 yards 7 yards,in each piece. by 3, we get 201 yards in 3 pieces; and 23 numbzer o0pieces. multiplying 67 by 2 tens, we get 134 tens, = 1340 yards in 20 pieces. Add 201 yards in 3 pieces. the two products together, and we get 134 " " 20 pieces. 1540 yards (No. of stars) in 23 nieces Ans. 1540 yards. 1541 " " 23pieces. Hence,- To perform multiplication when the multiplier exceeds 12,RULE. I. Multiply the multiplicand by each figure in the multiplier separately, first by the units, then by the tens, &c., remembering always to place the first figure of each product directly under its multiplier. II. Having multiplied in this manner by each figure in the multiplier, add these several products together, and their sum will be the answer. PRD0F.- Take the multiplicand for the multiplier, and the multiplier for the multiplicand, and if the product be the same as at first, the work may be supposed to be right. Questions. — ~ 23. How do you multiply when the multiplier exceeds 12? Wher3 do you write the first figure of each product i Why? What do you do with the several products? Repeat the rule What is the method of proof? A figure of any one order multiplied bi some figure of another order, the product willobe whla.-? 23. MULTIPLICATION OP SIMPLE NUMBERS. 39 EXA.PLES FOR PRACTICE. 3. There are 320 rods in a mile; how many rods are there in 57 miles? 320 X 57 how many? 4. It is 436 miles from Boston to the city of Washington; how many rods is it? 5. What will 784 chests of tea cost, at 69 dollars a chest? 784 X 69 - how many? 6. If 1851 men receive 758 dollars apiece, how many dol. lars will they all receive? Ans. 1403058 dollars. NOTE. -Proof by the excess of nines. Casting out the nines in the multiplicand, we have an excess of 6, which we write be1851 fore the sign of multiplication. Also, we find the excess 758 in the multiplier to be 2, which we write after the sign. The product of the nines cast out from each factor will be 1403058 an exact number of nines, since every nine multiplied by nine produces an exact number of nines. Hence, if there PROOF. is an excess of nines in the entire product, it must be from 3 an excess in the product of the excesses, 6 and 2, found 6 X 2 in the factors. Multiplying 6 by 2, and casting out nine 3 from the product, we write the excess, 3, over the sign; and casting out the nines from the product of the factors, we find the excess will be the same number 3, which we write under the sign, and presume that the work is right. 7. There are 24 hours in a day; if a ship sail 7 miles in an hour, how many miles will she sail in 1 day, at that rate? how many miles in 36 days? how many miles in 1 year, or 365 days? Ans. 61320 miles in 1 year. 8. A merchant bought 13 pieces of cloth, each piece containing 28 yards, at 6 dollars a yard; how many yards were there, and what was the whole cost? Ans. to the last, 2184 dollars. 9. Multiply 37864 by 235. Product, 8898040. 10. " 29831 - 952. " 28399112. 11. " 93956 " 8704. " 817793024. 12. The factors of a certain number are 25 and 87; what _s the number? Ans. 2175. 13. A hatter sold 15 cases of hats, each containing 24 hats worth 8 dollars apiece; how many hats did he sell, and to how many dollars did they amount? Ans. to the last, 2880 dollars. 14. A grazier sold 23 head of cattle every year for 6 years, at an average price of 17 dollars a head; how many head of cattle did he sell, to how much did they amount each year, and to how much did they amount in 6 years? Ans. to the last, 2346 dollars. 40 T~ULTIPLC]A'IOIUrt JF SIMPlPLoE NUMBERS. [T it Conotractions i n 1:tultpl1i ation qT'4. I. TPW n te multiplier is a composite number. Any number which can be produced by multiplying twvo or more numbers together, is called a Composite number, and The numbers which are multiplied together to produce it are called its Componeent parts, or Factors; thus, 15 can be produced by multiplying together 3 and 5, and is, therefore, a composite number, and the numbers 3-and 5 are its compon nt parts. So, also, 24 is a composite number. Its component parts may be 2 and 12, (2 X 12 = 24,) or 3 and 8, (3 X 8 =24,) or 4 and 6, (4 X 6=24,) or 2, 3, and 4, (2 X 3 X 4 = 24,) or 2, 2, 2, 3, (2 X 2 X 2 X 3-=24.) 1. What will 18 yards of cloth cost, at 4 dollars a yard? 3 X 6 c 18. It follows, therefore, that 3 and 6 are component parts of 18. OPERATION. SOLUetION. — If 1 yard cost 4 dol4 dollars, cost of 1 yard. lars, 3 yards will cost 3 times 4 dol3 yards. lars, = 12 dollars; and, if 3 yards cost 12 dollars, 18 yards (3 X 612 dollars, cost of 3 yards. 18) will cost 6 times as much as 3 6 (3 X 6 =) 1S yards. yards, that is, 6 times 12 dollars 72 dollars. Hence, 72 dollars, cost of 1S yards. To pe.form multiplication when the multiplier exceeds 12, and is a composite number,REULEo I. Separate the multiplier into two or more component parts, or factors. II. }Multiply the multiplicand by one of the component parts, and the product thus obtained by the other, and so on, if the component parts be more than two, till you have multiplied by each one of them. The last product will be the product required. Qestions. -- 2m4. What is a composite number? What are the zompoinent parts, or factors? Why is 15 a composite number? How many factors may a composite number have? Is 11 a composite number? Why not? Explain the Ist example. How do you multiply by a composite numb er? Does it matter by which factor you multiply first? Have you performed the 3 operations, (Ex 2 ) anda compared( theL products? ~ 25. MULTIPLICATION OF SIMPLE NUMBERS. 4. EXAM]PLE FOR PRACTICE. 2. WVhat will 136 tons of potash cost, at 96 dollars per ton? Let the pupil make 3 operations, and multiply, lst,by 12 and 8; 2dly, by 4, 4, ant 6; 3dly, by 96, and compare the operations; he will find the results to be the same in each case. Arns. 13056 dollars. 3. Supposing 342 men to be employed in a certain piece of work, for which they are to receive 112 dollars each; how much will they all receive? 8 X 7 X 2== 112. Ans. 38304 dollars. 4. How many acres of land in 48 farms, each containlng 367 acres? Ans. 17616 acres. 5. Supposing 168 persons to be employed in a woollen factory, at an average price of 274 dollars each per year; how much will they all receive? 8 X 7 X 3 168. Ans. 46,032 dollars. 6. Multiply 853 by 56. Product, 47,768. 7. " 18109 " 35. " 633,815. S. " 1947271 " 81. " 157,728,951. T 25. II. Whten the mualtiplier is 10, 100, 1000, 4-c. 1. What will 10 acres of land cost, at 25 dollars per acre? SOLUTION. - The price of 10 acres will be 10 times the price of 1 acre, or 10 times 25 dollars. Now if we annex a cipher to OPERATION. 25, the 5 units are, made 5 tens, 25 dollars, price of 1 acre. and the 2 tens are made 2 hun250 dollars, pr'ice of 10 acres. dreds. Each figure, then, is increased ten-fbld, and consequently the whole number is multiplied by 10. It is also evident that if 2 ciphers were annexed to 25, the 5 units would be made 5 hundreds, and 2 tens would be made 2 thousands each figure being increased a hundred fold, or multiplied by 100. If 3 ciphers were annexed, each figure would-be multiplied by 1000, &c Hence, WV/en the multiplier s 10, 100, 10009, or 1, with any num ber of cipher s annexed, — RULE. Annex as many ciphers to the multiplicand as there are Questions. - ~ 25. How are the figures of a number affected, by lacing one cipher at the right hand? two ciphers? three ciphers? &c. Wow, then, do you multiply by 1, with any number of ciphers annexed 4*~ 2, MIULTIPLICATION OF SIMPLE NUMBERlS. ~[ 26 ciphers in the multiplier, and the multiplicand, so increased, will be the prodcuct required. EXAMPLES FOR PRACTICE. 2. What wlL 76 barrels of flour cost, at 10 dollars a balrel? Ans. 760 dollars. 3. If 100 men receive 126 dollars each, how many dollars will they al. receive? Ans. 12600 dollars. 4. What will 1000 peces of broadcloth cost, estimating each piece at 312 dollars? Ans. 312000 dollars. 5. Multiply 5682 by 10000. 6. " 82134 " 100000 ~f 0'. III. IWhen there are czphers on the right hand oj ihe multiplicand, multiplier, either or both. 1. WVhat will 40 acres of land cost, at 27 dollars per acre 2 OPERATION. SOLUTION. - The price of 40 27 dollars, price of 1 acre. acres will be 40 times the price 4 of 1 acre. But 40 being a composite number, (4X 10=40,) we multiply by 4, one compo10S dollars, price of 4 acres. nent part, to get the price of 4 1080 dollars, price of 40 acres. acres, and then to multiply the price of 4 acres by 10, the other component part, we annex a cipher to get the price of 40 acres. 2. What will 200 acres of land cost, at 400 dollars an acre? FIRST OPERATION. SOLUTION. —The 200 acres 400 dollars, price of 1 acre. will cost 200 times the price 200 of 1 acre. WTe see in the'op--- eraticn that the product is 8 000 wvitl' 4 ciphers at the right 000 hand, the same number as in o00 the multiplicand and multiplier counted together. We 80000 dollars, price of 200 acres. may then shorten the opera tion, as follows:IECOND OPERATION. Multiplying the significant figures together, 400 we place their product, 8, under the 2. Then 200 we annex to this product 4 ciphers, the number in both factors. Hence, 800D0 To perform multipZication when there are ciphers on the "ight hand of eithe? or both the factors, ~ 97 MULTIPLICATION OF SIMPLE NUMBERS. 43 RULE. I. Set the significant figures under each other, placing the ciphers at the right hand. Ii. Multiply the significant figures together. III. Annex as many ciphers to the product as there are on the right hand of both the factors. EX.AMPLES FOR PRACTICE. 3. If 1300 men receive 460 dollars apiece, how many dol lars will they all receive? Ans. 598000 dollars. 4. It takes 200 shingles to lay 1 course on the roof of a barn, and there are 60 courses on each of the two sides; how many shingles will it take to cover the barn? Ans. 24000. 5. A certain storehouse contains 30 bins for storing wheat, and each bin will hold 400 bushels; hoNw many bushels of wheat can be stored in it? Ans. 12000 bushels. 9T 27. IV. WThen there are ciphers between the significant figures of the multh/lier. 1. What is the product of 37S, multiplied by 204? FIRST OPERATION. Multiplying by a cipher SECOND OPERATION 378 produces nothing. There- 378 204 fore, in the multiplication, 204 we may omit the cipher, 1512 and multiply by the sig- 1512 000 nificant figures only, as in 756 756 the second operation. Hence, to perform mul- 77112 77112 tiplication, Wizen there are ciphers between the significant figures of the multiplier, - RULE. Omit the ciphers, and multiply by the significant figures only, remembering to place the 1st figure of each product directly under its multiplier. Questi.,s-ns. - I 26. How do yen set down numbers for multiplcation, when there a:e ciphers on the right hand of the multiplicand, multiplier, either Jr ooth? How do you multiply? How many ciphers do you annex to the product? If there were 2 ciphers on the right hand of your mnlcipl'cand, and 5 on the right hand of your multiplier, how many woulJ Jou annex to the product? ~ 27. Wi~,n there are ciphers between the significant figures of the multiplier kow do you multiply? Where do you set the 1st figure:f she produet f 44 MULTIPLICATrION OF SIMPLE NUMBERS. ~ 2& EXAMPLES FOR PRACTICE. 2. Multiply 154326 by 3007. Product, 464058282, 3. Multiply 543 by 206. Product, 111858. 4. Multiply 1620 by 2103. Product, 3406S60 5. Multiply 36243 by 32004. Product, 1159920972. 6 Multiply 101,010,101 by 1,001,001. Product, 101,111,212,111,101. ql 2S. Other Methods of Contraction. 1. WThen the multiplier is 9, 99, or any number of 9's,Annex as many ciphers to the multiplicand as there are nines in the multiplier, and from the number thus produced, subtract the multiplicand; the remainder will be the product. Thus, Multiply 6547 by 999. OPERATION. 6547000 Let the pupil prove the operation by actual mul6547 tiplication. 6540453 Ans. II. Then the.multiplier is 13, 14, 15, 16, 17, 18, or 19,Multiply 32046375 by 14. Place the multiplier at the right of the mulOPERATION. tiplicand, with the sign of multiplication be32046375 X 14 tween them; multiply the multiplicand by the 128185500 unit figure of the multiplier, and set the product one place to the right of the multiplicand. This 44S649250 Ans. product, added to the multiplicand, makes the true product. NOTE. — If the multiplier be 101, 102, 4-c., to 109,Multiply 72530486 by 103. OPERATION. 725304S86 >( 103 Multiply as above, and set the product two 217591458 places to the right of the multiplicand, and add them together for the true product. 7470640058 Ans. Qucz - nas. - ~ 28. When the multiplier is 9, 99, &c., why does the contrac;on, as above directed, give the true product? Ans. [Multiplying by 9 repeats the multiplicand 9 times; annexing a cipher repeats or increases it 10 times, which is 1 tulne too many: hence th6 rule, subtract it 1 time, &c. When the multiplier is. 13, 14, &c., why When 101, 102, &c., why? When 21, 31, &c., why? ~ 29. MULTIPLICATION OF SIMPLE NUMBERS. 45 III. W4then the multiplier is 21, 31, and so on to 91,Multiply 83107694 by 31. OPERATION. Multiply by the tens' figure only of the mul83107694 X 31 tiplicand, and set the unit figure of the product 2493230S2 under the place of the tens, and so on; then add them together for the true product. 2576338514 A ns. TY 209. Review of Multiplication. Questions. - What is multiplication, and how defined? Explain th}e use of the diagramn of stars, and show its application to Ex. 1, ~ 22. What must the true multiplier always be? the product? Why can the factors exchange places? How do you multiply by 12, or less? by a number greater than 12? by a composite number? by 1 with ciphers annexed? by any number with ciphers annexed? when there are ciphers between the significant figures? When units of different orders are multiplied together, of what order is the product? EXERCISES. 1. An army of 10700 men, having plundered a city, took so much money, that, when it was shared among them, each man received 46 dollars; what was the sum of money taken 2 Ans. 492200 dollars. 2. Supposing the number of houses in a certain town to be 145, each house, on an average, containing two families, and each family 6 members, what would be the number of inhabitants in that town? Ans. 1740. 3. If 46 men can do a piece of work in 60 days, how many men will it take to do it in one day? Anzs. 2760. 4. Two men depart from the same place, and travel in op-:osite directions, one at the rate of 27 miles a day, the other 31 miles a day; how far apart will they be at the end of 6 days? Ans. 348 miles. 5. What number is that, the factors of which are 4, 7, 6 and 20? Ans. 3360. 6. If 18 men can do a piece of work in 90 days, how long will it take one man to do the same? Ans. 1620 days. 7. What sum of money must be divided between 27 men, so that each man may receive 115 dollars? 8. There is a certain number, the factors of which are 89 and 265; what is that number? 9. What is that number, of which 9, 12, and 14 are factors? 1(. ff a carriage wheel turn round.46 times in running 46 MUI,TIPLICATION OF SIMPLE NUMBERS. ~ 29 1 mile, how many times will it turn round in the distance from New York to Philadelphia, it being 95 miles? Ans. 32870. 11. In one minute are 60 seconds, how many seconds in 4 minutes? in 5 minutes? - in 20 minutes? in 40 minutes? Ans. to the last, 2400 seconds. 12. In one hour are 60 minutes; how many seconds in an hour? - ill two hours? how many seconds from nine o'clock in the morning till noon? Ans. to the last, 10S00 seconds. 13. Multiply 275S27 by 19725. Produzct, 5440687575. 14. Two men, A and B, start from the same place at the same time, and travel the same way; A travels 52 miles a day, and B 44 miles a day; how far apart will they be at the end of 10 days? Ans. 80 miles. 15. A farmer sold 468 pounds of pork at 6 cents a pound, and 48 pounds of cheese at 7 cents a pound, and received in payment 42 pounds of sugar at 9 cents a pound, 100 pounds of nails at'6 cents a pound, 108 vards of sheeting at 10 cents a yard, and 12 pounds of tea at 95 cents a pound; how many cents did he owe? Ans. 54 cents. 16. A boy bought 10 oranges; he kept 7 of them, and sold the others for 5 cents apiece;- how many cents did he receive? Ans. 15 cents. 17. The component parts of a certain number are 4, 5, 7, 6, 9, 8, and 3; what is the number? Ans. 181440. 18. In 1 hogshead are 63 gallons; how many gallons in 8 hogsheads? In 1 gallon are 4 quarts; how many quarts in 8 hogsheads? In 1 quart are 2 pints; how many pints in 8 hogsheads? Ans. to the last, 4032 pints. 19. The component parts of a multiplier are 5, 3 and 5, and the multiplicand is 118; what is the multiplier? what the product? Ans. to the last-the product is SS50. 20. An army consists of 5 divisions, each division of 8 bri gades, each brigade of 4 regiments, each regiment of 9 comn panics, and each company of 77 men, rank and file; the number of officers, &c., to the whole army is 42, the number belonging peculiarly to each division is 19, to each brigade 25, to each regiment 11, and to each canpany 14; how many men in the array? Ans 133937 ~ 30. DIVISION OF SIMPLE NUMBERS. 47 DIVISION OF SIMPLE NUMBERS. vT 30. 1. James has 12 apples in a basket, which he distributes equally among several boys giving them 4 apples each; how many boys receive them? SOLUTION. - He can give the apples to as many boys as the tirnme he can take 4 apples out of the basket, which is 3 times. Ans. 3 boys. 2. If a man travel 4 miles in an hour, in how many hours will he travel 24 miles? SOLUTION. -It will take him as many hours as 4 is contained times in 24. Ans. 6 hours 3. James divided 2S apples equally among 3 of his companions; how many did he give to each? SOLUTION. -The 28 apples are to be divided into 3 equal -parts, and one part given to each boy, who will thus receive 9 apples. It will require 27 apples to give 3 boys 9 appiles each, since 9 X 3= 27. There will be one apple left, which must be cut into 3 equal parts, and 1 part given to each boy. NOTE. - If a unit, or whole thingr, be divided into 2 equal parts, one of those parts is called one half; if into 3 equal parts, I part is called I third; two varts are called 2 thirds, &c. If divided into 4 equal parts, one part is called I fouzrth, or one quarter; 2 parts are called 2 fourtlhs, or 2 quairters; 3 parts, 3 fourths, or 3 quarters, &c. If divided into 5 equal part the part s are called 4fiths. If into 6 equal parts, the p)arts are called sixths, &c. 4. Seven men bouoht a barrel of flour, each man paying an equal share; for what part of the barrel did 1 man pay? 2 men? 3 men? - 4 men? 5 nen? 6 men? 5. Twelve men built a steamboat, each man onirig- an e>qual share of the'work; how much of the work didl 3 men do? - 5 emen? 9 men. 9?men. Il men? 6. A boy had two apples, and gave one half an alple to each of his companions; how many were his companions? Anrs. 4. 7. A boy divided four apples among his companions, by giving them one third of an apple each; among how many did he divide his apples? Ans. 12. Questions. - ~ 30. What do you understand by 1 half of any thing or number? 1 third? 2 thirds? 1 seventh? 4 sevenths? 6 flf teenths? 8 tenths? 5 twentieths? 9 twelfths? How many halves make a whole one? How many thirds? fourths? sevenl:hs? ninths? twelfths i tffteerths? twentieths? &c. How many thirds makle three whole ones I ,X DIVISION OFP SIMPLE NUMBERS. ~ 31 8. How many quarters in 5 oranges? SOLUTION. -In 1 orange there are 4 quarters, and in 5 oranges there are 5 X 4 = 20 quarters. Ans. 9. How many oranges would it take to give 12 boys one quarter of an orange each? Ans. 3 or. 10. How much is one half of 12 apples? Ans. 6 ap. 11. How much is one third of 12? 12. How much is one fourth of 12? Ans. 3. 13. A man had 30 sheep, and sold one fifth of them; how many of them did he sell? Ans. 6. 14. A man purchased sheep for 7 dollars apiece, and paid for them all 63 dollars; what was their number? Ans. 9. 9- 31. 1. How many oranges, at 3 cents each, may be bought for 12 cents? SOLUTION. - As many times as 3 cents can be taken from 12 cents so many oranges may be bought; the object, therefore, is to find how many times three is contained in 12. 12 cents. First orange, 3 cents. 9 We see, in this example, that 12 conSecond orange, 3 cents. tains 3 four times, for we subtract 3 from 12 four times, after which there is 6 no remainder; consequently, subtraction alone is sufficient for the operation; Thlird orange, 3 cents. but we may come to the same.: result by - a much shi6bter process, called.division 3 Ans. 4 oranges. Fourth orange, 3 cents. 0 We see from the above, that one number will be co;ained:r another as many times as it can be subtracted from it, and hence, that Division is a short way of performing many subtractions of the same number. The minuend is called the dividend, the number which is subtracted at one time is called the divisor, and the number which indicates the number of times the subtraction is performed is called the quotient. The cost of one orange, (3 cents,) multiplied by the number of oranges, (4,) is equal to the cost of all the oranges, (12 cents;) 12 is, therefore, a product, and 3 one of its factors; and to find how many times 3 is contaiimed il 12, is to find I 31. DIVISION OF SIMPLE NUMBERS. 49 the other factor, which, multiplied into 3, will produce 12. Hence, the process of division consists in finding one factor of a product when the other is known. 2. A man would divide 12 cents equally among 3 children; to-v many would each child receive? SOLUTION. - The numbers in this are the same as in the former example, but the object is different. In the former example, the object was to see how many times 3 cents are contained in 12 cents; in this, to divide 12 cents into 3 equal parts. Still the object is to find a number, which, multiplied into 3, will produce 12. This, as in the former example, is, Ans. 4 cents. Hence Division may be defined - I. The method of finding how many times one number is contained in another of the same kind. (Ex. 1.) Or, II. The method of dividing a number into a certain numiber of equal parts. (Ex. 2.) The Dividend is the number to be divided, and answers to the product in multiplication. The Divisor is the number by which we divide, and answers to one of the factors. The Quotient is the result or answer, and is the other factor. When anything is left, it is called the Remainder. NOTE.- In the first use of division, the divisor and dividend must be of the same kind, for it would be absurd to ask how many times a number of pounds of butter is contained in a number of gallons of molasses. In the second use, the quotient is of the same kind with the dividend, for if a number of acres of land should be divided into several parts, each part will still be acres of land. SIGN. - The sign of division is a short horizontal line between two dots, thus -. This shows that the number before it is to be divided by the number after it; thus 27 +. 9 = 3, is read, 27 divided by 9 is equal to 3; or, to shorten the expression, 9 in 27 3 times. Or the dividend may be written in place of the upper dot, and the divisor in place of the lower dot; thus zj_ shows that 27 is to be divided by 9 as before. Questions. - ~ 31. In what way is the first example performed? HLow might the operation be shortened? How often is one nuomber contained in another? What, then, is division? Show its relation to mnultiplication. What is the object in the first, and what in the second, example? Define division. What is the dividend? to what does it answer in subtraction, and to what in multiplication.? What the divisor, awid to what does it answer in subtraction and multiplication? What the quotient, antd to what does it answer? Explain the divisor and dividend in the first use of division. The dividend and quotient in the,vcona 50 DIVISION OF SIMPLE NUMBERS.':32. DIVISION TAiBLE. NOTE. - The expression used by the pupil in reciting the table may be, 2 in 2 one time, 2 in 4 two times, 4 in 12 three times, &c 2 = —-1 4 -1 - 1 6_ I _ 1 4- 2 6 2 8=2 150 2 J1 —2 _ 24 2,6,=3 9=-3 42 -3 15 3 8-=3 21- 3 8 4 12 4 16 4 2 4 24= — 4 2 8_ 4 o-= 5 5 I-5 - =5 25 5 _ - -6~=4 —5 A-5 2 47 4 6 9 2t ==6 8 6 2A_ =6 30 6 36 — 6 4 6 = 1? — = 7 %: L=7 2=7 -7 7 - =4 7.1Q8 _=S 2=8_ 4 0- 8 A 86 S X8-.- 9 --- 9 29 5 9 54 --— 9 63 - - 9 _ - 9 1 2TT -6 2 8 20 2 _ 2 24 2 2 _- 3 2 7 3 2~ V = 3 f l, 3 6 3 - 3 3g2 4 3-6 4 40 — 4 41 — 4 4 4 o_ 5 49 _ 5 --- 5 61 60 5 56 = 7 6gS _ 7 7 7 7 177 5 7 84 — 7'-A -- 6 6 -- 8 72 -- 8 -8-7 817 9 -- S' --' _9 1 IY T 1Y T' 72 -=- 9 - 9- 9 99 9 o 9 2S 7, or 01 z how many? 49 7, or 49 _how many? 42 + 6, or a —z hlow many? 32 4, or 32_- =how many? 54+ 9,or =4 — how many? 99 +11, or 99_ how many? 32 - 8, or =how many? 84 12, or ~4 =hoN many? 33 11, or 3- how many? 108.12, or -I0 —how many? NOTE.- The pupil should be thoroughly exercised in the foregoing lable. ~ 52. The principles of division will be made more plain to the pupil by turning his attention to the samne diagram to which it was directed while illustrating the principles of mumtiplication, since division is the reverse of multiplication. DIAGRAMI OF STARS. * * * *k In multiplication, we call the whole nuro.* * * * ber of stars a symbol of the produce. In division, a symbol of the dividend. * * 1 3,3-35. 1) VISION OF SIUPLE NUMBERS. 51 In multiplication, stars in a row, and number of rows, are symbols of the multiplicand and multiplier, which are factors of the product. In division, stars in a row, and 7rows of stars, are symbols of the divisor and quotient, which are factors of the dividend. 9' 33. When the object in division is to find how many times one number, or quantity, is contained in another number, or quantity, the divisor must be of the same kind as the dividend, (stars in a row,) and the quotient will be a number telling how many times (rows of stars.) On the other hand-when the object is to divide a number or quantity, into a given number of equal parts, the quotienz will be of the same name or kind as the dividend, (stars in a row.) If we divide 35 apples into 5 parts, the quotient, 7 apples, will be one part, (stars in a row,) and the divisor, 5, will be a number, that is, the number of parts, (rows of stars.) ~ 34. It has been remarked that division is a short way of performing many subtractions. How often can 3 be subtracted from 963? Ans. 321 times. To set down 963 and subtract 3 from it 321 times would be a long and tedious prozess; but by division we may decompose the number 963 thus; 963 = 900 + 60 + 3, and say 3 is contained in 9(00,) 3(00) times, in 6(0,) 2(0) times, and in 3, I time - 321 times, which brings us to the same result in a much shorter way. 9T 35. 1. How many yards of cloth, at 3 dollars a yard, can be bought for 936 dollars? SOLUTION. - As many yards as 3 dollars are contained times in 936 Questions. - ~T 32. What, in the diagram of stars, may be taken as a symbol of the dividend in division? Stars in a rolw and rows of starn are taken as symbols of what, in multiplication? of what in division? If the dividend be 108, the divisor 12, and the quotient 9, how would ycu make a diagram to correspond? ~ 33. When the object is to find how many times one number or quantity is contained in another, the divisor will be of what name ot kind? the quotient will be what? When the oiject Is to divide a number or quantity iwno a given number of parts he quotient will be of what name or kind? the divisor wll be what? 5p 34. How can you mlake it appear from the diagram that d'vision Is a shorter way of performing many sL utractions? Find on ihe blackboard the quotient of 963 divided by 3 How would this example be pertonned by subtraction? 62 DIVISION OF SIMPLE ]HUMBERIS. ] 35 dollars, or as many as 3 can be subtracted times from 936; 936 dollars is the dividend, (number of stars,) 3 dollars (stars in a row) the divisor. PREPARATION. Write the divisor on the left of the diviDividend, dend, separate them by a curved line, and )ivisor, 3) 936 draw a line underneath. OPERATION. We may decompose the dividend thus, 936 — 3 ) 936 900 + 30 -- 6, and divide each part separately. Beginning at the left hand, we say, 3 in 9, 3 Quotient, 312 times. This quotient 3 is 3 hundred, because the 9 which we divided is hundreds; therefore we -write it under the 9 in the place of hundreds. Preceeding to the next figure, we say, 3 in 3, 1 time, which, being 1 ten, we write it in tens' place. Lastly, 3 in 6, 2 times, which, being units, we write the 2 in units' place, arid tile work is done. The quotient (number of rows) is 3 hundred, (300,) 1 ten, (10,) and 2 onits, or 312 yards, Ans. NOTE. - The quotient figure will always be of the same order of units as the figure divided to obtain it. 2. 2846 -- 2 - how many? 840- 4= how many? 500 5 how many? 3. If you give 856 dollars to 4 men, how many dollars will fou give to each? OPERATION. SOLUTION. - Write down the numbers Divisor, Dividend, as before. Divide the first figure, 8, (hun4 men, ) S56 dolZars. dreds,) in the dividend as before. Proceeding to the next figure, 5, (tens,) 4 is Quotient, 214 dollars. contained 1 (ten) time in 4 of the tens, or 40, and there is 1 ten left, which, added Lo the 6 units, will make 16, and 4 in 16 units, 4 (units) times. Ans. 214 dollars. Here again we see that the 856 is taken in three parts, 800, 40, and 16, and each part is divided separately. When this decomposing into parts can be done in the mind, as in these examples, the uDroces2 is called Short Division. It can always be done when thie divisor does not exceed 12. 4. Answer the following questions after the same manner, viz., 650. 5- how many? 8490 -- 6; or, what expresses the same thing, - 4 - hows many? -~1. = how many? Ao > — how many? 5. What is the quotient of 14371 divided by 7? OPERATION. There are two other things to be learned in 7)14371 this operation. First, -he divisor, 7, is net contained r 1, the first figure of the dividend then Quotient, 2053 take two figures, or so many as shall contain.the ~ 35. DIVISION OF SIMPLE NUJMBERS. 5:'~ divisor, and say, 7 in 14, 2 times; we write 2 in the quotient, in thotusands' place, because we divided 14 thousands. Then, again, proceeding to the next figure, 3 in ti/e dividend will not contain the divisor, 7; to obviate this difficulty, we place a cipher in the quotient, joining the 3 to,he 7 tens, calling it 37 tens, and so proceed. Ans. 2053. Hence, for Short Division, this general RULE. I. Write the divisor at the left hand of the dividend, sep~ arate them by a line, and draw a line under the dividend, to separate it from the quotient. II. Find how many times the divisor is contained in the first left hand figure or figures of the dividend, and place the result directly under the last figure of the dividend taken, for the first figure of the quotient. III. If there be no remainder, divide the next figure in the dividend in the same way; baut, if there be a remainder, join it to the next figure of the dividend as so many tens, and then find how many times the divisor is contained in this amount, and set down the result as before. IV. Proceed in this manner till all the figures in the dividend are divided. EXAMPLES FOR PRACTICE. 6. A man has 256 hours' work to do; how many days will it take him, if he work 8 hours each day? Ans. 32 days. 7.:a__~2 _7_ = how many? Ans. 215477. 8. In 1 gallon are 4 quarts; how many gallons in 278, quarts? Ans. 696 gallons. 9. Seven men undertake to build a barn, for which they are to receive 602 dollars; into how many equal parts must the money be divided? How much will I part be? ---, parts? - 5 parts? (See ~ 30.) Ans. to the last, 430 dellars. Questions. - T 35. When the dividend is large, how must it be taken? how divided? How is it done when the divisor does not exceed 12? What is the preparation? Where do you begin the division? It' you divide units, what will the quotient hla? if tens, what? hundreds what? If at any time you have a remaiinder, what do you do with it In Ex. 5 there are two things to be learned; what is the first thing the second thing? What then is to be dcne? How do you obviate this difliculty? What does the cipher you write in the quotient show? What is short division? When emp.byed? Repeat the rule. 5*i f[4Z DIVISION OF SIMPLE NUMBERS. ~ 30 10. Divide 24108 by 12 Quotient, 2009. T. 1. A man gave 86 appies to 5 boys; how many apples did each boy receive? Dividend, SOLUTION. - here, dividing Divisor, 5)86 the number of the apples (86) by the number of boys, (5,) we Quotient, 17 1 lRemainder. findl that each boy's share would be 17 apples; but there is 1 apple left, and this apple, which is called the remainder, is a portion of'he dividend yet undivided. Wherefore this 1 apple must be divided equally among the 5 boys. But!when a thing is divided into 5 equal parts, one of the parts is called 1, (~ 30.) So each boy will have I of an apple more, or 17j- apples in all. Ans. 17-5 apples. NOTE 1. —The 17 (apples) expressing whole apples, are called Integers, that is, whole numbers. Integers are numbers expressing whole things; thus, 86 oranges, 4 dollars, 5 days, 75, 268, &c., are integers, or whole numbers. NOTE 2. — The 1 (1 fifth) of an apple given to each boy, expressing part of a divided apple, is called a Fraction, or broken nurm ber. Fractions are the parlts into which a unit or whole thing may be divided. Thus,. (1 half) of an apple, j (2 thirds) of an orange., (4 sevenths) of a week, are fractions. NOTE 3. - A number composed of a whole number and a fraction, is called a Mi~xed Number; thus, the number 17- (apples) in the above example, is a mixed number, being composed of the integers 17 and the fraction 4If we examine the fraction, we shall see, that it consists of the remainder (1) for its numerator, and the divisor (5) for its denominator Therefore,If there be a remainder, set it down at the right hand of the quotient for the numerator of a fraction, under which write the divisor for its denominator. 2. Eight men drew a prize of 453 dollars in a lottery, how many dollars did each receive? Dividend, Here, after carrying the division as far as Divisor; s) 453 possible by wholp numbers, we have a remainder of 5 dollars, which, written as above Quotient, 568 directed, gives for the answer 56 dollars and a (5 eighths) cf another dollar, to each man. Questions. - ~ 36. What are integers? fracticns? a mixed nuember? If there be a remainder after division, it is a porticn of what? What do you do with it? If yo have a quotient o' 23-9, what w, the rematuder? What was the divisor? ~ 37. DIVISION OF SIMPLE NUMBERS. i q[ T7. PRooF. 1. 16 X 5- 80, Product. } Multiplication and division 2. Dividend, 80 - 5 -16. are the reverse of each other. We see, in the 2d of the above examples, that the product 80 of the 1st example, divided by 5, one of its factors, brings out 16, the other factor, and hence that division may be used to prove multiplication. We see, also, in the 1st exampbe, that the divisor and quotient of the 2d example, multiplied together, reproduce the dividend, and hence that multiplication may be used to prove division. Hence the RULEg. To prove multiplication by To prove division by mu?division.- Divide the prod- tiplication. -Multiply the diuct by one factor, and, if the visor and quotient together, work be right, the quotient and if the work be right, the vill be the other factor. product will be equal to the dividend. NOTE 1. - To prove division, if there be a remainder. Multiply the integers of the quotient by the divisor, and to the product add the remainder. If the work be right, their sum will be equal to the dividend. Example. —Divide 1145 by 7. OPERATION. PROOF. 7 )1145 163 integers of the quotient. 7 divisor. 163. 1141 4 remainder added. 11145 - the dividend. NTTE 2.- Proqf by excess of nines. Find the excess of nines in thile divisor, write it before the sign of multiplication, also in the quotient, and write it after the sign; multiply together these excesses, and write the excess of nines in their product over the sign; subtract the remainder, if any, from the divil'end, and write the excess of nines in what is left under the sign. If the numnbers under and over the sign be alike, the work is presumed to be right, in accordance with principles explained in multiplicatior, ~ 23, note 3. Questions. - ~ 37. To u(hat, in multiplication, does the dividend Vn division answer? To what, the divisor and quotient? How, then, is multiplication proved by cdhi ision? How division by,multiplication * How, when there is a remainder? 56 DIVISION OF SIMPLI,] NUMBERS. I1 38 Let the pupil be required to prve t.e examples which follow. EXAMPLES FOR PRACTICE. 1. Divide 1005903360 by 2, 3, 4, 5, 6, 7, 8, 9, 10, 1 and 12. 2. If 2 pints make a quart, low many quarts in 8 pints i --- in 12 pints? - in 20 pints? - in 24 rints - in 248 pints?.- in 3764 pints - in 47632 pints? Ans. to the last, 23816 quarts. 3. Foul quarts make a gallon; how many gallons in 8 quarts? - in 12 quarts? - in 20 quarts? - in 36 quarts? - in 368 quarts? -- in 4896 quarts? --- in 6436144 quarts? Ans. to the last, 1359036 gallons. 4. There are 7 days in a week; how many weeks in 365 davs? Ans. 521 weeks. 5. When flour is worth 6 dollars a barrel, how many barrels may be bought for 25 dollars? how many for 50 dollars? - for 487 dollars? for 7631 dollars? 6. Divide 640 dollars among 4 men. 640 -4, or 6_4_0 160 dollars, Ans. 7. 678- 6, or 6 = how many? Ains. 113. 8. 5_54-_= how many? Ans. 1008. 9. 2_2_4 = how many? Ans. 1033-. 10. _49-A= how many? Ans. 384g. 11. 2 7 6 4 how many? 12. a4_O 1 how many? 13. 41j1_zl = how many? 9f 38. 1. Divide 4478 dollars equally among 21 men. SoLUvrIoN. -When, as in this example, the divisor exceeds 12, the decomposing into parts cannot be done in the mind as in short division, but the whole process must be written down at length in the following manner. OPERATION. We say, 21 in 44, (hundreds,) 2 (hundred) times, and write 2 on the Div'r. Div'd. Quot. right hand of the dividend for the first 21) 4478 (213k4r figure of the quotient. That is, we 42 1st part. have 2 hundred dollars for each of 21 men, requiring 21 X 2 (hundred) = 42 27 hundred in all., This is the first part 21 2d part. divides. The 42 hundred must now be subtracted from the hur:dreds in the dividend, anl we find 2 (hundred) re63 3d part. maining, to which, bringing down the 5 Rematinder 7 tens, the whole is 27 tens. 21 in 27, (tens,). (ten) time. Each man T 3S. DIVISION OF SIMPLE NUMBERS. 57 has now 10 dollars more, which require'21 tens, the second part, and taking this from the'27 tens, and bringing down the 8 units, we have 68 dollars yet to be divided. 21 in 68, 3 times, that is, each man will have 3 dollars, which will require 63 dollars, the third part, and there are 5 dollars left. This will not give each man a whole dollar, but Ai of. dollar. So each man has 2 hundred, I ten, 3 T dollars; that is, 213iTk dollars, Ans. The parts into which the dividend PROOF. is decomposed, are 42 hundreds, which 1st part, 4200 dollars. contain the divisor 2 (hundred) times; 2d-part, 210 " 21 tens, which contain the divisor 1 3d part, 63 " (ten) time; and 63, which contain Remainder, 5 " the divisor 3 (units) times, or 213 times in all, and the remainder 5. 4478 " We here see that the parts added mroje the whole sum. This method of performing the operation is called Long Division. It consists in writing down the whole work of di viding, multiplying, and subtracting. From the illustrations now given, we deduce the following RULE. To peformnz Long Division. 1. Place the divisor at the left hand of the dividend, and separate them by a curved line, and draw another curved line on the right of the dividend, to separate it from the quotient. II. Take as many figures on the left of the dividend as will contain the divisor one or more times; find how many times they contain it, and put the answer at the right hand of the dividend for the first figure in the quotient. III. Multiply the divisor by this quotient figure, and set the product under that part of the dividend which you divided. IV. Subtract this product from the figures over it, and to the remainder bring down the next figure in the dividend. V. Divide the number this makes up as before. Continue to bring down and divide until all the figures in the dividend have been brought down and divided. PROOF. — Long division may be proved by multiplication, by the excess of nines, by adding up the parts into which the Questions. - ~ 38. What cannot be done, when the divisor exceeds 12? Into what parts is the divl:lend, in the first example, decomposed? How, and for what, is 42 obtained? 21? 6? Explain the proof. What is long division, and in what does it consist? Give the rule. Name the different methods of proof. Give the substance of' note..; note 2; note 3. 611 DIVISION 3F SIMPLE NUMBERS. ~ 38 dividend is decomposed, or by subtracting the remainder from the dividend and &ividing what is left by the quotient, which if the work is right, will bring the divisor. NOTE 1.- Having brought down a figure to the remainder, if the number it makes up will not contain the divisor, write a cipher in the quotient, -ad bring down the next figure. NOTE 2. - When we multiply the divisor by any quotient figuife, and the product is greater than the number we divided, the quotient figuIe is too large, and must be diminished. NOTE 3 - If the remainder, at any time, be greater than the divior, or equal to it, the quotient figure is too small, and must be ire zxeased. EXABMPL4ES FOR PRACTHCE. 1. How many hogsheads of molasses, at 27 dollars a hogshead, may be bought for 6318 dollars? Ans. 234 hogsheads. 2. If a man's income be 1248 dollars a year, how much is that per week, there being 52 weeks in a year? Ans. 24 dollars per week. 3. What will be the quotient of 153598, divided by 29? Ans. 5296A-L. 4. How many times is 63 contained in 30131? Ans. 478 —7 times; that is, 478 times, and 1- of anothu time. 5. What will be the several quotients of 7652, divided by 16, 23, 34, 86, and 92? Ans. to the last, 83.6 6. If a farm, containing 256 acres, be worth 7168 dollars, what is that per acre? Ans. 28 dollars. 7. What will be the quotient of 974932, divided by 365? Ans. 2671lJxV. 8. Divide 322S242 dollars equally among 563 men; how many dollars must each man receive? Ans. 5734 dollars. 9. If 57624 be divided into 216, 5S6, and 976 equal parts what will be the magnitude of one of each of these equal parts? Anis The magnitude of one of the last of these equal parts will be 69 4o. 10. How many times does 1030603615 contain 3215? Ans. 320561 times. 11. The earth, in its annual revolution round the sun, is -said to travel 596088000 miles' what is that per hour, there oeing S76(a hours in a year? Ans. 68000 miles. 12. ~ = 56 how many? Ans. 9445S1, 5 2. 13. -o_ 2_7 =o2 how many? Ans. 5210a2.5 14. 9~ —7,6~Aa- hew many? Ans. 10824-73-1g1. ~ 39, Q40. DIVISION OF SIMPLE NUMBERS. 59 T 4@0. Contractions in Division. ~T 39. I. When the divisor is a composite number. 1. Bought 18 yards of cloth for 72 dollars; how much was that a yard? SOLUTION. - This example is the reverse of Ex. 1, ~[ 24. It was there shown that 3 and 6 are factors of 18 (3 X 6= 18.) If ti e 18 yards ble di OPERATION. vided into 3 pieces, thien 3) 72 dollars, cost of 18 yards. the cost of 1 piece A outd 6) 24 dollars, cost 1 piece 6 yards. be one third as much as, the cost of 3 pieces, the Ans. 4 dollars, cost of 1 yard. is, 72 - 3 = 24 dollars and the cost of I yare would be one sixth of the cost of 6 yards that is, 24 - 6 = 4 dollars. That is, we divide the price of 18 yards by 3, and get the price of one third of 18, or 6 yards, and divide the price of 6 yards by 6, and get the price of 1 yard. Ans. 4 dollars. lHence, To perform division when the divisor is a composite number, RULE. I. Divide the dividend by one of the component parts, and the quotient arising from that division, by the other. II. If the component parts be MORE than two. -Divide by each of them in order, and the last quotient will be the quo tient required. EXAM-PLES FOR PRACTICE. 2. If a man travel 28 miles a day, how many days will it take him to travel 308 miles? 4 X 7 — 28. Ans. 11 days. 3. Divide 576 bushels of wheat equally among 48 (8 X 6) men. Ans. 12 bushels each. 4. Divide 1260 by 63 (=7 X 9) Quotient 20. Ans. 5. Divide 2430 by S1 (- ) " 30. Ans. 6. Divide 448 by 56 (- ) " 8. Anr. v 40. It not unfrequently happens that there are remamz n ders after the several divisions, as in the following example, 1. A mall wished to carry 783 bushels of wheat to market, how many loads would he have, allowing 36 bushels to a load? Questions. - ~ 39. How ray you contract the operation in ditrision when the divisor is a composite number? Which factor sL uid you dcivide by first? Repeat the lrule 60 DIVISION OF SIMPLE NUMBERS. ] 40 SOLUTION. - First, s:ppose his wheat put into barrels, each barre. containing 4 bushels. It would take as many barrels as 4 is contained times in 783. 1 ) 7833 It would take 195 barrels, and leave a amainder 195 3 rem. of 3 bushels. Next, suppose he takes 9 barrels at each load; 9 (barrels) X I (the number of bushels in each barrel) = 36 bushels at a load, and he would have as many loads as the number of times 9 barrels are contained in 195 barrels 9)195 Hence we see, that he would have 21 loads, and leave a remainder of 6 barrels; also, a former re21 6 rem. mainder of 3 bushels. r 4)783 bushels. The whole operation 9)195 barrels, and 3 bushels remainder. stands thus: 21 loads, and 6 barrels remainder. Our object now is to find the true remainder. The last remainder, 6 barrels, multiplied by the first divisor, 4, which is the number of bushels in a barrel, gives a product of 24 bushels. To this add the first remainder, 3 bushels, and we have the true remainder, 27 bushels. Therefore, When there are remainders in dividing by TWO component parts of a number, to get the TRUE remainder, RULE. 1. Multiply the last remainder by the first divisor, and to the product add the first remainder; the sum will be the true remainder. II. When there are DiorE than Two divisors. — Multiply each remainder, except that from the first divisor, by all the divisors preceding the divisor which gave it; to the sum of their products add the remainder from she first divisor, if any, and the amount will be the true remaiader. 2. 5783. 108how many?.X 4X 3-108; hence, three divisors. Questions. - ~40. When. there are remainders in dividing by two component parts, how do you find ihe true remainder? When there are more than two, how? If there 1,e a remainder by the first divisol only, what is the true remainder? if by the 2d divisor and none by the 1st, how do you obtain the true rea.nninder? if by the 3d, and none by the 2d and 1st, how? if by the 1st a nd none bv the 2d, low? Rteeat the rile. ~ 41. DIVISION OF SIMPLE NUMBERS. 61 OPERATION 3) 5783 4) 1927 and 2, 1st rein. 9) 481 and 3, 2d rem. 53 and 4, 3d reinm. 3d remn. 4 X 4 (2d div.) X 3 (st div.; = 48 2d rein. 3 X 3 (1st div.) = 9 1st remn. 2 added 2 True rem. 59 Ans. 53 5-9 NMOTE. - The remainder by the 1st divisor, if there be no other, is tle true remainder. EXAMPLES FOR PRACT ICE. 3. Divide 26406 by 42 = 6 X 7; what will be the true remainder? Ans. 30. 4. Divide 64S23 by 3 component parts, the continued product of which is 96; what will be the true remainder? Ans. 23. 5. AWNhat is the quotient of 6811 divided by the component parts of 81? Ans. 84 7. 6 Divide 25431 by the component parts 3 X 4 X 8 -- 96, first, in the order here given; secondly, in a reversed order, 9, 4, 3; and lastly, in the order 4, 3, 8, and bring out the true quotient in each case. Quotient, 264-1. ~ 1ft. Whzen t/he divisor is 10, 100, 1000, sc. 1. A prize of 2478 dollars is drawn by 10 men; what is each man's share? OPERATION. SOLUTION. - It has been shown (~[ 25) that an 10) 2478 fnexing a cipher to any number is the same as multiplying it by 10; the reverse of this is equally true 947 8 if we cut off the right hand figure from any number 247 it is the same as dividing' it by 10; the figures at the left will be the quotient. Thle figure 8, at the right, Shorter way being an undivided part, is the remainder, and may QUOT. REMt. be written over the divisor (~ 36) thus, -o. We 247 1 8 see that 7, which was tens before, is m-ade units; 4, wlhich was hundreds, is tens~, &c. On the sama principle, if we cot ofl two figrures it is tile same as dividing by 100. i, three figures, the soine as dividing by 10()(, &e. f 62 DIVISION Ov SIMPLE NUMBERS. If 42 Hence, To divide by 1 with any number of ciphers annexed, RULE. Cut off, by a line, as many figures from the right hand of.he dividend as there are ciphers in the divisor. The figures at the left of the line will be the quotient, and those at the right the remainder. EXAMPLES FOR PRACTICE. 2. A manufacturer bought 42604 pounds of wool in 100 cays; how many pounds did he average each day? 42604 + 100 - 426 -U, or 426 104 --- 426,, pounds, Ans. 3. In one dollar are 100 cents; how many dollars in 42425 cents? Ans. 4242 -f5~; that is, 424 dollars, 25 cents. 4. 1000 mills make on0e dollar; how many dollars in 4000 mills? - in 25000 mills? in 845000? Ans. to the last, 845 dollars. 5. In one cent are 10 mills; how many cents in 40 mills? - in 400 mills? - in 20 mills? in 468 mills - in 4603 mills? Anzs. to the last, 460-3 cents. [ 4g~, III. When there are ciph/ers on the right hand of the divisor. 1. A general divided a prize of 749346 dollars equally among an arrmy of 8000 men; what did each receive? 81000) 7491 346 SOLUTION. - The divisor 8000 is a composite number, of which 93 5 r-em,. 8 and 1000 are componellet parts. Dividingc what 8000 men receive by 1000, which we do by cutting 5 (2d remi.) X 1000 - 5000 off the three right hand figures of and 5000 + 346 (lst rem.) the dividend, rwe get 749 dollars, 5346, true?remainder. which 8 men will receive, with s remainder of 346 dollars; and d. viding 749 dollars, which 8 men receive, by 8, we get what 1 man receives, which is 93 dollars, and a remainder of 5. The 5 must be multiplied by the first divisor, 1000, and the first remainder added to the product; or, which is the same thing, (~ 25,) the first remainder, 3146, may be annexed to the 5, and we have the Ans. 93s 4: dolls. Questions. -- 41. If we annex one cipher to any number, how does it affect it? if two ciphers, how? three? &c. If we remove the right hand figure fiom any number, what is the result? How do you divide by 1 with any number of ciphers annexed:? What will expresx the remainder? How do you divide by 10? by 100.? by 10?0? bw 10000 &c. ~ 43. DIVISION OF SIMPLE NUMBEiS. 63 Hence WMlen there art ciphers on the right ]Land of the divisor, RULE. 1. Cut them off, and also, as many figures from the right hand of thle dividend. II. Divide the remaining figures in the dividend by the remaining figures in the divisor. III. Annex the figures cut off from the dividend to the remainder for the true remainder. EXAMPLES FOR PRACTICE. 2. In 1 square mile are 640 square acres; how many square miles in 23040 square acres? Ans. 36 square miles. 3. Divide 46720367 by 4200000. Quot. 1 j2 -o3P675 4. How many acres of land can be bought for o46500 dollairs, at 20 dollars per acre? ALns. 17225 acres. 5. Divide 76428400 by 900000. Quot. 84 -2840oo 6. Divide 345006000 by 84000. Qulot. 4107-a.oo 7. Divide 4680000 by 20, 200, 2000, 20000, 300, 4000, 50, 600, 70000, and 80. Ans. to 9th, 6660 000 ~: 43. Review of Division. Questions. - What is division? In what does the process consist? Define it. The dividend answers to what in subtraction and multiplication, and why? the divisor? the quotient? In what ways is division expressed? Apply the diagram of stars to division. How does long division differ from short division? Why the difference? Rule for short division-for long division. Give the different methods of proof introduced in both. To what does a remainder give rise, and how written? What are fractions? When the divisor is a composite number, how do you proceed? How are the remainders treated? How divide by 10, 100, &c.? How, wnen there are ciphers at the right hand of the divisor? EXERCISES. 1. An army of 1500 men, having plundered a city, took 2625000 dollars; what was each man's share? Ans. 1750 dollars. 2. A certain number of men were concerned in the payment of 18950 dollars, and each man-paid 25 dollars; what was the number of men? Ans. 758. Questions. -~ 42. When there are ciphers on the right hand of the divisor, what do you do first? How do you dii ide? How do you find the true remainder 2 Repeat the rule. 64 DIVISION OF SIMPLE NUMBERS. ~ 43, 3. If 7412 eggs be parked in 34 baskets, how many in a basket? Ans. 218. 4. What number must...:,tlp1iy oy 135, that the product may be 505710? Ans. 3746. 5. LUght moves with such amazing rapidity, as to pass from the sun to the earth in about 8 minutes. Adnmitting the distance, as usually computed, to be 95,000,000 miles, at what rate per minute does it travel? Ans. 11875000 miles. 6. If 2760 men can dig a certain canal in one day, how many days would it take 46 men to do the same? How many men would it take to do the work in 15 days --- n 5 days? --- in 20 days? - in 40 days? - in 120 days? 7. If a carriage wheel turns round 32870 times in running from New York to Philadelphia, a distance of t95 miles, how many times does it turn in running 1 mile? Ans. 346. 8. Sixty seconds make one minute; how many minutes in 3600 seconds. --- in 86400 seconds? - in 604800 seconds? - in 2419200 seconds? 9. Sixty minutes make one hour; how many hours in 1440 minutes? -- in 10080 minutes? - in 40320 minutes? -. in 525960 minutes? 10. Twenty-four hours make a day; how many days in 168 hours? - in 672 hours? in 8766 hours? 11. How many times can I subtract forty-eight from four hundred and eighty? Ans. 10 times, 12. How many times 3478 is equal to 47854? Ans. 132644 times. 13. A bushel of grain is 32quarts; how many quarts must I dip out of a chest of grain to make one half (1) of a bushel 2 - for one fourth () of a bushel? - for one eighth (4) )f a bushel? Ans. to the last, 4 quarts. 14. Divide 9302688 by 648. Quot. 14356. 15. Divide 1030603615 by 3215. Quot. 320561. 16. Divide 5221580 by 68705. Quot. 76. 17 Divide 2764503721 by 83000. Quot..q 33307fi-72212 18. If the dividend be 27586S665090130, and the quotient 5P62916859, what was the divisor? Ans. 490070 'IF 44. MISCELLANEOUS EXERCISES, 60 MISCELLANEOUS EXERCISES, INVOLVING THE PRINCIPLES OF THE PRECEDING RtULES. ~T 44. The four preceding rules, viz., Addition, Subtraction, Multiplication, and I)ivision, are called the Fundamental Rules of Arithmetic, for numbers can hl neither increased nor diminished but by one of these rules; xence these four rules are the foundation of all arithmetical cperaticns. EXERCISES FOR THE SLATE. 1. A man bought a chaise for 218 dollars, and a horse for 1.4i2 dollars; what did they both cost? 2. If a horse and chaise cost 360 dollars, and the chaise zost 218 dollars, what is the cost of the horse? 3. If the horse cost 142 dollars, what is the cost of the chaise? 4. If the sum of two numbers be 487, and the greater number be 348, what is the less number? 5. If the less number be 139, what is the greater number? 6. If the minuend be 7842, and the subtrahend 3481, wha is the remainder? 7. If the remainder be 4361, and the minuend be 7842, what is the subtrahend? 8. If the subtrahend be 3481, and the remainder 4361, what is the minuend? 9. The sum of two numbers is 4S, and one of the numbers is 19; what is the othfer? 10. The greater of two numbers is 29, and their difference 10; what is the less number? 11. The less of two numbers is 19, and their difference is 10; what is the greater? 12. The sum of two numbers is 136, their difference is 28. what are the two numbers? Am. Greater number, 82. s Less number, 54. DMENTAL EXERCISES. 1. When the minuend and the subtrahend are given, how do you find the remainder? Ex. 6. NOTE.- The pupil may be required to give written answers to these mental exercises, or he may answer orally; in either case: let 6-6 MISCELLANEOUS EXERCISES. IT 45 him turn to the exercise for the slate to which reference is made, and let him apply it in illustration of the answer lhe gives. Thus - Ans. Subtract the subtrahend from the minuend, and the difference will be the remainder, as Ex, 6, (slate,) where the minuend and subtrahend are given to find the remainder,- we subtract the subtrahend 3481 from the minuend 7842, and the difference, 4361, is the remainder. 2. When the minuend and remainder are given, how do you find the subtrahend? Ex. 7. 3. When the subtrahend and the remainder are given, how do you find the minuend? Ex. 8. 4. When you have the sum of two numbers, and one of them given, how do you find the other? Ex. 9. 5. When you have the greater of two numbers, and their difference given, how do you find the less number Ex. 10. 6. When you have the less of two numbers, and their difference given, how do you find the greater number? Ex. 11. 7. When the sum and difference of two numbers are given, how do you find the two numbers? Ex. 12. EXERCISES FOR THE SLATE. 4g5. 1. If the multiplicand (squares in a row) be 754, and the multiplier (rows of squares) be 25, what will be the product (no. of squares)? See Diagram, page 69. 2. If the product (no. of squares) be 18850, and the multiplicand (squares in a row) be 754, what must have been the multiplier (rows of squares)? 3. If the product (no. of squares) be 18S50, and the mumltiplier (rows of squares) be 25, what must have been the multiplicand (squares in a row)? 4. If the dividend (no. of squares) be 144, and the divisor (squares in a row) be 8, what is the quotient (no. of rows)? 5. If the dividend (no. of squares) be 144, and the quotient (no. of rows) be 18, what must have been the divisoi (squares in a row)? 6. If the divisor (squares in a row) be 8, and the quotient (rows of squares) be 18, what must have been the dividend (no. of squares)? 7. The product of three numbers is 5a25, and two of th, numbers ars 5 and 7, what is the other number? Ans. 15. ~ 46 MISCELLANEOUS EXERCISES. 67 ME NTAL EXERCISES. When the factors are given, how do you find the product? Ex. 1. When the product and on,, factor are given, how do you find the other? Ex. 2 and 3. When the dividend and quotient are given, how do you find the divisor? Ex. 5. When the divisor and quotient are given, how do you find the dividend? Ex. 6. When the product of three numbers and two of them are given, how do you find the other? Ex. 7. EXERCISES FOR THE SLATE. fT 46. 1. What will be the cost of 15 pounds of butter, at 13 cents a pound? 2. A man bought 15 pounds of butter for 195 cents; what was that a pound? 3. A man buying butter, at 13 cents a pound, paid out 195 cents; how many pounds did he buy? 4. When rye is 75 cents a bushel, what will be the cost of 984 bushels? how many dollars will it be? 5. If 9S4 bushels of rye cost 738 dollars, (73800 cents,) what is the price of 1 bushel? 6. A man bought rye to the amount of 738 dollars, (73800 vents,) at 75 cents a bushel; how many bushels did he buy? 7. If 648 pounds of tea cost 284 dollars, (28400 cents,) what is the price of 1 pound? 28400. 64- = how many? MENTAL EXERCISES. 1. When the price of one pound, one bushel, &c., of any commodity is given, how d6 you find the cost of any number of pounds, or bushels, &c., of that commodity? Ex. 1 and 4. If the price of the 1 pound, &c., be in cents, in what will the whole cost be 2 - if in dollars, what? -- if in shillings? - if in pence? &c. 2. When the cost of any given number of pounds, or bushels, &c., is given, how do you find the price of one pound, or bushel, &c.? Ex. 2, 5, and 7. In what hind of money will the answer be? 3. When the cost of a nunmer of pounds, &c., is given, and also the price of one pound, &c. how do you find the number of pounds, &c.? Ex. 3 and 6. 68 MISCELL,ANEOUS EXERCISES. ~ 47 EXERCISES FOR THE SLATE. ~ 47. 1. A boy bought a number of apples; lie gave away ten of them to his companions, and afterwards bought thirty-four more, and divided one half of what he then had 1among four companions, who received 8 apples each; how tmany apples did the boy first buy? Let the pupil take the last number of apples, 8, and reverse the process. Ans. 40 apples. 2. There is a certain number, to which if 4 be added, and from the sum 7 be subtracted, and the difference be multiplied by 8, and the product divided by 3, the quotient will be 64, what is that number? Ans. 27. 3. If a man save six cents a day, how many cents would he save in a year, (365 days?) -- how many in 45 years? how many dollars would it be? how many cows could he buy with the money, at 12 dollars each? Ans. to the last, 82 cows, and 1 dollar 50 cents remainder. 4. A man bought a farm for 22464 dollars; he sold one half of it for 12480 dollars, at the rate of 20 dollars per aere; how many acres did he buy? and what did it cost him per acre? Ans. to the last, 18 dollars. 5. How many pounds of pork, worth 6 cents a pound, can. be bought for 144 cents? 6.- How many pounds of butter, at 15 cents per pound, must be paid for 25 pounds of tea, at 42 cents per pound? 7. A man married at the age of 23; he lived with his wife 14 years; she then died, leaving him a daughter 12 years of age; 8 years after, the daughter was married to a man 5 years older than herself, who was 40 years of age when the father died; how old was the father at his death? Ans. 60 years. 8. The earth, in moving round the sun, kavels at the rate of 68000 miles an hour; how many miles does it travel in one day, (24 hours?) how many miles in one year, (365 days?) and how many days would it take a man to travel this last distance, at the rate of 10 miles a day? how many years? Ans, to the last, 40800 yea's, [ 48. MISCELLANEOUS EXERCISES. 69 Problems in tihe Measnerment of Rectangles and Sol ids., NOTE. - A rectangle is a figure having four sides, and each of the four corners a square corner or right angle. PROBLEM I. QT 4S. The length and breadth of a rectangle given, to find the square contents. 1. How many square rods in a plat of ground 5 rods lon(r and 3 rods wide? D C SOLUTION. —A square rod is a square measuring 1 rod on each side like one of those in the annexed dia-.___ [_ I - gram. We see from the diagram that there are as many squares in a row as. _ _ there are rods on one side, and as many rows as there are rods on the other side; that is, 5 rows of 3 squares in a A - 3 - - - - _ row, or 3 rows of 5 squares in a row. We multiply the number of squares in one row by the number of rows; 5 X 3 = 15 square rods, Ans. Hence the RULE. AMultiply the length by the breadth, and the product will be the square contents. NOTE. -Three times a line 5 rods long is a line 15 rods long. Hence the pupil must not fail to notice, that we multiply the number of square rods in a piece of ground 1 rod wide and of the given length by the number of rods in the width. EXAMPLES. 2. How many square rods in a piece of ground 160 rods long (squares in a row) and 8 rods wide (rows of squares)? Ans. 1280 square rods. 3. How many sq-aare feet in a floor 32 feet long and 23 feet wide 2 Ans. 736. 4. How many yards of carpeting, 1 yard wide, will it take Questions. —T 48, Describe a rectangle; a square rod. How do you determine the number of squares in a row, and the number of rows? Give the r ile. What is the quantity really mllltiplied?' What absurdity in eonsideaing,, l —rwise? 70 MISCELLANEOUS EXERCISES. ~ 49,50. to cover the floors of two rooms, one 8 yards long and 7 yards wide, and the other 6 yards long and 5 yards wide? Ans. 86 yards. 5. How many square feet of boards will it take for the floor of a room 16 feet long and 15 feet wide, if we allow 12 quare feet for waste? Ans. 252. 6. There is a room 6 yards ong and 5 yards wide; how many yards of carpeting, a yard wide, will be sufficient to cover the floor, if the hearth and fireplace occupy 3 square yards? Ans. 27 PROBLEDI II. v 49. The square contents and width given, to find t e length. 1. What is the length of a piece of ground 3 rods wvide, and containing 15 square rods? SOLUTION.- In this example we have 15, the number e- squares in several rows, (see the diagram, problem I.,) and 3,the number of squares in 1 row, to find the number of rows. We divide tl M squares in the number of rows by the squares in 1 row. Hence, RULE. Divide the square contents by the width, and the quotient will be the length. NOTE. - Or, really, since the divisor and dividend must be of the same denomination, we divide the whole number of square rods by the square rods in a piece of land 3 rods long by 1 rod wide; thus, 15 4 3 = 5 xods in length,.ns. EXAMPLES. 2. A piece of ground containing 1280 square rods, is S rods in width; what is its length? Ans. 160 rods. 3. A floor containing 736 square feet, is 23 feet wide; what is its length Ans. 32 feet. PROBLEM III. ~ 50. The square contents and length given, to find the wzdth. 1. What is the width of a piece of ground, 5 rods long, and containing 15 square rods? Questions, -- 49. Repeat the 2d problern; the example. What wo things are given in the example, and what reluired? Give the rule. What is really the divisor, and why? 1 51. MISCELLANEOUS EXERCISES. 7~ SoLUTION. —-We divide the square contents by the length, or really by the square contents of a portion of the ground 5 rods Ilng and 1 rod wide. Ans. 3 rods. Hence, RULE. Divide the square contents by the length, and the quctienl will be the width. EXAMPLES. 2. A piece of ground containing 1280 square rods, is 160 rods in length; what is its width? Ans. 8 rods. 3. What is the width of a field 186 rods long, and containing 13392 square rods? Ans. 72 rods. PROBLEMI IV. J51. The length, breadth, and hight, or thickness given, to find the contents of a cubic body. 1. How many solid feet of wood in a pile 5 feet long, 3 feet wide, and 4 feet high? SOLUTION. - A solid foot is a solid 1 foot long, 1 wide, and 1 high. By carefully inspecting the diagram, we may see that a portion of wood 5 feet long, 1 foot wide, and 1 high, will contain 5 solid feet. i IMultiplying 5 solid feet by 3, we Ii li' hi!Iget the contents of a portion 5 feet iIlong, 3 feet wide, and 1 foot high. 5 X 3- 15 solid feet; and multiplying 15 solid feet by 4, we get the contents of the whole pile, 15 X 4 = 60 solid feet, Ans. These are the quantities multiplied, out for convenience we adopt the following RULE. Multiply the length by the breadth, and the resulting product by the hight. EXAMIPLES. 2. A laborer engaged to dig a cellar 27 feet long. 21 feet Questions. -.~ 50. Repeat the 3d problem; the example; rule What is really the divisor, and why? ~f 51. What is the 4th problem? the first example? Describe a solid foot. W hat quantity do you multiply in the first multiplication 7 in the second? What rule do ycu adopt for convenience? 12 M3SCELLANEOUS EXERCISES. ~ 52. wide, and 6 feet deep; how many solid feet must he remove? Ans. 3402 solid feet. 3. A farmer has a mow of hay 28 feet long, 14 feet wide, and S feet high; how many solid feet does it contain? Ans. 3136 solid feet. PROBLEMD V. ~f 52. The soZid contents, length, and breadth given, to find the hight, 1. A pile of wood 5 feet long and 3 feet wide, contains 60 solid feet; what is its hight? SOLUTION. - Since the divisor must be of the same denomination as the dividend, (solid feet,) we have given the solid contents of a. pile 5 feet long, 3 feet wide, and several feet high, which we divide by the solid contents of a portion having the same length and breadth, and 1 foot high, to get the number of feet in the hight of the pile. Thus, 5 X 3 = 15, and 60 + 15- =4 feet in hight, Ans. Hence, RULE. Divide the solid contents by the product of the length multiplied by the breadth. EXAMPLES. 2. A man dug a cellar 27 feet long, and 21 feet wide, and removed 3402 solid feet of earth; what was its depth? Ans. 6 feet. 3. A mow of hay, 28 feet long and 14 feet wide, contains 3136 solid feet; what is its hight? Ans. 8 feet. NOTE.-In a similar manner we may find the breadth or the length, when the solid contents and the other two dimensions are given. 4. A pile of wood, 4 feet wide and 6 feet high, contains 360 solid feet; what is its length? Ans. 15 feet. 5. A stick of timber, 78 inches long and 8 inches thick, contains 6864 solid inches; what is its width? Ans. 11 inches. Questions.- ~[ 52. What is the 5tl; problem? the first example? solution? rule? When tne soiid contents, width, and" bight are given, how may the length be found? When the solii contents, length, and hight are given, how may tite width be found? 11 53-55. MISCELLANEOUS EXERCISES. 73, 53., General quzestzans to be answe2red mentally, or by the Elate. 1. If the number of squares be 84, and the squares in a row be 14, how many will be the rows of squares? 2. If the number of squares be 9500, and the rows of squares be I76, how many will be the squares in a row? 3. Were you required to form an oblong field containing 96 square rods, what, and how many ways might you vary the figure, (rows of squares and squares in a row,) each figure to contain just 96 square rods? 4. There is a frame, 40 feet square and 18 feet hiogh, the sides of which are to be covered with boards 13 feet long, 1 foot wide; what number of these boards will it take, allowing only 7 feet waste? Ans. 222 boards. 5. A room, in a furniture warehouse, is 36 feet long and 29 feet wide; how many'tables, 3 feet square, can be set in it, leavipg a space 2 feet wide on one of the sides? Ans. 108 tables. 4. Definitions. In-tegers are distinguished as prime, composite, even, and odd. 1. A Prime number is one that cannot be divided by any integral number except itself and a unit, without a fraction in the quotient; as, 1, 2, 3, 5, 7. NioTi. - Two numbers are prime to each other when a unit is the only whole number that will divide them both without a fractional quotient as, 8 and 1-5, 25 and 36. 2. A Comnposite numbe?, see ~ 24. 3. An Even number is one which is exactly divisible by 2. 4. An Odd number is one which is not exactly divisible by 2. ] o 1. One number is a Measutre of another whe-n it diVides it with/out a remainder. Thus, 2 is a measure of 1S; 5 of 45; 16 of 64. 2. A number is a Caozmon Mieasure of two or more numrbers when it divides each of themr without a remainder. Thus 3 is a common measure of 6 and 18; 7 of 28 and. 42; 4 of 12, 20 and 32;5 of 10, 15, 20, f95. uet-ioie, o w s -'[ 5- 4. How are integers disttinguished? -Ybsl t is. I, rFime nummber? composite nutmber? even number odd number?,J 74 MISCELLANEOUS EXERCISES. I 56, 57 3. One rumber is a Ml~ultiple of another when it can be divided by it without a remainder. Thus 8 is a multiple of 2; 15 of 5; 33 of 11. 4. A number is a Common MultipZe of two or more nulmhers when it can be divided by each of them without a remainder. Thus, 15 is a common multiple of 3 and 5; 16 of 2, 4 and 8; 28 of 4 and 7; 54 of 2, 3, 6, 9, 18 and 27. 5. An Aliquot, or even part, is any number which is contained in another number exactly 2, 3, 4, 5, &c., times. Thus, 3 is an alih-uot part of 15, so also is 5. Each of the numbers, 2, 3, 4, 6,'S, and 12, is an aliquot part of 24. 6. The Reciprocal of a number is a unLit, or 1, divided by the number. Thus, ~ is the reciprocal of 2; ~ of 3;, of 4; 4 of 9, &c. NoTE. - Any number is a multiiple of all its measures andt a measure af c(ll its multiples., e. General Principles of Division. The value of the quotient in division evidently depends on the relative values of the dividend and divisor. EXAMPLE. —Let the dividend be 24, the divisor 6, and -the quotient irill be 4. Miultiplying the dividend by 2, we in effect multiply the quotient by 2. Thus, 24 X 2 =48, and 4S -- 6 - 8, which is 2 times 4, the quotient of 21 -- 6. Again, dividing the divisor by 2, we in effect multiply the quotient by 2. Thus, 6 -- 2 = 3, and 24 3 --- 8, which is 2 times 4, the quotient of 24 + 6, the same as before. Hence, PUINcIPLE I. MIultiplying the dividend, or dividing the divisor, by any number, is in effect -multiplying the quotient by that number. -f 5,. Example as above, namely, dividend 24, divisor 6, and c(iotient 4. Dividing the dividend by 2, we in effect divide the quotient by 2. Thus, 24 2= — 12, and 12- 6 W- 2, which is equal to I half of the quotient of 24 -. 6. Again, multiplying the divisor by 2, we in effect divide the quotient by 2. Thus, 6 X 2 = 12, and 24 +- 12 = 2, which is equal to 1 half of the quotient of 24 +- 6, the same as before. Hence, Questions. - 7[ 5. 7What is a mteasure? common measure? nu.tiple? common multiple? an aliquot part? the reciprocal of a quantity? ~i 56. On what does the value of the quotient in division ldepend? What is the Ist prihcipic? 5 -60. MISCELLANEOUS EXERCISES. 76 PRINCIPLE 11. Dividing the dividend, or multiplying the divisor by any number, is in effect dividing the quotient by,hat number. 58. Example, the same as before. AlTu]tiplying both lividend and divisor by 2 does not alter the:quotient. Thus, 24 X 2 = 48; 6 X 2 12; and 48 -- 12 _ 4, which is equal to the quotient of 24 -6. Again,' dividing both dividend and divisor by 2 does not alter the quotient. Thus, 24-. 2 -- 12; 62- 3; and 12- 3 =_4, which is equal to the quotient of 24 - 6, tho same as before. Hence, PRINCIPLE III. Multiplying or dividingr both dividend and divisor by the same number does not alter the quotient. -T 59. EXAMPLE. It is required to multiply 24 by 6, and divide the product by 6. 24 X 6 - 144, and the product 144 -. 6 24, which is equal to the number multiplied. Hence, PRINCIPLE IV. If a number be multiplied, and the producl divided by the same number, the quotient will be the numnber. This result depends upon the principle that if the product be divided by the multiplier, the quotient will be the multiplicand. ~l 60; Cancelation. 1. How many oranges, at 4 cents apiece, cart be bought for 4 dimes, or 4 ten cent pieces? SOLUTION. -WTe multiply 10 by 4 to get the number of cents, 10 X 4 = 40; then as many times as 4 is contained in 40 so many oranges can be bought. But multiplying 10 and dividinog the pro duct by the same number does not change it, (~I 59}') lhence, we may omit both operations, taking 10 for the result, as followrs: PERATIN. riting 10 and the multiplier 4 ablove, and the OPERATION. divisor 4 below a horizontal line, we strike out 4 1 X_ 10 above and below the line, and we have 10 for tile ~4 result. Ans. 10 oranges. NOTE. -This process of omitting 4 is called caneelation. Whena we cancel a number, we usually draw an oblique line across it. Questions. - 57. AVhat is the 2d principle? ~ 58. What is the 3d principle? v 59. What is the 4th principle? 76 MISCELLANEOUS EXEPtCISES ~ 60. 2. A farmer sold 15 cows for 24 dollars apiece, and took his Say in sheep at 5 dollars apiece; how many sheep did he receive? SOLUTION. - WT/e see that 24 is to be multiplied by the composite number 15 =3 X 5, and the product divided by 5. Using the component parts of the multiplier, we multiply 24 by 13. Now the product of 24 X 3 is to be multiplied and the result divided by 5, which operations we may omit, as follows: TWriting the numbers as already described, w3 OPERATlION. strilke out 5 below, and 15 = 3 X 5 above the line, 3 and above 15 set the factor 3, by which we multi[4 X Ai nly 24. Since there is no number by which to di72 vde this product, it is the result required. Ans. 72 sheep. 3. Multiply 165 by 33, and divide the product by 31; multiply the quotient by 16 and divide the product by 99; multiply the quotient by 62 and divide the product by 55; multiply the quotient by g and divide the product by 20. OPERATION. By closely ini$ 4 2 specting these num-. X X ~ X'~ X 3i X 3 Sat 4 bers,. we see that 4 - all the factors above ~ gX X 4)9 X,is X,0 5 5 the line are canceled,:~ 5 except 4, 2 and 3, which must be multiplied together; and that all the factors below the line are canceled except 5, by which the product of the remaining factors above the line is to be divided. NoTE 1. - It is plain that 16 above and 20 below the line have the factor 4 common, for 16 = 4 X 4 and 20 =4 X 5; we therefore cancel the factor 4 from 16 and 20; this we do if we erase the two numbers, and write 4 the other factor of 16 over it, and 5 the other factor of 20 iunder it. W~e see also that 3, the reserved factor of 165, can eels 3, the reserved factor of 99. NOTE 2. — if the pupil will perform the operations at length, of'm-ltiplying and dividing, in this example, he will see how much is unud by cancelation. Cazrcelation, then, is the method of erasing, or rejecting, a factor ci factors, from any number or numbers. It, may be applied fSO mhortening the operation where both multiplication and divisio, adre required, by rejecting equal factors fromn the nurnbers.o bL zaultipied and the divisors. ~ 60 MIISCELT,ANEOUS EXERCISES. 77 RULE. I Wri;e down the numbers to be multiplied together above, and the divisors below, a horizontal line. II. Cancel all the factors common to the numbers to be multiplied and the divisors. III. Proceed with the remaining numbers as required by the question. NOTE.- One factor en one side of the line will cancel only one like factor oI1 the other side. EXAMi'PLES FOR PRACTICE. 4. A man sold 35 barrels of flour at 5 dollars per barrel, arid took his pay in salt at 3 dollars per barrel; he sold the salt at 4 dollars per barrel, and took his pay in broadcloth at 7 dollars per yard; he sold the broadcloth at 8 dollars per yard, and took his pay in sheep at 2 dollars a head; he sold the sheep at 3 dollars a head, and took his pay in land at 15 dollars per acre; how nmany acres of land did he purchase? If likle factors be canceled from the numbers to be multiplied and the divisors, there will remain of the numbers to be multiplied 5 X 4 X 4 - 80, and of the divisors 3; and = 262. Ans. 26' acres. 5. What is the quotient of 36 X 8 X 4 X 8 X 2 divided by 6 X 5 X 3 X 4 X " NOTE. -The remaining factors of the numbers to be multiplied are 2, 8 and 8, and of the divisors, 5. 6. In a certain operation the numbers to be multiplied are 27, 14, 40, 8 and 6, and the divisors are 7, 10, 12 and 15; what is the quotient? 9 X 2 X 2 X S -28SS, and 2A -+ 5 =57}, Ans. 7. - hat is the quotient of 4X 7 X 18 X 10 X 8 X 9, divided by 24 X 72 X 3? NOTE. --- All the divisors cancel. Ans. 70. 8. If the numbers to be multiplied are 14, 5, 3 and 2S, aind the divisors 15 and 9; what is the quotient? NOTE. -The remaining factor of the divisors is 9. Ans. 43g. Questions. - ~ GO. If a number be multiplied and the product divided by the same number, what is he result? When sn:h operations are to be performed, how may they be contracted? What is this pro.. cess called? Hecw do you indicate that a number is c.arnceled? Wha' is cancelation? When may it be applied? Repeat tne rule. Explain the operation in Ex. 5; in Ex. 6, &c. 7* 7s MISCELLANEOUS EXERCISES. ~ 61, 6g' ST C!. To find a common divisor of two or more numbers 1. Find a common divisor of 6, 9 and 12. 6 = 3 X 2 The factor 3, which is common to the 9PERATION 9 = 3 X 3 several numbers, must be a common divi123>-3X 4 sor of them. IHence the RUJLE. Separate each number into two factors, one of which shal, be comsmon to all the numbers. The common factor will be their common divisor. EXAMPLES'oaR PRACTICE. 2. Find a common divisor of 4, 1i, 24, 36 and 8. Ans. 4. 3. Find a common divisor, or common measure, (which terms mean the same thing,) of 22, 44, 66, and 88. Ans. 11. 4. Required the length of a rod which will be a common measure of two pieces of cloth, one of them 25 feet, the other 30 feet long. Ans. 5 feet. ~T 6I. To find the greatest commnon. divisor, or measure, of two or mzore numbers. Find the greatest common divisor of 128 and 160. FIRST MIETHOD. OPERATION. SOLUTION. - The greatest 128 = 2 X 2 X 2 X 2 X 2 X 2 X 2 common divisor of two or more 160 = 2 X 2 X 2 X2 X 2 X 5 numbers is their greatest common measure, (9T 55,) and is the greatest factor common to them. By separating the numbers into their prime factors, we find the factor 2 occurs 7 times in 128, and 5 times in 160. As no number will contain another, unless it contains all its prime factors, the product of all1 the prime factors common to both numbers must be the greatest common divisor. 2X 2 X 2X2X2X2=-32,. ns. Or, SECOND MIETHOD. By a sort of trial. The greatest common divisor cannot exceed the less number, for it must contain it. We will try, therefore, if the less number, 128, which measures itself, will also divide, or measure, 160. 128)160(1 128 in 160, 1 time, and 32 re7nain; 128, therefore, 128 is not a divisor of 160. We will now try whether this reiaisnder be not the divisor sought; for, if 32 be a 32)128(4 divisor of 128, the former divisor, it must also be-of 128 160, which consists of 128 + 32; 32 in 128, 4 tiames, abitlhout any rems e-iSnder. Consequently it is contained in 160 = 128 + 32, just 5 times; that is, once more than in 128. Acnd as no number greater than 32, the difference of the two numbers, is contained once more in the greater, it is the greatest common divisor. Hence, ~ 62,. MISCELL ANEOUS EXERCISES. 79 UJL]E. I. Separate each number into its prime factors, and the product of all the prime factors common to the several num bers will be the greatest common divisor. Or, II. Dihide the greater number by the less, and that divisor by the remainder, and so on, always dividing the last divisor by the last remainder, till nothing remain. The last divisor will be the greatest common divisor required. NOTE 1 - -When we would find the greatest common divisor ol more than two numbers, we may first find the greatest common divisor of two numbers, and then of that common divisor and one of the other numbers, and so on to the last number. Then will the greatest common divisor last found be the answer. NOTE 2. - Two numbers which are prime to each other, of course, can have no common divisor greater than 1. EXAMPLES FORl. P:RACTICE. 1. Apply the foregoing rule to find the greatest common divisor of 21 and 35. 2. Find the greatest common divisor of 96 and 544. Ans. 32. 3. Find the greatest common divisor of 468 and 1184. Ans. 4. 4. What is the greatest common divisor of 32, 80, and.256? Ans. 16. 5. What:is the greatest common divisor of 75, 200, 625, and 150? Ans. 25. 6. A certain tract of land containing 100 acres, is 160 rods long and 100 wide; what is the length of the longest chain that will exactly measure both its length and breadth? Ans. 20 rods. 7. A has 2640 dollars, B 1680 dollars, and C 756 dollars, which they agree to lay out for land at the greatest price per acre that will allow each to expend the whole of his money; what was the price per acre, and how many acres- did each.nan buy? Ans. A bought 220 acres, B 140 acres, and C 63 acres, at 12 dollars per acre. Questions. -- 62o What is the greatest common divisor of two or more numbers? Describe the process of finding it for twc numbers I rule? How founA when the numbers are more than two? What is tb; grcatest common measure of numbers that are primne to each ether? SO COMMON FRACTIONS. I' 63. COMMON FRACTIONS. f 63. WThen whole numbers, which are called integers (~ 36,) are -ubjects of calculations in arithmetic, the operations are calked operations in whole numbers. But it is often necessary to make calculations in regard to parts of a thing or unit. We may not only have occasion to calculate the price of 3 barrels, 5 barrels, or 8 barrels of flour, but of one third of a barrel, two'fifths of a barrel, or seven eighths of a barrel. When a unit or whole thing is divided or broken into any number of equal parts, the parts are called fractions, or broken numbers, (from the Latin word, frango, I break.) If it be divided into 3 equal parts, the parts are called thirds; if into 7 equal parts, sevenths; if into 12 equal parts, twelfths. The fraction takes its name, or denomination, from the number of parts into which the unit or whole thing is divided. If the unit or whole thing be divided into 16 equal parts, the parts are called sixteenths, and 5 of these parts would be 5 sixteenths. Fractions are of three kinds, Common, (sometimes called Vulgar,) Decimal, and Duodecimal. Common fractions are always expressed by two numbers, one above the other, with a horizontal line between them; thus,,?,. The number belowz the line is called the Denominator, because it gives name to the parts. The number above the line is called the Numerator, because it numbers the parts. The denominator shows into how many parts a thing or unit is divided; and The numerator shows how many of these parts are contained in the fraction. Thus, in the fraction a, the denominator, 8, shows that the unit or whole thing is divided into 8 equal parts, and the numerator, 3, shows that 3 of these parts are contained in the fraction. The numerator, 3, numbers the parts; the denominator, 8, gives them their denomination or Questions. -~ 63. What are integers? What fractions, and whence their necessity? Whence do fractions take their name? How many kinds of fractions? Name them. How are common fractions written.? What is the lower numnber called, and why? What does it show? What is the upper number called, and why? What dtetermines the size of the parts, and why? What are the terms of a fractioa What are the terms of the fraction _- -? _ 2? &c. ~ 64, 65. COMMON FRACTIONS. 9 name, anti shows their size or magnitude; for if a thing be divided into 8 equa_ parts, the parts are birt half as large as if divided into but 4 equal parts. It will evidently take 2 eighths to make 1 fourth. The rumerator and denominator, taken together, are called the terms of the fraction. Thus, the terms of the fraction -a are 7 and 10; of s, 2 and 8. 9T 04. It is important to bear in mind, that fractions arise from division, and that the numerator may be considered a azvidend, and the denominator a divisor, and the value of the fraction the quotient; thus, 1 is the quotient of 1 (the numerator) divided by 2, (the denominator;) ~ is the quotient arising from 1 divided by 4; and 3 is 3 times as much, that is, 3 divided by 4; thus, 1 fourth part of 3 is the same as 3 fourths of 1. Hence, a common fraction is always expressed by the sign of divisioin, the numeraitor being written in the place of the upper dot, and the denominator in the place of the lower dot. 4 expresses the quotient, of which - is the divildeod or denuoerator. 4 is the divisodn or denontiuator. 1. If 4 oranges be equally divided among 6 boys, what part of an orange is each boy's share? A sixth part of I orange is a, and a sixth part of 4 oranges is 4 such pieces, --. Ans. 4 of an orange. 2. If 3 apples be equally divided among 5 boys, what part of an apple is each boy's share? if 4 apples, what? if 2 apples, what? if 5 apples, what? 3. What is the quotient of I divided by 3? --- of 2 by 3? -- of 1by 4 —of2 by 4?-of by 4 —- of 3 by 4? -- of 5 by'7? of 6 by 8? of 4 by 5 -- of 2 by 14? 4. What part of an orange is a third part of 2 oranges? one fourth of 2 oranges? - I of 3 oranges? of 3 oranges? of 4? of 2 + of 5 --- - of 3? of 2?' 65. A fraction being part of a whole thing, is properly.ess than a unit, and the numerator will be less than the denominator, since the denominator shows how many parts Questions. - ~ 04. From what do fractions alwaVS arise? What may the nlumerator be considered? the denominator? Wrhat is the value of the fraction? Of what is A the quotient? a? {-? I of 3 is what part of L? w of 7 is what part of 1? By what is a common fraction al ways expressed? c2 COMMON FRACTIONS, ~ 65. make a w.ole thing, and there must not be so many of the parts taken as will make a whole thing. But we call an expression written in the fractional form a fraction, though its numerator equals or exceeds the denomninator, and its value, consequently, equas or exceeds a unit' but since there is not a strict propriety in the name, it is called an improper fraction. Hence, A Proper Fraction is one that is less than a unit, its numerator being less than the denominator. An Ivmproper Fraction is one that equals or exceeds a unit, its numerator equaling, or exceeding the denominator. Thus, 3, 21, are improper fractions. A Simple Fraction is a single fraction, either proper or improper.'hus, 9, 9, j6, are simple fraction,]. A Compound Fraction is a fraction of a fraction, or several fractions connected by the word of. Thus, I of *, - of'-,- 2 of i of' %6, are compound fractions. A Complex Fraction is one which has a fraction, -eithei 9nrf)le or compound, or a mixed number, for its numerator, or for its denominator, or for both. Thus, 2 4$6, o 2 are 2' ~' 25 complex fractions. A Mixed Number, as already shown, is one composed of at whole numnber and a fraction. Thus, 14-, 13i, &c., ale mixed numbers. A father bought 4 oranges, and cut each orange into 6 equal parts; he gave to Samuel 3 pieces, to James 5 pieces, to Mary 7 pieces, and to Nancy 9 pieces; what was each one's fraction? Was James' fraction proper.or improper? Why? Was Nancy's fraction proper or improper? Why? If an orange be cut into 5 equal parts, by what fraction is 1 part expressed 2 2 parts? - 3 parts? -- 4 parts? -- 5 parts? How many parts will make unity or a whole orange? If a pie be cut into 8 equal pieces, and two of these pieces be given to Harry, what will be his fraction of the pie? if 5 Questions. - ~ 65. WVhat is a proper fraction, and why so called? ~ts value? What is an nnproper fraction, and why so called? When is its value a unit? When grea, er than a unit? Why? What is a sinlple fraction? a simple proper fraction? a simple improper fraction a compound fiaction? a complex fraction? a mLxed number? Whax kind of a fraction is - of i of - %? 3core qu2estions of this character. ~ 66 CC~MON FRACTIONS 8 pieces be griven to John, what will be his fraction? what frac tLon or part of the pie xc ill be left? l 66. 1Reduction of Fractions. Reduction of fractions is changin(r them from one form to another without altering their value. To reduce an inproper frac- To reduce a whole or mixed tion to a whole or 7mixed num- number to anz improlper fracber. tion. 1. In 4 halves (4) of an 2. In 2 whole apples how apple how many whole ap- many halves? ples? SOLUTION. - Since 2 halves SOLUTION.- In 2 apples are (2) of an apple are equal to 1 two times as many halves as there whole apple, 4 halves (~) are are in 1 apple. Since there are equal to as many apples as the 2 halves (2) in 1 apple, there are lumber of times 2 halves are con- 2 times 2 halves in 2 apples, = 4 rained in 4 halves, which is 2 halves, that is, ~, Ans. times. Ans. 2 apples. 3. In 6 of an apple how 4. In 3 apples how many mnany wthole apples? ~ in halves? in 4 apples? in 6 ap8? __ in - 2? - in 20 ples? in 10 apples? in 242 in 4s in _Lo0? in 60? in 170? in 492? - in 9s84? 5. How many yards in 3 6. Reduce 2 yards to of a yard - in ~ of a thirds. Ans.. Reduce 22 yard? - in? - in9? yards to thirds. Ans. ~. Rein At? in -lib duce 3 yards to thirds. - in 15? - in ]-? 31 yards. 3 yards. - in O-2?.~ in t_? -- 5 yards. 523 yards ----- 62 yards. 7. How many bushels in 8 S. Reduce 2 bushels to pecks.? that is, in of a bush- fourths. 2- 2 bushels. el? -- in a~-? -- in'-? _ 6 bushels. 6- I bushin 13? in? els. 73 bushels. - - in 10Q in 3.t? 252 bushels. 9. If I give 27 children 1 10. In 6` orangies how of an orange each, how many nmany fourths of ar. orange 2 aranges will it take? 64 COMMION FRACTIONS. ~ 66 OPERATION. It will take OPEARATION. 4)27 2'; and it is 64 orangzes. evident, that 4 Ans. 64 oranges. dividing the numerator, 24 fourths inz 6 oranges. 27, (= the number of parts con- 3 " cont'd ins the fractiot. tained in the fraction,) by the denominator, 4, (= the number of 27 - 247, A2s. parts in 1 orange,) will give the number of whole oranges, and the Since there are 4 fourths in 1 remainder, written over the de- orange, in 6 oranges there are nominator, will express the frac- 6 times 4 fourths - 24 fourths, tional part. Hence, and 24 fourths + 3 fourths = 27 fourths. Hence, To reduce an improper To reduce a?mzixed nzemnbei fraction to a whole or mixed to an improper frxaction, number, JRULE. RULE.o Divide the numerator by Multiply the denominator the denominator; the quo- of the fraction by the whole tient will be the whole or number; to the product add mixed number. the numerator, and write the result over the denominator. NOTE 1.- A whole number may' be reuced to the form of an inmproper fraction, by writing 1 under it for a denominator. NOTE 2. - A whole number may be reduced to a fraction having a specified denominator, by multiplying the whole number by the given denominator, and taking the product for a nu aerator. EXAlIPLES FOPR. PRACTICE. 11. In -lu, of a dollar, how 12. In 13.- dollars, hoN many dollars? many sixths of a dollar X 13. In 1-dtom of an hour, 14. What is the irpropel how many hours? fraction equivalent to 232-7 hours? 15. In -8763 of a shill]irg, 16. Reduce73013 shillinrgs 12, how many shillings? to an improper fraction. Questions, - ~ 66. What is reduction of fractions? To what is the value of a fraction equal? What is the rule for reducing anl improper fraction to a whole or mixed number? a mixed number to mul improper fraction? How may a whole number b re reuued to the1, ltirm of an ifrproper -IL.ztionT?'How to a f'raction having a. speci:ieai:.(t-V.ir.:inator?? ~ 67. COMMON FRACTIONS. 17. In 3jt1 of a day, how 18. In 1561j days, now many days? many 24ths of a day? Ans. 3 7 6 1 - 3761 hours. 19. In _31_t of a gallon, 20. In 3421 gallons, how how many gallons? many 4ths of a gallon? Ans. _34_l of a gallon 1371 quarts, 21. Reduce 3, 706, 7, 22. Reduce 1l6, 1726, 4786 3465twhl6, 31-t i', 0'344 __, to whole or mixed 1S-5, 4- 8zs6 and 7 3 to imnumbers. proper fractions.? 67. To reduce a fraction to its lowest or most sim:ple terms. If 2 of an apple be divided into 2 equal parts, it becomes s The effect on the fraction is evidently the same as if we had multiplied both of its terms by 2. In either case, the parts are made 2 times as MIANY as they were before; but they are only HALF AS LARGE; for it will take 2 times as many fourths to make a whole one as it will take halves; and hence it is that 2 is the same in value or quantity as I. 2 is 2 parts; and if each of these parts be again divided into 2 equal parts, that is, if both terms of the fraction be mul tiplied by 2, it becomes -. Now if we reverse the above operation, and divide both terms of the fraction 4 by 2, we obtain its equal, 2-; dividing again by 2, we obtain 2, which is the most simple form of the fraction, because the terms are the least possible by which the fraction can be expressed. Hence, - -=- and the reverse of this is evidently true, that4 -2 - I It follows, therefore, by multiplying or dividing both terms of thefraction by the same number, vwe chanlge its terms without alterinog its value. (~ 58.) The process of changing 4- into it- equal -, is called reducing the fraction to its lowest terms. A fraction is said to be in its lowest terms when no number greater than 1 will divide its numerator and denominator without a remainder. 1. Reduce u2g to its lowest terms. OPERATION. TWe find, by trial, that 4 will exactly measure both 128 and 160, and, dividing, 1.28 32 4 wt;we change the fraction to its equal -- )- = - c — Ans. Again, we find that 8 is-a divisor common 160 40 5 to both terms, and, dividing, we reduce 86 COMMON FRACTIONS. ~T 67 the fraction to its equal 4-, which is now in its lowest terms, for no greater number than I m ill again measure them. NOTE 1. —Any fraction may evidently be reduced to its lowest terms by a single division, if we use the greatest common divisor of the two terms. Thus, gwe may divide by 32, which we found (H1 62) to be the greatest common divisor of 128 and 160. 32) 1 2= ABns. Hencn, To reduce a fraction to its lowest terms, RULE. Cancel a11 the factors common to both terms of the fraction. NOTE 2.- By referring to IT 60, the pupil will perceive that to cancel all the factors common to both terms consists in dividing both terms by their greatest common divisor (ST 62), or by any common divisor (ju 61), and the quotients thence arising by any other, and so on, until the terms are prime to each other (~U 54, Note). EXAMrIPLES FOR PRACTICE. 2. Reduce 156, to its lowest terms. Ans. A. 3. Reduce 4 O, 4 64F 15-, and 21 to their lowest terms. Ans. 4 5 Xa NOTE 3. - Let the following examples be wrought by both ir ethods; by several divisors, and also by finding the greatest common divisor. 4. Reduce 4su,,9s9 140 and 1644 to their lowest terms. Ans. L, 3, -, and 3. 5. Reduce T3as4g to its lowest terms. Ans. ~. 6.'Reduce'-4, to its lowest terms. Ans. 2. 7. Reduce 4 6 8 to its lowest terms. Ans.. 7 S. Reduce,2a9 to its lowest terms. Ans. 1. NO.E 4. - The reduction of compound fractions to simple ones is presented in If 79; the reduction of fractions to a common denominator, in ~r 70, 71; and the reduction of complex fractions to simple ones, in ~T 85 (2), for the reason that the pupil is not sufficiently advanced to understand them at this place. Questions.- ~ 67. Give the illustraticn with the half apple. Re verse the operation. What follows? What is the process mentioned, and what is it called? When is a fraction in its lowest terms? Explain.Ex. 1. Hlow can a fraction be reduced by a single division? Rule. Give the note by which you determine by what number you divide. 6l 6S, 69. COIIMMON FRACTIONS. 87 Addition and Subtra&t'on of Fractions. COMMON DENONINATOR. T G. 1. A bov gave to one of his companions f of an orange, to another a, to another k; what part of an orange did he give to all + - -4 -how much' SOLUTION. - The adding together of 2, 4 and ~ of an orange is the same as the adding of 2 oran(es, 4 oranges, and I orange, which would make 7 oianges. The 8 is called the common denomninator, as it is common to the several fractions; and we write over it the sumn of the numerators, to express the answer. Ans. }. 2. A boy had -7 of a dollar, of which he expended W; Arhat had he left? SOLUTION. — w of a dollar is one dime, or ten cent piece. The operation, then, is to subtract 3 ten cent pieces from 7 ten cent pieces, which will leave 4 ten cent pieces, or, Ans. 4s' 3. ~ + 2 _- _ -9 how much? - how much? 3 1 - 13 4 4 4 I +, + 9 + 14 + 2_ho uch 1 3 i3 4. x +YZy sY: 2. -- how much?, how much? 5. A boy, having - of an apple, gave I of it to his sister; what part of the apple had he left? 1- - how much? ~] 69. 1. A boy, having an orange, gave 4 of it to his sister, and ~ of it to his brother; what part of the orange did he give, away? SOLUTION.- The fractions ia and X of an orange can no more be added than 3 oranges and 1 applQe which would make neither 4 oranges nor 4 apples, as they are of different kinds, (~ 12.) But if 1 orange made 2 apples, the 3 oranges would make 6 apples, and the 1 apple being added we should have 7 apples. Now I does make just -, and consequently ~ make 6-, to which if 8 be added we shall have the Ans, 7-. The denominator, 4, of the fraction -3, is a factor of 8, the denominator of the fraction,. And if each term of the fraction 4 be multiplied by 2, the remaining factor of 8 it will be reduced to 8t'1', (6,) without altering its value. (~ 67.) Hence, if the denominator of one fraction be a factor of the deznominzaiztor of anc ther fraction, and both its terms be 7m2lti. Questions. - [ 69. Like what. is the process of adding eighths What is the 8 called, and why? WXV-t is the tenth of a d llar? qS (JCOMMON FRACTIONS. ~ 70. plied by tie renaining factor, it will be reduced to the sainme denominator with the latter fraction, withozut altering its value. (I58.S)'or example: 2. How many 12ths in,? SOLUTION. - The factors of 12 are 3 and 4, the latter of which is the denominator of 4, and multiplying both terms of 4 by 3, the other factor, we have -9x, a fraction of the same value as 4, but having a different denominator. Ans. -2-. 3. A man has -,- of a barrel of sugar in one cask, A in lanother, and 4 in another; how much in all? SOLUTION. - The denominator 6, of the second fraction, is a factor of 12, the denominator of the first; and if both terms of ~ be multiplied by the other factor, 2, it will become a2 Also 4, the denominator of the third fraction, is a factor of 12, and if both its terms be multiplied by 3, the other factor, it will be 1,. And -1 l_~2 — = 1~1 barrels. Ans. 4. What is the amounlt of~, 2, and 5? SOLUTION. - As the denominators are not factors of each other, wc must take some number of which each is a factor. 36 is such a num her. The first denominator, 4, being a factor of 36, both terms ot } may be multiplied by 9, the other factor, and we shall have BIn like manner, both terms of - being multiplied by 6, we have la and both terms of ~ being multiplied by 4, we have 2;- then, 2+2 __. - - 1. Ans. The process in the above examples is called r'educing frac tions to a common denominator, and is necessary when we wish to add or subtract those of different denominators. The common,denominator, it will be perceived, mnust contain, as a factor, each of the other denominators. It is not always manifest what number will contain all the denominators. There are two methods of finding such a number. FIRST METHOD. T70. If several numbers are multiplied together, each will evidently be a factor of the product. We have, then, the following Questions. — 69. How can eighths and fourths be added? When, and how, can one fraction be reduced to the denominator of another? Explain thp third example; the fourth. What is the process called? When is this necessary? What must the common denominator contain? What is not manifest? How many methods of finding it? ~ 71. COMMON FRACTIONS. RJULE..M[ultiply.he numerator and denominator if each fraction by the product of the other denominators. NOTE 1. -The several new denominators will be products of the same nlumbers, and, therefore, will be alike; and the numerator and denominator of each fraction being multiplied by the same number, its value is not altered. See 91 58. NOTE. 2 The common denominator of two or more fractions is the common multiple of all their denominators; see'~ 55. EXAMPLES. 1. Reduce 2, -t and 4 to equivalent fractions having a (,ommon denominator. Each term of 2 being multiplied by 4 X 5. or 20, we have 4-0 t "< 3 " "( 3X.4, or 15, " " 54 it I it t 3 X 4, or 12, In It 4s The terms of each fraction are changed, while its value is not altered. 2. Reduce I, 2, 7, and 4 to equivalent fractions, having a 2' 3' 9' common denominator. Anns. 2o 160o 2 4. -923. Reduce to equivalent fractions, of a common denominator, and add together, ~, 3, and I. Ans. + fo6- 1 -1jl I I-, Amount. 6UU. TU T+ - 5_ __ 4. Add together 4I and e. Amount, 1i. 5. What is the amount,f 1 +- 1 + I — I? A7.s. 24 1-17:= 6. What are the fractions of a common denominator equiva lent to 4 and 5 z Ans. I and a, or -Z- and -. SECOND MIETHOD. ST 71. While we can always find a common denominator by the above rule, it will not always give us the least common denominator. In the last example, 12 as well as 24 is a common denominator of 4 and -W. Let us see how the 12 is obtained. One number will contain another having several factors, when it contains all these factors. For example, let 18 be resolved into the factors 2 X 3 X 3, which, multiplied together, will produce it. It contains 6, the factors of which, 2 and 3, are the first and second factors of 18. It also contains Questions. - [ 70. What is the rule in the first method? Whence'ts propriety? What is a common multiple? Explain the first example 90 1COMMION FRACTIONS. ~ 72, 9, the factcrs of which, 3 and 3, are the second and third factors of 18. But it vill not contain 8 -2 X 2 X 2, for 2 is only once a factor in 18. Now 12, the factors of which are 2 X 2 X 3, will contain 4 — 2 X 2, since these factors are the first and second factors of 12. It will in like manner contain 6. And it is the least number that will contain 6 and 4, for 2 must-be twice a factor, or it will not contain 4, and 3 must be a factor, or it will not contain 6. Hence, no one of these factors can be spared. But 24:2 X 2 X 2 X 3, has, it is seen, 2 three times as a factor; so one 2 can be omitted, and we have the factors of 12 as before. We have 2 as a factor once more than necessary, because it is a factor in both 4 and 6. Hence, when several of the denominators have the same factor we need retain it butt once in the common deonzinator. ~72, The process of omitting the needless factors is called getting the least common denominator of several fractions, and is as follows: 2 4.6 4 and 6 are each divided by 2; and & — the divisor and remainders being taken 2. 3 for the factors of the common denomina2 X 2 X 3 = 12. tor, we have rejected 2 once. 1. Find the least common denominator of 1, 4, a, a, 35.7 2 2.. 6.. 10 SOLUTION. -Ve write the denominators in a line, and divide as here seen. 2?1. 2. 3.4. 5 IBy the first division, 2 existing as a factor in each of tlhe five, numbers, is 1 3. 2. rejected four times, being retained once; as the divisor is substituted for 2 X 2 X 3 X 2 X 5 — 120 the fi've factors 2, which we should have had by multiplying all the numbers together. But 2 being a factor in two of the remainders, it is rejected once more by a second division. Anzs. 120. 2. Find the least common denominator of 7, -r, and v 3 Questions. - ~ q1. Why the necessity of a second method! When will one number contain another? What numbers will 18 contain, and why? What will it not contain? why? Why w?.l 12 contain the denominators of both a and i? WVhy is it the least nun Her that will cmntain them? Why is a factor in 24 once more than necessary What may then be done? ~ 73. COMMON FRACTIONS. 91 FIRST OPERATION. SECOND OPERATION. THIRD 31t RATION. 12 8. 12. 24 4 8. 1i 24 2 8. 12. 24 2 8. 1. 2 3 2. 3. 6 2 4. 6.12 4. 1. 1 2 2. 1. 2 3 2. 3. 6 12X2X496,Ans. 1. 1. 1 2 2 1. 2 4X3X2=24,Ans. 1. 1 A 2 2X 3 X 2 -24, Ans It may be seen that the product of the factors rejected by the first operation is 24, while it is 96 by the second and third. The answer by the first is consequently four times greater, and is not the least commQn denominator. Care must be taken to avoid.this error in practice. The divisor should not be too large. It may always safely, though not necessarily, be the smallest number that can divide any two or more of the denominators without a remainder. NEW NUMIERATORS. ~ 73. 1. Reduce the fractions -1, ~,,, and - to equiva lent fractions having the least common denominator. 2 1 2. 3. 4. 6 SOLUTION. -The new denominator being found, - as above, to be 12, the denominator, 2, of the first 3 1 3.2 3 fraction, has been really multiplied by 6, and to preserve the equality of the fraction, the numerator 1 2. 1 2 must be multiplied by the same number, and A be2 X3 X 212 comes 6. So - 9 and 2 — 4 and hence the fractions are,> - l, -s, -. NOTE 1. -The factor by which the numerator of any fraction is to be multiplied, may be found by dividing the commen denominator byv the denominator of this fraction. Hence, - For reducing fractions to their lowest terms. RULE. Write down all the denominators in a line, and divide by the smallest number greater than 1 that will divide two o1 more of them without a remnainder. Having written the quo Questions. - 772. What is the process called? In the first ex ample, what factors are omitted, and what substituted, by the first divv sion? W, hat by the second? Explain the second example. 92 COMMON FRACTIONS. s 73. tients and undivided numbers beneath, divide as before; and so on till there are no two numbers which can be divided by a number greater than 1. The continued product of the quotients and divisors will be the denominator required. Then multiply each numerator by the number by which its denominator has really been multiplied. NOTE 2.- The least common denominator of two or more fractions: the least common multiple of all their denominators. See ~ 55. 2. Reduce I, 8, and 4 to fractions having the least comnmon denominator, and add them together. NOTE 3.- In writing fractions for addition and subtraction which h.ve a common denominator, the numerators may be written in a line, connected by the appropriate signs, one line extended under them all,'and the denominator written under this line but once. Thus, in the last example, 2 8 6 2 4 mount 3. Reduce I and - to fractions of the least common denom inator, and subtract one from the other. Ans. 3s_ -_ T, difference. 4. There are 3 pieces of cloth, one containing 74 yards another 13 5 yards, and the other 157 yards; how many,yards in the 3 pieces? Before adding, reduce the fractional parts to tli-ir lea.- common denominator; this being done, we shall have,Adding- together all the 24ths, viz. 18 -I 20 +-`- 718~ 21, we obtain 59, that is, -4- 24. WTe rw-ta 134 = 1324 down the fraction.~4 under the other fractiocs, Andi 16 =1544) reserve the 2 integers to be carried to the amounr Arts. 3'77 yds Aof the other integers, making in the whole 374I4Y Ans. 5. There was a piece of cloth containing 343 yards, from which were taken 12at yards; how much was there left? 34- 3429 We cannot take 16 twenty-fourths, ({146,) 12 _ -1216- from 9 twenty-fourths, (); 9 e must, there Ats. 21417 yd fore, borrow 1 integer, = 24 twenty-fourths Ans z ycs.(4 (,) which, with 9~, malkes 3-3-; we can now Questions, - i 73. In getting 12 as the common denominator of the fractions in the first example, by what number has the denoriniatot of. been multiplied? By what, then, must the numerator be multiplied? The same questions in regard to i; in regard to 2. How is this mnul tiplier found? Give the rule. What is the least common multiple? What is done with the sum of the fractions in the fourth exampleI Explain the borrowing in the fifth example. ~ 74, 75. COMMON FRACTIONS. 93 subtract 46 from'2 and there will remain Ax; and taking 12 integers from 33 integers, we have 21 integers remaining. Ans. 21I;. 9' 74. We have, then, for the addition and subtraction Effractions, this general RULE. Add and subtract their numerators when they have a corn mon denominator; otherwise, they must first be reduced to a common denominator.:EXA PLES FORa PRACTICE. 1. What is; the amount of &-, 42, and 12? Ans. 17~t-. 2. A man bought a ticket, and sold a of it; what part of he ticket had he left? Ans. -. 3. Add together a, -, 4,, 1, and -'. Aimount, 24I. 4. What is the difference between 14- 1 and 16k7? Ans. 116 5. From 1 ltake I. Remainder, X 6. From 3 take 41. Rem. 22. 7. From 147~ take 484. Rem. 984. 8. Add together 1121, 3112, and 10004. Alns. 1424~i. 9. Add together 14, 11, 42, -, and i. AIs. 3047. 10. From 4 take. From 7 take a. 11. What is the difference between2 and 1? and.? md2? 4 and? and 4? 5 and 4 2 12. Hlow much is 1 1- 1 1I -4? 1 —? 2 -4 2-? 2-1 9. 34 -? 1000 - l?. Multiplication of Fractions. O 75. I. To vzultiply a Sraction by a whole number. 1. If 1 yard of cloth cost 1 of a dollar, what will 2 yards cost? X2 -- ho-w much? 2. If a cow consume 4 of a bushel of meal in I day, how much will she consume in 3 days? i X 3= how much? 3. A boy bought 5 cakes, at - of a dollar each; what did he give for the whole? 4 X 5 = how much? 4. How much is 2 times? — 3 times4? 2 times t Questions. - 74. Give the rule. How may j be reduced to the aenominator of i? i to the denominator." j 1? (~ 69. 94 COMMON FRACTIONS. ~ 75. 5. Multiply, by 3. - by 2. - by 7. 6. A woman gives to her son 3 of an apple, and to her daughter twice as much; what part of an apple does the daughter receive? SOLUTION. - She gives the son 3 pieces of an apple that had been cut into 8 pieces, and she may give to the daughter twice the number of the same size, that is, 6 pieces, 3 X 2 =-. We multiply the numerator without changing the denominator. Or, she may give the daughter 3 pieces of an apple that had been cut into half as many, that is, 4 pieces, each piece being twice as large. We divide the denominator by 2, without changing the numerator, showing that, as 2 small pieces make 1 large piece, the 8 small pieces will make 4 large oires. Ans. 6, or A. Hence, dividing the denominator, which is the divisor, has the same effect on the value of the fraction as multiplying the numerator, which is the dividend. (~[ 56.) Hence, there ar?- Two ways to multzipy a fraction by a whole nrumber - I. Divide the denominator by the whole number, (when it can be done without a remainder,) and over the quotient write the numerator.- Otherwise, II. Multiply the numerator by the whole number, and under the product write the denominator. If then the product be an improper fraction, it may be reduced to a whole or mixed-number. EyXAMPLES FOR PRACTICE. 1. If 1 man consume - of a barrel of flour in a month, how much will 18S men consume in the same time? - 6 men? - 9 men? Ans. to the last, 1J barrels. 2. What is the product of -T7zsI multiplied by 40? lady X 40 = how much? Ans.- 23-. 3. Multiply 13 by 12. by 18. ---- by 21. ty 36. by 48. - by 60. NOTE 1. When the multiplier is a composite number, we may first multiply by one component part, and that product by the other. (~f 2Q4.) Thus, in the last example, the multiplier, 60 is equal to 12 X 5; therefore, 3A X 12_ 3, and l+ X 5- 6=5 5, Ans. Questions. - ~[ 75. Repeat the sixth example. Why is a fraction multiplied by multiplying the numeratr? Why by dividing the denominator? Give the rule. How may we proceed when the multi plier is a composite number? _{ow is a mixed number multiplied? T 76. COMIMON FRACTIONS. 9b 4. lMBultip y 5- by 7. NOTE 2. TLe mixed number may be reduced to an improper frac tion, and multiplied, as in the _reccding examples; but the operation will usually be shorter to multiply the fraction and whole number seprately, an add add the results together. Thus, 7 times 5 are 35; anu 7 times 4 are 2 1 51, w lhich, added to 35, make 40~, Ans. Or, we may multiply thefraclion first, and, writing down the fraction, reserve the inteers, to be carried to the product of the whoI: number. 5. What will 91Z tons of hay come to, at 17 dollars per ton? Ans. 164 1~ dollars. 6. If a man travel 2T6-, miles in 1 hour, how far will he travel in 5 hours? in 8 hours? ~ in 12 hours - in 3 days, supposing he travel 12 hours each day? Ans. to the last, 772 miles. iff 76. II. To multiply a whole number by afraction. 1. If 36 dollars be paid for a piece of cloth, what costs X )f it? SOLUTION.- If the price of 1 piece of cloth had been given to find the price of several pieces, we should multiply the price of 1 piece by the number of pieces, and we must consequently multiply the price of 1 piece by the fraction of a piece where the price of a fraction is rekquired. The price of 1 piece, 36 dollars, must be multiplied by i. One fourth of the cloth would cost ~ of 36, or 9 dollars, and l would cost 3 times as much, 9 X 3= 27. Ans. 27 dollars. The product is v of the multiplicand, a part denoted by the multiplying fraction. Multiplication, tkerefore, when applied to fractions, does not always imply increase, as in whole numbers; for, when the multiplier is less than unzity, it will always require the produtt to be less than the multiplicand, to which it would be equal if the multiplier were 1. There are two operations, a division and a multiplication. But it is matter of indifference, as it respects the result, which of these operations precedes the other, for 36 X 3 - 4- 27, the same as 36 +-4 X 3 =27. Hence, To 7tmulGtiply by a fraction, we have this RULE. Divide the multiplicand by the denominator of the nmltiQuestions, - ~C 7T. Why must 36 be multiplied by l? How doam the product compare with the multiplicand, and why? Give the rule. r't COMMON FRACTIONS. If 77, 73. plying fraction, and multiply the quotient by the numerator; Dr, when there would be a remainder by division, first multi ply by the numerator, and divide the product by the denomi. nator. 2. What is the product of 90 multiplied by 2? Ans. 45. 3. Multiply 369 by 2. 4. Multiply 45 by 7. Product, 3121. 5. Multiply 210 by t. 6. Multiply 1326 by AT'. Prod. 241J-f. NOTE.- As either factor may be the multiplier, (~ 21,) we may multiply by the whole number, making the fraction the multiplicand. [Hence, the examples in this and O 75, may be performed by the samin rule SW 77. 1. At 40 dollars for 1 acre of land, what will 4 of an acre cost? 40 X 4 - how much? In this example, the price of 1 acre, 40 dollars, is multipliea by the fraction of an acre, 4. Ans. 32 dollars. Hence, When the price of unity is given, tofind the cost'Uf any quantity, less or more than unity, RULE. Multiply the price by the quantity. EXAMPLES FOR PRiACTICE. 2. If a ship be worth 1367 dollars, what is 2 of it worth? Ans. 303J dollars. 3. What cost Ad of a ton of butter, at 225 dollars per ton? Ans. 190-5 dollars.'f 7S. III. To multiply one fraction by another. 1. At i of a dollar for one bushel of corn, what willt of a oushel cost? 9 X = — how much? SOLUTION. - The price of one bushel, A, is to be multiplied by the fraction of a bushel,., (~ 77.) We first divide 4 by 3, to get the price of ~ of a bushel. This wre can do by multiplying the denominator by 3, thus making the parts of a dollar only one third as large, (15ths,) while the same number, 4, is taken..-+ -3==- of a dollar, the price of one Questions. - ~ 77. Explain the first example. What two things are given, and what required? Rule. 1 79 cOMnIMON FRACTIONS. 97 third; and -4 X 2 -= % of a dollar, price of 2 of a bushel Ans. r8 of a dollar. The denominator 5 of the multiplicand is multiplied by 3, the denominator of the multiplier, and 4, the numerator of the multiplicand, by 2, the numerator of the multiplier. Hence, To multiply onefraction by a?wzther, RTULE. Multiply the denominators together for the eroi-nator at the product, and the numerators for the numerator of tlhe product. By this process the multiplicand is divided by the denominator of the multiplying fraction, and the quotient multiplied by the numerator, as in ~ 76. ~EXAMPLES FOR PRACTICE. 2. Multiply { by -. Multiply 9 by 2. Product, 5. 3. At 6s of a dollar a yard, wvhat will ~ of a yard of cloth cost? 4. At 63 dollars per barrel for flour, what will 7- of a barrel cost? NOTE. -Mixed numbers must be reduced to improper fractions. 61 =- 5.l; then 51_ X 7- -3 5S7 - 91Q1 dollars, Anls. ~5. At -e of a dollar per yard, what cost V7 yards? Ans. 6,- dollars. 6. At 21 dollars per yard, what cost 6j yards? Ans. 14~29 dollars. ~1 79. 1. What will j of a yard of cloth cost at 4- of a dollar per yard? SOLUTION. - We multiply the price of 1 yard, 4, by -, the fraction of a~yard, (~ 78.) Getting the price of 2 of a yard is getting 2 of 4 of a dollar. ~ of i is an expression called a compound fraction, (~[ 65.) The reducing of a compound fraction to a simple cne is, then, the same as multiplying fractions together. 2 What is of? of 1 of? - 3. IHlow much is 2 of A of? o (of a we have found to be S, and j-f of l- by the above luleC is 46, Ans. Hence, Questions. - 7~. What is the first operation, Ex. 1? Whence.ts propriety? Second operation? Rule. What is done by the first a;1peration retquiredt i thl rrule? by the second operationl. 98 COMIMON FRACTIONS. g SO The word of between fractions implies their continued mui. tiplication. If there are more than two fractions, we multiply together the several Lumerators and the several denominators. 4. How much is ~ —h of j of 4 of 6 2 Ans. 17s8 = W-N —Tr 5. How much is 45 of 2 of 7 of 3? Ans. H2. ~U SO. 1. How much is -9 of - of 2 of 2? Since the numerators are to be multiplied together, and their product to be divided by the product of the denominators, The operationz may be shortened by Cancelation, (~[ 60.) OPERATION. SOLUTION. - Performing this operation, as described in ~ 60, we have cana i i > 1 celed all the factors of the numerators, of - Of - ofIf- -. and have the factors 2, 2, reuraininl of o0 of o 4' the denominators. But the numerator 2 2 - 2=2 X 1, the numerator 5=5 X 1, 3 =3X 1, &c., and we have in reality he factors 1, 1, 1, and 1, left in the numerators. I XlI 1 X 1 > = 1 ohr the new numerator, and 2 X 2 X 1 X 1 = 4 for the new denomina-,(t,. Hience, when all the factors excepting the l's in the numerators or (lenominators cancel, the new numerator or denominator, as the case may be, will be 1. EXAI'I.PLES FOR PRACTICE IN CANCELATION. 2. 2 of of 6 of f 9 f 7 f ho much? As. 3. 3. WThat is the continual product of 7, -, - of -- and 3? NOTE. -The integer 7 may be reduced to the form of an improper fraction, by writing a unit under it for a denominator, thus, 7. Ans. 2 1. 4. What is the continued product of 3,, of, 25, and +I of 6 of'- 2 Ans. 2im. 5. Reduce 4 of - of - of. of 22- to a simple fracti9n. Ans. 9.10 6, A horse consumed 6 of i of 8 tons of hay in one win t0r; how many tons did he consume? Ans. 2-' tons. 7 Reduce of - of ) of -of of of to a simple fraction. Ans..2 Questions. - ~ 79. How does it appear that we have a compottrai fraction in the first example? What does the word of between fractions imply? What is (lone when there are more than two fractions? T 8,0. Why can cancelation be applied to the tnultiplication of fraclons? Explain the pro:esr.,Vhat is dore vithl integers when occurring with fr'actions? 8 COMMBON FRACTIONS. 99 v9 80. (2.) PROMISCUOUS EXAMPLES IN THE MULTI.-, PLICATION OF FRACTIONS. 1. At 3 dollars per yard, what cost 4 yards of clotn? 5 yards? 6 yards? 8 vards - 20 yards? Ans. to the last, 15 dollars. 2. Multiply 148 lyr. - by;. - by 2. - by 3-a Last product, 44kii. 3. If 2-9 tons of hay keep 1 horse through the winter, how much will it take to keep 3 horses the same time? 7 horses? 13 horses? Ans. to last, 37 7 tons 4. What will 87-~ barrels of cider come to, at 3 dollars per barrel? Ans. 253 dollars. 5. At 14` dollars per cwt., what will be the cost of 147 cwt.? Ans. 21684 dollars. 6. A owned 3 of a ticket; B owned — 6 of the same; the ticket was so lucky as to draw a prize of 1000 dollars; what was each one's share of the money? Ans. A's share, 600 dollars; B's share, 400 dollars. 7. Multiply B of 2 by 3 of t. Product, 1. S. Multiply 7% by 21. " 15. 9. Multiply 7 by 22.,, 21. 10. Multiply 4 of 6 by a-. " 1. 11. Multiply 4 of 2 by 2 of 4. " 3. 12. Multiply continually together A of 8, 2 of 7,. of 9, and a of 10. Product, 20. 13. Multiply 1000000 by a. Product, 5555556. Division of Fractions. f Sl. I. To divide a fraction by a whole number. 1. If 2 vards of cloth cost 2 of a dollar, what does 1 yard cost? how much is 2 divided by 2? 2. If a cow consume 4 of a bushel of meal in 3 days, how much is that per day? i -, 3 = how much? 3. If a boy divide 4 of an orange among 2 boys, how much will he give each one? -- 2 - how much? 4. A boy bought 5 cake! for I- of a dollar; what did 1 cake cost? d -':- 5 6 how much? 5. If 2 bushels of apples cost J?f a dollar, what is tnat per bushel? 1 bushel is the half of 2 bushels; the half of J is -. Ans. ~ dollar 100 CQMMON FRACTIONS. ~ 81 6. If 3 horses consume j4 of a ton of hay in a month, what will 1 horse consume in the same time?'SCLUTION. —-12 are 12 parts; if 3 horses consume 12 such parts in a month, as many times as 3 are contained in 12, so many parts I horse will consume. Arns. x45 of a ton. lhence, we divide a fraction by dividing the numerator wit ihout changing the denominator, taking a less number of paits of the same size. 7. A woman w iuld divide J of a pie equally between her two children; how much does each receive? SOLUTION. - She cannot divide the 3 pieces into 2 equal parts anid leave them all whole. But as the denominator 4 shows into how many parts the pie is cut, multiplying it by 2 is equivalent to cutting the pie into twice as many, or 8 pieces of half the size. That is, we may cut each piece into 2 equal parts, and give 1 of them to each child, who will then have the same number of pieces, 3, only half as large. Ans. -. Hence, a fraction is divided by multiplying its denominator without changing its numerator, as the parts are made smaller, while the same number is taken. Multiplying the denominator, then, which is the divisor, has the same effect on the fraction as dividing the numerator, which is the dividend, (~ 57.) NOTE 1. -By comparing this, and ~ 75, we shall see that where either term of a fraction is to be multiplied or divided, the contrary operation may be performed on the other term. Hence, we have TWO ways to divide a fraction by a whole,zuCmber -- I. Divide the numerator by the whole number, (if it will contain it without a remainder,) and under the quotient write the denominator. — Otherwise, II. Multiply the denominator by the whole number, and over the product write the numerator. Questrions.-T~ 81. How are 2 divided by 4? What difficulty in dividing 3 pieces of pie among 2 children? How then may i be divided? Why does multiplyilg the denominator divide the fraction? _n what, two ways is a fractie a divided? Apply [ 57 to the operation. What appears from comparing this with ~ 75? Repeat the rule. How do you divide a fraction by a composite number? How divide a mixed Eamber? How, when the mixed number is large? ~, 82. COMMUisN FRACT1.IONS. 101 EXAMPLES FOR PRACTICE. 8. If 7 rounds of coffee cost 2- of a dollar, what is that per pound? A + 7 - how much? Ans. 23- of a dollar. 9. If 19 of an acre produce 24 bushels, what part of an acre will produce 1 bushel? 9 +t 24 = how much? 10. If 12 skeins of eilk cost jt of a dollar, what is that a skein { 1 = 12- how much? 11. Divide f by 16. NOTE 2.- When the divisor is a composite number, we can first divide by one component part, and the quotient thence arisirg by the other, (~ 39.) Thus, in the last example, 16 =8 X 2, and. -8-, and b ~-. 24n_,. ad 2An. 12. Divide A- by 12. Divide j7 by 21. Divide -EW by 24. 13. If 6 bushels of wheat cost 4] dollars, what is it per bushel? NOTE 3.- The mixed number may be reduced to an improper fraction, and divided as before. Ans. 3s= 13 of a dollar, expressing the fraction in its lowest terms. 14. Divide 41 1 dollars by 9. Quot. -7: of a dollar. 15. Divide 126 by 5. Quot. -: =. 16. Divide 14i by 8. Quot. le. 17. Divide 1841 by 7. Arns. 26-5,. NOTE 4. - When the mixed number is large, it will be most convenient, first, to divide the whole number, and then reduce the remainder to an improper fraction; and, after dividing, annex the quotient of the /raction to the quotient of the whole number; thus, in the last example, dividing 184. by 7, as in whole numbers, we obtain 26 integers with 21 = remainder, and, dividing this by 7, we have 5i:, aId 26 + =- 26Ai. Ans. 18, Divide 27861 by 6. Ans. 464g. 19. How many times is 24 contained in 7646~{? Ans. 318$-k. 20. How many times is 3 contained in 462~? Ans. i549. ~* 82. I. To iivide a whole number by a fraction. 1. A man would divide 9 dollars among some poor persons giving them. of a dollar each; Liow many will receive hc money? 102.OATcIMON FRACTIONS. ~ 98 SOLJ ON,, N- A wi h to see srow cany times a of a dollar is contained in 9 dollars. But,x'- ERATION. as the divisor is 4thS, (25 9 cen. pieces,) we must re4 duce'he dividend to 4th's, as both must be of the 4'st of a dollar, 3) 36 4't of a dollar. same denomination, (~ 33;) thus, we multiply 9 12 persons. by 4, to reduce it to' 4th, since there are 4 fourths in one dollar. Then, as many times as 3 fourths are contained in 36 fourths, so many persons will receive the money. We find the number to be 12 persons, a number greater than the dividend or number of dollars. Division, then, when applied to fraeLions, does not always imply decrease. The quotient is greater than the dividend when the divisor is less than 1, to which it is just equal when the divisor is 1. Hence, To divide a whole number by a fraction, RULE. Multiply the dividend by the denominator of the dividing fraction, (thereby reducing the dividend to parts of the same magnitude as the divisor,) and divide the product by the nuraerator. EXAMPLES FOR PRACTICE. 2. How many times is I contained in 7? 7 = how many? 3. How many times can I draw ~ of a gallon of wine out of a cask containing 26 gallons? 4. Divide 3 by -- 6 by a. - 10 by. 5. If a man drink -9 of a quart of beer a day, how long' will 3 gallons last him? Ans. 21~ days. 6. If 2j bushels of oats sow an acre, how many acres will 22 bushels sow. 22 - 23 - how many times? NOTE. —Reduce the mixed number to an improper fraction,!` r~t-d. Ans. 8 acres 7. How many times is a contained in 6? Ans. t of 1 time. 8, How many times is 8$ contained in 53? Ans. 6rP- times. Questions. — ~ 82. How is the principle that the divisor and divi Rend must be of the same denomination'applied to the first example When is the quotient greater than the dividend, and when equal to it Give the rule for dividing a whole number by a fraction. ~ 83, S4. COMMON FRACTIONS. 10l 9T 83. 1. At J of a dollar per yard, how much cloth can be bought for 12 dollars? SOLUTION. - As many times as. of a dollar is contained in (or car, De subtracted from) 12 dollars, so many yards can be bought. Ans. 18 yards. Hence, When the price of unity and the price of any quan fity are given, to find the quantity, RULE. Divide the price of the quantity by the price of unity. EXAMPLES FOR PRACTICE. 2. At 41 dollars a yard, how many yards of cloth may be bought for 37 dollars? Ans. S29 yards. 3. At -F96% of a dollar a pound, how many pounds of tea may be bought for 84 dollars? Ans. 90W pounds. 4. At ~ of a dollar for building 1 rod of stoge wall, how many rods may be built for 87 dollars? 87 - i- how many times? Ans. 104* rods. - S84. III. To divide onefraction by another. 1. At 2 of a dollar per bushel, how many bushels of oats can be bought for 5 of a dollar? SOLUTION. — We are to divide - by 2. (~ 83.) To divide by a fraction we multiply the dividend by the denominator of the dividing fraction, and divide the product by the numerator. (ff 82.) X X9=-49, and 45-_. i —- 3 bushels, Ans. Hence, RULE. Invert the divisor, and multiply together the two upper terms for a numerator, and the two lower terms for a denominator. NOTE 1.- In the above rule it will be seen that the numerator of the dividend is multiplied by the denominator of the divisor, and thus the dividend is multiplied by this denominator, and the denominator of the dividend is multiplied by the numerator of the divisor, and thus the dividend is divided by this numerator, as in IT 82. Questions. - ~ 83. What two things are given, and what required. Efx. 1? Rule. ~. 84. How do we divide by a fraction? iow multiply a fraction How divide a fraction? Rule for dividing one fraction by another Whiat thereby is done? 104 COMMON FRACTIONS. if[ Sb EXAMPLES FOR PRACTICE. 2. Divide 1 by 1. Quot. 1. Divide I by 4I. Quot. 2. 3. Divide 4 by 1. Quot. 3. Divide 7 by 19. Quot. S4. If 43 pounds of butter serve a family 1 week, how many weeks will 367 pounds serve them? NOrTE. 2 The mixed numbers, it will be recollected, may be reduced to improper fractions. Ans. 81 a weeks. 5. Divide 24 by 14.- Divide 103 by 24. Quot. 14. Quot. 444. 6. How many times is ad contained in? Ans. 4 times. 7. How many times is I contained in 44? Ans. 114 times. 8. Divide Z of i by i of Q. Quot. 4. ST S5. 1. If 7 of a yard of cloth cost 3 of a dollar, what is that per yard? SOLUTION.- Had the price of several yards been given, we wouln divide it by the number of yards, to find the price of 1 yard, and, in like manner, we must divide the price of the fraction of a yard (4 of a dollar) by the fraction of a yard, (7,) to find the price of 1 yard. Ans. 044 of a dollar per yard. Hence, When the price of any quantity less or more than unity is given, to find the price of unity, RIULE. Divide the price by the quantity. EXAMPLES FOR PRACTICE 2. At 4 of a dollar for 34 bushels of apples, what does 1 bushel cost? Ans. i of a dollar. 3. At 4 of a dollar for 423 bushels of oats, what does 1 bushel cost? Ans. A of a dollar. T S5. (2.) Reductiot of complex to simple fractions. 1. What simple fraction is equivalent to the complex frac, 2 tion 4? SOLUTION. — Since the numerator of a fraction is a dividend of Questions. - ~ 85. What two things aregivenr ana li hat required, Ex. 1? Give the rule. If L5. COMMON FRACTIONS. 105 which the lenominator is the divisor, we may dHiide j by A, by the rule, ~ 84. 2 - -- lg. Ans. 2. NVhat simple fraction is equal to 7 OPERATION. SOLUTION.- We reduce 43 to the im44 -_ 1, and proper fraction AV, which we divide by 7,. 7 i ==, Ans. according to the rule, ~ 81. The above illustrations are sufficient to establish the following RULE. Reduce any mixed number which may occur in the coinplex fraction to the fractional form, or any compound fraction to a simple one, after which divide the numerator by the denominator, according to the ordinary rules for the division of.ractions. EXAMPLES FOR PRACTICE. 3. Reduce the complex fraction V to a simple one. Nisi = 1Zs = 21g 4. What is the value of 6 Ans. 19. 3 5. What simple fraction is equal to 93. Ans. { 6. What simple fraction is equal to Ans. _g 7. What simple fraction is equal to 9Ans.'A 10 8. What simple fraction is equal to 1.? Ans. f =262. 9. What simple fraction is equal to 7- Ans. ~o. l',, What simple fraction is equal to Ans.. 11. What simlnle fraction is equal to I of t 2 3f-q X 2* Ans. 3. Questions. - ~ 85. (2.) How is the complex fraction, Ex. 1, re dtuced o a simple one? why? Give the rule for reducing complex to simple Tract icas. 106 COMMION FRACTIONS. s 85, S6 91 S5, (3.) PRONTLSCUOUTS EXAMPLES N THE DIVIlSION oF FRACTIONS. 1. If 7 lb. of sugar cost 6a,3 of a dollar, what is it pei pound? 6z 3 7 - how much?z of r63, is how much I 2. At - of a dollar for 3 of a barrel of cider, what is that per barrel? AnZs ~ of a dollar. 3. If 4 pounds of tobacco cost i of a dollar, what does I porund cost? Ans. 72 doll. 4. If 7 of a yard cost 4 dollars, what is the price per yard Ans. 4t dollars. 5. If 148 yards cost 75 dollars, what is the price per yard? Ans. 5~x dollars. 6. At 31- dollars for 10! barrels of cider, what is that per tarrel? Ans. 3 dollars. 7. How many times is R contained in 746 Ans. 19893 8. Divide 1 of i by 4. Divide - by $ of 2. Quot. a. Quot. 3GT. 9. Divide X of 4 by of 2. Quot. 4 10. Divide A of 4 by. Quot. 3. 11. Divide 45 by i of 4. Quot. 2-LF 12. Divide X of 4 by 4!. Quot. ~-. 13. Divide by a Quot. 9. 9 1. R geilew of Common Fractions. Questions, - What are fractions? Whence is it that the parts into which any thing or any number may be divid(ed, take their name? What determines the magnditude of the parts? Why? Hlow does increasing the denominator affect the value of the firaction? Increasing the numerator affects it how? How is an inlproper fraction reduced to a whole or mixed number? How is a mixed number reduced to an improper fraction? a whole number? How is a fiaction reduced to its most simple or lowest terms? I-low is a common divisor found? (I 61.) the greatest common divisor? (~ 62.) Whence the necessity of relucing fractions to a common denominator? When may one fraction be reduced to the denominator of another? AWrhat must the common denominator be? (~[ 69.) Give the first method of finding it, anti the principles on which it is founded; the seconod method, and the principles. What is understood by a m1ultiple? by a commonl?mutltiple? by the least common multiple? What is the process of finding it? (T 72.) How are fractions added and su'otracted. vHow many ways are there to multiply a fraction by a whole number? How does it appear, that dividing the denominator multiphles the fractionz? How is a nmixed number multiplied? What is implied in multiplying by a fraction? Of what operations does it consist?'When the multiplier is less than a unit, what is the prod let compared with the multiplicand? What two things are ~ 86. COMMiON FIRACTIONS. 107 given, and what required in ~Q 77? What in ~ 85? What in a[ 85 Explain the principle of multiplying one fraction by another. Of dividing one fraction by another.# How do you multiply a mixed number by a mixed number? How does it appear, that in multiplying both terms of the fraction by the same number, the value of the fraction is not altered? How many, and what are the ways of dividing a fraction by a whole number? How does it appear that a fraction is divided by multiplying its denominator? How does dividing by a fraction differ from multiplying by a fraction? When the divisor is less than a unit, what is the quotient compared with the dividend? How do you divide a whole number by a fraction? EXERCISES. 1. What is the amount of 6 and - of and 2 a 2 — of 121, 3a, and 43? Ans. to the last, 20L. 2. How much is 1 less a 3?. -?. - 14 -- 41? 6-4`? j4-0 of of i? Ans. to the last, Ah,. 3. What fraction is that, to which if you add a the sum will be 5? Ans. 13 4. What number is that, from which if you take i the remainder will be 4? Ans. 2fs 5. What number is that, which being divided by 3 the quotient will be 21? Ans. 153. 6. What number is that, which multiplied by 2 produces 1? Ans. 4. 7. What number is that, from which if you take 2 of itself the remainder will be 12? Ans.- 20. 8. What number is that, to which if you add 2 of A of itself the whole will be 20? Ans. 12. 9. What number is that, of which 9 is the i part? Anis. 13-. 10. At -i of a dollar per yard, what costs 4 of a yard of cloth? Ans..d of a dollar. 11. At 5S dollars per barrel, what costs 18I- barrels of flour? AZns. 108S dollars. 12. What costs 84 pounds of cheese, at fix of a dollar per pound? Ans. 1121 dollars. 13. What cost 45 yards of gingham, at i of a dollar per yard? Ans. 28s dollars. 14. What must be paid for 1-g of a yard of velvet, at 5 dollars per yard? Ains. 23T dollars. 15. If 7 of a pound of tea cost "-I of a dollar, what is that per pc undl? Ans.,s of a dollar. 16. If 74 barre.s of pork ctt 73t dollars, what is that per Qarrel? Anrs. 104 dollars. 108 DECIMAL FRACTIONS,. J 87 17. ]f 4 acres of land cost 82$9 dollars, what is that pe~ acre? Ans. 20t- dollars. 18. At 25 of a dollar for 3~ bushels of lime, what costs 1 oushel? Ans. h of a dollar. 19. Paid 4- dollars for coffee, at Z% of a dollar per pound, how many pounds did I buy? Ans. 291 pounds. 20. At 12 dollars per bushel, how much wheat may be bought for 82 dollars? Ans. 59 3 bushels. 21. If 84 yards of silk make a lress, and 9 dresses be made from a piece containing 80 yards, what will be the remnant left? Ans. 14 yards. NOTE. -Let the pupil reverse and prove this, and the following example. 22. How many vests, containing 7 of a yard each, can be made from 22 yards of vesting? what remnant will be left? Ans. 25 vests. Remnant, I yard. 23. What number is that, which being multiplied by 15 the product will be -? Ans. I. 0of -U 6 24. What is the product of 7 into 3 7 3A~ Ans. 3. 25. Which of the eleven numbers, 8, 9, 1, 12, 14, 15, 16, 8, 20, 22, 24, have all their factors the same as factors in 72? (I~ 61.) NorE. - The 72 must be resolved into the greatest number of factors possible, which are 2, 2, 2, 3, 3. In like manner, each of the other numbers must be resolved. Ans. 8, 9, 12, 18, 24. of 9 2 of 1 of 2A 26G. What is the quotient of 4- - divided by 1 f 19 Ans. 10- 9 5 69 DECIMAL FRACTIrONS. v S7. We have seen (~ 69) that fractions having differ cnt denominators, as thirds, sevenths, elevenths, &c., cannot be added and subtracted until they are changed to equal fractions, having a common denominator —a process xvwhich is *often somewhat tedious. To obviate this difficulty, Decimal fractions have been devised, founded on the Arabic system of notatio;. ' 87. DECIMAL FRACTIONS. 109 If a unit or whole thing be divided into'0 equal parts, each of those parts' will be I tenth, thus, I of 1 =- IT; and if each tenth be divided into 10 equal parts, the 10 tenths will make 100 parts, and each part will be 1 hundredth of a whole thing, thus, I of -1. In like manner, if each hundredth be divided into ten equal parts, the parts will be thousandths, 1,,of Tv1-=T —-, and so on. Such are called Decimal Fractions, from the Latin decem, meaning ten. Comnmon fractions, then, are the common divisions of a unit or whole thing into halves, thirds, fourths, or any number of parts into which we choose to divide it. Decimal fractions are the divisions of a unit or whole thing first into 10 equal parts, then each of these into 10 other equal parts, or hundredths, and each hundredth into 10 other equal parts, or thousandths, and so on. The parts of a unit, thereby;increase and decrease in a ten fold ratio in the same manner as whole numbers. The following examples will show the convenience of decimal fractions. 1. Add together', and 15, SOLUTION. -Since 1 tenth makes 10 hundredths, we may reduce the tenths to hundredths by annexing a cipher which, in effect, multiplies them by 10. Thus, - =-220 hundredths, (T203?,) and adding 15 hundredths, (fl,) we have 35 hundredths, (13%.) 2. From 1-f6u take x%7. SOLUTION.-We reduce the 36 hunf= 36~0 thousa~ndths. dredths to thousandths by annexing a 1S7 thousandths. 187 thousandths. cipher to multiply it by 10. Then subtracting and borrowing as in whole 173 thousandths. numbers, we have left 173 thousandths, (T 05~') Questions. — T 87. What are fractions? What occasions the chief difficulty in operations with common fractions? To what has this difficulty led? What are decimal fractions? Why are they so called. What are the divisions and subdivisions of a unit in decimal fractions. and what are the parts of the 1st, 2d, &c., divisions called? With what system of notation do these divisions of a unit correspond? What is the law of increase and decrease in the Arabic system of notation? What, then, do you say of the increase of the whole numbers, and of the parts? How do these divisions of a unit in decimal fractions differ from the divisions of a unit in common fractions Give examples st(wing the superiorty of decimnl fractions. 110 DECIMAL FRACTIONS. ~ 8 NOTE. -The pupil will notice, that in thus reducing the fraction 1-u6, the Tr{ makes 60 thousandths, and the 130- makes 300 thousandths. Notation of Decimal Fractions. ~T SS. 1. Let it be required to find the amount of 325k + 16T-178- +- 4Y-72 + 1-2 5 and express the fractions decimally. SOLUTION. -Since 1 hundred integers make 10 tens, 1 ten 10 units, 1 unit 10 tenths, 1 tenth 10 hundredths, &c., decreasing uniformly from left to right, we may write down the numerators of the fractions, placing tenths after units, hundredths after, tenths, and so on, in this way indicatinc their values without expressing their denominators. We jplace a point (') called the Decimal point, or Separalrix, on the left of tenths to separate the fraction from units, or w~hole numbers. s) u a1 As 10 in each right -3 t hand make 1 in the next,,j Xrj X,1 ) as left hand column, the hiZ U1 ~ =,, adding and carrying will:'E d o: a): ~'~ Zz be the same throughout 3 2 5'5 3 2 6;5 0 0 as in whole numbers. 3 2 5,5 3 2 5'5 0 0 1 6'7 8 or, 1 6T7 8 0 Thereducingtoacom4,3 7 9 413 7 0 mon denominator, it will'O 2 5'0 2 5 be seen, is simply filling up the vacant places with 3 4 6'6 8 4 3 4 6'6 8 4 ciphers, which are omitted in the first operation without affecting the result, each figure being written in its proper place. The denominator to a decimal fraction, although not expressed, is always understood, and is 1 with as many ciphers annexed as there are places at the right'hand of the point. Thus,'684 in the last example, is a decimal of 3 places; con sequently 1, with 3 ciphers annexed, (1000,) is- its propel denomrinator. Any decimal mnay be expressed in the form of a common fraction by writing under it its proper denominator. Thus,'684, expressed in the form of a common fraction, is 684 Questions, -~ 88. Have decimal fractions numerators and denominiators, and both expressed? How can you write the numerators so as to indicate the value of the fi mcti(mo, without expressing the denote inator? How can the denominataor be known, if it is r.ot expressed What is the separatrix and its use? How can a decirna. be,yxpressea in form of a common fraction? How mas:~. and what at dvanrulges 1have oleclnal overl con11iloll'n frattio;ls? S8, DECIMiAL FRACTIONS. 1 1 NOTE.- rTe decimal point can never be safely omitted in operations with decimals. Decimal fractions have two advantages over common frac, tions. First,-They are more readily reduced to a common denominator. Second, —They may be added and subtracted like whole numbers, without the formal process of reducing them to a common denominator. The names of the places to ten-millionths, and, generally how to read or write decimal fractions, may be seen froml the following TABLE. 1 c.. r 3d place. HCu 11 11 f l 1 I Hundreds. 2d place. cw t\ Tens. CD 1st place. ~ c, -.. Units. 1st place. o c oo m c oo cnRenths. cD 2d place. c cC o Co cA cA Hundredths. 3d place. o o o Thousandths. S 4th place. c O 1 w Ten-Thousandths. It S5th place. o o Hundred-Thousandths 6th place. o c Millionths. - 7th place. o Ten-Millionths - f 00 ~ C C ~ rn3 rn, mFc - a-.(Ec:r 11'2 DECIMAL FRACTIONS. ~ S9, 90 ~f 89. To read, Decimals.- As every fraction has a numnerator and a denominator, to read the decimal fraction, of which the denominator is not expressed, requires two enumerations,-one from left to right to ascertain the denominator, that is, the name or denomination of the parts, and another from right to left, to ascertain the numerator, that is, the num her of parts. Take, for instance, the fraction'003S7. We begin at the first place on the right hand of the decimal point and say, as in the table, tenths, hundredths, &c., to the last figure, which we ascertain to be hundred-thousandths, and that is the name or denomination of the fraction. Then, to know how many there are of this denomination, that is, to determine the numerator, we begin as in whole numbers, and say units, (7,) tens, (8,) hundreds, (3,) which being the highest significant figure, we proceed no further; and find that we have 387 hundred-thousandths, (T3gsy70W,) the numerator being 3S7. In this way a mixed number may be read as a fraction. Take 25'634. Beginning at the first place at the right of the point we have tenths, hundredths, thousandths, (the lowest denomination;) then beginning at the right with 4, we say, units, tens, &c., as in whole numbers, and find that we have 25634 thousandths, which, expressed as a common fraction, is 25634 ST 90. To zorite Decimal Fractions. I. Write the given decimal in such a manner that each figure contained in it may occupy the place corresponding to its value. II. Fill the vacant places, if any, with ciphers, and put the decimal point in its proper place. Forty-six and seven tenths = 461o -46'7. Write the following numbers in the same manner: Eighteen and thirty-four hundredths. Fifty-two and six hundredths. Nineteen and four hundred eighty-seven thousandths. Twenty and forty-two thousandths. One and five thousandths. 135 and 3784 ten-thousandths. 9000 and 342 ten-thousandths. Questions. - ~ 89. To read decimals requires what? how made and for what purposes? How may a mixed number be read as a frae tlon? go90 What is the rule for writing decimal fractions 2 ~ 91. DECIMAL FRACTIONS. 3 10000 and 15 ten-thousandths. 974 and 102 millionths. 320 and 3 tenths, 4 hundredths, and 2 tbousan& ns 500 and 5 hundred-thousandths. 47 millionths. Four hundred and twenty-three thousandths. Reduction of Decimal Fraction s 9r 91. The value of every figure is determinE,,m' i),:.'s place from units. Consequently, ciphers annexed to (I tL,,ij1s do rot alter their value, since every significant figur, continues to possess the same place from unity. Thus,'5, 50(),'500, are all of the same value, each being equal to 5, or I. But every cipher prefixed to a decimal diminishes it tenfold, by removing the significant figures one place further from unity, and consequently making each part only one tenth as large. Thus,'5,'05,'005, are of different value,'5 being equal to A-, or -;'05 being equal to a5r, or Ia; and'005 being equal to, or A whole number is reduced to a decimal by annexing ciphers; to tenths by annexing 1 cipher, since this is multiplyIng by 10; to hundredths by annexing two ciphers, &c. Thus, if 1 cipher be annexed to 25 it will be 25'0, (250 tenths;) if 2 ciphers, it will be 25'00, (2500 hundredths.) Several numbers may be reduced to decimals,- having the same or a common denominator, by annexing ciphers till ll have the same number of deck,-l nlaces. Thus, 15'7,'75, 12'183, 9'0236 and 17' are reducea bo ten thousandths, the lowest denominator contained in them, as follows: 15'7 - 15'7000, annexing three ciphers'75 -'7500, " two " Questions. - ~ 91. How is the value of every decimal figure de. termined? How do ciphers at the left of a decimal affect its value? al the right, how? In the fraction'026 t3, what is the value of the 4? of the 2? How does the 0 affect the vat ne of the fraction? In the fractioa,15012 is the value of each significant figure affected by the 0? if not, point out the difference, and wherefore? If from the fraction'8634 w( withdraw the 6, leaving the fraction to consist of the other three figures only, how much should we deduct from its value? Demonstrate by some process that you are right. How may integers be reduced to decimals? Reduce 46 to thousandths; how many thousandths does the number make? How may several numbers be reduced to decimals having a commoa denominator? To what denominator should they al be reduced? 10* 114 DECIMAL FRACTIONS. ~ 92, 12'183 - 12'1830, annexing one cipher. 9'0236 9'0236, already ten-thousandths. 17' 17'0000, annexing four ciphers. NOTE. - All the numbers should be reduced to the denominator oi lihe one having the greatest number of decimal places. EXAMPLES. 1. Reduce 7'25, 14'082, 243,'00083, and 25 to a common denominator. 2. Reduce 2'1, 3'02,'425, 32'98762, and'3000001, to a common denominator. ~T 92. To change or reduce Common to Decimal Fractzons. 1. A man has ~ of a barrel of flour; what is that, expressed in decimal parts? As many times as the denominator of a fraction is contained In the numerator, so many whole ones are contained in the fraction. We can obtain no whole ones in i, because the denominator is not contained in the numerator. We may, however, reduce the numerator to tenths, by annexing a cipher to it, (which, in effect, is multiplying it by 10,) making 40 tenths, or 4'0. Then, as many times as the denominator, 5, ts contained in 40, so many tenths are contained in the fraction. 5 into 40 eight times, and no remainder. Ans.'S of a bushel. 2. Express 4 of a dollar in decimal parts. OPERATION. The numerator, 3, reduced to IVm. tenths, is U-, 3'0, which, divided Denom. 4 ) 3r0 ('75 of a dollar. by the denominator, 4, the quo28 tient is 7 tenths, and a remainder of 2. This remainder must now be reduced to hundredths by an20 nexing another cipher, making 20 20 hundredths. Then, as many times as the denominator, 4, is contained in 20, so many hundredths also may be obtained. 4 into 20 5 times, and no remainder. 3 of a dollar, therefore, reduced to decimal parts is 7 tenths and 5 hundredths; that is,'75 of a dollar. Questions. - [ 92. To what is'he value of every traction equaL (f[ 64.) How is a common fraction reduced to a decimal? Of how many places must the quotient cor:ist? When there are not so many places how is the deficiency to be supplied? Repeat the rule. What is the caurse Jf reasoning advanced to establish this rule? ~ 92. DECIMAL FRACTIONS 11b 3. Reduce tg tc a decimal fraction. The numerator must be reduced to hundredths, by annexing two ciphers, before the division can begin. 66 ) 4'00 ('0606 +, the Anower. 396 ___.__ As there can be no tenths, a 400 cipher must be placed in the quo tient, in tenths' place. 396 4 NOTE. - 4 cannot be reduced exactly; for, however long the division be continued, there will still be a remainder.* It is sufficientiy exact for most purposes, if the decimal be extended to three or four places. * Decimal figures, which continually repeat, like'06, in this example, are called Repetends, or Circulating Decimals. If only one figure repeats, as'3333 or'7777, &c., it is called a single repetend. If two or morefigures cir culate alternately, as'060606,'234234234, &c., it is called a compound repetend. If other figures arise before those which circulate, as'743333,'143010101, &C., the decimal is called a mixed repetend. A single repetend is denoted by writing only the circulating figure with a point over it: thus,'3, signifies that the 3 is to be continually repeated, forming an finite or never-ending series of 3s. A compound repetend is denoted by a point over thefirst and last repeating figures: thus,'234 signifies that 234 is to be continually repeated. It may not be amiss, here, to show how the value of any repetend may be found, or, in other words, how it may be reduced to its equivalent vulgarfraction. If we attempt to reduce I to a decimal, we obtain a continual repetition of the figure 1: thus,'1I 1111, that is, the repetend'i. The value of the repetend'i, then, is.; the value of'222, &c., the repetend'2, will evidently be twice as much, that is, 2. In the same manner, 3 = a, and'4 = -, and'5 = i, and so on to 9, which - = -1. 1. What is the value of'? Ans. i. 2. What is the value of'6? Ans. 6 _. What is the value of'? -- of'7? of'4? - of'5? of'9? -of'i? If gA be reduced to a decimal, it produces'010101, or the repetend'bi. Tilhe repetend'02, being 2 times as much, must be 2 and'03, and'4, being, 4S times as much, mlst be 4, and'74 -- 7, &c. If Big be reduced to a decimal, it produces'0i; consequently,'002g-g, alld'037 = — 7W, and'425 = -9-, &c. As this principleswill apply to any number of places, we have this general RULE for reducing a circulating decimal to a common fraction. Make the given rep -tend the numerator, and the denominator xwil bs as aany 9s as there are repeatingfigures. 116 DECIMAL FRACTIONS. ~ 93. From the foregoing examples we may deduce the following general RUL;E. To reduce a common to a decimal friaction. -Annex one or more ciphers, as may be necessary, to the numerator, and divide it by the denominator. If then there be a remainder, -,nnex another cipher, and divide as before, and so continue to do, so long as there shall continue to be a remainder, or until the fraction shall be reduced to any necessary degree of exactness. Thb quotient will be the decimal required, which must consist of as many decimal places as there are ciphers annexed to the numerator; and, if there are not so many figures in the quotient, tile deficiency must be supplied by prefixing ciphers. EXAMPLES FOR PRACTICE. 4. Reduze,, I, -, and T9-T to decimals. Ans.'5;'25;'025;'00797-+. 5. Reduce a-, 5-FF, 65, and -IIs to decimals. Ans.'692 +;'003;'0028 +;'000183+-. 6. Reduce tit, ~6-~, 6 to decimals. -7. Reduce, 4, A-, s, 1g, a: T, -1- to decimals. 8. Reduce i, ], 5,, 6,., jo, 2 6,, to decimals. Federal Money. ST 93. Federal Money is the currency of the Ulnited States The unit of English money is the pound steeling, wvhih is divided into 20 equal parts, (twentieths,) called shiliingo, 3. What is the vulgar fraction equivalent to'704? Ans. -. 4. What is the value of'003? -'014? -'324? —'0102i? -'i463? 7-'02103? Ans. to last, H37sajsj7. 5. What is the value of'43? In this fraction, the repetend begins in the second place, or place of hundredths. The first figure, 4, is _4, and the repetend, 3, is ~ of' that i., 3; these two parts must be added together. +- - ~ - -, An. Hence, to find the value of a mixed repetend, —Find the value of the tw parts, separately, and add them together. 6. What is the value of'153? iJV -]- 23, Ans. 7. What is the value of'004? Ans. -t'.' 8. What is the vale of'138? -'16? -'4123? It is plain, that circulates may be added, subtracoed, multiplied, and divided by f.st reducing them to their equivalent vulgar fractions. ~T 93. DECIMAL FRACTIONS. 117 each shilling is divided into 12 parts called pence; a penny being Hz of a pcund. Each penny again is divided into 4 parts called farthings, a farthing being -}, of a pound. These divisions, therefore, are like those of common fractions, and the same difficulties occur in operations with English money as with commoL fractions. The unit of Federal money is the Dollar, divided into 10 parts called dimes, from a French word meaning tenth (of a dollar); each dime into 10 parts called cents, from the French for hundredth (of a dollar); and each cent into 10 parts called mills, from the French for thousandth (of a dollar). These divisions of the money unit are like those of decimal fractions. Our money, then, has this advantage over the English, viz., that operations in it are as in whole numbers, and we shall therefore consider it in connection with Decimals. The denominations of Federal money are eagles, dollars, dimes, cents, and mills. TABLE. 10 mills make 1 cent. 10 cents (- 100 mills) 1 dime. 10 dimes (- 100 cents - 1000 mills) 1 dollar. 10 dollars 1 eagle. NOTE. - Coin is a piece of metal stamped with certain impressions to give it a legal value, and also to serve as a guarantee for its weight and purity. The mill is so small that it is not usually regarded in business. The eagle is merely the name of a gold coin worth 10 dollars. Dimes are read as 10s of cents. Federal money, then, is calculated in dollars and cents, and accounts are kept in these denominations. A character, $, which may be regarded a contraction of U. S., placed before a number, signifies that it is Federal, or U. S. money. Questions. — f 93. What is Federal money? What is the unit of English money? What are its denominations? and what are they like? What is the unit of Federal money? how divided? and whence the name; of the divisions? What advantages has Federal over English money? Repeat the table. What is the eagle? How are dimes readi? In what then is Federal money calculated, and accounts kept? What is coin? What is the character for U. S. money, and where placed? Where is the decimal point placed? Where and how many are the places for cents? for mills? Why more places for cents than for mills? If the sum be but 8 cents how may it be written? if three mills only, ho1w? How are 5 mills usually written? 118 DECIB~IAL FRACTIONS. ~ 94 As the dollar is the unit of Federal money, the decimai point is placed at the right hand of dollars; and since dime(tenths) and cents (hundredths) are read together as cents the first two places at the right hand of the point expres, cents, and the third, mills, (thousandths.) Thus, 25 dollars 78 cents, and 6 mills, are written, $25'786. If there be no dimes, (tenths, or 10s of cents,) that is, if the cents are less than 10, a cipher is put in the piace of tenths; thus, 8 c(-nts are written $'08; and if there are only mills, ciphers must be put in the place of tenths and hundredths; thus, 5 mills are written $'005. But 5 mills are usually expressed as half a cent; thus, 12 cents 5 mills, are written 122 cents, or $'12~. Reduction of Federal Money, ~- 94. It is evident that dollars are reduced to cents in the same manner as whole numbers are reduced to huitdredths, by annexing two ciphers; To mills or thousandths, by annexing three ciphers. On the contrary, Mills are reduced to cents by cutting off the right hand figure; To dollars, by cutting off three figures from the right, which is dividing by 1000, (s 41.) Cents are reduced to dollars by cutting off two figures from the right, which is dividing by 100. EXAMIPLErS. 1. Reduce $34 to cents. 2. Reduce 48143 mills to Ans. 3400 cents. dollars. Ans. $48'143. 3. Reduce $40'06. to mills. 4. Reduce 48742 cents to Ans. 40065 mills. dollars. Ans. $487'42. 5. Reduce $16 to mills. 6. Reduce 125 mills to Ans. 16000 mills. cents. $'12C. 7 Reduce $'75 to mills. 8. Reduce 20641 cents to Ans. 750 mills. dollars. Ans. $20'64-. 9. Reduce $'007 to mills. 10. tReduce 9 cents to dolAns. 7 mills. lars. Ans. $'09. Questions. — ~ 94. How are dollars reduced to cents? to mills I,ents to mills? mills to cents? to dollars? cents to dollars? ~ 95. DECIMAL FRACTIONS. 119 Addition and Subtraction of Decimal Fractions. v 95. As the value of the decimal parts of a unit vary in a tenfold proportion like whole numbers, the addition and subtraction of decimal fractions, and of Federal money, may be performed as in whole numbers. 1. What is the amount 2. From 765'06 take 27of 14'68, 9i045, 38'5, and'6895. 9'0025? SOLUTION. -As numbers of the same denomination only can be added together, (~ 12,) or subtracted from each other, the several numbers in each of these examples must be reduced to the lowest denomination contained in any one of the numbers, (~f 91,) which is ten-thousandths, when the operation may be performed as in simple numbers. FIRST OPERATION. Or if only like denominations FIRST OPERATION 146t800 are written under each other, 765 0600 9'0450 these alone will be added to- 27'6895 38'5000 gether, or subtracted from 950025 each other, and the opera- 737'3705, Ans. tions may be performed with71,2275, Ans. out the formality of reducing to a common denominator, since the ciphers, by which the reduction is effected, make no difference in the result: thus, SEf'OND OPERATION. NOTE. -As the dleci- SECOND OPERATION 14'68 mal point is at the right 765.06 9'045 of units, which are writ- 27'6S95 38'5 ten under each other, the 9'0025 point in the result is di- 737'3705 rectly below the points in 71'2275 the several numbers. Hence, "'1o add or subtract decimal fractions, RULE. Write the numbers under each other, tenths under tenths, hundredths under hundredths, &c., according to the value of their places; add or subtract as in simple numbers, and point off in the result as, many places for decimals as are equal to Questions. —~ 95. Why can addition and subtraction of decimals be performed as in whole numbers? What numbers only can be addted and subtracted? How is this effibted by the first operations? How by the second? How do you write down decimals for add(ition? How for mubtraction? Where place the point in the results? Repeat the rule How lo you prove addition of decimals? Howu subtraction, 120 DECIMAL FRACTIO~ ~I 9b the greatest number of decimal places in any of the given n urnbers. PROOF. - The same as in the addition and subtraction of simple numbers. EXAMPLES FOR PRACTICE. 3. A man sold wheat at several times as follows, viz.,'3;25 bushels, 8S4 bushels, 23'051 bushels, 6 bushels, and'75 if a bushel; how much did he sell in the whole? Ans. 51'451 bushels. 4. What is the amount of 429, 21T-7vJ, 355- 3%, 1TO, and 1-AY? Ans. 808 -14. 3, or o08' 143. 5. What is the amount of 2 tenths, 80 hundredths, 89 thousandths, 6 thousandths,9 tenths, and 5 thousandths? Ans. 2. 6. What is the amount of three hundred twenty-nine and seven tenths, thirty-seven and one hundred sixty-two thousandths, and sixteen hundredths? 7. From thirty-five thousand take thirty-five thousandths. Ans. 34999'965. 8. From 5'83 take 4'2793. Ans. 1'5507. 9. From 480 take 245'0075. Ans. 234X9925. 10. What is the difference between 1793'13 and 817'05 693? Ans. 97607307. 11. From 41-g take 2A. Remainder, 198, or 1'98. 12. What is the amount of 29-f3, 374,9 9f, 975O, 315r 45 27, and 100-? Ans. 942'957009. Examples in Federal Money can evidently be performed in the same way. 1. Bought 1 barrel of flour for 6 dollars 75 cents, 10 poulds of coffee for 2 dollars 30 cents, 7 pounds of sugar for 92 cents, 1 pound of raisins for 121 cents, and 2 oranges for 6 cents' what was the whole amount? Ans. $10'155. 2. A man is indebted to A, $237'62: to B, $350; to C, SS6'12-; to D, $9'62j; and to E, $0'834; what is the amount of his debts? Ans. $684'204. 3. A man has three notes specifying the following sums viz., three hundred dollars, fifty dollars sixty cents, and nine dollars eight cents; what is the amount of the three notes? Ans. $359'68. 4. What is the amount of $56'18, $7'37~, $28O, 8$0287 817, and $90'413? i As. 8451'255. ~ 96. DECIMAL FRACTIONS. 121 5. Bought a pair of oxen for $76'50, a horse for $85, and a cow for $17'25; what was the whole amount? Ans. $178'75. 6. Bought a gallon of molasses for 2S cents, a quarter of tea for 371 cents, a pound of saltpetre for 24 cents, 2 yards of broadcloth for 11 dollars, 7 yards of flannel for 1 dollar 62. -ents, a skein of silk for 6 cents, and a stick of twist for 4 cents; how much for the whole? Ans. $13'62. 7. A man bought a cow for eighteen dollars, and sold her again for twenty-one dollars thirty-seven and a half cents; how much did he gain? Ans. $3'375. 8. A man bought a horse for S2 dollars, and sold him agfain for seventy-nine dollars seventy-five cents; did he gain or lose? and how much? Ans. He vzst $2`25. 9. A merchant bought a piece of cloth for $176, which proving to have been damaged, he is willing to lose on it $16'50; what must he have for it? Ans. $159'50. 10. A man sold a farm for $5400, which was $725'37{ more than he gave for it; what did he give for the farm? 11. A manl, having 500 dollars, lost 83 cents; how much had he left? Ans. $499'17. 12. A man's income is $1200 a year, and he spends $S00'35; how much does he lay up? 13. Subtract half a cent from seven dollars. Rem. $6'99J. 14. How much must you add to $16'82 to make $25? 15. I1ow much must you subtract from $250, to leave $87' 14? 16. A man bought a barrel of flour for $6'25, 7 pounds of coffee for $1'41, he paid a ten dollar bill; how much must he receive back in change? Ans. $2'34. Multiplication of Decimal Fractions. ~ 96. 1. Multiply'7 by'3.'7 -= -f7 and'3 - -A, then -;X -- 1-='21, Ans. We here see that tenths multiplied into tenths produce hundredths, just as tens, (70,) into tens, (30,) make hundreds, (2100.) We may write down the numerators decimally, thus: 122 DECIMAL FRACTIONS. ~ 96. OPERATION. The 21 must be hundredths as before. The number'7 cf figures in the product, it will be seen, is equal to the'3 number in the multiplicand and multiplier; hence, we - have as many places for decimals in the product as there'21, Ans. are in both the factors.'7 NOTE. - The correctness of the, above rule may be illustrated by the annexed diagram. The length of the jL3 P _ _ plot of ground which it represents may be regarded 10 feet and the breadth 10 feet, each division of a line, consequently, being one tenth of the whole line. Multiplying 10 by 10, we have 100 square feet i, the plot, each of the small squares being 1 square foot, or one hundredth of the whole plot. Now take the part encircled by the black lines, 7 feet ('7 of the whole line) long, and 3 feet ('3 of the whole line) wide. The contents are 21 square feet, or'21 of the whole plot; hence the product of'7 into'3 is'21 as above. 2. Multiply'125 by'03. Here, as the number of significant figures in the OPERATION. product is not equal to the number of decimals in'125 both factors, the deficiency must be supplied by pre-'03 fixing ciphers, that is, placing them at the left hand. The correctness of the rule may appear from the'00375 Prod. following process:'125 is 1?2 6, and'03 is 3 now, -12lr5w X Trt5 T3 - 5'00375, the same as before. Hence, To multiply decimal fractions, RULE. Multiply as in whole numbers, and from the right hand of the product point off as many figures for decimals as there are decimal places in the multiplicand and multiplier counted together, and if there are not so many figures in the product, supply the deficiency by prefixing ciphers. Questions. - T 96. Tenths X tenths produce what? Illustrate thlls by a diagrani. To what must the number of places in the product be equal? When the number of decimals in the product is less than the number in both factors what do you do? How can you tell of what aname or denomination will be the, product of one given decimal multiplied into another given decimal, without going through the process of multilrlication? Of what denomination, then, will be tole product o q46 )X'25? of'0005 X'07? ~ 97. DECIMAL FRACTIONS. 12m EXAMPLES FOR PRACTICE. 3, Multiply five hundredths by seven thousandths. Product,'00035. 4. What is'3 of 116? Ans. 34'8. 5. What is'85 of 3672? Ans. 312'I2. 6. What is'37 of'0563? Ans.'020831. 7. Multiply 572 by'58. Product, 331'76. S. Multiply eighty-six by four hundredths. Product, 3;44 9. Multiply'0062 by'0008. 10. Multiply forty-seven tenths by one thousand eighty-six hundredths. Prod. 51'042. EXAMPLES IN FEDERAL MONEY. SE 97. 1. If a melon be- worth $'09, what is'7 of it worth? (I77.) Ans. $S063. 2. What will 250 bushels of rye cost, at $'8$S per bushel? 3. What is the value of 87 barrels of flour, at $6'371 a barrel? Ans. $554'62.e 4. What will be the cost of a hogshead of molasses, containing 63 gallons, at 28~ cents a gallon? Ans. 817'955. 5. If a man spend 121 cents a day, what will that amount to in a year of 365 days? What will it amount to in 5 years? Ans. It will amount to $228'121 in 5 years. 6. If it cost $36'75 to clothe a soldier 1 year, how much will it cost to clothe an army of 17800 men? Ans. $654150. 7. Multiply $367 by 46. 8. Multiply $0'273 by 8600. Ans. $2347'S80. 9. At $5'47 per yard, what cost 8'3 yards of cloth? Ant $45'401 10. At $'07 per pound, what cost 26'5 pounds of rice? Ans. $1'855. 11. lWhat will be the ccst of thirteen hundredths of a ton of hay, at $11 a ton? Ans. $1'43. 12. What will be the cost of three hundred seventy-five thousandths of a cord of wood, at $2 a cord? $'75. 13. If a man's wages be seventy-five hundredths of a dollar a day how much will he earn in 4 weeks, Sundays excepted? Ans. $18 124 DECIMAL FRACTIONS. ~ 98. Division of Decimal Fractions. IV 98. 1. Divide'21 by'3.'21 -T2=, and'3 — =. Now 12 * -- T; or a of -a 0- f - 1 f. It appears, then, that hundredths divided by tenths give tenths, just as hundreds (2100) divided by tens (30) give tens, (70.) The numerators may be set down decimally, and the division perfornmed as follows:OPERATION. The 7 must be tenths as before. The dividend, "1)'21 which answers to the product in multiplication, con-...- tains two decimal places; and the divisor and quotient,'7 which answer to the factors in multiplication, (~ 31,) together contain two decimal places. Hence, we see that the number of decimal places in the quotient is equal to the dif ference between the number in the dividend and divisor. 2. At 4'75 of a dollar per barrel, how many barrels of flout can be bought for $31? OPERATION. The 4'75 are 475 hundredths, and, since 1'75) 31'00(6'526 + the dividend and divisor must be of the 2850 same denomination, we annex 2 ciphers to 31 and it becomes 3100 hundredths, (~ 91.) 2500 Then there can be as many whole barrels 2375 bought as the number of times 475 hundredths can be subtracted from 3100 hun1250 dredths. The 6 barrels thus found will 950 cost 2850 hundredths of a dollar, and as 3000 250 hundreths or cents remain, itvwill buy part of another barrel, which we find by 2850 annexing ciphers, and continuing the op150;:r eration. We now see that there are 5 decimal places in the dividend, counting all the ciphers that are annexed, and as there are but two in the divisor, we point off 3 in the quotient. There is'still a remainder of 150, which, written over the divisor, (~f 36,) gives 5 of a thousandth of a barrel, a quantity so small that it may be neglected. But we place + at the right of the last quotient figure, to show that there is more flour than indicated by the quotient. Ans. 6'526 -+barrels. NOTE. -It is sufficiently exact for most practical purposes to carry he division to three decimal places. 3. Divide'00375 by 1125. OPERATION. The 9_ivisor, 125, in 375, goes 3 times, and 125)'00375 ('03 no remainder. We have only to place the deci375 meal point in the quotient, and the work is done. There are five decimal places in the dividend; 000 consequently there must be five in the divisor ~ 98. DECITAL FItACTIONS. 1 i and quotient counted together; and as there are three in the divisor, there must be two in the quotient; and, since we have but one figure in the quotient, the deficiency must be supplied by prefixing a cipher. The operation by vulgar fractions will bring us to the same result. Thus,'125 is 1-ao-95, and'00375 is w7-sr~ no3Iw, -ff 1-6-J5+ = 375 oo = - = 3 --'03, the same as before. 4. Divide'75 by'005. OPERATION. SOLUTION.- We cannot divide hundredths'75 by thousandthls, until the former are reduced to 10 thousandths by multiplying by 10, or annexing 1- -- one cipher, when the divisor and dividend will'005 )'750 be of the same denomination; and'005 is con-.-.- tained in (can be subtracted from)'750, 150 150, Quot. times, the quotient being a whole number. These illustrations will establish the following RU11IE. I. Reduce, if necessary, the dividend to the lowest denomination in the divisor, divide as in whole numbers, annexing ciphers to a remainder which may occur, and continuing the operation II. If the decimal places in the dividend with the ciphers annexed exceed those in the divisor, point off the excess fiom the right of the quotient as decimals; but if the excess is more than the number of places in the quotient, supply the deficiency by prefixing ciphers. EXAMPLES FOR PRACTICE. 5. Divide 31566293 by 25'17. Quot. 12513-. 6..Divide 17394S by'375. Quot. 463S61 -. NoTE.- The pupil will point off the decimal nlaces in the alnotient of this and the following example, as directed by the rule. 7. Divide 5737 by 13'3. Quot. 431353. S. What is the quotient of 2464'8 divided by'00S? Anas. 308100. Questions. - ~ 98. HuncdrPdths, divided by tenths, give what? Ifow is it in integers? Exhibit on the blackboard the process of dividing 7 by 1"25. Why do you annex ciphers to the 7? What is the quotient? Why pointed thus? Give a denemonstration by common fractions, as after Ex. 3, and sho-w. that this placing of the point is right. The sign of addition, annexed to the quotient, is an indication of what? When there are remainders, to how many places should the division be carried? Why niot to mrre places? Repeat the rule for division b. i [lZws DECIMAL FRACTIONS. ~ 99 9. Divide 2 by 53'1. Quot.'037 -. 10. Divide'012 by'005. Quot. 2'4. 11. Divide three thousandths by four hundredths. Quot.'075. 12. Divide eighty six tenths by ninety-four thousandths. 13. How many times is'17 contained in 8? EXAMPLES IN FEDERAL MONEY. ~ 99. 1. Divide $59'387 equally among 8 men; how much will each man receive? OPERATION. 8) 59'387 Ans. $7'423R, that is, 7 dollars, 42 cents, 3 mills, and { of another mill. The g is tbE -emnainder, after the last division, written over the divisor, and expresses such fractional part of another mill. For most purposes of business, it will be sufficiently exact to carry the quotient only to mills, as the parts of a mill are of so little value as to be disregarded. 2. At $'75 per bushel, how many bushels of rye can be bought for $141? Ans. 1SS bushels. 3. At 121 cents per lb., how many pounds of butter may be bought for $37? Ans. 296 lbs. 4. At 61 cents apiece, how many oranges may be bought for $8? Ans. 128 oranges. 5. If'6 of a barrel of flour cost $5, what is that per bar rel? Ans. $8'333 +. NOTE. — If the sum to be divided contain only dollars, or dollan and cents, it may be reduced to mills, by annexing ciphers before dividing; or, we may first divide, annexing ciphers to the remainder, if there shall be any, till it shall be reduced to mills, and the result will be the same. 6. If I pay $468'75 for 750 pounds of wool, what is the value of 1 pound? Ans. $0'625; or thus, $'62~. 7. If a piece of cloth, measuring 125 yards, cost $181'25, vvhat is that a yard? Ans. $1'45. 8. If 536 quintals of fish cost $1913'52, how much is that quintal? Ans. $3'57 9. Bought a farm, containing 84 acres, for $3213; what d it cost me per acre? Ans. $38'25. "U At $954 for 3816 yards of qannel, what is that a yard I Ans. $0'25. ~ 100. DECIMAL FRACTIONS. 127 11. Bought 72 pounds of raisins for $8; what was that a pound? Ans. $0'111,-; or, $0'111 —. 12. Divide $12 into 200 equal parts; how much is one of the parts?. --- how miuh? Ans. $'06. 13. Divide $30 by 750. T3 _- how much? 14. Divide $60 by 1200. 60 = how much? 15. Divide $215 into 86 equal parts; how much will one of the parts be.? _2 - how much? ST 140. Review of Decimal Fractions. Questions. -WWhat are decimal fractions? How do they differ from common fractions? How can the proper denominator to a decimal fraction be known, if it be not expressed? What advantages have decimal over common fractions? How is the value of every figure determined? Describe the manner of numerating and reading decimal fractions? of writing them? How are decimals, having different denoml nators, reduced to a common denominator? How may any whole number be reduced to decimal parts? How can any mixed number be read together, and the whole expressed in the form of a common fraction? What is federal money? What is the money unit, and what are its divisions and subdivisions? How is a common fraction reduced to a decimal? To what do the denominations of federal money correspond? What is the rule for addition and subtraction of decimals? - multiplication? — division? EXERCISES. 1. A merchant had several remnants of cloth, measuring as follows, viz.: 7, yds. ) How many yards in the whole, and what would 6 " [ the whole come to, at $3'67 per yard? 1 2 " t NOTE. -Reduce the common fractions to decimals. 9 Do the same wherever they occur in the examples which 8 4, follow. 3- (" J Ans. 36'475 yards. $133'863 -, cost. 2. From a piece of cloth, containing 36J yards, a merchant sold, at one time, 7-03 yards, and, at another time, 125 yards; how much of the cloth had he left? A.ns. 16'7 yds. 3. A farmer bought 7 yards of broadcloth for $334>1, two barrels of flour for $14-r5, three casks of lime for $7, and 7 pounds of rice for $5; what was the cost of the whole? NOTE.- The follbwing examples are to be psrformed according to the rule in ~ 77, or in ~f 83, or in ~T 85. 4. At 121 cents per lb., what will 37; lbs. of blatter cost? Ans. $4'7181. 128 DECIMAL FRACTIONS. ~ 100, 5. At $17'37 per ton for hay, what will 11 tons cost? Ans. $201'92-. 6. The above example reversed. At $201'92J for 11 tons of bay, what is that per ton? Ans. $17'37. 7. If'45 of a ton of hay cost $9, what is that per ton? Ans. $20. 8. At'4 of a dollar a gallon, what will'25 of a gallon of molasses cost? Ans. $ 1. 9. What will 2300 lbs. of hay come to, at 7 mills per lb.? Ans. $16'10. 10. What will 765k lbs. of coffee come to, at 18 cents per lb.? Ans. $137'79. 11. Bought 23 firkins of butter, each containing 42 pounds, for 16- cents a pound; what would that be a firkin? and how much for the whole? Ans. $159'39 for the whole. 12. A man killed a beef, which he sold as follows, viz., the hind quarters, weighing 129 pounds each, for 5 cents a pound; the fore quarters, one weighing 123 pounds, and the other 125 pounds, for 4-2 cents a pound; the hide and tallow, weighing 163 pounds, for 7 cents a pound; to what did the whole amount? Ans. $35'47. 13. A farmer bought 95 pounds of clover seed at 11 cents a pound, 3 pecks of herds grass seed for $2'25, a barrel of flour for $6'50, 13 pounds of sugar at 121 cents a pound; foi which he paid 3 cheeses, each weighing 27 pounds, at 8- cents a pound, and 5 barrels of cider at $1'25 a barrel. The balance between the articles bought and sold is 1 cent; is it for or against the farmer? 14. A man dies, leaving an estate of $71600; there are demands against the estate, amounting to $39876'74; the residue is to be divided between 7 sons; what will each one receive? Ans. $4531'8942. 15. How much coffee, at 25 cents a pound, may be had for 100 bushels of rye, at 87 cents a bushel? Ans. 348 pounds. 16. At 121 cents a pound, what must be paid for 3 boxes of sugar, each containing 126 pounds? Ans. $47'25. 17. If 650 men receive $86'75 each, what will they all receive? Ans. $56387'50. 18. A merchant sold 275 pounds of iron at 61 cents a pound, and took his pay in oats, at $0'50 a bushel; how nmany bushels did he receive? Ans. 34'375 bushels. 19. How many yards of cloth, at $4'66 a yard, must be giveon for 18 barrels of flour, at $9'32 a barrel? Ans. 26 ryaads 20. What is the price of three pieces of cloth, the first containing 16 yards, at 3I'75 a yard; the second, 21 yards, at $4'50 a yard; and the third, 35 yards, at $5' 121 a yard? Ans. $333'871. BILLS, ~T 101. A Bill, in business transactions, is a written list ol the articles bought or sold, and their prices, together with the entire cost or amount footed. No. 1. —Bill of Sale. Payment received. Boston, May 25th, 1847. James Brown, Esq. Bought ol Hastings & Belding, 6 yards black broadcloth, a $3'00 18'00 21, cambric, "'14'35 2 dozen buttons, "'15'30 4 skeins sewing silk, i"'04'16 25 lbs. brown sugar, 6"'09 2'25 Received payment, $21'06. Hastings & Belding. No. 2.-Bill of Sale. Charged in account. New Orleans, Aug. 1st, 1847. Gen. Z. Taylor, To Daniels & Thomas, Dr. To 278 bbls. beef, m $9'75' 191 " pork, " 12'00, 250 " flour, " i70 {c 500 sacks Indian meal, "'621 Charged in acc't. Amount, $6741'25 Daniels & Thomas. No. 3.- Barter Bill. Buffalo, Sept. 15th, 1847. MNr. D. F. Standart, To O. B. Hopkins & Co., Dr. To. 15 lbs. brown sugar a $'10 "2 " Y. H. tea, "'871 "24 A" mackerel, "'041 " 3 gal. molasses, i"'42 S16 yds. sheeting, " 09 130 BILLS. i 101. Cr. By 4 doz. eggs, a'08 "8 lbs. butter, "'14 "40 " ch6ese, 4"'074 "note at 30 days, to balance, 2'59 $7'03 O. B. Hopkins & Co. by L. D. Swift. No. 4. — Bill of goods sold at wholesale. New-York, April 5th, 1847. Davis & Horton, Bought of Barnes,Porter & Co. 3 hhds. molasses, 118 gal. each, c $'31 2 " brown sugar, 975 and 850 lbs.'091 3 casks rice, 205 lbs. each, "'041 5 sacks coffee, 75 "' "'11 I chest H. tea, 86 " " "'92 $431'16 Rec'd payment, by note, at 60 days. For Barnes Porter & Co. James D. Willard. It is sometimes practised, in collecting and settling accounts to make a copy of each individual account, and -present it to the person for his inspection. No. 5.- Copy of an individual account. Frank H. Wright, In acc't with Edward F. Cooper, 1847. Dr. Jan. 7. To 125 bushels corn, ~ $'50 it "d " 20 " apples, "'31 March 13. " 12 " rye, "'62 " 20, " 153 lbs. cast steel, 6'24 Questions.-J 101. What isabill? If the amount of the bill be paid at the time, how is it shown? Which oill is an example of thLs! if charged in account, how is it shown? example? How does a batter bill differ from a bill of sale? In what order are the articles bought and sold arrangedt? What is practised in collecgting and settling accounts? flow does such a copy differ from a barter bill? To which of the bills must the bill to be made out confbrm? and what will it be called? ~ 102. COMPOUND NUMBERS. 131 1847. Cr. Feb. 15. By 3 cows, i $17'00 " 22., "5 sheep, " 2'50 Amount due me, $16,42 Edward F. Cooper. Baltimore, May 9th, i847. The pupil is required to make out a bill from the statement contained in the following example. Winm. Prentiss sold to David S. Platt 780 lbs. of pork, at 6 cents per lb.; 250 lbs. of cheese, at 8 cents per lb.; and 154 lbs. of butter, at 15 cents per lb.; in pay he received 60 lbs. of sugar, at 10 cents per lb.; 15 gallons of molasses, at 42 cents per gallon; i barrel of mackerel, $3'75; 4 bushels of salt, at $1'25 per bushel; and the balance in money: how much money did he receive? Ans. $68'85. COMPOUND NUMDERS. ~T 102. When several abstract numbers, or several denominate numbers of the same unit value, are employed in an arithmetical calculation, they are called simple numbers, and operations with such numbers are called operations in simple numbers. Thus, if it were required to add together 7 gallons, 9 gallons, and 5 gallons, the numbers are simple numbers, being denominate numbers of the same unit value, (1 gal.,) and the operation is an addition of simple'numbers. We have had, also, subtraction, multiplication, and division of simple numbers. But when several numbers of different unit values are employed to express one quantity, the whole together is called a compound number. Thus, 12 rods, 9 yards, 2 feet, 6 inches employed to express the length of a field, is a compound number. So also, 9 gallons, 2 quarts, 1 pint, employed to express a quantity of water, is a compound number. NOTE. - The word denomination is used in compound numbers to Questions. -I 102. What are simple numbers? Examples. What are operations in such numbers called? What is a compound number? Give examples other than those in the book. WI-at is meant by the word denomination? 132'OMNPOUND NUMBERS. ~ 103, 104 denote the -aine of the unit considered. Thus, bushel and peck arte names or d6 ominations of measure; hour, minute and second are denominations Jf time. I~ 103. Tue fundamental operations of addition, subtraction, multiplication and division, cannot be performed on comnpound numbers till we are acquainted with the method of changing numbers of one denomination to another without altering their value, which is called Reduction. Thus, we wish to add 2 bushels 3 pecks, and 3 bushels 1 peck, together. They will not make 9 bushels nor 9 pecks, (adding together the several numbers,) since some of the numbers express bushels, and some express pecks. But 2 bushels equal 8 pecks, (2 times 4 pecks, the number of pecks in a bushel,) and 3 pecks added make 11 pecks; 3 bushels equal 12 pecks, and 1 peck added make 13 pecks. Then, 11 pecks-L 13 pecks = 24 pecks. Hence, before proceeding further, we must attend to the Reduction of Compound Numbers. STERLING OR ENGLISH MONEY. IF 104. Money is expressed in different denominations, and 4 dollars, 3 dimes, 7 cents, 5 mills = $4'375, employed to express one sum in Federal money is a compound number. But as the denominations in Federal money vary uniformly in a tenfold proportion, (~ 93,) being conformed to the Arabic notation of whole numbers, the operations in it are as in whole numbers. The denominations in English (called, also, sterling) money, Oounds, shillings, pence and farthings, do not vary uniformly, but according to the following TABLE. NOTE 1.-All the tables in Reduction of Compound Numihere must be carefully committed to memory by,he pupil. 4 farthings (qrs.) make 1 penny, marked d. 12 pence (plural of penny) 1 shilling, " s. 20 shillings 1 pound, " ~. NOTE 2. - Farthings are often written as the fraction of a penny, tl;us, I farthing = id., 2 farthings -d., 3 farthings = -d. Questions. — ~ 103. What is reduction? Whence its necessity t Explain by the example of adding bushels and pecks. To what, then, must we attend before proceeding further? ~ 104 COMPOUND NUMBERS. 133 NOTE 3. -The value of these denominations in Federal money is nearly as follows: lqr. - of 1 cent. There is in England Id. -- 2 cents. id. = 2 - cents. a gold coin, called a Is. 2$4 t84 5 sovereign, the value LC. $4'184~of which is ~l. 4s. ld. 2a1*2qrs. -$1'00 1. How many farthings in 2. In 20 farthings, how 5 pence? many pence. IST OPERATION. SOLUTION. - SOLUTION. - WV 4 We may multi- OPERATION- have given the numply the number 4)20 ber of farthings in 1 of farthings (4) - penny to find the in I penny by 5d. number of pence in a 2~qrs. the number of given number of far2D OPERATION. pence, (5.) (~ things, (20,) and we divide the 5 46.) Or, as ei- number of farthings in the num4 ther factor may ber of pence by the number in I be made the penny, (T 46.) Ans. 5d. 20qrs. multiplicand, (~ 21,) we may multiply the number of pence (5) by the number of farthings in 1 penny. Arns. 20qrs. 3. How many farthings in 4. How many pence in 12 3 pence? 6 pence? - farthings? 24 farthinrgs? 9 penc? 7 pence? -.- 36 farthings? - 28 2 pence? 10 pence? farthings? 8 farthings? 11 pence? - 12 pence -- 40 farthings? 44 I shilling? farthings? -- 48 farthings? 5. How many pence in 3 6. How many shillings in shillings? - 5 shillings? 36 pence? - 60 pence? 5s.8 d.? -7s.? - 68d.? 84d.? Ss.'4d.? 12s.?- -- 15s. 100d.? - 144d.? 6d.? 186d.? 7. How many shillings in S. How many pounds in 3 -.. 5M.? -- 4~. 2s.? 60s.? 100s.? — 82s.? 6~. 11s.?. 131s.? Questions. - ~ 104. What is said of operations in Federal money? Wha. are the denominations of English money? the signs? How do they vary differently from those of Federal money? Give the tablie. How are farthings written? What is the value of a pound sterling in Federal money? Explain the first operation of Ex. 1; the second operation Explain Ex. 2. Of how many kinds is reduction? what are they What is recu ttibn desccnding? - reduction ascending? 134 COMPOUND NUMBERS. ~ 105 The changing of iigher The changing of lower cte. denominations te lower, as nominations to higher, as shil pounds to shillings, is called lings to pounds, is called ReReduction Descending, and is duction Ascending, and io performed by multiplication. performed by division. REDUCTION DESCENDING. RED)UCTION ASCENDING. ~ 105. 1. In ~17 13s. 2. In 16971 farthings, hr 6Wd., how many farthings? many pounds? OPERATION. 17~ 13s. 6d. 3qrs. 20 OPERATION. Fthg iny, 4 ) 16971 353s. in 17~ 13s. Pence in 12)4242d 12 1 illing, 42d. 3qrs 4242d. in 17~ 13s. 6d. *'illnfe2 )3513s. 6d. 4 17~ 13s. 16971qrs. Ans. Ans. ~1' 13s. 6d. 3qrs. In 17~ 13s. 6d. 3qrs. SOLUTION. - We multiply 17~ SOLUTION. — We divide th. by 20, the shillings in 1~, and whole number of farthing, by 4 add in the 13s. to get the number the number in id., to get the of shillings in 17~ 13s., which is number of pence; for as many 353. This number we multiply times as 4 can be subtracted from by 12, adding in the 6d. given, to 16971, so many pence there will get the number of pence, 4242, be, which is 4242d. and 3qrs. rewhich we multiply by 4, adding maining. On the same principle, in the 3qrs. given, to get the num- dividing the 4242 by 12, the quober of qrs. or farthings, which is tient, 353, is shillings, and the 16971qrs. remainder, 6, is pence, and dividing 353s. by 20, the quotient, 17, is pounds, and the remainder, 13, is shillings. hence, for Reduction De- Hence, for Reduction Asstending, cending, RUJLE. RULE. Multip.y each higher de- Divide each lower denominomination by the number nation by the number which which it takes of the nl6xt less it takes of it to make one of Questions.- O105. Explain the first example. Give the rule for reduction descending. Ex. 2. Give the rule for reduction ascending ~ 106. COMPOUND NUMBERS. 135 to make 1 of this higher, in- the next higher. Proceed in creasing the product by the this way till the work is done. given number, if any, of this lower denomination. Proceed in this way till the work is done. EXAMPLES FOR PRACTICE. 3. Reduce 32~. 15s. 8d. to 4. Reduce 31472 farthings qrs. to pounds. 5. Reduce 7~. 14s. 6d. 1 6. Reduce 7417 qrs. to qr. to qrs. pounds. 7. In 91~. 11s. 31d., how 8. In 87902 farthings, now many farthings? many pounds? 9. In 40~. 12s. 8d., how 10. In 9752 pence, how many pence? many pounds? 11. In 1C. 18s. 4~d., how 12. In 921 half pence, how many half pence? many pounds? Weight. I. AvoIRDUPOIS WEIGHT. ST 106. Avoirdupois Weight is employed in all the ordinary purposes of weighing. The denominations are tons, pounds, ounces, and drams. TABLE. 16 drams (drs.) make 1 ounce, marked oz. 16 ounces " 1 pound, 6" lb. 2000 pounds " 1 ton, " T. Or, as was formerly reckoned, 28 lbs. 1 quarter, qr. 4 qrs. (_ 112 lbs.) 1 hundred weight' cwt. 20 cwt. (-2240 lbs.) 1 ton, " T. By the last talle, 2240 lbs. make 1 ton, which is sometimes called the " long ton;" while the ton of 2000 lbs. is called the " short ton." The long ton is still used in the U. S. cusQuestions. - ~ 106. what is the use of avoirdupois weight? the denominations? the signs? Repeat the table; the table by the old method. Explain the difference between the long and short ton. When is the long ton used? the short ton? 136 COSMPOUND NUMBERS. ~ 107 tom-house operations, in invoices of English goods, and of coal from the Pennsylvania mines. But in selling coal in cities, and in other transactions, unless otherwise stipulated, 2000 lbs. are called a ton. EXAMPLES FOR PRACTICE. 1. In 14 tons 607 lbs. 6 oz. 2. In 7323500 drams, how 12 drs., how many drams? many tons? SOLUTION. — As there are 2000 SoLuTION.-Dividing the drams br. in a ton, we multiply 14 ly by 16, the number in an oz., the 2000, to get 14 tons to lbs., and quotient is oz. and the remainder add the 607 lbs. to the product. drs. Dividing the oz. by 16, the The lbs. we multiply by 16 to get quotient is lbs. and the remainder them to oz., adding in 6 oz., and oz., and dividing the lbs. by 2000, the oz. by 16, adding in 12, and the quotient is tons and the rethe whole are in drams. mainder lbs. 3. In 7 tons 665 lbs. of su- 4. In 14665 lbs. of sugar, gar, how many lbs.? how many tons? 5. In 12 T. 15 cwt. 1 qr. 19 6. In 7323500 drams, how lbs. 6 oz. 12 drs. of glass, re- many tons? ceived from an English house, how many drams? 7. Received from Birming- 8. In 470 packages of ham; England, 5 T. 9 cwt. 12 screws, each containing 26 lbs. of iron screws, in pack- lbs., how many tons? ages of 26 lbs. each; how many were the packages? II. TROY WEIGHT. I 1107. T.oy Weight is used where great accuracy is required, as in weighing gold, silver, and jewels. The Je nominations are pounds, ounces, pennyweights, and grains. TABLE. 24 grains (grs.) make 1 pennyweight, marked pwt. 20 pwts. 1 ounce, " oz. 12 oz. 1 pound, " lb. NOTE. - A lb. Troy = 5760 grs., and 1 lb. avoirdupois = 7000 grs. Troy. Hence a quantity expressed in one weight, may be changed to the denominations of the other. Questions. - a 107. For what is Troy weight used? What are the denominations?' -the signs? Repeat the table. What difference oetween the pound Troy and the pound avoirdupois.? ~ 108S. COMPOUND NUMBERS. 137 EXAMPLES FOR PRACTICE. 1. Inr 210 lbs. 8 oz. 12 2. In 50572 pwts., how pwts., how many pwts.? many lbs.? SOL JTION. - Multiply the lh,. SOLUTION.-Dividing the pwts. by. 12, adding the 8 oz. to the by 20, the quotient is oz., and the product, and the sum is oz., remairner pwts., and dividing the which, multiplying by 20, adding oz. by 12, the quotient is lbs., in the 12 pwts., the sum is pwts. and the remainder oz. 3. In 7 lbs. 11 oz. 3 pwts. 4. In 45681 grains of sil9 grs. of silver, how many yer, how many lbs.? grains? 5. Reduce 11 oz. 13 pwts. 6. Reduce 5605 grs. of 13 grs. of gold to grains. gold to ounces. 7. Reduce 28 lbs. avoirdu- 8. Reduce 34 lbs. 6 pwts. pois to the denominations of 16 grs. Troy to lbs. avoirdu Troy weight? pois. (Consult Note.) III. APOTHECARIES' WEIGHT. 1'f 10S. Apothecaries' Weight is used by apothecaries and physicians, in mixing and preparing medicines. But medicines are bought and sold by avoirdupois weight. The denominations are pounds, ounces, drams, scruples, u.nd grains. TABLE. 20 grains (grs.) make 1 scruple, marked E. 3 E) 1 dram, " 5. 8 5 1 ounce, " 12 % 1 pound, " ib. NOTE. — The pound and ounce, Apothecaries' and Troy weight, are the same, but the ounce is differently divided. EXAMPrLES FOR PRACTICE. 1. In 9 lbs. 8 S. 1 5. 2 E). 19 2. Reduce 55799 grs. to grs. how many grains 2 lb. Questions. - ~ 108. To what is apothecaries' weight limited the denominations? Give the file. Make the sign for each denomii nation. What is said of the pound and ounce? 1.2* 138 COMPOUND NUMBERS. ~ 109 Measures of Extension. Extension has three dimensions, length, breadth, and thick. ness. I. LINEAR MEASURE. f 109. Linear Measure (the measure of lines) is used when only one dimension is considered, which may be either the length, breadth, or thickness. i'he usual denominations are miles, -furlongs, rods, yards, feet, inches, and barley-corns. TABLE. 3 barley-corns (bar.) make 1 inch, marked in. 12 inches 1 foot, " ft. 3 ft. 1 yard, " yd. 5! yards, or 16~ ft., 1 rod, " rd. 40 rods 1 furlong, " fur. 8 furlongs, or 320 rods, 1 mile, " mi. 69% common miles, 1 degree, deg., or oan the erttuath.. creumfer. 3 geographical miles, 1 league, L., used in measuring distances at sea. 60 geographical miles, 1 degree of latitude. 6 feet, 1 fathom, in measuring depths at sea. NOTE. —The geographical mile, used in measuring latitude, is not quite uniform, but is always a little less than 1 common miles, as the degree varies from 685 to 69~ miles. The degree of longitude grows shorter towards the poles, where it is nothing. Instead of the barley-corn, inches are now divided into eighths and tenths. EXAMPLES FOR PRACTICE. 1. How many inches in the 2. In 1577664000 inches, equatorial circumference of how many miles? How many the earth, it being 360 de- degrees of the equatorial cir. grees? cumference? Questionis. - 109. How many dimensions has extension? What are they? What is linear measure? Give the denominations, and the sign of each. Repeat the table. How is the inch usually divided? For what is the fathom used? For what is the geograph:zal mile used? - its length? What is said of the degree of longitude, and Its length? - of a degree of latitude? What causes the difference in the length of degrees? Fcr what is the league used, and about what is its ength hi common miles? r 110. COMPOUND NUMBERS. 139 3. How many inches from 4. In 30539520 inches, Boston to Washington, it be- how many miles? ng 482 miles? 5. How many times will a 6. If a wheel, 16 feet 6 rvheel, 16 feet 6 inches in cir- inches in circumference, turn eumference, turn round in go- round 12800 times in going ing from Boston to Provilence, from Boston to Providence, it being 40 miles? what is the distance? 7. If a man step 2 feet 6 8. A man walked 90816 inches at once, how many steps, of 2 feet 6 inches each, steps will he take in walking in a day; how many miles 43 miles? did he walk? CLOTH MEASURE. T lO10. Cloth Measure is a species of linear measure, being used to measure cloth and other goods sold by the yaid in length, without regard to the width. The denominations are ells, yards, quarters, and nails. TABLE. 4 nails; (na.,) or 9 inches, make 1 quarter, marked qr 4 qrs., or 36 inches, 1 yard, " yd. 3 qrs. 1 ell Flemish," E. Fl. 5 qrs. 1 ell English," E. E. 6 qrs. 1 ell French, " E. Fr. NOTE. - Eighths and sixteenths of a yard are now used instead of nails. EXAMPLES FOR PRACTICE. 1. In 573 yds. 1 qr. 1 na., 2. In 9173 nails, how many how many nails? yards? 3. In 296 E. E. 3 qrs., how 4. In 5932 nails, how many many nails? E. E.E. 5. In 151 E.E., how many 6. In 188 yds. 3 qrs., how yards? many E. E.. 7. In 29 pieces of cloth, 8. In 783 yds., how many each containing 36 E. Fl., E. Fl.? how many yards? Questions. — ~ 110. What is cloth measure.? How used? Why a species of lin, ar measure? G've the denominations, and the sign of each; the table. What is used instead of nails? How may yards be reduced to E. E.? to E. Fr.? t> E. Fl.? How each of these to yards t 140 COMPOUND NUMBERS. ~ 111 I1. LAND OR SQUARE MEASURE.' ~ 11oi Square Measure is used in estimating the aTea of land, and other things wherein length and breadth are considered I linear yard. _ NOTE. -It takes 3 feet in length to make 1 1 1 2 1 3 i linear yard. 3 feet- 1 yard. But it requires a square, 3 feet = 1 linear T —I-V —-- 1 yard in length, and 3 feet = 1 linear yard in e I I [ breadth, to make 1 square yard. 3 feet in $, _ P + j length and 1 foot in width, make 3 square feet, (3 squares in a row, ~ 48.) 3 feet in i1 __I I length and 2 feet in width make 3 X 2 — 6 square feet, (2 roaws of squares, ~ 48.) 3 feet a L i I {in length and 3 feet in width make 3 X 3=9 L _ 1__l____-.___Asquare feet, (3 rows of squares.) 9 sq. ft. I 1 sq. yd. It is plain, also, that 1 square foot, that is, a square 12 inches in length and 12 inches in breadth, must contain 12 X 12= 144 square inches, (12 rows, of 12 squares each.) The denominations of square measure are miles, acres, roods, rods or poles, yards, feet, and inches. TABLE. 144 square inches (sq. in.) make 1 square foot, marked sq. ft. 9 square feet 1 square yard, " sq. yd. 301 sq. yds.- 52 X 5-,or 1 sq. rod, perch, }" sq. rd. 2724 sq. ft. — 16 X 16-, or pole, P 40 square rods 1 rood, " R. 4 roods, or 160 square rods 1 acre, " A. 640 acres 1 square mile, " M. EXAMPLES FOR PRACTICE. 1. In 17 acres 3 roods 12 2. In 776457 square feet, poles, how many square feet? how many acres? 3. Reduce 64 square miles 4. In 1,784,217,600 square to square feet. feet, how many square miles? 5. There is a town 6 miles 6. Reduce 23040 acres to square; how many square square miles. miles in that town? how many acres? Questions. - - 111. What is square measure? What is a square? Draw or describe a square inch; a square yard; a figure showing the square inches contained in a square foot. How many square inches in a row, and how many rows of square inches would it contain? MIultiply questioas at pleasure. What are the denominations of square measure? Repeat the table How does square measure differ from linear measure? 1112, 113. COMPOUND NUMBERS. 141 7. How many square feet 8. In 5510528179200000 on the surface of the globe, square feet, how many square supposing it contain 197,663,- miles? 000 square miles? ~f 112. The Surveyor's,, or what is called Gunter's Chain, is generally used in surveying land. Tt is 4 rods, or 66 feet in length, and consists of 100 links TABLE FOR LINEAR MEASURE. 7925 inches make 1 link, marked l. 25 links 1 rod, " rd. 4 rods, or 66 feet, 1 chain, " C. 80 chains 1 mile, " mi. 1. In 5 mi. 71 C., how 2. In 471 chains, how many chains? many miles? 3. Reduce 2 mi. 15 C. 3 4. Reduce 17593 links to rds. 18 1. to links. miles. 5. In 75 C., how many 6. In 4950 feet, how many feet? chains TABLE FOR SQ UARE MEASURE. 625 square links (sq. 1.)make 1 square rod, marked sq. rd. perch,orpole, P. 16 square poles 1 square chain, " sq. C. 10 square chains 1 acre, " A. NOTE. - Land is generally estimated in square miles, acres, roods and square poles or perches. 7. Reduce 8 A. 2 sq. C. 7 8. In 824831 sq. 1., how P. 456 sq. 1. to square many acres? links. 9. In 80 A., how many 10. In 8000000 sq. 1., how square chains? how many many square chains? In 800 square links? sq. C., how many acres? III. CUBIC OR SOLID MEASURE. ST 113. Cubic or Solid Measure is used in measuring things that have length, breadth, and thickness; such as timber, wood, earth, stone, &c. Questions. - ~ 112. What is generally used in surveying land What is its length? Qf how many links does it consist? Repeat the tdble for linear measure for square measure. How is land generally estimated I 142 COMPOUND NUMBERS. ~ 113. NOTE 1. -It has been shown (~ 111,) that 1 square yard contains 3 X 3 9 square feet. A block 3 feet long, 3 feet wide and 3 feet thick, is a ~ li i I ~ figure represents such a block. Iii \VllllWere a portion 1 foot in thicki' [1 iji jljiillll~ Illness cut off from the top of this,.;. block, the part cut off would be }~i1~iL 9ti/ 3 feet long, 3 feet wide, and 1 foot thick, and would contain 3 X 3 yd 3feet long. X 1 = 9 cubic feet. The bottom part being 3 feet long, 3 feet wide, and 2 feet thick, would contain 3 X 3 X 2 =- 18 cubic feet. But the entire block being 3 feet long, 3 feet wide, and 3 feet thick, contains 3 X 3 X 3 _ 27 cubic feet. It is plain also, that a cubic foot, that is, a solid body, 12 inches long, 12 inches wide, and 12 inches thick, will contain 12 X 12 X 12 — 1728 cubic or solid inches. The denominations of cubic measure are cords, tons, yards5 feet and inches. TABLE. 1728 cubic inches, (cu. in.) =12 X 12 X 12, that is, 12 inches in length, 12 make 1 cubic foot, marked cu. ft in breadth, and 12 in" thickness, J 27 cubic feet, 3 X 3 X 3, 1 cubic yard, " cu. yd. 40 feet of round timber, or 1 ton, T 50 feet of hewn timber, 42 cubic feet 1 ton of shipping, h T Used in measuring the capacity oY ships, 16 cubic feet ( 1 cord foot, or C. ft. 1 foot of wood, 8 cord feet, or 1 cord of wood, C. 128 cubic feet, Questions.- ~ 113. What is cubic measure? What distinctions do you make between a line, a surface and a solid? What is a cube? a cubic inch? a cubic foot? a cubic yard? For what is cubic or solid measure used? What are its denominations? Repeat the table. For what is the cubic ton used?'What do you understand by a ton of round timber? What are the dimensions of a pile of wood containing l cord t What is a cord foot? If 114. COMPOUND NUMBERS. 14[, NOTE 2 - A cubic ton is used for estimating the cartage and transportation of timber. A ton of round timber is such a quantity (about 50 feet) as will make 4( feet when hewn square. NOTE 3.- A pile or load of wood 8 feet long, 4 feet wide, and 4 feet high, contains 1 cord. 8 X 4 X 4 = 128 cubic feet. A cord foc t is 1 foot in length of such a pile. 1. Reduce 9 tons of round 2. In 777600 cubic inches, timber to cubic inches. how many tons of round timber? 3. In 37 cord feet of wood, 4. In 592 solid feet of how many solid feet? wood, how many cord feet? 5. Reduce 8 cords of wood 6. In 64 cord feet of wood, to cord feet. how many cords? 7. In 16 cords of wood, 8. 2048 solid feet of wood how many cord feet? how how many cord feet? how many solid feet? many cords? 9. In 25 C., 5 C. ft., 9 cu. 10. In 5684967 cubic ft., 1575 cu. in. of wood, how inches, how many cords many cubic inches? Measures of Capacity. I. WINE MEASURE. ~ 1 14. Wine Measure is used in measuring all liqulds except ale, beer, and milk. The denominations are tuns, pipes, hogsheads, tierces, bar. rels, gallons, quarts, pints, and gills. TABLE. 4 gills (gi.) make 1 pint, marked pt. 2 pints 1 quart, " qt. 4 quarts 1 gallon, " gal. 31- gallons 1 barrel, " bar. 42 gallons 1 fierce, " tier. 63 gallons, or 2 barrels, 1 hogshead, " hhd. 2 hogsheads 1 pipe, " P. 2 pipes, or 4 hogsheads, 1 tun, " T. No'rE. -The wine gallon contains 231 cubic inches. A hogshead of molasses, &c, is no definite quantity, out is estimated by the gal Ion. Questions. -- 114. For what iswine measure used? What are Its denominations? Repeat the table. How many cubic inches in a wine gallon? How many gallons in a hogshead of rmolasses 7 &c .44 COMPOUND NUMBERS. ~ 115, 116. 1. Reduce 12 pipes of wine 2. In 12096 pints of wine to pints. how many pipes? 3. In 9 P. 1 hhd. 22 gals. 4. Reduce 39032 gills to 3 qts., howv many gills? pipes. 5. In 25 tierces, how many 6. In 33600 gills, how gills? many tierces? II. BEER MEASURE. 1' 115. Beer Measure is used in measuring beer, ale and milk. The denominations are hogsheads, barrels, gallons, quarts, and pints. TABLE. 2 pints (pts.) make 1 quart, marked qt. 4 quarts 1 gallon, " gal. 36 gallons 1 barrel, " bar. 54 gallons, or 1- barrels, 1 hogshead," hhd. NOTE. - The beer gallon contains 282 cubic inches. 1. Reduce 47 bar. 18 gal. 2. In 13680 pints of ale, of ale to pints. how many barrels? 3. In 29 hhds. of beer, 4. Reduce 12528 pints to how many pints? hogsheads. III. DRY MEASURE. T 116. Dry Measure is used in measuring all kinds of grain, fruit., roots, (such as potatoes and turnips,) salt, charcoal, &c. The denominations are chaldrons, quarters, bushels, pecks, quarts, and pints. TABLE. 2 pints (pts.) make 1 quart, marked qt. 8 quarts 1 peck, " pk. 4 pecks 1 bushel, " bu. 8 bushels 1 quarter, " qr. 36 bushels 1 chaldron, " ch. NOTE 1.- The dry gallon contains 268t4 cubic inches. The Winchester bushel, which is adopted as our standard, contains 2150~ cubic inches. It is 18 inches in diameter, and 8 inches deep. The quarter of 8 bushels is an English measure. Questions.. —T 115. What is the use of beer measure? What are its denominations? Repeat the table. How many cubic inches in a beer gallon? ~ 117. COMPOUND NUMBERS. 146 NOTE 2. -The Imperial gallon, adopted in Great Britain in 1826i for all liquids and dry substances, contains 277 o4, cubic inches. 1. In 75 bushels of wheat, 2. In 4800 pints, bow how many pints? many bushels? 3. Reduce 42 chaldrons of 4. In 6048 pecks, how coal to pecks. many chaldrons? 5. In 273 qrs. 6 bu. 3 pks. 6. In 140223 pints, how 7 qts. 1 pt. of wheat, how many quarters? many pints? Time. ~ 117.o Time is the measure of duration. The denominations are years, months, weeks, days, hours, minutes, and seconds. TABLE. 60 seconds (s.) make 1 minute, marked m. 60 minutes 1 hour, " h. 24 hours 1 day, " d. 7 days 1 week, " w 52 weeks 1 day 5 hours 48 min- ) utes 48 seconds, or 365 days 5. 1 year, " yr. hours 48 minutes 48 seconds, ) NOTE 1. - As there is nearly i of a day more than 365 days in a year, we add 1 day to February of certain years, thus giving them 366 days. If the excess was just j of a day, we would add 1 day to every 4th year, thus making the years average 365 days 6 hours, the odd day being added to every year exactly divisible by 4. But as the excess is not quite 6 hours, lacking about i of a day in 100 years, a year divisible by 100, though divisible by 4, has only 365 days, unless it be divisible by 400, when it has 366 days. Thus, 1844 and 1600 had 366 days each, but 1845 and 1700 had only 365 days each. A year of 366 days is called Bissextile, or Leap year. The calendar months, into which the year is divided, are from 28 to 31 days in length. Questions. -- 116. For what is dry measure used? What are its denominations? Repeat the table. How many cubic inches in a dry gallon? Describe the Winchester bushel. What measure is the quar. per? What is said of the Imperial gallon? Which is the larger quan. tity, a quart of milk, or a quart of salt? -- a quart of milk, or a quart of vinegar?- a quart of oats, or a quart of cider? 13 146 COMPOUND NUMIBERS. 1 118. The number of days in each month may easily be remembered from the following lines: Thirty days hath September, April, June, and November; All the rest have thirty-one, Save February, which alone Hath twenty-eight; and one day more We add to it, one year in four. EXAMPLES FOR PRACTICE. 1. How many seconds from 2. In 446947200 seconds Jan, 1, 1790, till March 1, how many weeks? 1804, including the two days named, and making allowance for leap years? 3. How many minutes from 4. On what month, day, July 4th, M., to Sept. 29th, and hour, will 125640 minutes 6 o'clock, P. M.? past 12 o'clock, M., July 4th, expire? 5. At Boston, on the long- 6. In 55020 seconds, how est days, the sun rises at 23 many hours? min. past 4 o'clock, and sets 40 min. past 7; how many seconds in such a day? 7. How many minutes from 8. In 4079520 minutes the commencement of the war how many years i between America and England, April 19th, 1775, to the settlement of a general peace, wnich took place Jan. 20th, 1783? NOTE 2. - The pupil will notice that the years 1776 and 1780 were leap years. Circular Measure. ~ llo8 Circular Measure is used in computing latitude and longitude; also in measuring the motions of the earth, and other planets round the sun. Questions. -- 117'. Of what is time the measure? WVhat are the denominations? Repeat the table. Why is 1 day added to Feb. of certain years? Why is it not added to every 4th year? What years have 365 days, and what 366 days? What is a year of 366 days called Name ihe calendar months, and the number of days in each. Ir 119; 120. COMPOUND NUMBERS. 14M The denominations are circles, signs, degrees, minutes, and seconds. TABLE. 60 seconds (") make 1 minute, marked'. 60 minutes 1 degree, " o 30 degrees 1 sign, " S. 12 signs, or 360 degrees, 1 circle. 1. Reduce 9s. 130 25 to 2. In 1020300", how many seconds. degrees? 3. In 3 signs, how many 4. In 5400', how many ininutes? signs? T i119. IMiscellaneous Table. 20 units make 1 score. 100 lbs. of raisins make 1 cask. 5 score 1 hundred. 100 lbs. of fish 1 quintal. 12 units 1 dozen. 100 lbs. 1 hundred. 12 doz. = 144 1 gross. 18 inches 1 cubit. 12 gross - 144 doz. 1 great gross. 22 inches, nearly, 1 sacred cubit. 200 lbs. of beef 1 barrel 1 gallon of train oil 7~ lbs. pork, or fish, 1 gallon of molasses 11 lbs. 196 lbs. of flour 1 barrel. 24 sheets of paper 1 quire 8 bushels of salt 1 hogshead. 20 quires 1 ream. 280 lbs. of salt at 2 reams 1 bundle, the salt works 1 barrel. 5 bundles 1 bale. in N. Y. A sheet folded in two leaves, or 4 pages, is called a folio A sheet folded in four leaves, or 8 pages, a quarto, or 4to. A sheet folded in eight leaves, or 16 pages, an octavo, or 8vo. A sheet folded in twelve leaves, or 24 pages, a duodecimo, or 12mo. A sheet folded in 18 leaves, or 36 pages, an 18no. A sheet folded in 24 leaves, or 48 pages, a 24mo. 5 points make line, used in measuring the length of the rods of 12 lines 1 inch, clock pendulums. 4 inches I hand, used in measuring the hight of horses. 6 feet 1 fathom, used in measuring depths at sea. Reduction of Fractional Compound Numbers. 9T 120. There are four particular cases in the reduction of fractional compound numbers. 1st, To reduce a fraction of a higher denomination to one of a lower. 2d, To reduce a fraction of a lower denomination to one of a higher. 3d, To Questions. - S118. What are the uses of circula measure What are the denominations? Repeat the table. 148 COMPOUND NUMBERS,'~ 120 reduce a fraction of a high denomination'o Integers of lower denominations. 4th, To reduce integer.'f lZoer denominations to a fraction of a higher. We will consider them in their order. I. To reduce a fraction of II. To reduce a fraction of a higher denomination to one a lower denomination to one of a lower. of a higher. 1. Reduce D —y of a pound 2. Reduce 6 of a penny to to the fraction of a penny. the fraction of a pound SOLUTION. -W e must reduce SOLUTION. - We must reduce O.F of a pound to the fraction - of a penny to the fraction of a of a shilling by multiplying it by shilling, by dividing it by the 20, since 20 shillings make ~1. composite number 12, since 12 This done by ~ 75, gives Y: of pence make 1 shilling. This a shilling, which multiplied by the done by ~ 81, note 2, gives. composite number 12, (~ 75, note of a shilling, which divided by 20, 1,) is reduced to the fraction 6 (multiplying the denominator,) of a penny. Hence, is reduced to the fraction E-B of a pound. Hence, RiJLE. RULE. Multiply as in the reduction Divide as in the reduction of whole numbers, according of whole numbers, according to the rules for the multiplica- to the rules for the division of tion of fractions. fractions. EXAMPLES FOR PRACTICE. 3. Reduce FEW of a pound 4. Reduce Adz of a grain of gold to the fraction of a of gold to the fraction of an grain. ounce. 5. Reduce y;W of a hogs- 6. Reduce,lon of a pint of head of milk to the fraction milk to the fraction of a hogs of a pint. head. 7. Reduce A8 of a hogs- 8. Reduce -8- of a barre' head of ale to the fraction of of ale to the fraction of a a barrel. hogshead. 9. Reduce hja-r of a tun 10. Reduce I2s of a gill of oil to the fraction of a gill. of oil to the fraction of a tun. Questions. -~ 120. How many cases in the reduction of fractiona, compound numbers? Give the first. How are integers reduced fromir a higher denomination to a lower? - from a lower denomination to a higher? How are fractions reduced from a higher denomination to a lower? Give the example and its explanation. Rule. How are they reduced from a lower denomination to a higher? Give Ex. 2 and the sd.utiol. Rule. I 121. COMPOUNE NUMBERS. 149 11. Reduce: rvFf 12. Reduce'-6 8 - of a of a square mrle to the frac- square inch to the fraction of tion of a squares inch. a square mile. 13. Reduce rs5-gs of a 14. Reduce A of a pint te bushel to the fraction of a pint. the fraction of a bushel. 15. Reduce AriW-m of a 16. Reduce " of an hour week to the fraction of an to the fraction of a week. hour. 17. A cucumber grew to 18. A cucumber grew ta the length of I,, of a mile; the length of 1 foot 4 in:hes what part is that of a foot? A _ - of a foot; wha.t part is that of a mile. 19. Reduce 2 of - of a 20..r of a shilling is 2 of pound to the fraction of a what fraction of a pound.? shilling. 21. Reduce j of 2l of 3 22. W of a penny is Q of pounds to the fraction of a what fraction of 3 pounds? penny?. M of a penny is PT of what part of 3 pounds? -rY~- of a penny is j of ) 1925 same manner as by any simple proportion. 830 5 days, Ans. f 9. Hence, questions znvolving a compound proportzon, may be peiformed by the followingRULE. I. Having' made that number the third term which is of the same kind as the answer sought, we form a ratio, either direct or inverse, as required, from any two remaining numo bers which are of the same kind, and place it for one couplet -f the compound ratio. Form each two like remaining numbers into a ratio, and so on, till all are used. II. Having reduced the compound to a simple ratio, by multiplying together the antecedents and the consequents, shortening the process by cancelation, the operation will be the same as in simple proportion. EXAMIPLES FOR PRACTICE. 1. If 6 men build a wall 20 ft. long, 6 ft. high, and 4 ft. thick, in 16 days, in what time will 24 men build one 200 ft. long, 8 ft. high, and 6 ft. thick? 2. If the freight of 9 hhds. of sugar, each weighing 1.200 lb., 20 miles, cost S 16, what must be paid for the freight of 50 tierces, each weighing 250 lbs., 100 miles. Questions. -- T 193. What is the rule for compound proportion? How many, and what circumstances in Ex. 2? How else may the examples be performed? ~ 194 PKROPORTION. z24 NOTE.'- The price of freight depends on three circumstances, the numbse of casks, the size of the casks. and the distance. Ans. $92'59 -{. 3. If 56 lbs. of bread be sufficient for 7 men 14 days, how much bread will serve 21 men 3 davs? Ans. 36 lbs. T'he same by analysis. If 7 men consume 56 lbs., 1 man will consurne I of 56-8 lbs. in 14 days, and IA of 8 lbs., = = of a 1b., in 1 day. 21 men will consume 21 times what 1 man will consume. that is, 21 times 18 168 12 lbs. in 1 day, and 3 times 12 lbs 36 lbs. in 3 days. NToTE 2. - Having wrought the follorilng examples by proportion, le" the pupil be required to do the same by analysis. 4. If 4 reapers receive $11'04 for 3 days' work, how many men may be hired 16 days for $103'04? Ans. 7 men. 5. If 7 oz. 8 drs. of bread be bought for $'06 when wheat is $'76 per bushel, what weight of it may be bought for $18 when wheat is $"90 per bushel? Ans. 1 lb. 3 oz. 6. If $100 gain $6 in 1 year, what will $400 gain in 9 months. NOTE 3. - This and the three following examples reciprocally prove each other. 7. If $100 gain $6 in 1 year, in what time will $400 gain $18? 8. If $400 gain $18 in 9 months, what is the rate per cent. per annum? 9. What principal, at 6 per cent. per annum, will gain $18 in 9 months? 10. A usurer put out $75 at interest, and at the end of 8 months, received(, for principal and interest, $79; I demand at what rate pet cent. he received interest. Ans. 8 per cent. T 1941. Review of Proportion. Questions. —What is ratio? Between what quantities does it exist? What is inverse ratio? - compound ratio? What is proportion? - rule of three? How is it shown that the product of the ex tremes equals the product of the means? What inversions of the ratios ale place? How is it shown that the operation of the rule of three depen-ds on analysis? What is inverse proportion? - direct propor tirl? -compound proportion? EXERCISES~ 1. If 1 buy 76 yds. of cloth for $113'17, what does it cost per ell English 1 Ans. $1'861 t. 2. Bought 4 pieces of IIolland, each containing 24 ells English, for $96; how much was that per yard? Ans.$0'80. 3. A garrison hadi provision for 8 months, at the rate of 15 ounce - c7X 246 ALLIGATION. 1T 195 to each person per day; how much must be allowed per day, in order that the provision may last 9- months? Ans. 12 — cGz. 4. How much land, at $2'50 per acre, must be given in exchange for 360 acres, at $3'75 per acre? Ans. 540 acres. 5. Borrowed 185 quarters of corn when the price was 19s.; hot much must I pay when the price is 17s. 4d.. Ans. 2024- qrs. 6. A person, owning -3 of a coal mine, sells 3 of his share for.~171; what is the whole mine worth I Ans. ~380. 7. If R of a gallon cost, of a dollar, what costs -a of a tun? Ans, $140 8. At ~1i per cwt., what cost 3~ lbs. IAs. As. d. 9. If 4~ cwt. can be carried 36 miles for 35 shillings, how many pounds can be carried 20 miles for the same money T Ans. 9071- lbs. 10 If the sun appears to move from east to west 360 degrees in 24 hours, how much is that in each hour? - in each minute. in each second I Ans. to the last, 15" of a deg. 11. If a family of 9 persons spend $450 in 5 mrnths, how much would be sufficient to maintain them 8 months, if 5 persons more were added to the family? Ans. $1120. AJLLIGATION -- eIedial.:~ 19g. 1. A farmer mixed 8 bushels of corn, worth 6G cents per bushel, 4 bushels of rye, worth O8 cents per bushel, and 4 bushels of oats, worth 40 cents per bushel; what was a bushel of the mixture worth? When several simples of different values are to be mixed the process of finding the average price, is called Alligation Medial. The average price is called the mean price. OPERATION. SOLUTION. - IMultiply 8 bushels, at $'60;* 60 X 8= 8480. ing 8'60, the price of I 4s buAshe's, at $80;'s0 X >4 *320. bushel of corn, by 8, tho product is the price of 8 4 bushels, at $'40; 40 X 4- $1'60. bushels. In like mani-er 16 bushels are worth 90 we find the price of the bls r orh rye, and the oats in the 1 bushe is worth $'60. mixture, and adding to gether the prices of ih corn, the rye, and the oats, we have $9'60, the price of 8 + 4 + 4 = 16 bushels, which are contained in the whole mixture, and dividing Lhe price of 16 bushels by 16, we get the price of 1 bushel. Ai. 60 cents. Hence, the luestions,. —t 195. What is alligation medial? — xmean rate? What > he rule? To ~what does it apply? T 196. ALLIGATION, ~247 RULE. Find the prices of the severa, simples, and add them to gether for the price of the whole compound, which divide by the number of pounds, bushels, &c., to get the price of 1 pound or bushel. NOTE. -The principles of the rule are applicable to many exam{les not embraced by the above definition of Alligation Medial. EXAMPLES PO f P]RACTICE. 2. A grocer mixed 5 lbs. of sugar worth 10 cents per lb., 8 lbs. worth 12 cents, 20 lbs. worth 14 cents; what is a pound of the mixture worth l Ans. $'121-. 3. A goldsmith melted together 3 ounces of gold 20 carats fine, and 5 ounces 22 carats fine; what is the fineness of the mixture l Ans. 214 carats. 4. A grocer puts 6 gallons of water into a cask containing 40 galions of rum, worth 42 cents per gallon; what is a gallon of the mixture worth? Ans. 36-{2 cents. 5. On a certain day the mercury was observed to stand in the thermometer as follows: 5 hours of the day it stood at 64 degrees; 4 hours, at 70 degrees; 2 hours, at 75 degrees; and 3 hours at 73 degrees; what was the mean temperature of that day l Ans. 698; degrees. 6. A farm contains 16 acres of land worth $90 per acre, 22 acres worth $75, 18 acres worth $64, 10 acres worth $55, 30 acres worth $36, and 42 acres worth $25 per acre; what is the average value of the farm per acre! Ans. $50'16 nearly per acre. 7. A dairyman has 20 cows, 3 of which are worth $35 each, 4 are worth $30, 6 are worth $24, 4 are worth $20, 2 are worth $18 each, and 1 is worth $13; what is the average value! Ans. $24'90. Alligation Alternate. ~f 196. 1. A farmer has 1 bushel of corn, worth 60 cents. How many bushels of oats, worth 40 cents, must be put with it, to make the mixture;v1,rth 42 cents? SoWTIoN. -- The 1 bushel of corn is worth 8 ctnts more than the price of the mixture, and as each bushel of oats is worth 2 cents less, he must take 4 bushels of oats Ans. When the price of several simples, (corn and oats,) and the price of a mixture to be formed from them are given, the method of finding the quantity of each simple is called Alliua'i n Alternate. 248 ALLIGATION. ~T 196 2. How many bushels of oats must the farmer take to mix with 2 bushels of corn, prices the same as above? SOLUTION. - Evidently, twice as many as were required for 1 bushel) or, Ans. 8 bushe!d NOTE 1. —We see that 2, the number of bushels of corn, equals the number of cents that the oats are worth less than the mixture, and 8, the number of bushels of oats, equals the number of cents that the corn is worth more than the mixture. Then, if the three prices had been given, we might have taken 2, the difference between thile price of the oats and of tile mixture for the numbter of bushels of corn, and 8, the difference between the price of the corn and of the mixture, for the number of bushels of oats, and we should have such a mixture. That is, we take the differcnce between the price of each simple and of the mixture for the number of bushels of the other simple. NOTE 2. - By this process, the sum of the excesses, found by multiplying 8 cents, the excess of 1 bushel of corn, by 2, the number of bushels. equals the sum of the deficiencies, found by multii)lying 2, the deficiency of 1 bushel of oats, by 8, the number of bushels. Or, 8 X 2 = 2 X 8, since the factors are the same in each. 3. A merchant has two kinds of sugar, worth 6 cents and 17 cents per pound, of which he would make a mixture worth 10 cents; how much of each must he take? OPERATION. SOLUTION. - Take the. 6 7 lbs. difference between 10 Price of mixture, 10 cents. 17 4 lbs. and 17 for the nulmber of lbs. at 6 cents, and the difference between 6 and 10 for the number of lbs. at 17 cents. The sum of the deficiencies, 7 times 4 cents, equals the sum of the excesses, 4 timles 7 cents. NOTE 3. - This gives a mixture of 11 pounds, and if 3 times, 5 times, one half; or any other proportion of the mixture be required, the same proportion of each simple must evidently be taken. The same would be clone if 3 times, 5 times, &c., one simple were given. 4. A merchant has two kinds of sugar, worth 8 cents and 13 cents; how much of each must he take for a mixture worth 10 cents per pound? OPERA'rION. Price of mixture, 10 cents. 8 13 3 lbs at 8 cents 13 2 lbs. at 13 cents. NOTE 4. - The price of the mixture in the last two examples is the same. 5. A merchant has four kinds of sugar, worth 6, 8, 13, and 17 cents per pound; how much of each must he take for a mixture worth 10 cents? ~ 197. ALLIGATION. 249 OPERATION. SOLUTIoN.-Connecting i( 6A 7 6 with 17, to show that t I asugar at those prices are Price of mixture, 10 cents. -J to be mixed we have 7 13 2. lbs. at 6 cents and 4 lbs. 17 —% 4. at 17 cents, forming a mixture of 11 lbs. -worth _0 cents. In like manner, we have a mixture of 5 lbs. worth 10 cents, by connecting S and 13. And 11 + 5 3 16 lbs. in all. INOTE 5.- We may see in this and all examples, that the sum of the defi, ciencies equals the sum of the excesses. 6. A merchant has 3 kinds of sugar, worth 6, 8, and 15 cents; how much of each must he take for a mixture worth 11 cents per pound? OPE RATION. SOLUTION. - (6- 4. We have not two pairs, as Price of mixture, 11 cents. S- 4 i4. o pairs, asbme ( 15- 5+3 8. example, but4 lbs. at 6 cents, may be mixed with 5 lbs. at 15 cents, and 4 lbs. at 8 cents, with 3 lbs. at 15 cents. The 15 is connected with both, because we take two portions at this price, 5 lbs. to mix with 4 lbs. at 6 cents, and 3 lbs. to mix with 4 lbs. at 8 cents. We have, then, 8 lbs. at 15 cents. ~ 19.o From the examples explained, we have the fol lowing RULE. I. WTrite the prices of the simples directly under each other, beginning with the least, and the price of the mixture at the left hand. II. Connect each price less than the mean price with one or more greater, and each price greater with one or mctre that is less. III. Write the difference between the price of the mixture and of each simple opposite to th3 price or prices with which it is connected; the number or numbers opposite to the price of each simple will express its quantity in the mixture. IV. If any measure or multiple, as one third, or three times one of the simples is to be taken, the same measure or multiQuoestions. - 196. Why 4 bushels of oats to I of corn, Ex. 1 and 2'. What equalities appear from Ex. 2 H-low could we halve made such a mixtuie as we have, if the prices onlly had been klnown? Give the general imethod of forming such mixtures. What are equal, as explained in inote 2? Why? What must be done if a greater or less quantity of the mixture be required? - of one simple? Why are numbers connected? What is done, in Ex. 6, when Here are not two pairs? What is alligation alternate ~T 197. What is the rule? 250 ALLIGATION. ~ 397 ple of each of the other simples will be required. Or, if any measure or multiple of the whole compound be required, the same measure or multiple of each simple must be taken. EXAMPLES FOR PRACTICE* 1. What proportions of sugar, at 8 cents, 10 cents, and 14 cents per pound, will compose a mixture worth 12 cents per pound? Ans. In the proportion of 2 lbs. at 8 and 10 cents to 6 lbs. at 14 cents. 2. A grocer has sugars, worth 7 cents, 9 cents, and 12 cents per lb., which he would'mix so as to form a compound worth 10 cents per pound; what must be the proportions of each kind! Ans. 2 lbs. of the first and second, to 4 lbs. of the third kind. 3. If he use 1 lb. of the first kind, how much must he take of the others? -~ if 4 lbs., what? -- if 6 lbs., what -? if 10 lbs., what. -~ if 20 lbs., what? Ans. to the last, 20 lbs. of the second, and 40 of the third. 4. A merchant has spices at 161., 20d., and 3'2d. per pound; he would mix 5 pounds of the first sort with the others, so as tG foim a compound worth 24d. per pound; how much of each sort must he use? Ans. 5 lbs. of the second, and 7A lbs. of the third. 5. I-low many gallons of water, of no value, must be mixed with SO gallons of rum, worth 80 cents per gallon, to reduce its value to 70 cents per gallon? Ans. 84- gallons. 6. A man would mix 4 bushels of wheat, at $1'50 per bushel, with rye at $1'16, corn at $'75, and barley at $'50, so as to sell the mixture at $'84 per bushel; how much of each must he use? 7. A goldsmith would mix gold 17 carats fine with some 19, 21, and 24 carats fine, so that the compound may be 22 carats fine; what proportions of each must he use? Ans. 2 of the first 3 sorts to 9 of the last. 8. If he use 1 oz. of the first: kind, how much must he use of the others?' What would be the quantity of the compound? Ans. to the last, 7A ounces. 9. If he would have the whole compound consist of 15 oz., how nuch must he use of each kind --- if of 30 oz., how much of each kind t if of 37. oz., how much? Ans. to the last, 5 oz. of the first 3, and 22. oz. of the last. 10, A man would mix 100 pounds of sugar, some at 8 cents, some at 10 cents, and some at 14 cents per pound, so that the compound may be worth 12 cents per pound; how much of each kind must he use9 20 lbs. at 8 cts. ) 20 lbs. at 10 cts. Ans. 60 lbs. at 14 ets. 3 11. A grocer has currants at 4d., 6d., Md., and 1ld. per lb., and he ivould make a mixture of 240 lbs., so that the mixtuie may be sold it 8d. per lb.; how many pounds of each sort may he take? Ans. 72, 24, 48, and 96 lbs., or 48, 48, 72, 72, &o. NOTE. - This question may have fiv 2 different answers. ~ 198, 199. EXCHANGE.,251 EXCHANGE. ~1 198. If a farmer, A, has corn, and a manufacturer, B) has cloth, each mo:e than he needs himself, while he wants some of the article possessed by the other, an exchange will be made for mutual accommodation. But if B done not want A's corn, while A still wants the cloth, the latter must find a third pnrson, if possible, who wants his corn, and can give him something for it which B may want. This might ]en difficult, unless some article was settled on, which every one would take; then A might exchange his corn for it, as B would part with his cloth for such an article, since, if every one would take it, he could procure with it whatever he might desire, though he did not want it himse.f. Such an article is called money. Gold and silver, contain ng great value in little space, are employed for money among civilized nations, and sometimes copper, to represent small values. This exchange between individuals is called trade, or commerce. NOTE 1. - Since gold and silver, in their pure state, are too flexible for the purposes of a circulating medium, nine parts of pure gold and one part of silver and copper, in equal quantities, are used by the U. S. government, for gold coins, nine parts (f silver and one of copper, for silver coins. The baser metal, in each instance, is cahed alloy. The English government uses only one part of alloy to eleven of the gold, and a little less alloy in silver coins. Hence, English coins are more valuable than ours of the same weight. Again, as coins are troublesome, and sometimes expensive to transport, and also suffer loss by wear, bank bills are much used for circulation, which, though valueless themselves, are readily taken as money, being payable in specie, on demand at the banks which issued them. The coins are called specie, in distinction from paper money, and together they form what is called the circulating medium, or currency. NOTE 2. - By the fineness of gold, is meant its purity, a twenty-fourth part of any quantity being called a carat. When, for example, there are two parts of alloy to twenty-two of' pure gold, it is said to be twenty-two carats fine. IT 1199. Bank bills can be used in trade by individuals of the same country, but are not convenient in trade between those of different countries, since they would be removed too far from the place where payable. But the cost and risk of Questions. - IT 195. What trade are A and B supposed to male? What will A do, if B does not want his corn? What is exchange' What is money' What are used for money, and why? How much do coins want of being 1llmre gold and silver? What is the value of English coins compared with ours, and why? Why are bank bills used? What do you understand to be the differ ence between a bank bill now described, and a bank note, 1 174? What is specie? What is currency? What is meant by fineness of gold? — by carat? - by carats fine? Illustrate.. 252 EXCHANGE. I 199. transporting specie 3 considerable. If, for instance, Boston merchants purchase goods in HaTrn-mbur'h to the amount of $2,050,000 a year, and the expense of transporting specie was 3 per cent., which is only a moderate allowance, this expense would be $61,500 to make payments for one year. If, on the other hand, Han1burgh merchants should purchase S2,000,000 worth of goods, it would cost them $60,000 to make the pay.. ments in specie. To reduce this expense, the following method has been devised. A B, a Boston merchant, has $10,000 due him in Hamburgh, from C D. He writes an order for the sum, and finds some one who owes $10,000 to another merchant in Hamburgh. He sells to him this order, and the purchaser sends it to his creditor, who goes to C D, A B's debtor, and receives the money. In this way, $2,000,000 of the Boston purchase might be balanced by the $2,000,000 purchased in Boston by Hamburgh merchants, and the Boston merchants would have to send only $50,000, the balance, thus reducing the expense of making payments between the two ports from $121,500 to S1500! Such an order is called a Bill of Exchcazge. NOTE 1.- Lest a bill of exchange may be lost, or delayed, three copies are sent, by different conveyances, and when one is paid, the others are canceled. The form of one bill, to which the others agree, except in the numbers of the bills, is here given. Exchange for $10,000. Boston, Jan. 1, 1848. Three months after date, pay this, my first of exchange, (second and third of the same tenor and date not paid,) to the order of Flemming and Johnson, ten thousand dollars, value received, with, or without further advice from me. A B. C D, Merchant at Hamburgh. NOTE 2. If A B had to be at the expense of bringing from Hamburgh the opecie on his due, he could afford to sell his bill for less than'10,000, on account of freight. This he would have to do when more was to be paid from Hamburgh to Boston than from Boston to Hamburgh, that is, if the balance of trade were in favor of Boston, as there would not be dem-and for all tis Questions. — T 199i. Why are not bank bills convenient in extha,,-. between difdierent countries? What diftlcultI in playing with specie? I lustrate by the exanlple of trade between Boston and Hamburgh. How is A B supposed to get his pay from Hallm:urgh, on a debt of $10,0c3? Hoxv mucti is saved by this means, in the supposed trade between the two ports? Why must $50,000 of specie be sent to Harnburgh? What is a bill of exchange 7 What care is taken in sending bills? Give a form. What would be the form of the second bill? - of the third? When, and why, is exchange elaw,ar? -- above par? Why do blokers deal in exchange? i1 200. EXCHANGE. 253 orders on Hamburlgh, and specie wound have to be brought on some. ExEhange is then said to be below par. But if the pulchaser had to pay fior freighlting the monev he owes to Harmburgh, he could afiord to pay more than 10o,000 for A B's bill, which he would have to (do when the balance of trader waS- against Boston, that is, when Boston owed more to lHalnlnurgh than it blad owing, for then there would not be orders on Hamblurgll suflicient to pay the debts, and some must se)nd specie. Exchange is then said to be above pat, or at a premium. NOTE 3.- The broker is the medium between the seller and purchaser of bills, since the former might not Ikow to whom he could sell, or the latter of cwhom he could buy. Brokers purchase bills, or take them to sell on coinmisSion. The illustrations of the principles of exchange will be applied to exchange with England and France. Exchange with England. v 200. The nominal value of the pound sterling is $4'444-, consequently, a bill of exchange for ~1000 is said to be worth $4444'44~. But by comparing the materials of the English sovereign, a gold piece representing a pound, and the eagle of our currency, the former is worth about $486 6. Sovereigns, however, are more or less worn by use, those dated as far back as 1821 being worth no more than $4'80. It is pre sumed that a quantity will average $4'84 each, and at thi;, value they are taken in payment of duties. If, now, the nominal pound, $4'44~-, be multiplied by'09, 84'44t-X'09 ='40, and the product be added to it, $4444 +'40 $4'844 it will be changed to about its value, in the custom-house*es timations. If'09k be added to the nominal pound, it will become $4'862, nearly its commercial value. When, then, sterling exchange is quoted at 9 or 91 per cent. advance, we must understand that bills sell for their par value. When above these rates, they are at a premium; when below, at a discount. NOTE 1.- In the following examples we shall consider 9. per cent. abtovt the nominal, the par value. The pupil must not suppose, however, that 10 per cent. above the nominal, would be l per cent. above the real value of a bill. 1. A nerelhant sells a bill of exchange for ~5000 at its par valu e; what does he receive? Ans. 924333'33-. 2. A merchant sold a bill of exchange for ~7000 sterling, at 11 per cent. advance; what did he receive more than its real value? Ans. $406'666. 3. A merchant sells a bill on London for ~4000 at 8 per cen. above its nominal value, instead of importing specie at an expense of i per cent.; what doeghe save? Ans. $122'6i6. 254 EXCHANGE. 20 L 4. A Mroker sold a bill of exchange for ~2000, on conwnission, at 10 per cent. above its nominal value, receiving a commission of -io per cent. on the real value, and 5 per cent. on what he obtained fob the bill above its real value; wha, was his commission. Ans. $11'915-. NOTE 2.-ThouIgh dollars and cents are the denominations of U. S. money, shillings and pence are much used in common calculations. But the dollar nas different values in different states, as expressed in shillings; thus. il. New York, Ohio, and Michigan, it is 8 shillings; in North Carolina, 10.slillings; in New Jersey, Pennsylvania, Delaware, and Maryland, it is 7 shillings 6 pence; in Georgia and South Carolina it is 4 shillings 8 pence, while in the other states it is 6 shillings. The change of monev from one of these currencies to the other is not now worthy of a formal discussion, as a method will readily suggest itself to the practised arithmetician, and the custom of using these denominations, it is hoped, will be speedily given up for the sim pler system of our federal currency. Thus, as 6 shillings in New E glane equal 8 shillings in New York, add one third of any number of shillings N. E currency to the number, and we have the value expressed in shillings N. Y nurrelncy. Exchange with France. [ A@d1. The unit of French money is the franc, the value of which is $'18;. In the quotations of French exchange, wc have the number of francs that the dollar is rated at..As $1'00 is equal to 5 186 francs = 5'376 +, hence, when a dollar 18 6 is worth 5337 i francs, it is at par. I1 A New York merchant sold a bill of exchange for $2500 on lIavre, at 5'4 francs per dollar; what did he obtain for it more than its value? Ans. $11. 2. A merchant bought a bill on Havre of $2800 at 5'31 francs per dollar; what did he give less than its value Ans. $34'552. Questions. -S T 200. What is the nominal value of the English pound? ~-the real value? What are sovereigns of 1821 worth? Why no more? What is the average value of sovereigns supposed to be, and where are they taken at this value? How is the pound changed from its nominal to its real value? What is added to the nominal value in the examples? What is said of 10 per cent.? What denominations are still used in common calculations? Whllat are the different values of the dollar in different states? 9 201. What is the unit of French money? -- its value? What Is,ha far value of the dollar, as expressed in francs ~1 202. EXCHANGE. 265 202. Value of Gold Coins According to the Laws passed by Congress, May and Ji ne, 1834.] NAMES OF COINS. NAMIES OF COINS.' d. c.m. d. cm. UNITED STATES. HANOVER. Eagle, coined before July 31, Double George d'or, single in 1834, 10 66 5 proportion, 7 87 9 Shares in proportion. Ducat, 2 29 6.Gold Florin, double in proporF'OREIGN GOLD. tion, 1 67 0 AUSTRIAN I OtMIINIONS. HO,LLAND. Souverein, 3 37 7 Douhle Ryder, 12 20 5 Double Ducat, 4 58 9 Ryder, 6 04 3 Hungarian, do., 2 29 6 Ducat, 2 27 6 BAVARIA. Ten Guilder piece, 5 do. in proCarolin, 4 95 7 portion, 4 03 4 Max d'or, or MIaximilian, -3 31 8 MAL'rA. Ducat, 2 27 5 D)ouble Louis, 9 27 8 BERNE. Lou;s, 4 85 2 Ducat, double in proportion, 1 98 6 Demi Louis, 2 33 6 Pistole, 4 54 2 MEXICO. BRAZIL. Doubloons, shares in proporticn, 15 53 5 Johannes, I in proportion, 17 6 4 MILAN. )obraon, 32 70 6 Sequin, 2 29 0 Dobra, 17 30 1 Doppia, or Pistole, 3 80 7 Moidore, ] in proportion, 6 55 7 Forty Livre Piece, 1808, 7 74 2 Crusade, 63 5 NAPLES. BRUNSWICK. Six Ducat Piece, 1783, 5 24 9 Pistole, double in proportion, 4 54 8 Two do., or Sequin, 1762, 1 59 1 Ducat, 2 23 0 Three do., or Oncetta, 118, 2 49 0 COI,OGNE. NETHEIRLANDS. Ducat, 2 26 7 Gold Lion, or Fourteen Florin COLOMTBIA. Piece, 5 04 6 Douhiloons, 15 53 5 Teo Florin Piece, 1820, 4 01 9 LENIMIARK. PARMiA. Ducat, Current, 1 81 2 Quadruple Pistole, double in Ducat, Specie, 2 26 7 proportion, 16 62 8 Christian d'or, 4 02 1 Pistole or Doppia, 1787, 4 19 4 EAST INI)IES. do. do., 1796, 4 13 5 Rupee, Bombay, 1818, 7 09 6 Maria Theresa, 1818, 3 86 1 Rupee, Madras, 1818, 7 11 0 PIEDMONT. Pagoda, Star, 1 79 8 Pistole, coined since 1785, half ENGLAND. in proportion, 5 41 1 Guinea, half in proportion, 5 07 5 Sequin, half in proportion, 2 2S 0 Sovereign, do., 4 84 6 Carlino, coined since 1785, half Seven Shilling Piece, 1 69 8 in proportion, 27 34 0 FRANCE. Piece of 20 francs, called Ma. Douhle Louis, coined before reno, 3 56 4 1786, 9 69 7 POLAND. Louis, do., 4 84 6 Bucat, 2 27 5 I)ouble Louis, coined since 1786, 9 15 3 PORTUGAL. Louis, do. do., 4 57 6 Dobraon, 32 70 6 Doulle Napoleon, or 40 francs, 7 70 2 Dobra, 17 30 1 Napoleon, or 21 do., 3 85 1 Joliannes, 17 06 4 FRANKFORT ON THE MTAIN. MVIoidore, haf in proportion, 6 55 7 Ducat, 2 27 9 Piece of 16 Testoons, or 1600 GENEVA. Rees, 2 12 1 Pistole, old, 3 98 5 Old Crusade, of 400 Rees, 58 5 Pistole, new, 3 44 4 New do., 480 do., 63 5 GENOA. Milree, coined iil 1775, 78 0 Sequin, 2 30 2 PRUSSIA. HAMBURG;. Ducat, 1748, 2 27 9 Ducat, double in proportion, 2 27 9 do.,'787, 2 21 7 246- DUODECIMALS. ~ 203 NAMES OF COINS. NAMES OF CoiNSs. d. c. n. d. c. nm. Frederick, double, 1769, 7 95 5 Coronilla, Gold Dollar, or Vinm do. do., 1800, 7 95 1 tern, 1801, 98 3 do. single, 1778, 3 99 7 SWET)EN. do. do., 1800, 3 97.5 Ducat, 2 23 5 ROME. SWITZERLAND. Sequ a., coined since 1760, 2 25 1 Pistole of Helvetic Republic, Scudc if Republic, 15 81 1 1800, 56 RUS 1JSIA. TREVES. Ducat, 1796, 2 29 7 1)ucat, 2 26. do., 1763, 2 26 7 TURKEY. Gold Ruble, 1756, 96 7 Sequin Fonducli, of Constantido., 1799, 73 7 nople, 1773, 1 86 d'). Poltin 1777, 35 5 do., 17S9, 1 84 In,perial, 1801, 7 82 9 Half Misseir, 1818, 52 Half (ldo., 1801, 3 93 3 Sequin Fonducli, 1 83 3 t iItl)INIA. Yeerrneeblekblek, 3 02 8 Carlino, half in proportion, 9 47 2 TUSCANY. S>,XONY. Zechino, or Sequin, 2 31 8 Ducat, 1784, 2 26 7 Ruspone of the kingdom of do., 1797, 2 27 9 Etruria, 6 93 8 Augustus, 1754, 3 J20 5 VENICE. do., 1784, 3 97 4 Zechino, or Sequin, shares in SICILY. proportion, 2 31 0 OuLnce, 1751, 2 50 4 WIRTEMBURG. Double do,, 1758, 5 04 4 Carolin, 4 89 8 SPAIN. Ducat, 2 23 5 Doubloon, 1772, double and sin- ZURICH. gle, and shares in proportion, 16 02 8 Ducat, double and half in proDoubloon, 15 53 5 portion, 2 26 7 Pistole, 3 88 4 DU0@DECIIM'ALS ~ 0,@ 3. Duodecimrnals are fractions of a foot. The word is derived from the Latin word duodecim, which signifies twelve. A foot, instead of being divided decim2ally into ten equal parts, is divided duodecimally into tzelve equal parts, called primes, marked thus ('). Again, each of these parts.s conceived to be divided into twelve other equal parts, called seconds, ("). In like manner, each second is conceived to be divided into twelve equal parts, called thirds ("'); each third into twelve equal parts, called fourths (""); and so on t: any extent.'In this way of dividing a foot, it is obvious, that 1 prime is....... of a foot. second is I of ].. T. - of a foot. 1 third is I- of I of IT, - T.2 of a foot, 1 fourth is T of -: of 1 of, - r+TT of a foot. 1 fifth is -z of l of of z of ~ f, =z-s: of a foot, &c. IT 204. DUODECIMALS. 2f57 TABLE. 12"" fA urths make 1"'third, 12"' thirds... 1" second, 12" seconds'.. 1' prime, 12' primes,.. foot. NOTE 1. -The marks,', ", %"', ", &C., which distinguish the different parts, are called the indices of the parts or denominations. NOTE 2.-The divisions of a unit in duodecimals are uniform, just as in decimal fractions, —with this difference: they decrease in a twelvefold proportion, 12 of a lower denomination making 1 of a higher. Operations in them are consequently the same as in whole numbers or decimals, exceut that 12 is the carrying number instead of 10. MBultiplication of Duodecirmals. - @~04. Duodecimals are used in measuring surfaces and solids. 1. How many square feet in a board 16 feet 7 inches long and 1 foot 3 inches wide? NOTE. -- Length X breadth = superficial contents, (ST 48.) OPERATION. SOLUTION. -7 inches, or primes, - 7 ot ft. a foot, and 3 inches = - of a foot; conse Len zth, 16 7' X 2 Lngh 1' quently, the product of 7' X 3'- -T of a Br3ealth, 1 3' foot, that is, 21"/- 1' and 9"; wherefore we set down the 9", and reserve the 1' to be car 4 1' 9" ried forward to its proper place. To multi16 7' ply 16 feet by 3', is to take T3 of 4l6 =4 that is, 4S'; and the 1' which we reserved As$. 20 -,' 9,, makes 49', = 4 feet 1'; we therefore set down Ans. 20 8 9. the 1', and carry forward the 4 feet to its proper place. Then, mrultiplying the mu-Lltipllcand by the 1 foot in the multiplier, and adding the two products together, we obtain the Answver, 20 feet 8' and 9-. NOTE 1. - In all cases the product of any two denominations will always be of the denomination denoted by the sum of their INDICES. Thus, in the above example, the sum of the indices of 7'X3' is "; consequently, the product is 21'; and thus primes multiplied by primes, produce seconds; priimes multiplied by seconds, produce thirds; fourths multiplied by fifths, produce ninths, tC. Questions. — ~ 203. V'hat are duodecimals? Explain the duodecimal divisions and subdivisions of a foot. Repeat the table. What are indices? What part of a foot is 1'? - 1"? - 1' - I""? - 1"" What difference between the decimal and duodecimal divisions of a unit? How are operations 1n duedecimals performed? 22: 258 DUODE: IMALS. ~ 204 2. HIow many solid feet.n a block 15 ft. 8' long 1 ft. 5' wide, and 1 ft. 4' thick. OPERATION. ft. Length, 15 8' Breadth, 1 5' 6 6' 4" 15 8' The _ength multiplied by the breadth, arid that product by the thickess, gives the solid contents, (~[ 51.) 22 2' 4" Thickness, 1 4' 7 4' 9" 4"' 22 2' 4" Ans. 29 7' 1" 4"' Hence, To multiply diuodecimals, I. Write the multiplier under the multiplicand, like denominations under like, and in multiplying, remember that the product of any two denominations will be of that denomination denoted by the sunm of their indices. II. Add the several products together, and their sum will be the product required. EXAIP LES FOr PRACTICE. 3. How many square feet in a stock of 15 boards, each of which is 2 ft. 8' in length, and 13' wide? Ans. 205 ft. 10'. 4. What is the product of 371 ft. 2' 6" multiplied by 181 ft. 1' 9"T Ans. 67242 ft. 10' 1" 4"' 6"". 5. There is a room plastered, the compass of which is 47 ft. 3', and the hight 7 ft 6'; what are the contents? Ans. 39 yds. 3 ft. 4' 6". 6. What will it cost to pave a court yard, 26 ft. 8' long by 24 ft. 9 wide, at $'90 per square yard T A ns. $66 7. There is a house containing two rooms, each 16 ft. by 15 ft. 4'; a hall 24 ft. by 10 ft. 6'; three bed-rooms, each 11 ft. 4 by 8 ft.; a pantry 7 ft. by 9 ft. 6'; a kitchen 14 ft. 2' by 18 ft., and two chamQuestionso,- 201,o For what are duodecimnals used? Of what denomination is the prodt.ct of any two denominations? Repeat the rule for the multiplicationl of duodecimals. How do you carry firm one denominationl to another? How is masons' work estimated? What is understood by girt: and for wllat used? ~l 205. DUODECIMALS. 259 bers, each 16 ft. by 2C l. 8'; what did the work cf flooring cost, at $'02 per square foot? Ans. $39'95. NOTE 2. — Masons' w)Jrk is estimated by the perch of It2 feet in length, 1I feet in widthi, andl 1 foot in hight. A perch contains 24'75 cutic feet. If allny wall be 1i feet thick, its contents in perches may tbe found by divi(ling its superficial contents by 161; but if it be any other tlicrkness than 1i feet, its cubic contents must be divided by 24'75, (_24-4,) to reduce it to perches. Joiners, painters, plasterers, brick-lnyers, and nmasons, make no allowance for w indows, doors, &c. Bricl-layers and inasons rnake no allowance for corners to the walls of houses, cellars, &c., but estimate their work by the girt, that is, the length of the wall on the outside. 8. The side walls of a cellar are each 32 ft. 6' long, the end walls 24 ft. 6', and the whole are 7 ft. high, anti 1. ft. thick; how many perches of stone are required, allowing nothing for waste, and for hox, many must the mason be paid? As. 459 perches in the wall. The mason must be paid for 48-4T perches. 9. How many cord feet of wood in a load 7 feet lontg, 3 feet wide, and 3 feet 4 inches high, and what will it cost at $'40 per cord foot? Ans. 4h cord feet, and it will cost $1'75. 10. H-ow much wood in a load 10 ft. in length, 3 ft. 9' in width and 4 ft. 8' in hight? and what will it cost at $1'92 per cord? Ans. 1 cord and 21- cord feet, and it will cost $2'62. ~T 1@25. By some surveyors of wood, dimensions are taken in feet and decimals of a foot. For this purpose, make a rule or scale 4 feet long, and divide it into feet, and each foot into ten equal parts. Such a rule will be found very convenient for surveyors of wood and of lumber, for painters, joiners, &c.; for the dimensions taken by it being in feet and decimals of a foot, the casts will be no other than so many operations in decimal firactions. 1. How many square feet in a hearth stone, which, by a rule, as above described, measures 4'5 feet in length, and 2'6 feet in widths and what will be its cost, at 75 cents per square foot? Ans. 11'7 feet; and it will cost $8'775. 2. How many cords in a load of wood, 7'5 feet in length, 3'6 feet iil width, and 4'8 in hight? Ans. 1 cord l-6 ca. ft. 3.' How many cord feet in a lead of wood 10 feet long, 3'4 feet wide, and 3'5 high Ans. 7i 7 Questions. -- 205. Hvow do some surveyors of wood take dimen. s.ons'! Explair th-s rule ased in measuring. How are dimensions taken bh i; estimated I 260 INVG LUT1ON. ~ 206 INVOLUTION. First power. ST.@0. Three feet in length (~T 111) are a yard, linear measure; 3 in length and 3 in width, 3 X 3 -9 square feet, are a yard, Second power. square measure; 3 in length, 3 in width, and 3 in bight, 3 X 3 X 3 27 solid feet, are a yard, cubic measure, (T113.) When a number, as 3, is multiplied into itself, and the product by the original number, and so on, the several numbers produced are called powers, and the process of producing them is called Involution. T'zird power. The first number, represented by a line, is called the first power, or root; the second, represented by a square, is called the square, or 2d power; the third, represented by a cube, is,COLT~ called the cube, or 3d power. Fourth power. The 4th power of 3 is 3 times the 3d power, 3 blocks like- that employed to represent the 3d power, and mray be rep-!i]D' 3 l 1 Jresented by a figure 3 times as large, that is, 3 feet wide, 3 feet high, and 9 feet long. Fifth power. The 5th power, by 3 times such a figure, or one 3 high, 9 wide, and 9 long. The 6th power, 3 times this, by I la figure 9 long, 9 wide, and 9 high, or a cube. Sixth pouwer Thus it may be shown that the 9th, 12th, 15th, 1Sth, &c., powers,; may be represented by cubes; the 7th, 10th, 13th, 16th, &c., by fig. i i width and hight; the Sth, 11th. [.i/jjijij1~ili iIi'11 14th, 17th, &c., by figures having 11iiliI greater length and width than To involve a number, take it as a factor as many times as is indicated by the required power. NOTE. 1 —The number denoting the power is called the index, or expa Irat; thus, 54 denotes that 5 is raised or involved to the 4th power. ~ 206. INVOL JTION. 261 EXAMPLES FOR PRACTICE. 1, What is the square, or 2d power, of 7. Ans. 49 2. What is the square of 30? Ans. 900. 3. What is the square of 40001 Ans. 16000000. 4. What is the cube, or 3d power, of 4. Ans. 64. 5. What i3 the cube of 800. Ans. 512000000. 6 What is the 4th power of 60 1 Ans. 12960000. 7. What is the squareof f 1 -- of 2 -- of 3 - of 4 1 Ans. 1, 4, 9, and 16. 8. What is the cube of 1 - of 2. - of 31 of 41 Ans. 1, 8, 27, and 64. 9. What is the square of 2 -- of 4 - of. Ans. 4 {6, and 49 10. What is the cube of? -- of 5 of 7 f Ans., 64 and 3U 11. What is the square of ----- the 5th power of A? Ans. I and 12 12. What is the square of 1'5 -- the cube 4 Ans. 2'25, and 3'375. 13. What is the 6th power of 1'2 1 Ans. 2'985984. 14 Involve 291 to the 4th power. NOTE 2. - A mixed number, like the above, may be reduced to an im proper fraction before involving; thus, 24 =4; or it may be reduced to a decimal; thus, 2-1 2'25. Ans. 6 5 6 1 n- 251-6' ~5F- I 5 15. What is the square of 4z- 7 Ans. -iu- =234. 16. What is the value of 74, that is, the 4th power of 7 1 Ans. 2401. 17. How much is 93. -- 651 - 1041 Ans. 729, 7776, 10000, 18. How much is 27-.- 36 - 531 65 -. 10s.t Ans. to the last, 100000000. NOTE 3. - The powers of the nine digits, from the first power to the fifth may be seen in the following TA]BLES. Roots.. or lst Powers Il 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 Squares. or 2d Powers i 1 f 4T 9 1 16 1 25 1 36 1 49 1 64 1 81,Cubes.. f or 3-1 Powers 1 1 8. 271 64f 125 f 216 { 343 1 512 1 729 Biquadrates f or 4th Powers I 1 16 81 1 256 1 625 1 11296 1 2401 1 4096 6561 Sursolids. or 5th Powers I 1 1 32 243 1 1024 1 312'5 1 7776 1 1i680 7 1 32768 1 59049 Questions, --' 206. What are powers? How is the first power represenited? 2Why is the second power called the square? Why the third called the cube? How is the fourth power represented? - the fifth? - the sixth? - thirtenth? - the fourteenth? - the twenty-first? - the twentythird? - the twenty-fifth? What is involution? How is a number involved to any power? What is tl e index, and how written? How is a mixed number involved? 262 EVOLUTION. ~ 207, 208 WEVLUTION. ~ 2'.o Evolution, or the extracting of roots, is the method of finding the root of any power or number. The root, as we have seen, is that number, which, by a continual multiplication into itself, produces the given power, and to find the square root of a number (one side of a square when the contents are given) is to find a number, which, being squared, will produce the given number; to find the cube root of a number (the length of one side of a cubic body when the solid contents are given) is to find a number, which, being cubed or involved to the 3d power, will produce the given number: thus, the square root of 144 is 12, because 122 -- 144; and the cube root of 343 is 7, because 73, that is, 7 X 7 X 7 = — 343; and so of other numbers. NOTE. — Although there is no number which will not produce a perfect power by involution, yet there are many numbers of which precise roots can never be obtained. But, aby the help of decimals, we can approximate, or approach, towards the root to any assigned degree of exactness. Numbers, whose precise roots cannot he obtained, are called surd numbers, and those whose roots can be exactly obtained are called rational numbers. The square root is indicated by this character A placed before the number; the other roots by the same character, with the index of the root placed over it. Thus, the square root of i5 is expressed / 16; and the cube root of 27 is expressed 4/ 27; and the 5th root of 7776, 57767. When the power is expressed by several numbers, with the sign + or -- between them, a line, or vinculum, is drawn from the top of the sign over all the parts of it; thus, the square root of 21 -5 is A21 —5.:xtraction of the Square Root. ~T 0S~. 1. Supposing a man has 625 yards of carpeting, a-.yard wide, what is the length of one side of a square room, the floor of which the carpeting will cover? that is, what is one side of a square, which contains 625 square yards? SOLUTION.- We may find one sidle of a square containing 625 square yards, that is, the square root of 625, by a sort of trial; and, Questions. - -T 207. What is evolution? What is a root? - the square root, and how found? - the cube root, and how found? Give exanmpies. What do you say of perfect powers and perfect roots? Give the distinction between surd and rational numbers. How is the square root indi rted? - the cube root? Describe the manner of using the vlnculum. f 208. EVOLUTION. 263 1st. AWe will. endeavor to ascertain how many figures there will be in the root. This we can easily do, by pointing off the number, from units, into periods of two figures each; for the square of and root always contains just twice as Jr: any, or one figure less than twice as many figures, as are in the root. the square of 3 (3 X 3 = 9) contains 1 figure, the square of 4 (4 X 4= 16) contains 2 figures; the square of 9 (9 X 9= 1) contains 2 figures; the square of 10 OPERATION (10 X 10 = 100) contains 3 figures; the e2 j0 square of 32 (32 X. 32- 1024) contains 625 (2 4 figures; the square of 99 (t99 X 994 9801) contains 4 figures; the square of 100 (100 X 100 = 10000) containl 5 figures, and so of any number. Pointing 225 off the number, we find that the root will consist of two figures, a ten and a FIG. I. unit. 2d. We iwxill now seek for the first figure, that is, for the tens of the root, which we must extract from the left hand periA od, 6, (hundreds.) The greatest square in 6 (hundrbds) we find to be 4, (hundreds,) the root of which is 2, (tens, - 20; therefore, we set 2 (tens) in the root, 20 Since the root is one side of a square, let 20 us form a square, (A, Fig. I.,) each side 400 of which shall be regarded 2 tens, = 20 yards long. L__ _ The contents of this square are 20 X) a~ 2 o ~ 20 - -400 yards, now disposed of, and which, consequently, are to be deducted from the whole number of yards, (625,) leaving 225 yards. This deduction is most readily performed by subtracting the square number, 4, hlundreds,) or the square of 2, (tens,) from the period 6, (hundreds,) and bringing down the next period to the remainder, making 225. 3d. The square A is now to be enlarged by the addition of the 225 remaining yards; and in order that the figure may retain'its square form, the addition must be made on two sides. Now, if the 225 yards be divided by the length of the two sides, (20 $- 20 = 40,) the quotient will be the breadth of this new addition of 225 yards to the sides c d and b c of the square A. But our root already found, = 2 tens, is the length of one side of the figure A; we therefore take double this root, - 4 tens, for a divisor. The divisor, 4, (tens,) is in reality 40, and we are to seek how many times 40 is contained in 225, or, which is the same thing, we may seek how many times 4 (tens) is contained in 22, (tens,) rejecting the right hand figure of the dividend, because we have rejected the cipher in the Questions. - 208, How may one side of a sqtuare, when the contenta are given, be found? Why, in the trial, must we point off the nuirther into periods of two figures each? Illustrate. Why is the first root figure 2 tens? Of what numbex' is 2 tens the root? What may now be formed? Hlow large' HIow must it be increased? What will be the divisor? - tile di viderld? -- the quotient? Why is the divisor too small? What is the entire divisor't How are the contents of the addition found? What does Fig. II. represent'! What is the first method of proof? - the second method? it 64al EVOLUTION. ~r 209. OPERATION - CONTINUED. divisor. We find our quotient that is, the breadth of the addi 625(25 tion to be 5 yards; but, if we 4 look at Fig J1., we shall perceive that this au.dition of 5 yards to 45) 225 the two sides does not complete the square; fobr there is stil. 225 wanting, in the corner D, a small square, each side of which is equal to this last quotient,; we FIG II must, therefore, add this quotient, 5, to;he divisor, 40, that is, place 20 yds. 1 yds. it at the right hand of the 4, 20 1 5 (tens,) making the length of the B 5 D 5 whole addition formed by 225 100 l 25X square yards, 45 yards; and then 100 25 _the whole length of the addition, d c 45 yards, multiplied by 5, the A C number of yards in the width, will give the contents of the whole v; ~20 20 w addition around the sides of the 20 5. figure A, which, in this case, be-- 4 lo ing 225 yards, the same as our,y 400 100 dividend, we have no remainder, and the work is done. Conseca b quently, Fig. II. represents the floor of a square room, 25 yards 20 yds. 5 yds. on a side, which 625 square yards of carpeting will exactly cover. The proof may be seen by adding together the several parts of the figure, thus: - The square A contains 400 yards. Or we may prove it by involuThe figure B " 100 " tion, thus — 25 X 25 = 625, as " "cc C 100 " before. " " D " 25 c Proof, 625 " f0T 29. From this example anid illustration we derive the Following general FOR THE EXTRACTION OF TtIE SQUARE ROOT. I. Point off the given number into periods of two figures each, by putting a dot over the units, another over the hundreds, and so on. These dots show the number of figures of vhich the root will consist. II. Find the greatest square number in the left hand period, and write its root as a quotient in division. Subtract the square number from the left hand period, and to the remainder bring down the next period for a dividend. 1 209. EVOLUTION. 265 III. Double the root already found for a divisor; seek how many times the divisor is contained in the dividend, excepting the right hand figure, and place the result inthe root, and also at the right hand of the divisor; the divisor thus increased will be the length of the whole addition now made to two sides of the square; multiply the divisor, or length of the addition, by the last figure of the root, (the breadth of the addition,) and subtract the contents of the addition thus obta;ned from the dividend, (and to the remainder bring down the next period for a new dividend. IV. Double the root already found for a new divisor, and continue the operation as before, until all the periods are brought down. NOTE 1. — As the value of figures, whether integers or decimals, is dete,mined by their distance from the place of units, we must always begin at units' place to point off the given number, and if it be i mixed number, we must point it off both ways from units, and if there be but one figure in any period of decimals, a cipher must be added to it. And as the root must always consist of as many integers and decimals as there are periods belonging to each in the given number, when it is necessary to carry the operation to a greater degree of exactness by decimals in the root, after all the periods are brought down, two ciphers, a whole period, must be annexed for every decimal figure which we would obtain in the root. EXAMPLES FOR PRACTICE. 2. What is the square root of 61504? ILLUSTRATION. OPERATION. 240 X 8 _ 19290 64 61504(248 Ans' - -40~ 4 40 200 X 40 =8000 40 44)215 ~i __ 1600 44)215 176 iO X 200 X 48S)3904 200 II 3904 41000 II o co0 0000 ___L_________ __o The pupil wvill easily illustrate the 200 +40- 8 operation by the annexed diagram. PorF. - 40000 + 8000 4- 8000 +- 1600 + 1920 + 1920 + 64 9515J4, or 248 X 248 61504. Qcestions. -i 2,69. What is the general rule for extra.cting the square trot? Where must we begin to point off? What is done whten one decimyal p.ace is wanting!'IJw is the operation continued, when all the periods are brought down? Why cannot the precise root be found when there is a re. mainder? 7How is the root of a fraction obtained? Why? What is done when the terms of the fraction are not exact squares? 23 266 EVOLUTION. ~ 210 3. What is t? e square root of 43264? Ans. 208. 4. What is the square root of 998001? Ans. 999. 5. What is the square root of 234'09 1 Ans. 15'3. 6. What is the square root of 964'5192360241? 4ns. 31'05671. 7. What is the square root of'001296? Ans.'036. 8. WVhat is the square root of'2916? Ans.'54. 9. What is the square root of 36372961? Ans. 60131. 10. What is the square root of 164? Ans. 12'8TNOTE 2. - In the last example, there was a remainder, after as. the figures werl brought down. In such cases, the precise root can never h)e obtained. FY't, as the operation is continued by annexing ciphers, the last figure of ever dividend must be a cipher. But the root figure obtained from this cdividenc, is also placed at the right hand of the divisor, and consequently is multiplied into itself, and the last figure of the product placed under the cipher, which is the last figure of the dividend, to be subtracted from it. And as lae product of no one of the significant figures ends in a cipher, there will always be a remainder. 11. What is the square root of 3? Ans. 1'73-. 12. What is the'square root of 10? Ans. 3'16 —. 13. What is the square root of 184'2? Ans. 13'57+-. 14. What is the square root of 4? NOTE 3.- Since, from the rule for multiplying one fraction by another, a fraction is involved by involvinu, its numerator and its denominator, the root cf a fraction is obtained by finding the root of its numerator, and of its denom-:nator. Ans. -. 15. What is the square root of 4. Ars. 25. 16. What is the square root of 16 Alns. 4 17. What is the square root of -a14 Ans. 79~ - 18. What is the square root of 204. Ans. 4a. NOTE 4.- When the numerator and denominator are not exact squares, the reaction may be reduced to a decimal, and the approximate root found. 19. What is the square root of = -'75? Ans.'866 +-. 20. What is the square root of s_. Ans.'912 -. PRACTICAL EXERCISE]S IN THIE EXTRACTION OF THEI SUAIE ROOT. 9T 21e 1. A general has 4096 men; how many must he place in rank and file to form ther into a square? Ans. 64. 2, If a square field contains 2025 square rods, how many rods does it measure on each side? Ains. 45 rods. 3. How many trees in each row of a square orchard containing 5625 trees? Ants. 75. 4. There is a circle whose area, or superficial contents, is 5184 feet; what will be the length of the side of a square of equal area? V5184 -72 feet, Ans. 5. A has two fields, one containing 40 acres, and the other oontaining 50 acres, for which B offers him a square field containing the ~1 210. EVOLUTION. 267 same nlmber of acres as both of these; how many rods must each side of this field measure? Ans. 120 rods. 6. If' a certain sqluare field measure 20 rods on each side, how much will the side of a square field measure, containing 4 times as much? V 20 X 20 X = 4 = 40 rods, Ans. 7. If the side of a square be 5 feet, what will be the side of one 4 times as large? - 9 times as large? - 16 times as large? - 25 times as large -- 36 times as large? Answers, 10 ft.; 15 ft.; 20 ft; 25 ft., and 30 ft. 8. It is required to lay out 288 rods of land in the form of a parallelogram which shall be twice as many rods in length as it l in width. NoTE 1. — If the field be divided in the middle, it will form two equal squares. Ans. 24 rods long, and 12 rods wide. 9. I would set out, at equal distances, 784 apple trees, so that my orchard may be 4 times as long as it is broad; how many rows of trees must I have, and'how many trees in each row? Ans. 14 rows, and 56 trees in each row. 10. There is an oblong piece of land, containing 192 square rods, of which the width is a as much as the length; required its dimensions. Ans. 16 by 12. 11. There is a circle, whose diameter is 4 inches; what is the diameter of a circle 9 times as large? NoTE 2. -A square 4 inches on one side, contains 16 square inches one twice as long, or 8 inches on each side, contains 64 saquare inches, 4 times 16; one 3 times as long, or 12 inches on each side, contains 144 9 times 16 sqluare inches. It may also be shown by geometry, that if the diameter of a circle be doubled, its conitents Aill be increased 4 times; if the diameter be trebled, the contents vill be increased 9 times. That is, the contents o0,vtuares are to each other as the squares of their sides, and the contents of circles are to each other as the squares of their diameters. Hence, to perform the above example, square the diameter, multiply the square by 90,;-and extract the square root of the product. Ans. Mlinches. 12. There are two circular ponds in a gentleman's pleasure ground; the diameter of the less is 100 feet, and the greater is 3 times as large; what is its diameter? Anis. 173'2 +L feet. 13. If the diameter of a circle be 12 irches, what is the diameter of one 4 as large? Ans. 6 inches. 14. A carpenter has a large wooden square; one part of it is 4 feet long, and the other part 3 feet loneg; what is the length of a peole, which will just reach from one end to the other NoTr 3. - A figure of 3 sides is called a triangle, and if Fig. I. one of the corners be a square corner, or right angle, like the angle at B in the aunexed figure, it is called a right angled triangle. It is proved by a geometrical demnonstration that the square contents of a square formed on the longest side, A C, are equal to the square contents of the two squares, one formed on each of the other two sides, A B, and C B. Thus, Fig. 2, a square formed on A B, the shortest side, wvill co:tain 9 square feet, the square on C B ~8 aS EVOLUTION. ~f 210. Fig. 2. will contain 16 square feet, 9 + 16 = 25 square feet, in bcth squares. The square on A C contains 25 small squares of the same size as the squares on the other two sides are divided into, or 25 square feet and the square root of 25 will be the length of the longest side, or, Ans., 5 feet. Hence, if the length of the two short sides are given, square each, add the s quares together, and ex~ract the square B - 0-C square root of the sum; the root will be the length of the long side. If the long side, and one of the short sides are given, square each, subtract the square of the short side from the square q f the long side; the square root of the remainder will be the other short side. EXAMPLES. 15. II', from the corner of a square room, 6 feet be measured off one way, and 8 feet the other way, along the sides of the room, what will be the length of a pole reaching from point to point! Ans. 10 feet. 16. A wall is 32 feet high, and a ditch before it is 24 feet wide; what is the length of a ladder that will reach from the top of the wall to the opposite side of the ditch? Ans. 40 feet. 17. If the ladder be 40 feet, and the wall 32 feet, what is the width of the ditch? Ans. 24 feet. 18. The ladder and ditch given, required the wall. Ans. 32 feet. 19. The distance between the lower ends of two equal rafters is 32 Feet, and the hight of the ridge, above the beam on which they stand, s 12 feet; required the length of each rafter. Ans. 20 feet. 20. There is a building 30 feet in length and 22 feet in width, and the eaves project beyond the wall 1 foot on every side; the roof terminates'iqa point at the centre of the building, and is there supported by a posftthe top of which is 10 feet above the beams on w!z;ch the rafters rest; what is the distance from the foot of the post to the corners of the eaves't and what is the length of a rafter, reaching to the middle of one side? -- a rafter reaching to the middle of one end., and a rafter reaching to the corners of the eaves! Answers, in order, 20 ft.; 15'62 +- ft.; 18'86-+- ft.; and 22'36 -{ feet. 21. There is a field 800 rods long and 600 rods wide; what is the distance between two opposite corners Ans. 1000 rods. 22. There is a square field containing 90 acres; h(w many rods Questionss,- I 210, How does it affect the contents ol a square to double its length? - to tretle its length? How does it affect the contents of circles to cot0L)le or treble their diameters? How will you find the diameter of a circle nine times as large as one of a given 6iamneter? What is a right angled triangle? What is said of the squares on its sides? How shown Dy Fig. 2? When both short sides are given, how do you find the long side When the long side, and one short side are given, how do you find the other 3 n 211. EVOLU'rTION. 2 in length is each side of the field and how many rods apart are the opposite corners t Anszwers, 120 rods, and 1C9'7-4- rods. 23. There is a square field containing 10 acres; what distance is the centre from each corner 1 Ans. 28'28- + rods. Extraction of the Cube Root. ~T 2 1. 1. How many feet in length is each side of a cubic block, containing 125 solid feet? SOLUTiON.- As the solid contents of a cubical body are found, wheone side is known, by involving the side to the third power, or cube% (If 206,) so when the solid contents are known, we find the length of cne side by extracting the cube root, a number, which, taken is a flctor 3 times, will produce the given number. (ET 207.) The cube root of 125 we find by inspection, or by the table, V 20(6, to be 5. Ans. 5 feet. A. What is the side of a cubic block, containing 64 solid feet - 27 solid feet. -_ 216 solid feet! - 512 solid feet! Answers, 4 ft., 3 ft., 6 ft., and 8 ft. 3. Supposing a man has 13824 feet of timber, in separate blocks of 1 cubic foot each; he wishes to pile them up in a cubic pile; what will be the length of each side of such a pile? OPERATION. SOLUTION. — It iS evident that, as in S13 242 the former examples, we must find 13824 ( the length of one side of a cubical 8 pile which 13824 such blocks will 6824 make by extracting the cube root of 13824. But this number is so large, FIG. 1. that we cannot so easily find the root FIeG. I. as in the former examples; - we wvil C 20 D endeavor, however, to do it by a sorf of trial; and, I sW. We will try to ascertain the B. I I number of filures, of which the root 20 will consist. This we may do by pointing the number off into periods of thnree figures each. For the cube of any figure will contain 3 times as E many, or 1 or 2 less than 3 times as many figures as the number itself. The cube of 2 contains 1 figure; the 20 F cube of 5 contains 2 figures; the cube 20 Df 9 contains 3 figures; the cube of 10 400 contains 4 figures, and so on. 420 Pointing off, we see that the root will consist of two figures, a ten and 8000 feet, onitentl a unit, Let us, t50n, seek for the first 2-3~'4' P97' EVOLUTION. ~l 211. figure. or tens of the root, which must be extracted from the left hand period, 13, (thousands.) The greatest cube in 13 (thousands) we find by inspection, or by the table of powers, to be 8, (thousands,) the root ot which is 2, (tens;) therefore, we place 2 (tens) in the root. As the root is one side of a cube, let us form a cube, (Fia. I.,) each side of which shall be regarded 20 fe,;i, expressed by the root now obtained. The ontents of this cube are 20 X 20 X 20 - 8000 solid feet, which are now disposed ot; and which, consequently, are to be deducted from the whole nurnmber of feet, 13824. 800C taken from 13824 leave 5824 feet. This deduction is most readily performed by subtracting the cubic number, 8, or the cube of 2, (the figure of the root already found,) from the peri)d 13, (thousands,) and bringing down the next period by the side of the remainder, making 5824, as before. 2d. The cubic pile A D is now to be enlarged by the addition of 5824 solid feet, and, in order to preserve the cubic form of the pile, the addition must be made on one half of its sides, that is, on 3 sides, a, b, and c. Now as each side is 20 feet square, its square contents are 400 square feet, ant[ the square contents of the 3 sides are 1200 square feet. HIence, an addition of 1 foot thi-k would require 1200 solid feet, and dividing 5824 solid feet by 1200 OPERATION -CONTINUED. solid feet, the contents of the addition 1 foot thick, and we get the 13824 (24 Root. thickness of the addition. It will be seen that the quotient figure must not always be as- large as it can be. There might be enough, for Divisor, 1200) 5S24 Dividend. instance, to make the three additions now under consideration 5 48030 feet thick, when there would not then be enough remaining to complete the additions. 64 The divisor; 1200, is contained in the dividend 4 times; conse5824 quently, 4 feet is the thickness of the addition made to each of the three sides, a, b, c, and 4 X 12(0Q 0000 - 4800, is the solid feet contained in these additions; but there are still 1024 feet left, and if we look FIG. II. at Fig. II., we shall perceive that 20 this addition to the 3 sides does not complete the cube; for snere = l n1 ars deficiencies in the 3 corners, mm, sn, a. Now the len i/th ot each of these deficiencies is the same as the length of each side, that is, 2 (tens)=-20, and their width and thickness are each equal to the last quotient figure, (4;) their con /III)~~j ~ tents, therefore, or the number of 20 ~~~feet required tofil these deficicn cies, will be found by multipiI..g the suare of the last quotient figure. (42= -— 16. by 20; 16 X 20= ~ 212. EVOLUTION. 271 FIG. Ill. 320 solid feet, required for one deficency, and multiplying 320 by 3, 320 X 3 960 solid feet, required 20 - -q~ 4 for the 3 deficiencies, n, n, n. 4 a- - i- Looking at Fig. III., we perceive i4 hi I i there is still a deficiency in the cori ner where the last blocks meet. i'V IL 20 This deficiency is a cube, each side of which is equal to the last quotient figure, 4. The cube of 4, therefore, (4 X 4 X 4 = 64,) will ic the. solid contents of this corner, 20 -which in Fig. IV. is seen filled. Now, the sum of these several 20 + 4 additions, viz., 4800 + 960 + 64 = 5824, will make the subtrahend, FIG. IV. which, subtracted from the dividend, leaves no remainder, and the 24feet. work is done. rive __.__ ~ Fig. IV. shows the pile which 13824 solid blocks of one foot each e ~1I I would make, when laid together, and the root, 24, shows the length of one side of the pile. The correctness of the work mray be ascertained by cubing te side now found 243, thus, 24 X 24 X 2413824, the given number; or it d4? may be proved by adO,ng together the contents of all the several parts. 24feet. thus, F"et. 830)0 - contents of Fig. I. 4800 addition to the sides, a, b, and c, Fig. I. 960 - addition to fill the deficiencies i, n. I, Fig. II. 64 = addition to fill the corner, e, e, e, Fig. IV. 13824 contents of the whole pile, Fig. IV., 24 feet on each side. 6212.. From the foregoing example and illustration we erive the follo7wing1 Questions.- -~ 211. How is the length of one side of a cube found, Wrhen the contents are knowvr? Why, Ex. 3, is the number pointed off as it is? How many figures in the cube of any number? illustrate by cubing iome numbers. Wlihat is 2, the first figure of the root? Of what is it the root? For what is the subtraction? What is to be done with the remainder? On how many sides is it to be added, and why i What is the divisor, 1200? W;hat is the object in dividing7? The quotient expresses what? Why should it not be made as large as it can lie? What additions are next made and what are the contents of each? How are the contents found! What deficiency yet remains, and how large? Of what parts of the last figule does the subtrahend consist? Describe Fig. I.; Fig. II.; - Fig. II.; - Fig. MV. How is the work proved? 272 EV JLUTION. If 212. RULE FOR EXTRACTING THE CUBE RotOT. I. Place a point over the unit figure, and over every third figure at the left of the place of units, thereby separating the given number into as many periods as there will be figures in the root. II. Find the greatest complete cube number in the left hand period, and place its cube root in the quotient. III. Subtract the cube thus found from the period taken, and bring down to the remainder the next period for a divi dend. IV. Calling the quotient, or root figure now obtained, so many tens, multiply its square by 3, and use the product for a divisor. V. Seek how many times the divisor is contained in the dividend, and diminishing the quotient, if necessary, so that the whole subtrahend, when found, may not be greater than the dividend, place the result in the root; then multiply the divisor by this root figure, and write the product under the dividend. VI. Multiply the square of this root figure by the former figure or figures of the root, regarded as so many tens, and the resulti g product by 3, add the product thus obtained, together with the cube of the last quotient, to the former product for a subtrahend. VII. Subtract the subtrahend from the dividend, and to the remainder bring down the next period for a new dividend, with which proceed as before, till the work is finished. NOTE. 1.- If it happens that the divisor is not contained in the dividend a cipher must be put in the root, and the next period brought down for a div — dend. NOTE 2.- The same rule mulst be observed for continuing the operation, old pointing off for decimals, as in extracting tLe square root. EXAMPLES FOR PRACTICE. 4. What is the cube root of 1860867? Questions -'ff 212, What is the general rule - noe 1? -? note 2' -- note 3? ~ 213. EVOBUTION. 273 OPE RA TION. 1860867 (123 Am. 102 X 3 - 300.) 860 first Dividend. 600 22,X 10 X 3 120 23 _ 8 728 first Subtrahend. 1202 X 3 -43200) 132867 second Dividend. 129600 32 X 120 X 3 3240 33= 27 132867 second Subtraheid. 000000 5 What is the cube root of 373248. Ans- V. 6. What is the cube root of 21024576. Ans.'6. 7. What is the cube root of 84'604519 Ans.. 4 39. 8. What is the cube root of'000343 Ans. 37. 9. What is the cube root of 2. Ans. 1'25 +. 10. What is the cube root of s? Ans. -. NOTE 3. - The cube root of a fraction is the cube root of tbh numeral w divided by the cube root of the denominator. (~ 209.) 11. What is the cube root of 125 s? Ans. J. 12. What is the cube root of Ng-o? Ans. -i. 13. WVhat is the cube root of a.? Ans.:125 -+-. 14. What is the cube root of Ado? Ans. j PRACTICAL EXERCISES IN EXERCISES IN E ACTING THE CUBE ROOT. ~]''I13. 1. What is the side of a cubical mound, equai to one 288 feet long, 216 feet broad, and 48 feet high? Ans 144 feet. 2. There is a cubic box, one side of which is 2 feet; how many solid feet does it contain! Ans. 8 feet. 3. How many cubic feet in one 8 times as large! and what would be the length of one sid-. Ans. 64 solid feet, and one side is 4 feet. 274 EVOLUTION ~ 214L 4. There is a cubical box, one side of which is 5 feet, what would be the side of one containing 27 times as mluch? 64 times as much? - 125 times as much? Ans. 15, 20, and 25 feet. 5. There is a cubical box, measuring 1 foot on each side; what i2 the side of a box 8 times as large - 27 times.? 64 times? Ans. 2, 3, and 4 feet. NOTE. — It appears from the above exanmples that the sides of cubes are As the cube roots of their solid contents, and their solid contents ms tie cubes ot their sides. It is also true that if a globe or ball have a certain contlents, tile contents of one whose diameter is double are S times as great, o. hLaving a treble diameter are 27 times as great, and so on; that is, the contents are t)n'portional to the cubes of their diameters. The samne proportion is true of the similar sides, or of the diameters of all solid figures of similar forns. 6. If a ball, weighing 4 pounds, be 3 inches in diameter, what wiL be the diameter of a ball of the same metal, weighing 32 pounds? 4: 3:: 33 63 Ans. 6 inches. 7. if a ball 6 inches in diameter weigh 32 pounds, what will be the weight of a ball 3 inches in diameter? Ans. 4 lbs. 8. If a globe of silver, 1 inch in diameter, be worth $6, what is thu value of a globe 1 foot in diameter? Ans. M10368. 9. There are two globes; one of them is 1 foot in diameter, and the other 40 feet in diameter; how many of the smaller globes would it take to make 1 of the larger? Atns. 64000. 10. If the diameter of the sun is 112 times as much as the diameter of the earth, how many globes like the earth would it take to make one as large as the sutn s Ans. 1404928. 11. If the planet Saturn is 1000 times as large as the earth, and the earth is 7900 miles in diameter, what is the diameter of Saturn! Ans. 79000 miles. 12. There are two planets of equal density; the diameter of the ess is to that of the larger as 2 to 9; what is the ratio of their solidities; Ans. or, as 8 to 729. i 214. -Review of Involution and Evolution. Questions. — What is involution? What are powers? Hiow are the different powers represented 2 How is a number involved' What is evolution? What is a root? How do you find the square root, ox the cube root of a number? What is a rational, and what a sued number? How is the square Abit indicated? - the cube root? Give briefly the solution of the example in the extraction of the square root; - rule. Ilcw are decimlals pointed off? How is the operation continued, when there is a remainder? Why cannot the precise root be ascertained? tlow is the square root of a vulgar fraction found? Wrhat is said of the relation between the sides and contents of squares? - the diameters and Questions. —T 213. What proportion exists between the sides of cubes, and their solid contents? Illustrate. What Jetweer the diameters of globes and their contents? If you increase the dinmeter of a ball 5 times, how much are its contents increased ~ 215. ARITHMaETICAL PROGRESSION. ~27 contents of circles? - the squares on the sides of a right-angled riaangle? Repeat briefly the solution of the example in cube root; -the rule. What is said of the relation of the sides of cubes to the conm'nts? — of the diameters of globes to their contents? EXERC]SES. 1 What is the difference of the contents of 6 fields, each 20 rods square, and 1 field 50 rods square? Ans. 100 square rods. 2. What is the difference between 56 cubical stacks of hay, each 10 feet on a side, and 1 stack 40 feet on a side? Ans. 8000 solid feet. 3. How many times larger is a circular pond, 1 mile in diameter, than one that is 40 rods in diameter? Ans. 64 times. 4. What is one side of a cubical pile of wood which contains 4 cords? Ans. 8 feet. 5. What is one side of a cubical pile of bricks which will lay up the walls of a house 33 feet high and 16 inches thick for the first 12 feet, 12 inches the next 12, and 8 inches the upper 12, the house being 60 feet long and 34 wide on the outside, no allowances being made for windows, doors, &c.. What are the so.id contents? Ans. to the last, 6C13 cu. ft., 576 cu. in. NOTE. - The principal object in evolution is to find one side of a square or of a cube, when the contents are known, or to extract the square and cube roots. There are methods of demonstrating these operations different from those here given, which are preferable in somle respects, but they are deficient in one important particular -iintelligibleness to those for whom they are designed. In a " higher arithmetic" they might be appropriate. Other roots may be extracted arithm'etically, but the methods of demonstrating the operations, even where any are given, are difficult of comnplrehension. The fourth root, however, may be found by taking the square root of the square root, the sixth root by taking the square root of the cube root, and so of many other roots. Any root is easily taken by what are called logarithms, used in the more advanced departments of mathematics. _ARITHMETICAL PROGRESSION. qT 15. 1. 1. A teamster starts with 5 barrels of flour; he pastes by 4 mills, at each of which he takes on 3 barrels; how many barrels has he then? SOLuTION. — He has 8 barrels after the first addition, 11 after the second, 14 after the third, and 17 after the fourth. Arts. 17 barrels. 2. Apeddler having 17 hats, sold 3 at each of 4 stores how many had he left? SOLUTION.- He had 14 after the first sale, 11 after the seeond, 8 after the third, and 5 after the fourth. Ans. 5 hats. A series of numbers increasing by a constant addition no 276 ARITHMETICAL PROGRESSION. ~ 216. decreasing by a constant subtraction of the sam, eiumber, is called an Arithmetical Progression or series. The first of the above examples is called an ascending, the second a descending series. NOTE 1. - The nunmbers which form the series are called the terms of tie series. The first and lasit terms are the extremes, and the other terms are called the means. There are five things in an arithmetical progression, any three of whicti being given, the other two may be found:1st. Thefirst terrm. 2d. The last term. 3d. The number of terms. 4th. The common difference. 5th. The sum of all the terms. NOTE 2. -The common difference is the number added Q.: subtracted at one time. ~q f16. One of the extremes, the common difference, and the number of terms being given, to find the other extreme. 1. A man bought 100 yards of cloth, giving 4 cents for the first yard, 7 cents for the second, 10 cents for the third, and so on, with a common difference of 3 cents; what was the cost of the last yard? SOLUTION. - We add 3 to 4 cents, (4 - 3 - 7,) to get the price of the second yard, 3 to 7 to get the price of the third yard, and so on, thus making 99 additions to 4, of 3 cents each; or, we may take 3, 99 times, (the multiplication being a short way of performing the 99 additions,) and add the product to 4, for the price of the last yard, 3 X 99; or, since either factor may be the multiplier, 99 X 3 = 297, and 4 + 297 = 301 cents, the price of the last yard. Ans. 301 cents. NOTE 1. —The prices, 4, 7, 10, 13 cents, &c., are an ascending series, which has as many terms as there are yards, namely, 100; 3 is the common difference, and 4 the first term, to which 99 times 3 must be added to find the price of the last yard, or the last term. It is added 1 time less than the number of terms, since 4 is the price of the first yard without any addition. Hence, To find the last term of an ascending series wshen the first term, common difference, and number of terms are given, RULE. Multiply the common difference by the number of terms less one, to get the sum of the additions, and add this sum to the first term; the amount will be the last term. Questions. - S 215. How is Ex. 1 explained? - Ex. 2? What are the extremes? - the means? - the terms? How many, and what things are there, of which, if t-ree are given, the others may be found? What is the common difference' [ 217. ARITHMIE'TICAI, PROGRESSION. 277 NOTE 2.- If the same things are given of a descending ser es, we must evidently take the surm of the subtractions from the first term to find the last. In the samle nmannler we may fitnd tile first term of an ascending series when the last termn and the other things namned are given; but having these things given of a descending series, we find the first term by the rule above for find ing the last term of an ascecndin series. EXAMPLES FOR PACT CE.o 2. There are 23 pieces of land, thle first containing 95 acres, the second 91, the third 87, and so on, decreasing by a common difference of 4; what is the number of acres in the last piece? Ans. 7. 3. The first term of a series is 6, the common difference is 3, and thle number of terms is 57; what is the last term? Ans. 174. 4. The last term of a series is 117, the common difference is 8, anti the number of terms is 15; what is the first term? Ans. 5. 5. The last term is 6, the number of terms 21, and the common difference 10; what is the first term? Ans. 206. Simple Interest by Progression. T. ~17o 1. A man puts out $10, at 6 per cent., simple interest; to what does it amount in 20 years? SOLUTION. - The first sum is $10, the amount at the end of the fire year is $10'60, at the end of the second year $11'20, increasing each year by the constant addition of $'60. Hence, simple interest is a case of arithmetical progression, the principal being the first term, the inter est for one year being the common difference, the number of' terms one more than the number of years, since there is one term, the principal, at the commencement of the first year, and one term, the amount for a year, at its close, and the last term, which we wish to find, is the amount for the number of years. To find the last term, or this amount, multiply the interest for 1 year by the number of years, (one less than the num ber of terms,) and add the product to the first term. Ans. $22. 2. Two lads, at 14-years of age, commence labor for themselves; the one lays up nothing, but the other, by prudence, lays up $300 by the time he is 20 years old, which he puts out at 7 per cent. simple interest; afterwards, each earns his lving, and no more; at the age of 70 the one is worth nothing, and comes upola public charity; what is the other worth at that age 1 Ans. $1350. Questions. - 91 216. What things are given, and what are required, in ~ 216? How many cases may there be, ansd what are they? What are given in Ex. 1? How much is the first termn increased to make the last'? Why ale only 99 times 3 added? Give the rule. To what case does it apply 7 Wlhat is done in each case, when other thilngs are givel? ~T 217. HIw does it appear that simple interest is a case of progressionli What things, according to ~I 216, are given, and what is required? Wh," is there one more term than t le number of years? How is simple interest p.r. formed by progressionI't 27}8 ARITHMETICAL PROGRESSION. ~ 218, 21. 3. What will a watch, purchased at 21 for $25, cost an individuat by the time he is 75, reckoning nothing for repairs but simple interest at 6 per cent, on the purchase money --- at 8 per cent.? Ans. to the last, $133. ~U 21 S. The extremes and thkz number cf tErms given, to find the common difference. 1. The prices of 100 yards are in arithmetical progression, the first being 4, the last being 301 cents; what is the common increase of price on each succeeding yard? SOLUrTION. -As the first yard costs 4 cents, 297 cents have been added to 4 for the price of the last yard, at 99 times, and dividing the number added at 99 times by 99, we get the number added at 1 time. Hence, RULE. Divide the whole number added or subtracted, by the num ber of additions or subtractions, that is, the difference of the extremes by the number of terms less 1, and the quotient is the number added or subtracted at one time, or the common difference. EXAMPLES FOR PRACTICE. 2. If the extremes be 5 and 605, and the lumber of terms 151, what is the common difference. Ans. 4. 3. A man had 8 sons, whose ages differed alike; the youngest wvas 10 years old, and the eldest 45; what was the common difference of their ages? Ans. 5 years. NOTE. - If the extremes and common difference are given, we may find the number of terms bv dividing the difference of the extremes by the common difference, and adding 1 to the quotient. 4. The extremes are 5 and 1205, and the common difference 8; what is the number of terms? Ans. 151. v 219. The extremes and thie number of terms being given, to find the sum of all the terms. 1. What is the amount of the ascending series, 3, 5, 7, 9, 11, 13, 15, 17, 19? SOLUTION. - The sum may be found by adding together the terms, but in an extended series this process would be tedious. We will therefore seek for a shorter method; and first, will write down the terms of Questions. —~ 218. What are given and what required, I 218? Explain how the common difference is found, Ex. 1. Give the rule. How is lhe number of terms found wh sn the extremes e:t the common difference are given? 7[ 220. ARITIHIMETICAL PROGRESSION. 279 the series in order, ard beginning with the last, write the erns of the same series under these, placing the last term under the first, the next to the last under the second, the third from the last under the third. and so on, thus: — 3 5 7 9 11 13 15 17 19 19 17 15 13 11 9 7 5 3 22 22 22 22 22 22 22 22 22 Adding together each pair, we see that the sums are alike, and the aniount of the whole is as many times 22, the first, sum, as there are terms in either series, which is 9. 22 X 9 = 198, the number in both serles, and 198 2 - 99 must be the sum of the first series, which we wish to find. But 22 is the sum of the extremes of the series; hence when the extremes and the number of tsrms are given to find the sum of the terms, RULE. Multiply the sum of the extremes by the number of terms, and half the product will be the sum of the terms. EXAMPLES FOR PRACTICE. 2. If the extremes be 5 and 605, and the number of terms 151, what is the sum of the series? Ans. 46055. 3. What is the sum of the first 100 numbers, in their natural order, that is, 1, 2, 3, 4, &c.? Ans. 5050. 4. How many times does a common clock strike in 12 hours? Ans. 78. Annuities by Arithmetical Progression. ~ 220. An annuity, (from the Latin word annus, meaning a year,) is a uniform sum, due at the end of every year. When payment is not made at the end of the year, the annuity is said to be in arrears, and the sums of the annuities should draw interest just as any other debts not paid when due. When on simple interest, the several years' annuit es, with the interest on each, form an arithmetical progression, and the calculation to ascertain the whole sum due, is - finding the sum of an arithmetical series; thus:Questions.- ST 219. What are given, and what is required, IT2197 How might the s-lm be found? What difficulty in this method? What is the process by wh ch a shorter method is found? What fact (lo we discover from tne additions, Ex. 1? What does the product express, and why is it divided by 2? What is the quotient? Why is 22 the sum of the extremes l Give the rule. 2,0 ARITHMETICAL PROGRESSION. ~ 221. i. A martn, whose salary is $100 a year, does not receive anytning till the end of S years; what was then his due, sir pie interest on the sums in arrears at 6 per cent.? SOLUTION. - The first year's salary not being paid till 7 years after it is due, since it was due at the end Of the first year, is on interest years. The interest of $100, 7 years, is $142, and $100 +- 42=$142 which he should receive on account of his first year's salary. Hiis second year's salary, on interest 6 years, will amount to $136, his thirc year's sakary swill amount to $130, and so on, decreasing uniformly by $6, the interest of $100 for a year, till the last year, when the salary, being paid at the end of the year for which it has accrued, will not be on interest, but will yield him $100. The sums, $142, $136, $130 8124, $118, $112, $106, $100, form a descending arithmetical progression, and to find the sum due, multiply the sum of the extremes by the number of terms, and take half the product, thus:-'$142 + $100 -= $242; and $242 X 8 = $1936, which - 2 = $968, Ans. EXAMPLES FOR PRFACTICE. 2. A soldier of the revolution did not establish his claim to a pension of $96 a year till 10 years after it should have begun; what was then his due, simple interest on the sums in arrears at 6 per cent.? Anrs. $1219'20. 3. A man uses tobacco at an expense of $5 a year from the age of 18 till the age of 79, when he dies, leaving to his heirs $300; what might he have left them if he had dispensed with the worse than useless article, and loaned the money, which it cost him, at the end of each year, for 7 per cent., simple interest? Anrs. $1245'50. 4. A and B have the same income, but the expenses of A are $210 a year, and those of B are $250; at the end of 40 years B is worth $1500; what is A worth, havingf loaned what he saved more than B at 7 per cent. simple interest at the end of each year? Ans. $5284. 5. Refer to ~ 164, Ex. 8: what would the merchant gain if he continued in trade 31 years to borrow money instead of purchasing on credit, loaning the money saved at the end of each year at 7 per cent. simple interest. Ans. $20151'70 nearly. EXERCISES. ~] 2 l. 1 1. If a triangular piece of land, 30 rods in length, be 20 rods wide at one end, and come to a point at the other, what number of square rods does it contain? Ans. 300. Questions. — T 220. What is an annuity? Why so called? Whenr, are tlnnuities in arrears'? Why should they then draw interest? When will they form an arithmetical progression9? 7Whlat is the calculation to find the whole sum due? Why, Ex 1i, was the first year's salary on interest 7 years? Show how long each year's salary was on interest. Why was not the last year's salary on interest? Why dl.o the sums form a descendiL-g series?.I-low,nay the sum be found? Why is not the nulmber of terms ons more than the number of years, as in ~ 217? ~ 2`22. GEOMETRICAL PROGRESSION. 231 2. A debt is to be discharged at 11 several payments, in ai'thmetica] series, the first to be $5, and the last $75; what.is the whole debt. - the common difference between the several payments. Ans. Whole debt, $440; common difference, $t. 3. What is the sum of the series 1 3, 5, 7, 9, &c., to 1001 1 Ans. 251001. NOTE. - The number of terms must first oe found. 4. A man bought 100 yards of cloth in arithmetical series; ioe gave 4 cents for the first yard, and 301 cents for the last yard; what was the amount-of the whole 1 Ans. $152'50. 5. What annuity, in 20 years, at 6 per cent. simple interest, mwill amount to $1570. Ans. $50. 6. What is the sum of the arithmetical series, 2, 2A, 3, 3., 4, 44, &c., to the 50th term inclusive'! Ans. 7124. 7. What is the sum of the decreasing series, 30, 294, 29Y, 29, 28j, &c., dow n to 0 o Ans. 1365. S. A laboring female was ahle to put $30, at the end of each year, in the savings bank, at 5 per cent., simple interest, from the age of 18 till 77, when she died; how much had she become worth? Ans. $4336'50. GEOMETRICAL PROGRESSION. ~ 222. 1. A man, having 5 acres of land, doubled the quantity at the end of each year for 4 years; how many acres had he then? SOLUTION. - Having 5 acres at first, he had 2 times 5, or 10, at the end of the first year, 2 times 10, or 20, at the end of the second year, 2 times 20, or 40, at the end of the third year, and 2 times 40, or 80, at the end of the fourth year. Ans. 80 acres. 2. A lady, having $80, traded at 4 stores, expending one half at the first, half of what she had left at the second, and so on, expending half the money in her possession at each store till the last: what had she left? SOLUTION. - She leaves the first store, to which she came with $80 + 2 = $40, the second with $40- 2 = $20, the third with $20 - 2 $10, the fourth with $10 -2 =$5. Ans. $5. Any series of numbers like 5, 10, 20, 40, 80, increasing by the same multiplier, or like 80, 40, 20, 10, 5, decreasing by the same divisor, is called a geometrical progression. The multiplier or divisor is called the ratio. The first and last terms are called the extremes. 24* 282 GEOMETRICAL PROGRESSION. ~1 220 The first is called an increasing, the second is called a decreasing geometrical series. NOTE. -As in arithmetical, so also in ge,::-letrica- progression, there Mare five things, any three of which being given, thl other two may be found. — 1st. The first term. 2d. The last term. 3d. The nuznbcr of terms. 4th. The ratio. 5th The sum. of all the terms. IT V 3o The first term, number of terms, and ratio of am inctreasingz geometr'ical series being given, to find the last te -m, 1. A man agreed to pay for 13 valuable houses, worth $5000 each, what the last would amount to, reckoning 7 cents for the first, 4 times 7 cents for the second, and so on, increasing the price 4 times on each to the last; did he gain or lose by the bargain, and how much? SOLUTION. - We multiply 7 cents, the sum reckoned for the first house, by 4, to get the sum reckoned for the second house, and'this by 4 to Set the stum reckoned for the third house, and so on, multiplying 12 times by 4. But to multiply twice by 4 is the same as to multiply once by 16, which is the second power of 4, or 4 times 4. Thus, 7 X 4=- 28 and 28 X 4 = 112. So, also, 7 X 16n 112. Hence, multiplying 7 by the twelfth power of 4 is the same as multiplying by 4 12 times. And 412, that is, the twelfth power of 4, is 1i677721(; and 7 X 16777216, (using, if we choose, the larger factor for the mul~tiplicand, [ 21,) produces 1174 10512 cents, = $1174405'12; the houses were worth $5000 X 13 _ $65000, and $1174405'12 - $G5000 = $1109405'12, loss, Anls. Hence, w;hen the first term, number of terms, and ratio of an increasing series are given, to find the last term, RULE. Mlultiply the first term by the ratio raised to a power one less than the number of terms. NOTE I. - To get a high power of a number, it is convenient to write down a few of the lower powers, and multiply them together, thus: powers of 4. Ist power. 2d power. 3dpower. 4th power. tlh power. 6th power. 7th power. 4.16 64 256 1024 4096 16384, &c. Now the 7th power, multiplied by the 5th power, will produce the 12th power, as also the tih by the 4th, and the product by the 2d; the 5th by the 4th, and the product by the 3d; the 7th by the 4th, and the product by the 1st, &c Iultiiplyin f all the powers now written, will produce the 28th*; all but the 5th, will produce the 23d; all but the 2 1, will produce the 26th, &c. Questions. —. If 222. What is a geometrical progression? - an inTreasing series? - a decreasing series? - the extremes? - the ratio WYhat five thii gs are there, of which, if three are given, the others may be -ululd ~ 2241. GEOMETRICAL PROGRESSIO N 283 EXAiMPrJ.ES FOR PRACTICE. 2. A man plants 4 kernels of corn, which, at harvest, produce 32 kernels: these he plants the second year; now, supposing the annual increase to continue 8 fold, what would be the produce of the 15t*i year, allowing 1000 kernels to a pint? NOTE 2. - The 4 kernels planted is the first term, and the 32 kernels larvested the second term, both within the first year. Ans. 2199023255'552 bushels. 3. Suppose a man had put out one cent at compound interest in 1620, what would have been the amount in 1824, allowing it to double once in 12 years? 27- 131072. Ans. $1310'72. NOTE 3.- When the ratio, the number of terms, and the last term of a de creasing series is given, the first term is evidently found by the same rule. But when the same things are given of a decreasing series, as those of the ascending series required by the rule, that is, the first term, flatio, and number of terms, we must divide the first term by the ratio raised to a power which is one less than the number of terms. In the sanme way we may f.nd the first term of anll increasing series, when the ratio, number of terms, and last term are given. 4. If the last term of a decreasing series be 5, the ratio 3, and the number of terms 7, what is the first term T Ans. 3645. 5. If the first term of a decreasing series be 10935, the ratio 3 and the number of terms 8, what is the last term? Ans. 5. 6. If the last term of an increasing series be 196608, the numbex of terms 17, and the ratio 2, what is the first term't Ans. 3. NOTE 4. - When the first and last terms, and the ratio are given, to find the number of terms, we may divide the greater termn by the less, the quotient b5 the ratio, alnd so on, continually dividing by the ratio till nothig remains the number of divisions will be equal to the number of terms. 7. The first term is 7, the ratio 10, and the last term 700000000; whatis the number of terms? Ans,9 Compound Interest by Progression., [24e 1. To what will $40 amount in 4 years, comp-und interest at 6 per cent. Qluestions. - I 223. What are given, and what is requited, in'T 223? How is the last term found, as lirst described'? What may be done instead of this? Why? -low may a high power be obtained? Hlow the 20ta power? - the 16th power? - the 27.h power.? - the lth power? W.tm Nq the rule? Give the substance cf no e 3; - of note 4. 2S4 GEOMETRICAL PROGRESSION ~ 224, OPERATION. SOLUTION. - The amount of $40', prin., or st tgerm $40 for one year is once $40-+ 1'06 rf~ of $40, or 1'06 (T+a) of $40, and to obtain it, we multiply $40 by 1'06. This pro 240 duct, multiplied by 1'06, giveq 40 the amount for 2 years. Hence, compound interest is a case of a;(40, 4Q d term an increasing geometrical progresslin, of which there are 1'06 given the first term, or principal, the ratio, or the amount of 2544;0 $1 for 1 year, and the number 42'40 of terms, which is one more than the number of years, there being 2 terms the first year, 44'9440, 3d term. the principal at the commence1'06 ment, and the amount with one. —------- year's interest at the close; and 2696640 we are to find the last termthe amount for the time - by 449440 the rule in -the last ~f. The several terms, it will be seen, 47'640640, 4th term. increase by the common multi106 plier, 1'06. Ans. $50'499+ - 285843840 NOTE 1. —The powers of the 47640640 ratio may usually be found in the table, ~ 161, since they are the same with the amounts of $1 for 50'49907840, 5th term. the number of years which indicates the pjower of the ratio that we wish. It appears also that the only difference in finding the amount of a sum at compound interest by the table, and by progression, is the order in which we take the factors. By ~ 161t, we multiply the amount of $1 for the time by the number of dollars; by progression, we multiply the number of dollars by the ratio raised to a power denoted by the nulmber of years, or the amount of $1 for the time. EXAMPLES FOR PRACTUICE. 2. What is the amount of 40 dollars, for I1 years, at 5 per cent. compound interest? Ans. $68'412 +-. 3 What is the amount of $6, for 4 years, at 16- per cent., corn pound interest! Ans. $8'784: 6. 4. In what time will $1000 amount to $1191'016, at 6 per cent compound interest NOTE 2. - The case is evidently one of finding the number of terms, (one more than the number of years,) when the ratio and the first and last terms are given, ~ 223, note 4. 4Ans. 3 years. Questions. - 224. How does it appear from the illustration that:ompound interest is a case of progression? What are the things given? - to find what? What, is the ratio, Ex. 1? Why? How may ti.e powers of the ratio usaall.y tb fo-unil Wh7y? I What does it appear are i-; len, and wrhat is require,. Es, 4? ~1225, 226. GEOMETRICAL PROGRESSION. 285 Compound Discount. ~.T 1. What is the rresent worth of $304'899, due 4 years hence withcut interest, money being worth 6 per cent. compound interest? SOLUTION. - We find, by the table, XT 161, that $1, in 4 years, amounts to $ 1'26247, and as many times as this amount of $1 is contained in the given sum, so many dollars it will be worth; for it is worth a sum, which, put at compound interest 4 years, would amount to it, and dlh it. ing the amount of the number of dollars by the amount of one clolar, - we have the number of dollars, or, Ans. 241'509 -. NOT-E. - The case is evidently one of a geometrical progression, in whicn the ratio, (1'06,) the number of terms, (5,) and the greater term are given, to find the less' as in ~U 223, note 3. TABLE Showing the present worth of $1, or ~1, from 1 year to 40, allowing compound discount, at 5 and 6 per cent. Years. 5 per cent. 6 per cent. Years. 5 per cent. 6 per cent. 1'952381'943396 21'358942'294155 2'907029'889996 22 341850'277505 3'863838'839619 23'325571'261797 4'822702'792094 24 310068'246979 5'783526'747258 25 295303'232999 6'746215'704961 26'281241'219810 7'71 06S'665057 27'267848'207368 8'676839'627412 28'255094'195630 9'644609'591898 29'242946'184557 10'613913'558395 30 231377'174110 11'584679'526788 31'220359'164255 12'556837'496969 32'209866'154957 13 530321'468839 33'199873'146186 14'505068'442301 34'190355'137912 15'481017'417265 35 181290'130105 16'458112'393646 36'172657'122741 17'436297'371364 37 164436'115793 18'415521'350344 38'156605'109239 19'395734'330513 39'149148'103056! 20'317689'311805 40'142046'097222 -26. The extremes and the ra(Io given to find th/e sumi vf the series. Questi ons.- -T 225. What is compound discount? How is the pres. ent worthl found? Like what case in a geometrica. progression is it I 286 GEOMETRICAL PROGRESSION.'F 226 1. A man bought 4 yards of cloth, giving 2 cents for the first yard, 6 for the seconid, 18 for the third, and 54 for the fourth; what does he pay for all? SoLUTIoN. -We may add together the prices of the several yards thus: 2+6+18+54=80. But in a lengthy series, this process would be tedious; we will therefore seek for a shorter method. Writing down the terms of the series, we multiply the first term by the ratio, and place the product over the second, to which it will be equal, since the second term is the product of the first into the ratio. Multiply, also, the second term, placing the product over its equal, the third; multiply the third, placing the product over the fourth; multiply the fourth, and place the product at the right of the last product thus: Second series, 6 18 54 162 First series, 2 6 18 54 020 162 2 2) 160, twlice the first serzes. 80, sum of the first series. Th~ hecond series is three times the first series, and subtracting the first fi,)n it, there will remain twice the first series. But the terms balance each other, except the first term of the first series, the sum of which we iwish to find, and the last term of the second, which is 3 times the last terni of the series whose sum we wish. Subtracting the former from the latter, we have left 160, twice the sum of the first, which dividing by 2, the quotient is 80, sum of the series required. Hence, Multiply the larger term by the ratio, and subtract the less term from the product, divide the remainder by the ratio less 1; the quotient will,\b the sum of the series. EXAMPLlES FOR PRACTICE. 2. If the extremes be 4 and 131072, and the ratio 8, what is the sum of the series! Ans. 149796. 3. What is the sum of the decreasing series, 3, 1,,1, -,., &e extended to infinity? Questions. - I 226. What are given, to find w lat, r 226? How might the sum be found? What difficulty in this? What is the manner of proceed. ing to find a shorter method? Give the rule. Explain the reason of the rule What is an infinite series, and what its last term? 9 227. GEOMETRICAL PROGRESSION. 283 NOTE. - Such a series is called an infinite series, the last term of which is so near nothing that we regard it 0; hence when the extremes are 3 and 0, and the ratio 3, what is the sum of the series? Ans. 4A. 4. What is the value of the infinite series, 1 + 1, -F+,- + 1', &c.. Ans. 1~. 5. Wrhat is the value of the infinite series, ~-} —+ T~+ T1yo T+f,~g-, &c., or, what is the same, the decimal'11111, &c.. continually repeated. Ans. ~. 6. What is the value of the infinite series, T2 — + 2, &C decreasing by the ratio 100, or, which is the same, the repeating decimal'020202, &c.? Ans. 2. ~[ 227. The first term, ratio, and number of terms given to find the sum of tihe series. 1. A lady bought 6 yards of silk, agreeing to pay 5 cents for the first yard, 15 for the second, and so on, increasing in a three fold proportion; what did the whole cost? SOLUTION. - We may find the prices of the' several yards, and add them together, or, having found the last term by ~ 223, we can find the sum by the last M. But our object is to find a still more expeditious method. Let us find the several terms and write them down as a first series, and below it write a series which we will call the second, having 1 for the first term, and the same number of terms, thus: First series, 5 15 45 135 405 1213 6tn power of ratio. Third series, 3 9 27 81 243 729 Second series, 1 3 9 27 81 243 729 - 1 728, which 2 = 364, and 364 X 5 -= 1820 cents. Now multiplying the second series by the ratio, 3, and writing the prod ucts as directed in the last ~[, we have a series three times the second The last term of the third series, it must be carefully noticed, is the 6tl power of 3, the ratio, the power denoted by the number of terms. Sub tracting the second series from the third, which is done by taking 1 from the last term of the third, the other terms balancing, 729 - 1= 728, we have twice the second series, and dividing 728 by 2, 728 -- 2 = 364, we have once the second series. Now the first series, the sum of which is required, is 5 times the second, since, as the first term is 5 times greater, each term is 5 times greater than the corresponding term of the second series; and multiplying 364, the sum of the second, by 5, we have the required sum, or 1820 cents = $18420, Ans. Hence, the first term, ratio, and number of terms being given to find the sum'of tire series, RULE. KRaise the ratio to a power whose index is equal to the 8tq GEOMETRICAL PROGRESSION. ST 22S. number of terms, from which subtract 1, and divide the remainder by the ratio less 1; the quotient is the sum of a series with 1 for the first term; then multiply this quotient by the first term of any required series; the product will be its amount. EXA1MPPI ES FOR PRACTICE. 2. A gentleman, whose daughter was married on a new year's day, gave her a dollar, promising to triple it on the first day of each month in the year; to how much did her portion amount. 531441-1 Applying this rule to the example, 312= 531441, and 31 X - 265720. Ans. $265,720. 3. A man agrees to serve a farmer 40 years without any other reward than 1 kernel of corn for the first year, 10 for the second year, and so on, in tenfold ratio, till the end of the time; what will be the amount of his wages, allowing 1000 kernels to a pint, and supposing he sells his corn at 50 cents per bushelS 104-l 1 1= 1,111,111,111,111,111,111,111,111, 10 - 1 X 111,111,111,111,111 kernels. Ans. $8,680,555,555,555,555,555,555,555,555,555,555'555T 5 %%. 4. A gentleman, dying, left his estate to his 5 sons; to the youngest $1000, to the second $1500, and ordered that each son should exceed the younger by the ratio of 1A; what was the amount of the estate. NOTE. - Before finding the power of the ratio I~, it may be reduced to an improper fraction,=, or to a decimal, 1'5. a~5 — 1 ~ 1'5s- 11 X 000=$1 3187 or, 1 X 1000 =$13187450 Answer. Annuities at Compound Interest. ~ WS. 1. A man rented a dwelling-house for $100 a year, but did not receive anything till the end of 4 years, when the whole was paid, with compound interest at 6 per cent., on the sums not paid when due; what did he receive? Questions. - T 227. What is the first method of finding the sum, when the things are given, named in T 227? - the second? Describe the process for finding a third method. What terms constitute the first series? - the second? How is the third found? What do you say of its last term'? How does it appear to be so? How much is the remainder, after subtracting the second from the third series? How, then, is the sum of the second series found? How the sum of the first, and why? Give the rale. Why raise the ratio, &c.? 2'29 GEOMETRICAL PROGRESSION. 2S SOLUTION. - As annuities in arrears at simple interest to!m an arithmetical series, so the several years' rents with compound interest on those in arrears, are so many terms of a geometrical series. The last, year's rent is $100 only, since it is paid when due, at the end of the year; the third year's rent is on interest 1 year, and is found by multiplying $100 by 1'06, producing $106', and this product multiplied by 1'06, will give the second year's rent, paid 2 years after it is due, and so on. The first term, $100, the number of terms, 4, and the ratio, 1'06, are given to find the sum, as in the last ~, and we may apply the same rule, thus: - 1'064-1'0 X 100- 43745. ns. 437'45. NOTE. - The powers of the ratio, see ~F 224, may be found in the table ~ 161. EXAMPLES FOR PRACTICE. 2. What is the amount of an annuity of $50, it being in arrears 20 years, allowing 5 per cent. compound interest! Ans. $1653'29. 3. If the annual rent of a house, which is $150, be in arrears 4 years, what is the amount, allowing 10 per cent. compound interest? Ans. $696'15. 4. To how much would a salary of $500 per annum amount in 14 years, the money being improved at 6 per cent. compound interest? -- in 10 years! - in 20 years? - in 22 years! -- in 24 years? Ans. to the last, $2.5407'75. 5. Two men commence life together; the one pays cash down, $200 a year to mechanics and merchants; the second gets precisely the same value of articles, but on credit, and proving a negligent paymaster, is charged 20 per cent. more than the other; what is the difference in 40 years; compound interest being calculated at 6 per cent.. Ans. $6190'478 -4-. 6. A family removes once a year for 30 years, at an expense and loss of $100 each time; what is the amount, 6 per cent. compound interest being calculated! Ans. $7905'818 -. Present Worth of Annuities at Compound Interest. ~ 229. 1. A man, dying, left to his nephew, 21 years old, the use of a house, which would rent at $300 a year for 10 years, after which it was to come in the possession of his own children; the young man, wishing ready money to commence business in a small shop, rented the house for 1( Questions.- ~ 228. How does it appear that an annuity is an example )f geometrical progression? Why is the number of terms only equal to the aumber of years? What is to be found,_ and by what rule? 25 'J90 GEOMETRICAL PROGRESSION. ~1230. years, receiving in advance such a sum as was equivalent te W300 paid at the end of each year, reckoning omrnpound dis count at 6 per cent.; what did he receive? SOLUTION. —First, we find what he would receive at the end of 10 years; if nothing had been paid before, by the last I. Now what he should receive at the commencement of the 10 years, is a sum, which, on compound interest at the rate given, would amount to this in 16 years, and we divide it by the amount of $1, found as above, for the present worth. Ans. $2208'024. EXAMPLES FOR PRACTICE. 2. What is the present worth of an annual pension of $100, to continue 4 years, allowing 6 per cent. compound interest I Ans. $346'503 —. 3. What is the present worth of an annual salary of $100, to continue 20 years, allowing 5 per cent.. Ans. $124i'218-~. ~T 230. The operations under this rule being somewhat telious, we subjoin a TABLE,.howing the present worth of $1 or ~1 annuity, at 5 and 6 per cent. compound interest, for any numbLer of years from i to 40. Years. 5 per cent. 6 per cent. Years. 5 per cent. 6 per cent. 1 0'9.5238 0'94339 21 12'82115 1 1'76407 2 1'85941 1'83339 22 13'163 12'04158 3 2'72325 2'67301 23 13'48807 12'30338 4 3'54595 3'4651 24 13'79864 12'55035 5 4'32948 4'21236 25 14'09394 12'78335 6 5'07569 4'91732 26 14'37518 13'00316 7 5'78637 5'58238 27 14'64303 13,21053 8 6'46321 6'20979 28 14'89813 13'40616 9 7'10782 6'80169 29 15'14107 13,59072 10 7'72173 7'36008 30 15'37245 13'76483 11 8'30641 7'88687 31 15'59281 13,92908 12 83'86332.5 8'38384 32 15'80268 14 08398 13 9:39.357 8'85268 33 16'00255 14,22917 14 9.89864 9'29498 34 16'1929 14'36613 15 10'37966 9'71225 35 16'37419 14'4;S24 16 10'83777 10'10589 36 16'54685 14,62098 17 11'274)7 10'47726 37 16'71128 14'73678 18 11'68958 10,8276 38 16'86789 14,84601 19 12'08532 11115811 39 17'01704 14,94907 ~20 12'46221 11'4Q992 40 17' 15908 15'04629 IT 231. GEOMIETRICAL PROGRESSION. 291 NOTE 1, —From the table it appears that, instead of $1 a year for 30 years, paid at the end of each, which would be,30. one would receive at the comiencement, $15'37245, at 5 per cent., or $13'76483, at 6 per cent compound discount, and for $50 a year 50 times as much. Ience, for finding the present worth at compound discount by the table, RULE. Multiply the present worth of $1 by the number of dollars. ]EXAiMIPLES FOR PIRACTI3CE. 1. What ready money will purchase an annuity of $150, to continue 30 years, at 5 per cent. compound interest? _/ns. $2305'8675 2. What is the present worth of a yearly pension of $40, to continue 10 years, at 6 per cent. compound interest? -- at 5 per cent.? - to continue 15 years? — 20 years? — 25 years? - 34 years? J/ns. to last, $647'716. NOTE 2. - The practised arithmetician will have no difficulty in calculating the present worth of annuities at simple interest, from principles heretofore presented. Annuities at Compound Interest in Reversion. T 31. NOTE.- An annuity is said to be in reversion when it does not commence immediately. 1. In Ex. 1, ~ 229, supposing the uncle had reserved the use of the house to his sister for 2 years after the young man was 21, and given it to him for 10 years after this time should have expired, how much could he have obtained with which to commence business? SOLUTION. - If he should wait till he is 23 years old, he could obtain $2208'024, as already found, and he can, at 21, obtain a sum which, at compound interest, would amount, in two years, to $2208'024, or the present worth-of this sum paid two years before due, found by ~ 225 to be Ans. $1965i'13+ fence, tofind the present worth of an annuity in reversion, RULE. Find the present worth, were it to conmmence now, and the present worth of this sum for the time in reversion. Questions. —-IT 229. Give the first example. What sum shdu.d no receive now? How is it found? a 230. What apgears from the table? How is the present worth of OsO found? Rule. 292 GEOMETRICAL PROGRESSION. ~ 232 1EXAMPLES FOR PRACTICE. 2. What ready money will purchase the reversion of a lease of $60 per annulm, to continue 6 years, but not to commence till the end of 3 years, allowing 6; per cent. compound interest to the purchaser. The present worth, to commence immediately, we find to be 295''039 $295'039, and l -247'72. Ans. $247'72. 3. What is the present worth of $100 annuity, to be continued 4 y wars, but not to commence till 2 years hence, allowing 6 per cent ccmpound interest T Ans. $308'392-+ 4. What is the present worth of a lease of $100, to continue 20 years, but no" to commence till the end of 4 years, allowing 5 per cent.? -- what, if it be 6 years in reversion? -- 8 years? - 10 years? - 14 years?. Ans. to last, $629'426. 5. The revolutionary war closed in 1783; one of the soldiers commenced receiving, in 1817, a pension of $956 a year, which continued till 1840; what was the pension worth to him at the close of the war, the rate being 6 per cent. compound interest? Ans. $162'89 -+. Perpetual Annuities. 32 1'. A farm rents for $60 a year, at 6 per cent.; what is its value? SOLUTION. - This is a perpetual annuity, since the owner is supposed to receive $60 a year forever. On every dollar which the farm is worth he receives 6 cents, and consequently the farm is worth as many dollars as the number of times 6 cents are contained in $60. $60 - $'06 -$1000, Ans. Hence, to find the worth of a perpetual annuity, RCULE. Divide the annuity by the rate per cent.; the quotient will oe the perpetual annuity. 2. A city lot is rented 999 years, at $800 a year; what is it worth, tbh rate being 7 per cent.? NOTE 1. - This -is the same as a perpetual annuity. Ans. $11428'57 + 3. What is the worth of $100 annuity, to continue forever, allowing to the purchaser 4 per cent. ----- allowing 5 per cent.? - -- per cent.? - 10 per cent.? -- 15 per cent.' -. 20 per cent. Ans. to last, $500. 4. A farm is left me which will rent for $60 a year, but is Questions. — ~ 231. What do you understand by annuities in refer *iou 1 How is the worth of an annuity in reversion found? ~T 233. PERMUTATiON. 29B not to come into my possession till the cd oGf P year; what is it Nworth to me, the rate being 6 per cunt. compound interest? SOLUTIO)N.- The farm will be wrrth $1000 to me 2 years hence, and it is now worth. sum which, put at compound interest 2 years. will amount to $1000. $100- $889'996, Ans. i'062 5. What is the present worth of a perpe.ual annuity of $100, to commence 6 years hence, allowing the purchaser 5 per cent. compound interest? -- what, if 8 years in reversion! -- 10 years! -4 years? 15 years?. 30 years. Ans. to last, $462'755. NOTE 2. -The foregoing examples, in comtpound interest, have been con fined to yearly payments; if the payments are half yearly, we may take hal the principal or arnnuity, half the rate per cent., andl twice the number of years and work as before, and so for any other part of a year. PERMUTATION.'IF 233. Permutation is the method of finding how many different ways the order of any number of things may be varied or changed. 1. Four gentlemen agreed to dine together so long as they could sit, every day, in a different order or position; how many days did they dine together? SOLUTION. - Had there been but two of them, a and b, they could sit only in 2 times I (1 X 2 =2) different positions, thus, a b, and b a. Had there been three, a, b, and c, they could sit in 1 X X X 3 = 6t different positions; for, beginning the order with a, there will be 2 positiins, viz., a b c, anti a c b; next, beginning with b, there will be 2 positions, b a c, and b c a; lastly, beginnring with c, we have c a b, and c b a, that is, in all, 1 X 2 X 3 = 6 different positions. In the same manner, if there be four, the different positions will be 1 X 2 X 3 X 4 = 24. Ans. 24 days. Hence to find the Eumber of different changes or permutations, of which any number of different things is capalle, — Multiply continually together all the terms of the natural Questions. - T 232. What do you understand by a perpetual annuity 7 How is its value found? Rule. When it does not begin immediately, how Is its worth calculated? What do you say of other than yearly uaymnents I ~1 233. What is permutation? Illustrate by the first example. What is the rulb? 25* 294 MISCELLANEOUS EXAMPLES. f 234 series of numbers, from 1 up to the given number, and the last product vwill be the answer. 2. How many variations may there be in the position of the nine digits. Ans. 362880. 3. A man bought 25 cows, agreeing to pay for them 1 cent foi every different order in which they coul: all be placed; how much d;:l the cows cost hrn?t Ans. $1551 1210043330985984000000. 1. Christ Church, in Boston, has 8 bells; how many changes may be rung upon them. Ans. 40320. MISCELLANEOUS EXAMPLES. ~ 231. 1.7-4 —2+3-40X 5= how many? Ans.230. NOTE. - A line drawn over several numbers, signifies that the whole are to be taken as one number. 2. The sum of two numbers is 990, and their difference is 90; what are the numbers? 3. There are 4 sizes of chests, holding respectively 48, 76, 87 and 90 lbs.; what is the least number of pounds of tea that will exactly till solme number of chests of either of the 4 sizes? Ans. 396720 lbs. 4. How many bushels of wheat, at $1'50 per bushel, must be given lbr 15 yards of cloth worth 2s. 3d. sterling per yard? Ants. 52839 bushels. 5. If oats, worth $:30 per bushel, are sold for $'35 on account, for what ought cloth to be sold on account, worth $3'75 per yard cash? Ans. $4'37A. 5. Bought a book, marked $4'50, at 33k per cent. discount for cash, what did I pay? Anls. $3'00. 7. Bought 120 gallons of molasses for $42; how must I sell it pet gallon to gatin 15 per cent.? Ans. $'40i. 8. What stin, at 6 per cent. interest, will amount to $150 in 2 years ahd 6 months? Afls. $130'434 +. 9. What is the present \orth of $1000, payable in-4 years and 2 months, discounting at the rate of 6 per cent.? Ans. $800. 10. Bought cloth at $3'50 per yard, and sold it for $4125 per yard; what did I gain per cent.? Ans. 21~ per cent. 11. If 20 men can build a bridge in 60 days, how many would be renjired tc build it in 50 days? Ans. 24 men. 12. How much Siles'ia, li yards wide, will line 12 yards of plaid, gd. wide? Ans. 5 yards. 13. A cistern, holding 400 gallons, is supplied by a pipe at the rate of 7 gallons in 5 minutes, but 2 gallons leak out in 6 minutes; in what tinme wili it be illed? Ans. 6 hours 15 minutes. 14. A. ship has a leak which would cause it to sink in 10 hours, but t could be cleared by a pulnp in 15 hours; in what time would it sink I Ans. 30 hours. 15. How long must I keep $300, to balance the use of $500, which I Ent a friend 4 months? Ans. 6t months ~ 2:34. MISCELLANEOUS EXAMPLES. 9 6. If 800 men have provisions for 2 months, how many must leave that the remainder may subsist 5 months on the same? Ans. 480. 17. Bought 45 barrels of beef, at.3S50 per barrel, except 16 barrels, for 4 of which I pay no more than for 3 of the others; what do the whole cost? Ais. $143'50. 18. A hare, running 36 rods a minute, has 57 rods the start of a dog; how far must the dog run to overtake him, running 40 rods per minute? Ans. 570 rods. 19. The hour and minute hands of a watch are together at 12 o'clock; when are they next together? Ans. 1 h. 5 m. 27-XT s. P. MI. 20. Three men start together to trave. the same way around an islamn 20 miles in circumference, at the rate of 2, 4, and 6 miles per hour; ii. what time will they be together again? Ans. 10 hours 21. Two boats, propelled by steam engines 8 miles an hour, start at the same time, the one up, the other down a river, from places 300 miles apart; at what distance from the'..place where each started will they meet, if the one is retarded, and the other accelerated 2 miles an hour by the current? Ans. 112k miles from the lower, 187J from the upper place. 22. The third part of an army were killed, the fourth part taken prisoners, and 1000 fled; how many in the army? Ans. 2400: 23. A farmer has his sheep in 5 fields: i in the first, A in the second, * in the third, -Tr. in the fourth, 450 in the fifth; how'many sheep has he? Ans. 1200. 24. If a pole be _ in the mud, 5 in the water, and 6 feet out of the water, what is its length? Ans. 90 feet. 25. If go of a school study grammar, I geography, 3d arithmetic, 9 learn to write, and 9 read, what number in the-school? Ans. 80. 26. A man being asked how many geese he had, replied, if I had ^ as many as I now have, and 2[ geese more, added to my present nun.ber, I should have 100; how many had he? Ans. 65. 27; In a fruit orchard, j the trees bear apples, 4 pears, ~ plums, 100 peaches and cherries; how many in all? Ans. 1200. 28. The difference between I and i of a number is 6; required the number. Ans. 80. 29. What number is that, to which, if 4 and 4 of itself be added, the sum will be 84? Ans. 48. 30. B's age is 14 times the age of A, C's age 2T- times the age of both, and the sum of their ages is 93 years; required the age of each. Ans. A 12 years, B 18 years, C 63 years. 31. If a farmer had as many more sheep as he now has, J, J, 4, and * as many, he would have 435; how many has he? Ans. 120. 32. Required the number, which, being increased by j and j of itself, atd by 22, will be three times as great as it now is. Ans. 30. 33. A and B commence trade with equal sums; A gained a sum equal to 5 of his stock, B lost $200 when A's money was twice B's; what stock had each? Ans. $500. 34. A man was hired 50 days, receiving $'75 for every day he worked, and forfeiting $'25 for every day he was idle; he received $27'50; how many days did he work? Ans. 40. 35. A and B have the same income; A saves j of his; B, spending 296 MISCELLANEOUS EXAMPLES.' 234. $30 a year more'han A, is $4C in debt at the end of 8 years; what die B spend each year? Ans.- $205. 36. A man left to A A his Property, wanting $20, to B i, to C the rest, which was $10 less than A's share; what did each receive? Ans. A received $80, B $50, C $70. 37. The head of a fish is 4 feet long, the tail as long as the head and A the length of the body, the body as long as the head and tail; what is the lerngth of the fish? Ans. 32 feet. 38. A can build a wall in 4 days, B in 3 days; in what time can both together build it? Ans. 1i days. 39. A and B can build a wall in 4 days, B and C in 6 days, A and C in 5 days; required the time if they work together. Ans. 3-9h days. 40. A and B can build a wall in 5 days; A can build it in 7 days; in how many days can B build it? Ans. 17J days. 41. A man. left his two sons, one 14, the other 18 years old, $1000, so divided, that their shares, being put at 6 per cent. interest, should be equal when each should be 21 years old; what was the share of each? Anzs. $546'153 +; $453'846 +. 42. What is paid for the rent of a house 5 years, at $60 a year, in arrears for the whole time at 6 per cent. simple interest? Ans. $336. 43. If 3 dozen pairs of gloves be equal in value to 40 yards of calico, and 100 yards of calico to 90 yards of satinet, worth $'50 a yard, how many pairs of gloves will $4 buy? Ans. 8'pairs. 44. A, B, and C divide $100 among themselves, B taking $3 more than A, C $4 more than B; what is C's share?' Ans. $37. 45. A man would put 30 gallons of mead into an equal number of I pint and 2 pint bottles; how many of each? Ans. 80. 46. A merchant puts 12 cwt. 3 qrs. 12 lbs. of tea into an equal nunber of 5 lb., 7 lb., and 12 lb. canisters; how many of each? Ans. 60. 47. If 18 grs. of silver make a thimble, and 12 pwts make a teaspoon, how many of each can be made from 15 oz. 6 pwts.? Ans. 24. 48. If 60 cents be divided among 3 boys so that the first has 3 cents as often as the second has 5 and the third 7, what does each receive? Ans. 12, 20, and 28 cents. 49. A gentleman paid $18'90 among his laborers, to each boy $'06, to each woman $'08, to each man $'16; there were three women for each boy, and 2 men for each woman; how many men were there-? Ails. 90. 50. A man paid $82'50 for a sheep, a cow, and a yoke of oxen; for the cow 8 times, for the oxen 24 times as much as for the sheep; what did he pay for each? Ans. $2'50, $20, and $60. 51. Three merchants accompanied; A furnished ~ of the capital, B i, and C the rest; what is C's share of $1250 gain? Ans. $281'25. 52. A puts in $500, B $350, and C 120 yards of cloth; they gain $332,50, of which C's share is $120; what is C's cloth worth per yard, and what is A's and B's share of the gain? Anis. C's cloth $4'per yd., A's share $125, B's do. $87'50. 53. A, B, and C bought a farm, of which the profits were $580'80 a vear; A paid towards the purchase $5 as often as B paid $7, and B $4 as often as C paid $6; what is each one's share of the gain? Ans, A's share $129'066., B's $18D'693}, CGs $271'04. ~ 234. MISCELLANEOUS EXAMPLES. 97 54. A gentleman divided his fortune among his sons, giving A $9 as bften as B $5, and C $3 as often as B $7; C received $7442'1(); what was the whole estate? Ans. $56t063'857-. 55. A and B accompany; A put in $1200 Jan. 1st, B put in such a sum, April Ist, that he had half the profits at the end of the year; how much did B put in? Ans. $1600. 56. Three horses, belonging to 3 taen, do work to the amount of $26:45; A and B's horses are supposed to do I of the work, A and C's 9,f B and C's Hi, on which supposition the owners are paid proporiionally; what does each receive? Ans. A $11'50, B $5'75, C $9'20. 57. A gay fellow spent 2 of his fortune, after which he gave $7260 for a commission, and continued his profusion till he had only $2178 left, which was i of what lie had after purchasing his commission; what was his fortune? Ans. $18295'20. 58. A younger brother received ~1560, which was ~- of his elder brother's fortune, and 5{ times the elder brother's fortune was I of twice as much as the father was worth; what was he worth? Ans. ~19165 14s. 3fd. 59. A gentleman left his son a fortune, la of which he spent in 3 months; i of t of the remainder lasted him 9 months longer, when he had only ~537 left; what was the sum bequeathed him by his father? Ans. ~2082 18s. 2-l Td. 60. A general, placing his army in a square, had 231 men left, which number was not enough by 44 to enable him to add another to each side; how many men in the army? Ans. 19000. 61. A military officer placed his men in a square; being reinforced by three times his number, he placed the whole again in a square; again being reinforced by three times his last number, he placed the whole a third time in a square, which had 40 men on each side; how many men had he at first? Ans. 100. 62. Suppose that a man stands 80 feet from a steeple, that a line to him from the top of the steeple is 100 feet long, and that the spire is three times as high as the steeple; what is the length of a line reaching from the top of the spire to the man? Ans. 197 feet nearly. 63. Two ships sail from the same port; one sails directly east at the rate of 10 miles, the other directly south at the rate of 7j miles an hour; how far are they apart at the end of 3 days? Ans. 900 miles. 64. How many acres in a square field measuring 70'71 rods between the opposite corners? Ans. 15{ acres. 65. Supposing that the river Po is 1320 feet wide and 10 feet deep and runs 4 miles an hour; in what time will it discharge a cubic mile of water into the sea? NOTE. -A linear mile is 5280 feet. Ans. 22 days. 66. If the country which supplies the river Po with water be 380 miles long and 120 broad, and the whole land upon the surface of the earth be 62,700,000 square miles, and if the quantity of water d:scharged by the rivers into the sea be everywhere proportional to the extent of land by which the rivers are supplied, how many times greater than the Po wil_ the whole amount of the rivers be? Ans. 1375 times. 67. Upon the sams supposition, what quantity of water, altogether will be discharged br all the rivers into the sea in a year of 365 days I Ans. 22812i cubic miles. 298 MEASUREMENT. ~F 235 68. if the ratio of sea to land be as 10A to 5, ant, the average depth of the sea- be 11. miles, in how long time, if the sea were empty, would it bc filled? OIs. 8657 years 27S days. 69. If a cubic foot of -water weigh 1000 oz., and nmercury be 13, times heavier than -vater, and the hight of tle mercury in the barometor (which weighs the same as a column of air on the same base and ex tending to the top of the atmosphere) hbe 30 inches, what will the ail weigh on a square foot? - on a square mile? What will the whole atmosphere weigh? Ans., in order, 2109'375 lbs., 58806000000 lbs., 11430122220000000000 lbs. 70. A traveler who had set a perfectly accurate watch by the sun in'Boston, 71~ 4' W. Ion., being in Detroit, 820 58' W. lon., 3 days after, was surprised to find it wrong, when compared with the sun; was it too. fast or too slow? how much, and why? 71. A building fell in Portland, Me., 70~ 20' W. lon., at 9 o'clock, A. 5., and in 3 minutes the intelligence of the event reached St. Louis, Mo., 90~ 15' W. Ion., by magnetic telegraph; when was it known at St. Louis? Ans. At 43 mn. 20 sec. past 7 o'clock, A. M. 72. At the battle of Bunker Hill'the roar of cannon was distinctly heard at Hanover, N. H, and business was suspendled for a time; in what time did the sound pass, the distance being supposed 120 miles? NOTE. - Sound moves 1142 feet in a second. Ans. 9 m. 14 sec. -. 73. Seeing the flash of a rifle in the evening, it was 8 seconds before I heard the report; what was the distance? Ans. 1 mi. 3856 ft. 74. A man in view on a hill opposite is chopping, at the rate of a blow in 2 seconds; I saw, him strike 4 blows before I heard the first; what is his distance from me? Anls. 1 mi. 1572 ft. 75. A laborer dug a cellar, the length of which was 2 times the width, and the width 3 times the depth; he removed 144 cubic yards of earth; what was the length? Ans. 36 fect. 76. A owes B $750, due in 8 months; but receiving $300 ready money, he extends the time of paying the remainder, so that B shall lose nothing; when must it be paid? Ans. In 1 yr. 1 mo. 10 days. 77. The sum of two numbers is 266fi, and the product of the greater multiplied by 3, equals the product of the less multiplied by 5; what are the numbers? Ans. 100, and 166fi. 78. A park 13 rods square is surrounded by a walk which occupies 1r9uy of the whole park; what is its width? Ans. 8 ft. 3 in. 79. A, B and C commence trade with $3053'25, and gain $610'65; A's stock -+- B's, is to B's +- C's, as 5 to 7; and C's stock - B's, is to C's + B's, as 1 to 7; what is each one's part of the gain? Ans. A's gain $135'70, B's $203'55, C's $271'40. MEASUREMENT OF SURFACES ~F 235. Tofind the area of a paraZlelogram, mnltiply the length by the shortest distance between the sides. ~ 235. MEAUTR9EM ENT. 299 An Jn0gle is the space comprised between two lines that meet in a point. A Right Angle is formed by one line meeting. another perpen dicularly. An Obtuse Angle is greater, and an A.cute.Angle is less, than a right angle. 0 D NOTE 1.- A parallelogram has its opposite sides equal, hut its adja.enit sidces unequal, like the figure A B C D, or E F C D. The former ix / called a rectangle, see T 48. The second is called. A F B a rhoinmboid, and is equal in size to the first. 1. What are the superficial contents of an oblique angled piece of ground, measuring 80 rods in length ano 20 rods in a perpendicular line between its sides? Ans. 1600 sq. rods. NOTE 2. -To find the contents ol a rhombus, which, like the annexed figure, has its sides equal, but its anglis;',ot right angles; multiply the length of one side by the shortest _ distance to the side opposite. To find the area of a trapezoid, multiply half the sum of the parallel sides by the shortest distance between them. NOTE 3. -A trapezoid is a figure, like the one in the an-: nexed diagram, bounded by four straight lines, only two of.... which are parallel. 2. What is the area of a piece of ground in the form of a trapezoid, one of whose parallel sides is 8 rods, the other 12 rods, and the perpendicular distance between them 16 rods? s+1a X 16 — 160 sq. rods, Ans. 3. How many square feet in a board 16 feet long, 1'8 feet wide at one end, and 1'3 at the other? Ans. 24'8 feet. To find the area of a triangle, multiply the base by half the altitude. a r NOTE 4. -The figure A B C is a triangle, of which the sice A B is the base, D C the altitude. The triangle is evidently half the parallelogram A B C F, the area of which equals A B X D C. A D B 4. The base of a triangle is 30 rods, and the perpendicular 10 rods; what is the area? Ans. 150 rods. 5. If the contents are 600 rods, and the base 75 rods, what is the altitude i Ans. 16 rods. 6. Required the base, the area being 400, the altitude 40 rode:. Ans. 20 rods. 7. How many square feet in a board 18 feet long, 1A feet wide at one end, and running to a point at the other T Ans. 13. feet. To find the circumference of a circle when the diameter is known, multiply the diameter by 34, or accurately by 3'14169 300 MEASUREMENT. ~ 236 To find the area, multiply the circumference by one fourth of the diameter; or multiply the square of the diameter by'7854. 8. What is the circumference of a circular pond, the diameter of which is 147 feet What is the area? Ans. to the last, 1697 l feet. 9. If the circumference be 22 feet, what is the diameter t Ans. 7 feet. 10. If the diameter of the earth is 7911 miles, what is the circumference? Ans. 24853 miles. 11. How many square inches of leather will cover a ball 39 inches in diameter t NOTE 5. - The area of a ball is 4 times the area of a circle having the same diameter. Ans. -38' square inches. 12. How many square miles on the earth's surface? Ans. 196,612,083. Measurement of Solids. T 236. NOTE 1.- The general principle for finding the contents of cubic bodies is to multiply the length bly the breadth, and the product by the thickness, but the rule applies directly only to the culbe or right prism, being subject to modifications as applied to solid figures of other forms. See T 51. 1. How many solid inches in a globe 7 inches in diameter? NOTE 2. - The solid contents of a globe are found by multiplying the area of its surface by Q part of its diamieter, or the cube of its diameter by'5236. Ans. 179] solid inches. 2. What number of cubic miles. in the earth? Ans. 259,233,031,4351. 3. What are the solid contents of a log 20 feet long, of uniform size, the diameter of each end being 2 feet? NOTE 3. -A figure like the above is called a cylinder. To find the solid contents, we find the area of one end by a foregoing rule, and multiply the area thus fqund by the length. Ans. 62'83 - cu. ft. 4. A bushel measure is 18'5 inches in diameter, and 8 inches deep; aiow many cubic inches, does it contain? Ans. 2150'4 -. NvTE 4. -Solids having bases bounded )by straight lines, and decreasing uniformly till they come to a point, are called pyranids. Solids which thus decrease, with circular bases, are called cones. Pyramids and cones are just one third as large as cylinders, of x; hich the area of each end is equal to the trea of the bases of these solids. Hence, if we multiply the area of the base by the hight, and divide the product by 3, the quotient will be the Solid con-,ents. 5. What are the solid contents of a pyramid, the base of which is 4 feet square, and the perpendicular hight 9 feet? Ans. 48 solid feet. ~T 237, 238. MEASUREMlENT. ~301 6. What are the solid contents of a cone, the hight of which is 27 feet, and the diameter of the base is 7 feet?.//ns. 346? solid feet. 7. What are the solid contents of a stick of timber 18 feet long one end of which is 9 inches square and the other end 4 inches square, uniformly diminishing throughout itsevhole length? NOTE 5. - Such a figure is called the frustum of a pyramid, and the solid contents are found by adding to the areas of the ends the square root of theii product, and multiplying the sum by one third of the hight. The pupil must notice that th6 diameters are expressed in inches, while the length is in feet Jns. 5 solid feet, 936 solid inches. 8. What are the solid contents of a round log of wood, 36 feet long, 1,6 feet in diameter at one end, and diminishing gradually to a diameter of'9 of a foot at the other? NOTE 6. - Such a figure is called the frustum of a cone, and the solid con tents are found by adding to the squares of the two diameters the square root of the product of those squares, multiplying the sum by'7854, and the resulting product by one third of the length. klns. 45'333 + solid feet. Gauging, or Measuring Casks. 1T 237. 1. How many gallons of wine will a cask contain the head diameter of which is 25 inches, and the bung diameter 31 inches, and the length 36 inches? How many beer gallons? NOTE. - Add to the head diameter 2 thirds, or if the staves curve but slightly, 6 tenths of the difference between the head and bung diameters the sum will be the average diameter. The cask will then be reduced to a cylinder, the contents of which may be found by a foregoing rule, in solid inches. The solid inches may be divided by 231, (IT 114,) to find the number of wine gallons which the cask will contain, and by 282, (IT 115,) to find the number of beer gallons. A.ns. 102'93+ wine gallons, 84'77+ beer gallons. 2. How many wine gallons in a cask, the bung diameter of which is 36 inches, the head diameter 27 inches, and the length 45 inches?.ns. 166'617 Mechanical Powers. 1 t23S. 1. A lever is 10 feet long, and the fulcrum, or prop, on which it turns is 2 feet from one end; how many pounds weight at the short end will be balanced by 42 pounds at the other end? NOTE 1. - In turning round the prop, the long end will evidently pass over a space of 8 inches, while the short end passes over a space of 2 inches. Now, it is a fundamental principle in mechanics, that the weight and power will exactly balance each other, when they are inversely as the spaces they pass over. Hence, in this example, 2 pounds, 8 feet from the prop, will balance 8 pounds 2 feet from the prop; therefore, if we divide the distance of. the rowEn from the prop by the distance of the WEIGHT fromr the prop, the quotient will always express the ratio of the WEIGHT to the Pow.ER; 8 = 4, tiit s,. the weight will be 4 times as much as the power. 42 X 4 =- 168../tns. 168 lbs 302 MEASUREMENT. ~ 238 2. Supposing the lever as above, what power would it require to raise 1000 pounds Ans. 4o-0 _~ 250) pounds. 3. If the weight to be raised be 5 times as much as tile power to be applied, and the distance of the weight from the prop he 4 feet, how far from the prop must the power be applied. Ans. 20 feet. 4. If the greater distance be 40 feet, and the less a of a foot, and the power 175 pounds, what is the weight? Aiws. 14000 pounds. 5. Two men carry a kettle, weighing 200 pounds; the kettle is suspended on a pole, the bale being 2 feet 6 inches from the hands of one, and 3 feet 4 inches from the hands of the other; how many pounds does each bear A 1142 pounds. 85# pounds. 6. There is a windlass, the wheel of which is 00 inches in diameter, and the axis, around which the rope coils, is 6 inches in diameter; now many pounds on the axle will be balanced by 240 pounds at the wheel; NOTE 2. - The spaces passed over are as the diameters, or the circumfernces; therefore, 6.Q = 10, ratio. Ans. 2400 pounds. 7. If the diameter of the wheel be 60 inches, what must be the diameter of the axle, that the ratio of the weight to the power may be 10 to 1 Ans. 6 inches. NOTiE 3.- This calculation is on the supposition, that there is nofriction. for which it is usual to make allowances. 8. There is a screw, the threads of which are 1 inch asunder; if it is turned by a lever 5 feet, =-60 inches, long, what is the ratio of the weight to the power NOTE 4. - The power applied at the end of the lever will describe the circumference of a circle 60 X 2 = 120 inches in diameter, while the weight is raised 1 inch; therefore, the ratio will be found by dividin g the circumrnfer-ence of a circle, whose diameter is twice the length of the lever, by the distance between the threads of the screw. 377120 X 3 =- 3777 circumference, and -- -'- 377,ratio, Ans. 9. There is a screw, whose threads are 4 of an inch asunder; if it be turned by a lever 10 feet long, what weight will be balanced by 120 pounds power? Ans. 36205771 pounds. 10. There is a machine, in which the power moves over 10 feet, while the weight is raised 1 inch; what is the power of that machine, that is, what is the ratio of the weight to the power? Ans. 1'20. 1 I. A man put 20 apples into a wine gallon measure, which was afterwards filled by pouring in 1 quart of water; required the contents of the apples in cubic inches. A': s. 1731 inches. 12. A rough stone was put into a vessel, whos capacity was 14 wine quarts, which was afterwards filled with 24A quarts of water; what was the cubic content of the stone Ans. 664k inches. NOTE 5. - For a more full consideration of the foregoing subjects, the pupil Is referred to a more extended treatise on Mensuration in connection with fth " series." FORMS OF NOTES, ETC. 303 FORMS OF NOTES, RECEIPTS, AND ORDERS. Notes. No. I. Keene, Sept. 17, 1840. For value received, I promise to pay OLIVER BOUNTIFUL, or order, sixty. dhree dollars, fifty-four cents, on demand, with interest after three months. WILLIAM TRUSTY. No. II. Ludlow, Sept. 17, 1846. For value received, I promise to pay to 0. R., or bearer, - dollars -- zents, three months after date. PETER PENCIL. No. III. By two per sons. Yates, Sept. 17, 1846. For value received, we, jointly and severally, promise to pay C. D., or 2rder, - dollars - cents, on demand, with interest. Attest, PETER SAXE. ALDEN FAITHFUL. JAMES FAIRrACE. Receipts. Boston, Sept. 19, 1846. Received from Mr. DURANCE ADLEY tell dollars in full of all accounts. ORVAND CONSTANCE. Newark, Sept. 19, 1846. Received of Mr. ORVAND CONSTANCE five dollars in full of all accounts. DURANCE ADLEY. Receipt for Money received on a Note. Rochester, Sept. 19, 1846. Received of Mr. SIMPSON EASTEY (by the hand of TITtrs TRUSTY) sixteen dollars twenty-five cents, which is endorsed on his note of June 3, 1846. PETER CHEERFUL. A Reccipt for Money received on Account. Hancock, Sept. 19, 1846 Received of Mr. ORLAND LANDIKE fifty dollars on account. ELDRO SLACKLEY. Receiptfor Money receivedfor another Person. Salem, August 10, 1846. Received from P. C. one hundred dollars for account of J. B. ELI TRUMAN, Receiptfor Interest due on a Note. Amherst, July 6, 1846. Received of I. S. thirty dollars, in f'lll of o:-' year's interest of 8500, due II me on the - day of --- last, on note from the said I. S. SOLOMON Ga AY. 304 FORMS OF BILLS. Receipt for Money paid before it becomes due. Hillsborough, May 3,1846. IReceived of T. Z. ninety dollars, advanced inl full for one year's rent of my farm, leased to the said T. Z., ending The first day of April next, 1t47. HoNEsrTU JAMES. NOTE. -There is a distinction between receipts given in full of all' accounts, and others in full of all demands. The former cut off accounts nmly; the lat. ter cjat off not only accounts, but all obligations and right of action. Orders. Utica, Sept. 9, 1846. Mr. STEPHEN BuRGEss. For value received, pay to A. B., or brder, ten dollars, and place the same to my account. SAMUEL SKINNER. Pittsburgh, Sept. 9, 1846. Mr JAMES ROBOTTOM. Please to deliver to Mr. L. D. such goods as he may call for, not exceeding the sum of twenty-five dollars, and place the same to the account of your humble servant, NicHOL As REUBENS. FORMS OF BILLS.' Before you build, sit down and count the cost." Simeon Thrifty built a house for Thomas Paywell, according to a plan agreed upon between them, for the sum of $1500. The cellar of the house is 24 by 28 feet, and is dug 4 feet deep below the top of the ground. The cellar walls ire 7 feet high. There is a wing at one end of the main building, 20 by 24 feet, which is underpinned with a wall 3 feet high. As one side of the wing joins the main building, for the underpinning of it but 3 walls are required, one 24, and two 20 feet long on the outside. The walls of the cellar, and the underpinning of the wing are 1~ feet thick. To the cellar there is a door 4 by 7 feet, and 2 windows, each 2 by 21 feet. Simeon Thrifty, wishing to know how much it cost him to build the house, kept an accurate account of all the materials used, the labor employed, and the cost of each. The following are his bills:Bill of Timber. 6 sticks for posts to up.right part..... each 14 ft. long, ard I by 10 in. 6 " " wing,........ " 11. 4" 10 " 2 " sills to upright part, 28 { 7 "A 8 5 " and sleepers to upright and Willng, " 24 " 7 " 8 " 4 " plates and side girts to upright, " 28 i 6 "C 7 7 " and beams to upright and wing,....... " 24 " 6" c 7 cc 2 " eave gutters,.......... " 30 " 6" 10 " 3 " "......... "20 " 6 10 " 52 rafters........ "...... ( 15 " 3 4 " 96 studs........... " 12 " - 2 " 4 " 175 partition planks,............ 10 " 2 4 96 scantlings......" 12 4 4 4" 10 " forbraces,:' 12 " 4 " 4 " 75 joists,."...... 14 2 7 " 80 "..............10 " 2 7 u FORMS OF BILLS. 305 Bill of Luzmber. 800 It best quality pine, for best doors, and other nice joiner work 10000 " common " " door and window casi.gs, staizs, base orwds common doors, &e, &c. 3850 " white wood siding. 2160 " bass " flooring. 1500 roof boards. 6000" lath. 26 bunches shingles, 500 in each bunch. Bill of Mraterials for WTindows. Sash and glass for 14 windows of 24 panes each,.:y 9 inches. " "c 4 " " 20 " 7" 9 " It " 5' I 6' 7 " 4 " " " 1 window, " 16 " 7 "9 " It it 1 " " 12 " 7" 9 30 lb. putty. Hardware Bill. 4 casks nails, 100 lbs. each. 22 pairs 3 inch door hinges, with screws. 2 " 4 " " " 20 door handles, " 2 outside door knobs and locks. 3 cupboard fastenings. 63 ft. tin eave conlductors, including 4 elbows. 3 stove-pipe crocks. 3 " thimbles. 3 ft. tin pipe for sink-spout. 20 window springs. 4 papers 1 inch brads. Bill o, Materials for Chimneys and Plastering. 1600 bricks. 27 loads sand. 200 bush. lime. 10 bush. hair. Bill of Prices of Materials. Stone for cellar walls and underpinning,.$'25 per perch. All the timber except the eave gutters reduced to board measure, that is, 1 inch thick,... 10 "M. Eave gutters,............. 15' " " Pine lumber, best quality....... 20' 4" " " " common,..10' "l " Siding, flooring, and roof boards,. 0' " " Lath,................... 5' " " Shingles,....... 1'50 " bunch. Window sash,..'03 " pane. "glass,. 2'50 " box of 114 MaeP Putty,...................'07 " b. Nails,'051" " 3 inch door butts, with screws,.'12~" pair. 4 " "'"'15" " Door handles,.............' 124 each. Outside door knobs and locks,.........'50 " Clipboard fastenings...'12 " Eave conducters,..'124 per ft. Extra for elbows,.'06i each. Stove-pipe crocks...... 37 thimbles,1.... r. *. " 4 306 FORMS OF BILLS. Si.tk spout... $ 44. Window springs,....'06* each. Brads,........'10 per papc Bricks, 10' " M. Lilae,e.................'121" bush. Sandl,......... 1'00 " load. Hair,...........'25 "bush. Bill of Prices of Lcbor. Digging cellar............... per cu.......... per yd 5 stone masons, 6 dayseach,............ 200 " day. 2 carpenters, 12 "..... 1'50 " 3 joiners, 40 "...... " " Painting and glazing... 100'00. Furring ready for lathing,.. 12'00. Lathing,................. 15'00. 2 plasterers, 7 days each,... 2'50 per day. 3 brick-layers, I day".. 2'75 " Team and hired man, 4 months of 26 days each,.... " 2'00 " Simeon Thrifty commenced the house on the Ist day of May, and completed it on the 3d day of Sept.; allowing him $1'50 per week for his board, how mutsh did he get for his own labor? Ans. $281'64T4T. 75,-6T perches stone,.. $ 18'88l- Brads,........'40 600 ft. eave -utters,. 900 1600 bricks..... 9'60 8287" other timner,. 82'7 200 bush. lime,. 25'00 800 " best pine lumber, 16'00 27 loads sand,... 2700 17510 " other lumber,. 1750 10 bush. hair,. 2.50 6000 1 lath,.:..... 30'00.Digging cellar,. 4'48 26 bunches shingles, 3900 Laying stone work,. 6000 Window sash,........ 13'68 Carpenters' work,. 36'00 4 boxes window glass, ~ 1000 Joiners' ". 210'00 30 lbs. putty,.2'10 Painting and glazing,. 100'00 300 lbs putty....... 2110 Furring......... 12'00 400 " nails,. 22 00 Furring..1200 Butt hinges,.......'0 Lathig, 15'00 Door hanldles,. 2 i0 Plastering.. 35'00 Outside door knobs & locks, 3'00 Brick-laying,. 750 Cup)bloard fastelin~gs,..'37* Team and hired ma,. 208'00 Tin eave conductors, 7'87 18 weeks' board.. 27'00 Extra on elbows,'25 Stove-pipe crocks,.... 1'12~ Amount,...*1218'35671 " thimbles,'37. Sink spout,...44 Errors excepted, Window springs, 1'26