> 
 
 i 
 
 i 
 
 1 
 
 ■k 
 
 :< '■' 
 
 r^< ^ 
 
 (,; 
 
 -, ( 
 
George Washington Flowers 
 Memorial Collection 
 
 DUKE UNIVERSITY LIBRARY 
 
 ESTABLISHED BY THE 
 
 FAMILY OF 
 
 COLONEL FLOWERS 
 
Digitized by tine Internet Arciiive 
 in 2010 witii funding from 
 Dul<e University Libraries 
 
 littp://www.arcli ive.org/details/intellectualaritOOcolb 
 
INTELLECTUAL 
 
 ARITHMETIC, 
 
 VPON THE 
 
 INDUCTIVE METHOD OF INSTRUCTION. 
 
 BY WARREN COLBURN, A. M. 
 
 WITH AN INTRODUCTION TO WRITTEN ARITHMETIC, BY HIS SON, 
 WARREN COLBURN. 
 
 REVISED AND ADAPTED TO THE USE OF SCHOOLS IN THE 
 CONFEDERATE STATES. 
 
 BY THOS. O. SUMMERS, D. D. 
 
 -«•»- 
 
 ^^^r 
 
 Nasfibtlle, ^tnn. : 
 SOUTHERN METHODIST PUBLISHING HOUSE. 
 
 18G2. 
 
 u 
 


 j^Tr^ INTRODUCTION,. ^ 
 
 The firet instructions piven to the child in Arithmetic, are usu- 
 ally given on the supposition that the child is already able to count. 
 This indeed seems a sufficiently low requisition ; and if children 
 were taught to count at home in a proper manner, they would have 
 this power in a sufficient degree when they enter the primary 
 scliool. But it will ))e found on trial, that most children, wlien 
 they begin to go to school, do not know well how to count. This 
 may be proved by requiring them to count 20 beans or grains of 
 corn. Few of them will do it without mistake. The difficulty is, 
 they have been taught to repeat the numerical names, one, two, 
 three, in order, without attaching ideas to them. They learn to 
 count without counting things. This point, then, calls for the 
 teacher's first attention— to lead the child to apprehend the. mean- 
 ing of each numerical word by using it in connection with objects. 
 
 The kind of objects to be employed as counters should, of course, 
 be similar, as marks on the blackboard, beans, pieces of wood, or 
 o^cork, or the balls in a numeration frame. Provided they are 
 similar, and large enough to be seen without effort by all the class, 
 it is of little consequence wliat they are : the simpler the better, 
 and those which the teacher devises or makes, will, other things 
 being equal, be best of all. Not more than ten should be used or 
 exhibited to the children in the first few lessons. 
 
 LESSON I. 
 
 Let the class have their attention called to the teacher ; and 
 when he lays down a counter, where all can see it, let them say 
 one; let the teacher lay down another, and the class say two; and 
 so on up to ten. If any of the class become inattentive, let the 
 teacher stop- at once: and, after the attention is fully centred on 
 him, let him begin again. 
 
 After going through this addition a few times in this form, it 
 may be varied thus : The teacher laying down the counters, one 
 by one, as before, the class may be led to say, one and one are 
 two, two and one are three, three and one are four, etc. 
 
 The above mode of adding may be shortened by leading the 
 class to say as follows : One and one are two, and one are three, 
 and one are four, etc. 
 
 At any time the word designating the counter may be used along 
 with the number, as beans, balls, pieces, marks, or books, as the 
 case may be. 
 
 At times, it will be well to give some fictitious designation to 
 the counters, such as the teacher, or still better, such as some one 
 of the class may choose, calling them men, sheep, horses, etc. 
 
 f^^^V 
 
IV INTRODUCTORY LESSONS. 
 
 Next to Addition, as illustrated above, should come Subtraction. 
 Having counted ten, let the teacher take away one, and the class 
 be made to say, one from ten leaves nine, one from nine leaves 
 eight, etc. In Subtraction, the same variations may be introduced 
 as in Addition. No further illustrations of this operation need be 
 given, as the teacher's discretion will supply all that is necessary. 
 
 In connection with these exercises, let the pupil be taught to re- 
 peat in reversed order the numerical words they have employed, 
 counting from one up to ten, and then in reverse order from ten to 
 one. 
 
 It is not to be supposed that the whole of the foregoing lesson 
 can be learned at one exercise. It is only a small part of it that 
 children will at first have sufficient power of attention to go over 
 with profit. The same remark may be made respecting the fol- 
 lowing Introductory Lessons. 
 
 LESSON II. 
 
 Let the teacher call the attention of the class, and require them 
 to count, and then lay down, one by one, a small number of count- 
 ers, say, for example, five ; then let him separate them into two 
 parts, as one and iour, thus, • • • • •, and say, " one and four 
 are five, " and require the class to say the same. Then let him 
 divide the number into different parts, as two and three, three and 
 two, four and one, one and one and three, etc., requiring the clasa 
 with each division to name the parts and make the addition. Let 
 them always begin at the left end of the line of counters as they 
 face them. Having exhausted the combinations of five, let the 
 number six be taken, giving combinations like the following : • • 
 
 • •• • ••-; etc. 
 
 It may be found that a lower number than ftve should be made 
 the first step in this exercise. 
 
 After the combinations of six have been exhausted, the number 
 seven may bo taken, and then successively, eight, nine, and ten. 
 
 As a part of this Lesson, each question in addition should be 
 converted into a question in subtra(;tion ; thus five and three are 
 eight; then, liaving put the two parts together, which make eight, 
 remove the three, and lead the class to say, '■'three from eight 
 leaves five,'' 
 
 The following exercise is important in this connection. Let the 
 teacher select s^me number, and give one part of it, and require 
 the class as quick as possible to name the complementary part. 
 Thus let six be the number, the exercise will be as follows : 
 Teacher: "Now attend, six is the numl)er; I am going to name 
 one part of it ; when you hear me name it, do you all name the 
 other part as quick as you can; now be ready; five." Class: 
 ''One." Teacher: "FoMr." Class: "Tmjo." Teacher: "TArec." 
 Class: "T/irfe." Teacher : "One " Class : "F?'w," etc. 
 
 This exercise should not be pressed too fast, but carried on gra- 
 dually, as the pupil's strength of mind will allow. Special pains 
 should be taken that the number ten be perfectly mastered in this 
 form of combining its parts. This will give the pupil the most im- 
 portant aid in all his calculations in larger numbers. 
 
INTRODUCTOEY LESSONS. 
 
 LESSON III. 
 
 For a number of days after becrinning the foregoing exercises, 
 the child should not have the book at all in his hands. If the child 
 has the book in his possession, it will be well for the teacher to 
 take it for a few days, and let the pui)il employ himself at his seat 
 in writing on a slate, or with ot])er books. In this way, the child 
 has awakened within him the idea of calculation in numbers, with- 
 out having become wearied with the reading of what excites no 
 interest. After a few days, however, the book may be put into tlie 
 pupil's hands, aud he may be directed to get a lesson in Section I. 
 In the meanwhile, the Introductory Lessons should be continued, 
 and form a part of each day's exercise tiil the}' are finished. In this 
 way, the pupil, in studying his first lesson from the book, will al- 
 ready have learned the use of counters, and will naturally resort to 
 them at his seat, using beans or marks on his slate for this purpose. 
 It will be far better for him to come to the use of counters in this 
 natural way. than to be enjoined to use them before he has been 
 interested in witnessing their application. 
 
 The pupil, in the preceding lef«sons, has become acquainted with 
 all the nujnbers as far as ten, regarding them either as units, or as 
 grou])ed into i)art8 of a larger whole. The next step is to carry 
 him through the numbers from ten to twenty. 
 
 First, let the class count with the objects before them from one 
 up to twenty ; then, removing all but ten, let the ten be grouped 
 in a pile ; or if they are marks on the board, let them be enclosed 
 by a line drawn around them, and begin to count upward from ten, 
 " One and ten are eleven ; two and ten are twelve ; three and ten 
 are thirteen." Here pause, and examine the composition of the 
 word, thirteen — three ten, or three and ten. Show how the three 
 is spelt in thirteen, and also how the ten is spelt. Then proceed 
 "four aud ten are fourteen," examining the word as in the former 
 case; '"five and ten are fifteen; six and ten are" — perhaps some 
 one in the class will now be able to give the compound word ; then 
 go on, "seven and ten, eight and ten, nine and ten, ten and ten." 
 
 When they can give the compound words readily from the sim- 
 ple ones, then give them the compound word, and let the class 
 separate it into its two component words ; thus : Teacher : 
 "Seventeen." Class: "Seven and ten," etc. Thus far let the 
 teacher be careful to present the name of the smaller of the two 
 numbers first, for that is the order in which the compound word 
 presents them ; let the teacher say four and ten, and not ten and 
 four. After the class have caught the analogy between the simple 
 words, and the compounds which they form, so that one instantly 
 suggests to them the other, then the order of the words may be 
 changed, and the ten put first. The caution here suggested may 
 seem to some unnecessary ; but a careful observation of the men- 
 tal habits of children will not fail, I think, to show its importance. 
 
 In the analysis of the compound words from ten to twenty, 
 eleven and twelve should be omitted till the last; for as the simple 
 words of wliich the}' are foiimed are disguised or obsolete, they 
 tend to obscure, rather than elucidate the subject to the mind of a 
 child. Having obtained the idea through the other words in the 
 series, he may take the statement respecting these on trust. 
 
VI INTRODUCTORY LESSONS. 
 
 LESSON IV. 
 
 Having counted twenty, and grouped the number in two tens, 
 let the class count ten more, making in all thirty, or three tens. 
 Keeping the tens separate, let the class count ten more, making 
 forty, or four tens. Let the class then answer such questions as 
 the following: Twenty are how many tens? Thirty are how many 
 tens? Forty are how many tens? Four tens are what number? 
 Three tens are what number? Two tens are what number? 
 
 After this, they may proceed with the higher multiples of ten, 
 fifty, sixty, seventy, eighty, ninety, a hundred. 
 
 Through the whole of this exercise, each multiple of ten should 
 be presented in groups of ten, so as to aid the idea by the visible 
 representation. 
 
 The pupils should be led to see the significancy of each numer- 
 ical name ; that thirty- seven, for example, means three tens and 
 seven; fifty-six means live tens and six. 
 
 In this way the pupil may be led to understand the Decimal 
 Ratio, at this early stage, and no farther trouble need be taken in 
 that direction. When in a latter stage of study, he comes to the 
 Decimal notation in written Arithmetic, he will find it only a natural 
 mode of expressing ideas already rendered familiar in practice. 
 
 LESSON V. 
 
 Let the teacher stand at the board, and call the attention of the 
 class to what he shall write ; then, making two f | 
 marks, ask, "How many marks on the board?" | 
 
 When the class have answered, let the teacher j 
 
 write two more, and ask, "How many now?" I 
 
 and so on to the number of twelve or more. | \ 
 Then take a writing-book or sheet of paper, and | 
 
 covering all but two of the marks, let the class j 
 
 repeat the same process while the teacher re- [ I 
 moves the book, so as to bring two more into view at each re- 
 move ; the numbers read by the class being two, four, six, eight, 
 ten, etc. Then let the process be reversed, subtracting two 
 successively, which gives, beginning with sixteen, the following : 
 sixteen, fourteen, twelve, ten, etc. 
 
 Again, the teacher may say to the class, "When I made those 
 marks, how many did I make at a time?" Class: "Two." 
 Teacher : "Did I make two more than once?" Class: "Yes, sir, 
 a good many times." Then the teacher, covering up all but two : 
 "Now look, how many times two are there?" Class: "Once." 
 Teacher: "Once two are how many?" Then, after the class 
 have answered, showing two more, "How many times two do you 
 see?" "Twice two are how many?" Then go on in the same 
 way with three twos, four twos, etc., to the end. 
 
 At this point the pupils may be taught the distinction between 
 even and odd numbers, and be trained to repeat rapidly the even 
 numbers, from two up to twenty. 
 
 The pupils may derive important aid in adding and multiplying, 
 by grouping the numerical names with the voice, in something 
 like the following manner: Teacher: "Listen now to me; one^ 
 
ISTEODUCTOBY LBSSONS. Vll 
 
 two— three, four— five, six. How many twos did I count?" Class : 
 "Three twos." Teacher: '• Count three twos just as I did." 
 Then let the teacher ask, "Three times two are how many?" 
 Then require them to count four twos, five twos, etc. This method 
 may be brought in aid of all their earlier attempts in adding and 
 muitiplving. 
 
 Care 'should be taken in this exercise tliat the words be pro- 
 nounced in a quick and neat manner, with a distinct pause between 
 the groups. If rightly conducted, this exercise furnishes one of 
 the best aids for overcoming the habit of counting, instead of ad- 
 ding—that fatal clog whicli keeps back many scholars in arith- 
 metic from making any rapid progress. 
 
 rr,, ■,,..■ r J. 1 • • r I I eleven. 
 
 The successive additions of two, beginning . j ^^^^ 
 
 with one, furnish the series of odd numbers. ! j ggyg'u^ « 
 
 This and the preceding exercise exhibit all the ■! . | five 
 results made by the additions of the number j . ^]^yQQ 
 
 t^o- [ I one. ' 
 
 The pupil should, in connection wMth this exercise, be taught to 
 group the numerical names, by twos, beginning from one, so as to 
 furnish the series of odd numbers, thus : one— two, three— fooir, 
 five, etc. 
 
 Finally, let him count the even series, up to twenty, and then 
 down, till he can do it readily. 
 
 Let him count the odd series up to twenty-one, up and down. 
 
 LESSON VL 
 To teach the pupils to add the number three, let the teacher make 
 
 three marks on the board, and require the 
 class to name the number; then three 
 
 I I I fifteen. 
 I I I twelve. 
 
 more, and so on. Then, covering all but < 111 nine 
 the lower three, let the class name the I I I six. 
 
 number in view, as the teacher exhibits i III three, 
 successively three more at each remove, or three less. 
 
 The pui)ils may then be led to count in groups of three; thus: 
 one, two, three— four, five, six — sevei\, eight, nine, etc. 
 
 From this, they may be readily led to name the multiples of 
 three ; three, six, nine, twelve, etc. 
 
 The first variation from the above method in the addition of three 
 may be made by beginning with two, which f || i fourteen, 
 gives the following numbers. In order to j | | eleven, 
 
 secure the necessary amount of repetition ■ i i i eight. 
 in these exercises, and yet to avoid weari- i i i five, 
 
 ness, it will be well often to change the [ t I two. 
 manner of adding. Thus, in the accompanying example, the pupils 
 may simply name the series made by the successive additions ; as 
 two, five, eight, eleven, fourteen ; or, they may state the process 
 more fully, thus: two and three are five, and three are eight, and 
 thi'ee are eleven, etc. A still fuller statement of the operation 
 would be thus: two and three are five, five and three are eight, 
 eight and three are eleven, etc. Tiie only remaining variety in 
 
viii 
 
 INTBODUOTORT LESSONS. 
 
 adding three will be exhibited by beginning with one, which may 
 be illustrated in the same way as the others, giving the numbers, 
 one, four, seven, ten, etc. 
 
 LESSON vn. 
 
 After the numbers have been mastered in the manner above sug- 
 gested, thev should be combined in a variety of ways by means of 
 marks on the board. The columns of marks subjoined are giveu 
 merely as examples and hints on this point. 
 
 I I 
 I 
 
 1 I 
 I I 
 
 1 I 
 I 
 
 A single column may be made to furnish exercise suflScient for a 
 recitation ; for, by adding or erasing a single mark at the begin- 
 ning, a change is made through the whole. 
 
 1 1 1 
 1 III 
 1 III 
 
 1 1 1 
 
 111 I 
 II .11 
 i 1 i 
 
 III III 
 
 1 1 1 
 
 1 1 i 
 
 1 1 1 
 
 1 1 
 
 III III 
 
 i Mi 
 ill 1 
 
 I 1 1 
 
 1 III 
 1 III 
 
 1 1 
 
 III 1 
 1 1 1 
 
 II 11 
 11 III 
 
 1 1 1 
 
 1 1 
 
 1 1 1 
 
 1 1 III 
 III 1 
 
 
ARITHMETIC. 
 
 PART I. 
 
 SECTION I 
 
 A.* 1. How many thumbs have you on your right 
 hand? how many on your left? how many on both 
 together ? 
 
 2. How many hands have you? 
 
 3. If you have two nuts in one hand, and one in the 
 other, how many have you in both * 
 
 4. How many fingers have you on one hand ? 
 
 5. If you count the thumb with the fingers, how 
 many will it make ? 
 
 6. If you shut your* thumb and one finger, and leave 
 the rest open, how many will be open ? 
 
 - 7. If you have two cents in one hand, and two in 
 the other, how many have you in both ? 
 
 8. James has two apples, and William has three ; 
 if James gives his apples to William, how many will 
 William have ? 
 
 9. If you count all the fingers on one hand, and 
 two on the other, how many will there be ? 
 
 10. George has three dimes, and Joseph has four; 
 how many have they both together ? 
 
 11. Robert gave five cents for an orange, and two 
 for an apple ; how many did he give for both ? 
 
 12. If a custard costs six cents, and an apple two 
 cents, how many cents will it take to buy an apple and 
 a custard ? 
 
 13. If you buy a pint of nuts for five cents, and an 
 orange for three cents, how many cents would you give 
 
 * The first questions in this section are intended for very young 
 children. It will be well for the instruotor to give a great many 
 more of this kind. Older pupils may prqjt tjiese. 
 
 (9) 
 
10 ARITHMETIC. [part i. 
 
 for both ? how many more for the nuts than for the 
 orange ? 
 
 14. If an ounce of figs is worth six cents, and a half 
 a pint of cherries is worth three cents, how much are 
 they both worth ? 
 
 15. Joseph had five plums, and John gave him four 
 more ; how many had he then ? 
 
 16. How many fingers have you on both hands ? 
 
 17. How many fingers and thumbs have you on both 
 hands ? 
 
 18. If you had six marbles in one hand, and four in 
 the other, how many would you have in the one 
 more than in the other ? how many would you have 
 in both hands ? * 
 
 19. David had seven nuts, and gave three of them 
 to George ; how many had he left? 
 
 20. Two boys, James and Robert, played at marbles ', 
 when they began, they had seven apiece, and when 
 they had done, James had won four; how many had 
 each then ? 
 
 21. A boy, having eleven nuts, gave away three of 
 them ; how many had he left ? 
 
 22. If you had eight dimes and your papa should 
 give you five more, how many would you have ? 
 
 23. A man bought a sheep for eight dollars, and a 
 calf for seven dollars ; what did he give for both ? 
 
 24. A man bought a barrel of flour for eight dollars, 
 and sold it for four dollars more than he gave for it; 
 how much did he sell it for. 
 
 25. A man bought a hundred weight of sugar for 
 nine dollars, and a barrel of flour for seven dollars; 
 how much did he give for the whole ? 
 
 26. A man bought three barrels of cider for eight 
 dollars, and ten bushels of apples for nine dollars; 
 how much did he give for the whole ? 
 
 27. A man bought a firkin of butter for twelve 
 dollars, but, it being damaged, he sold it again for 
 eight dollars ; how much did he lose ? 
 
SEO. i.J ARITHMETIC. H 
 
 28. A man bought three sheep for fifteen dollars, 
 but could, not; sell them ^gain for so much by ei"-ht 
 dollars ; how much did he sell them for ? 
 
 29. A man bought sixteen pounds of coffee, and 
 lost seven pounds of it as he was carrying it home • 
 how much had he left? ' 
 
 30. A man bought nineteen pounds of sugar, and 
 having lost a part of it, he found he had nine pounds 
 left; how much had he lost? 
 
 31. A man, owing fifteen dollars, paid nine dollars 
 of it ; how much did he then owe ? 
 
 32. A man, owing seventeen dollars, paid all but 
 seven dollars ; how much did he pay ? 
 
 B. 1. Two and one are how many? 
 
 2. Two and two are how many ? 
 
 3. Three and two are how many ? 
 -4. Four and two are how many ? 
 
 6. Five and two are how many ? 
 
 6. Six and two are how many? 
 
 7. Seven and two are how many ? 
 
 8. Eight and two are how many ? 
 
 9. Nine and two are how many ? 
 
 10. Ten and two are how many ? 
 
 11. Two and three are how many? 
 
 12. Three and three are how many? 
 
 13. Four and three are how many ? 
 
 14. Five and three are how many ? 
 
 15. Six and three are how many ? 
 
 16. Seven and three are how many? 
 
 17. Eight and three are how many? 
 
 18. Nine and three are how many ? 
 
 19. Ten and three are how many ? 
 
 20. Two and four are how many ? 
 
 21. Three and four are how many ? 
 
 22. Four and four are how many ? 
 
 23. Five and four are how many ? 
 
 24. Six and four arc how many ? 
 
12 ARITHMETIC. [parti. 
 
 25. Seven and four arc how many ? 
 
 26. Eight and lour are how many? ■ 
 
 27. Nine and four are how many ? 
 
 28. Ten and four arc how many? 
 
 29. Two and five are how many ? 
 
 30. Three and five arc how many ? 
 
 31. Four and five are how many? 
 
 32. Five and five are how many ? 
 38. Six and five are how many? 
 
 34. Seven and five are how many ? 
 
 35. Eight and five are how many ? 
 
 36. Nine and five arc how many ? 
 
 37. Ten and five are hoAV many ? 
 
 38. Two and six are how many? 
 
 39. Three and six are how many? 
 
 40. Four and six are how many ? 
 
 41. Five and six are how many? 
 
 42. Six and six are how many ? 
 
 43. Seven and six are how many? 
 
 44. Eight and sik are how many? 
 
 45. Nine and six are how many ? 
 
 46. Ten and six are how many ? 
 
 47. Two and seven are how many ? 
 
 48. Three and seven are how many ? 
 
 49. Four and seven are how many? 
 
 50. Five and seven are how many ? 
 
 51. Six and seven are how many ? 
 
 52. Seven and seven are how many ? 
 
 53. Eight and seven are how many ? 
 
 54. Nine and seven are how many ? 
 
 55. Ten and seven are how many ? 
 
 56. Two and eight are how many ? 
 
 57. Three and eight are how many ? 
 
 58. Four and eight are how many? 
 
 59. Five and eight are how many? 
 
 60. Six and eight arc how many ? 
 
 61. Seven and eight are how many? 
 
 62. Eight and eight are how many ? 
 
SEC. i] ARITHMETIC. 13 
 
 63. Nine and eight are how many? 
 
 64. Ten and eight are how many ? 
 
 65. Two and nine are how many ? 
 
 66. Three and nine are how many ? 
 
 67. Four and nine are how many? 
 
 68. Five and nine are how many? 
 
 69. Six and nine are how many ? 
 
 70. Seven and nine are how many? 
 
 71. Eight and nine are how many? 
 
 72. Nine and nine are how many? 
 
 73. Ten and nine are how many ? 
 
 74. Two and ten are how many ? 
 
 75. Three and ten are how many ? 
 
 76. Four and ten are how many ? 
 
 77. Five and ten are how many ? 
 
 78. Six and ten are how many ? 
 ,79. Seven and ten are how many? 
 
 80. Eight and ten are how many ? 
 
 81. Nine and ten are how many ? 
 
 82. Ten and ten are how many ? 
 
 C. 1. Two and one are how many? 
 
 2. Two and two are how many ? 
 
 3. Three and two are how many ? 
 
 4. Five and two are how many ? 
 
 5. Four and two are how many ? 
 
 6. Six and two are how many ? 
 
 7. Eight and two are how many ? 
 
 8. Five and three are how many? 
 
 9. Seven and three are how many? 
 
 10. Four and three are how many? 
 
 11. Two and three are how many? 
 
 12. Two and six are how many ? 
 
 13. Two and eight are how many? 
 
 14. Six and three are how many ? 
 
 15. Three and four are how many ? 
 
 16. Three and six are how many? 
 
 17. Two and seven are how many? 
 
ARITHMETIC. [paut i. 
 
 18. Ten and two are how many ? 
 
 19. Two and four are how many ? 
 
 20. Three and seven are how many ? 
 
 21. Four and four are how many ? 
 
 22. Five and four are how many ? 
 
 23. Seven and two are how many ? 
 
 24. Two and five are how many ? 
 
 25. Three and three are how many ? 
 
 26. Four and five are how many ? 
 
 27. Nine and two are how many? 
 
 28. Three and five are how many? 
 
 29. Two and ten are how many ? 
 
 30. Three and eight are how many ? 
 
 31. Ten and three are how many? 
 
 32. Two and nine are how many ? 
 83. Four and six are how many ? 
 
 34. Eight and three are how many ? ^ 
 
 35. Seven and four are how many ? 
 
 36. Nine and three are how many ? 
 
 37. Six and four are how many ? 
 
 38. Five and five are how many ? 
 
 39. Three and nine are how many ? 
 
 40. Four and seven are how many ? 
 
 41. Six and five are how many ? 
 
 42. Three and ten are how many ? 
 
 43. Eight and four are how many ? 
 
 44. Five and eight are how many ? 
 
 45. Four and nine are how many ? 
 
 46. Five and six are how many ? 
 
 47. Ten and four are how many? 
 
 48. Seven and five are how many ? 
 
 49. Six and six are how many ? 
 
 50. Nine and four are how many ? 
 
 51. Eight and five are how many ? 
 
 52. Five and nine are how many ? 
 
 53. Four and ten are how many ? 
 
 54. Six and seven are how many ? 
 
 55. Four and eight are how many ? 
 
BKO. 1.] ARITHMETIC. ^- 16 
 
 56. Nine and five are how many ? 
 
 57. Six and eight are how many ? 
 
 58. Ten and five are how many ? 
 
 59. Seven and six are how many ? 
 
 60. Eight and seven are how many ? 
 
 61. Six and nine are how many ? 
 
 62. Seven and seven are how many ? 
 
 63. Eight and six are how many ? 
 
 64. Ten and six are how many ? 
 
 65. Eight and eight are how many ? 
 
 66. Nine and seven are how many? 
 
 67. Ten and eight are how many ? 
 
 68. Six and ten are how many ? 
 
 69. Five and seven are how many ? 
 
 70. Nine and six are how many ? 
 
 71. Seven and eight are how many ? • 
 
 72. Eight and nine are how many ? 
 
 73. Nine and nine are how many ? 
 71. Five and ten are how many ? 
 
 75. Seven and nine are how many ? 
 
 76. Nine and eight are how many ? 
 
 77. Eight and ten arc how many ? 
 
 78. Ten and nine are how many ? 
 
 79. Seven and ten are how many ? 
 
 80. Nine and ten are how many? 
 
 81. Ten and ten are how many ? 
 
 D. 1. Three boys, Peter, John, and Oliver, gave 
 some money to a beggar. Peter gave seven cents, John, 
 four cents, and Oliver three cents ; how many did they 
 all give him ? 
 
 2. How many did Peter give more than Oliver ? 
 
 3. Frank had nine pears, and gave three of them to 
 Harry ; how many had he left ; and how many more 
 than Harry had he then ? 
 
 4. John had ten peaches, Harry twelve, and Charles 
 thirteen; John gave three to Stephen, Harry gave 
 him six, and Charles gave him five ; how many had 
 Stephen ? and how many had each left ? 
 
16 ARITHMETIC. [part i. 
 
 5. A boy had twenty apples, and gave them to his 
 companions, as follows : to one he gave three ; to an- 
 other, two; to another four; and to another, five; how 
 many did he give away ? and how many had he left ? 
 
 6. A boy gave to one of his companions eight 
 peaches; to another, six; to another, four; and kept 
 two himself; how many had he at first? 
 
 7. A boy went to the confectioner's and bought 
 three cakes of gingerbread, for which he gave a cent 
 apiece ; two buns, for which he gave three cents 
 apiece; one custard for four cents, and one orange 
 for six cents ; how many cents did he spend for the 
 whole ? 
 
 8. A boy having twenty-five cents, bought one 
 quart of cherries for eight cents, one orange for six 
 cents, and gave away three cents; how many cents 
 had he left ? 
 
 9. A boy bought a box for eighteen cents, and gave 
 eight cents to have it painted, and then sold it for 
 thirty-two cents; how much did he gain by the 
 bargain ? 
 
 10. A man bought a sleigh for seventeen dollars, 
 and gave nine dollars to have it repaired and painted, 
 and then vsold it for twenty-three dollars ; how much 
 did he lose by the bargain r* 
 
 11. Eleven and two are how many? 
 
 12. Eleven and three arc how many ? 
 
 13. Eleven and four are how many? 
 
 14. Eleven and five are how many? 
 
 15. Eleven and six are how many? 
 
 16. Eleven and seven are how many? 
 
 17. Eleven and eight are how inany? 
 
 18. Eleven and nine arc how many? 
 
 19. Eleven and ten are how many? 
 
 20. Twelve and two are how many? 
 
 21. Twelve and three are how many? 
 
 22. Twelve and four are how many? 
 
fiBC. I.] AttlTHMETIC. 17 
 
 23. Twelve and five are how many? 
 
 24. Twelve and six arc how many? 
 
 25. Twelve and seven are how many? 
 
 26. Twelve and eight are how many? 
 
 27. Twelve and nine are how many? 
 
 28. Twelve and ten are how many? 
 
 29. Thirteen and two are how many? 
 
 30. Thirteen and three are how many? 
 
 31. Thirteen and four are how many? 
 
 32. Thirteen and five are how many? 
 
 33. Thirteen and six are how many? 
 
 34. Thirteen and seven are how many 2 
 
 35. Fourteen and two are how many? 
 
 36. Fourteen and three are how many? 
 
 37. Fourteen and four are how many? 
 
 38. Fourteen and five are how many? 
 
 39. Fourteen and six are how many ? 
 
 40. Fifteen and two are how man}'? 
 
 41. Fifteen and three are how many? 
 
 42. Fifteen and four are how many? 
 
 43. Fifteen and five are how many? 
 
 44. Sixteen and two are how many? 
 
 45. Sixteen and three are how many? 
 
 46. Sixteen and four are how many? 
 
 47. Seventeen and two are how many? 
 
 48. Seventeen and three are how many? 
 
 49. Eighteen and two are how many? 
 
 E. 1. A man hought a sheep for nine dollars, and 
 to pay for it he gave five bushels of corn worth four 
 dollars and the rest in money; how much money did 
 he pay? 
 
 2. If a barrel of flour is worth eight dollars, and a 
 hundred weight of sugar is worth twelve dollars; how 
 much more is the sugar worth than the flour? 
 
 3. If a man had eleven dollars, and should buy three 
 bushels of corn for five dollars; how much money would 
 he have left? 
 
18 ARITHMETIC. [part i. 
 
 4. A man bought a jBrkin of butter for fifteen dol- 
 lars, but it being damaged he was willing to sell it 
 again for eight dollars less than he gave for it; what 
 did he sell it for? 
 
 5. A man bought three barrels of flour for eighteen 
 dollars, and sold it again for eleven dollars; what did 
 he lose by the bargain? 
 
 6. A man bought a keg of tobacco for thirteen 
 dollars, and sold it again for eighteen; what did he 
 gain by the bargain ? 
 
 7. Five less two are how many ? 
 
 8. Seven less three are how many? 
 
 9. Three less three are how many? 
 
 10. Nine less three are how many? 
 
 11. Six less two are how many? 
 
 12. Seven less four are how many? 
 
 13. Eight less three are how many? 
 
 14. Five less four are how many? 
 
 15. Seven less five are how many? 
 
 16. Nine less five are how many? 
 
 17. Eight less six are how many? 
 
 18. Eleven less two are how many? 
 
 19. Twelve less four are how many? 
 
 20. Ten less seven are how many ? 
 
 21. Thirteen less five are how many? 
 
 22. Fourteen less eight are how many? 
 
 23. Twelve less seven are how many? 
 
 24. Seventeen less five are how many? 
 
 25. Eighteen less ten are how many ? 
 
 26. Thirteen less seven are how many? 
 
 27. Sixteen less seven are how many? 
 
 28. Fifteen less seven are how many? 
 
 29. Nineteen less six are how many? 
 
 30. Eighteen less five are how many? 
 
 31. Seventeen less eight are how many? 
 
 32. Fourteen less nine are how many? 
 
 33. Sixteen less five are how many? 
 
8EC. I.] ARITHMETIC. 19 
 
 3-4. Fifteen less eight are how many? 
 
 35. Fourteen less nine are how many? 
 
 36. Sixteen less ten are how many? r 
 ,^37. Seventeen less nine are how many? 
 
 38. Eighteen less seven are how many ? 
 
 F. 1. How many are nine and two? Nineteen and 
 tw(J? Twenty-nine and two? Thirty-nine and two? 
 Forty-nine and two? Fifty-nine and two? Sixty-nine 
 an^^ two? Seventy-nine and two? Eighty-nine and 
 two? Ninety-nine and two? 
 
 2. How many are nine and three ? Nineteen and 
 three? Twenty-nine and three? Thirty-nine and 
 three? Forty-nine and three? Fifty-nine and three? 
 Sixty-nine and three ? Seventy-nine and three ? 
 Eighty-nine and three? Ninety-nine and three? 
 
 8. How many are nine and four? Nineteen and four? 
 Twenty-nine and four? Thirty-nine and four? Forty- 
 nine and four? Fifty-nine and four? Sixty-nine and 
 four? Seventy-nine and four? Eighty-nine and four? 
 Ninety-nine and four? 
 
 4. How many are nine and five? Nineteen and five? 
 Twenty-nine and five? Thirty-nine and five? Forty- 
 nine and five? Fifty-nine and five? Sixty-nine and 
 five? Seventy-nine and five? Eighty-nine and five? 
 Ninety-nine and five? 
 
 5. How many are nine and six? Nineteen and six? 
 Twenty-nine and six? Thirty-nine and six? Forty-nine 
 and six? Fifty-nine and six? Sixty-nine and six? 
 Seventy-nine and six? Eighty-nine and six? Ninety- 
 nine and six? 
 
 6. How many are nine and seven? Nineteen and 
 seven ? Twenty-nine and seven? Thirty-nine and seven ? 
 Forty-nine and seven? Fifty-nine and seven? Sixty- 
 nine and seven ? Seventy-nine and seven ? Eighty-nine 
 and seven? Ninety-nine and seven? 
 
 7. How many are nine and eight? Nineteen and 
 eight? Twenty-nine iind eight? Thirty -nine and eight? 
 
20 ARITHMETIC. [part i. 
 
 Forty-nine and eight? Fifty-nine and eight? Sixty- 
 nine and eight? Seventy-nine and eight? Eighty-nine 
 and eight? Ninety-nine and eight? 
 
 8. How many are nine and nine? Nineteen a^d 
 nine? Twenty-nine and nine? Thirty-nine and nine? 
 Forty-nine and nine? Fifty-nine and nine? Sixty-nyie 
 and nine? Seventy-nine and nine? Eighty-nine and 
 nine? Ninety-nine and nine? 
 
 9. How many are nine and ten? Nineteen and ten? 
 Twenty-nine and ten ? Thirty-nine and ten ? Forty-n4ne 
 and ten? Fifty-nine and ten? Sixty-nine and ten? 
 Seventy-nine and ten? Eighty-nine and ten? Ninety- 
 nine and ten ? 
 
 . 10. How many are eight and three? Eighteen and 
 three? Twenty-eight and three? Thirty-eight and 
 three? Forty-eight and three? Fifty-eight and three? 
 Sixty-eight and three? Seventy-eight and three? 
 Eighty-eight and three ? Ninety-eight and three ? 
 
 11. How many are eight and four? Eighteen and 
 four? Twenty-eight and four? Thirty-eight and four? 
 Forty-eight and four ? Fifty-eight and four ? Sixty- 
 eight and four? Seventy-eight and four ? Eigkty-eight 
 and four ? Ninety-eight and four. 
 
 12. How many are eight and five ? Eighteen and 
 five? Twenty-eight and five? Thirty-eight and five? 
 Forty-eight and five ? Fifty-eight and five? Sixty- 
 eight and five ? Seventy-eight and five ? Eighty-eight 
 and five ? Ninety-eight and five ? 
 
 13. How many are eight and six ? Eighteen and 
 six? Twenty-eight and six? Thirty-eight and six? 
 Forty eight and six ? Fifty-eight and six ? Sixty-eight 
 and six? Seventy-eight and six? Eighty-eight and 
 six? Ninety-eight and six? 
 
 14. How many are eight and seven? Eighteen and 
 seven? Twenty-eight and seven? Thirty-eight and 
 seven? Forty-eight and seven? Fifty-eight and seven ? 
 Sixty-eight and seven? Seventy-eight and seven? 
 EightyTcight and seven ? Ninety-eight and seven ? 
 
SEC. I ] ARITHMETIC. 21 
 
 15. How many are eight and eiglit? Eighteen and 
 eio'ht? Twenty-eight and eight i* Thirty-eight and 
 eight? Forty-eight and eight? Fifty-eight and eight? 
 Sixty-eight and eight? Seventy -eight and eight? 
 Ninety-eight and eight? 
 
 16. How many are eight and nine? Eighteen and 
 nine? Twenty-eight and nine? Thirty-eight and nine? 
 Forty-eight and^nine? Fifty-eight and nine? Sixty- 
 eight and nine ? Seventy-eight and nine? Eighty-eight 
 and nine? Ninety eight and nine? 
 
 17. How many arc seven and four? Seventeen and 
 four? Twenty seven and four? Thirty-seven and four? 
 Forty-seven and four? Fifty-seven and four? Sixty- 
 seven and four? Seventy-seven and four? Eighty seven 
 and four? Ninety-seven and four? 
 
 18. How many are seven and five? Seventeen and 
 five? Twenty-seven and five? Thirty-seven and five? 
 Forty-seven and five? Fifty-seven and five? Sixty- 
 seven and five? Seventy-seven and five? Eighty-seven 
 and five? Ninety-seven and five? 
 
 19. How many are seven and six? Seventeen and 
 six? Twenty-seven and six? Thirty-seven and six? 
 Forty-seven and six? Fifty-seven and six? Sixty-seven 
 and six? Seventy-seven and six? Eighty-seven and 
 six? Ninety-seven and six? 
 
 20. How many are seven and seven ? Seventeen and 
 seven? Twenty-seven and seven? Thirty-seven and 
 seven ? Forty-seven and seven ? Fifty-seven and seven ? 
 Sixty-seven and seven? Seventy-seven and seven? 
 Eighty-seven and seven? Ninety-seven and seven ? 
 
 21. How many are seven and eight? Sev'enteen and 
 eight? Twenty-seven and eight? Thirty-seven and 
 eight? Forty-seven and eight? Fifty -seven and eight? 
 Sixty-seven and eight? Seventy-seven and eight? 
 Eighty-seven and eight? Ninety seven and eight? 
 
 22. How many are seven and nine? Seventeen and 
 nine? Twenty-seven and- nine? Thirty-seven and nine ? 
 Forty-seven and nine? '^ifty-seven and nine? Sixty- 
 
22 ARITHMETIC. [part i. 
 
 seven and nine? Seventy-seven and nine? Eighty- 
 seven and nine? Ninety-seven and nine? 
 
 23. How many are six and five? Sixteen and five? 
 Twenty-six and five? Thirty-six and five? Forty-six 
 and five? Fifty-six and five? Sixty-six and five? 
 Seventy -six and five? Eighty-six and five? Ninety-six 
 and five ? 
 
 24. How many are six and six? Sixteen and six? 
 Twenty-six and six? Thirty-six and six? Forty-six and 
 six? Fifty-six and six? Sixty-six and six ? Seventy-six 
 and six? Eighty -six and six? Ninety -six and six? 
 
 25. How many are six and seven? Sixteen and 
 seven? Twenty-six and seven? Thirty-six and seven? 
 Forty-six and seven? Fifty-six and seven? Sixty-six 
 and seven? Seventy -six and seven? Eighty-six and 
 seven ? Ninety-six and seven? 
 
 26. How many are six and eight? Sixteen and 
 eight? Twenty-six and eight? Thirty-six and eight? 
 Forty-six and eight? Fifty-six and eight? Sixty -six 
 and eight? Seventy-six and eight? Eighty-six and 
 eight? Ninety-six and eight? 
 
 27. How many are six and nine? Sixteen and nine ? 
 Twenty-six and nine? Thirty-six and nine? Forty-six 
 and nine? Fifty -six an'd nine? Sixty-six and nine? 
 Seventy-six and nine? Eighty-six and nine? Ninety- 
 six and nine? 
 
 28. How many are five and six? Fifteen and six? 
 Twenty-five and six? Thirty-five and six? Forty-five 
 and six? Fifty-five and six? Sixty-five and six? 
 Seventy-five and six? Eighty-five and six ? Ninety-five 
 and six? ''"■ 
 
 29. How many are five and seven? Fifteen and 
 seven ? Twenty-five and seven ? Thirty-five and seven ? 
 Forty-five and seven? Fifty-five and seven? Sixty-five 
 and seven? Seventy-five and seven? Eighty -five and 
 seven? Ninety-five and seven? 
 
 30. How many are five and eight? Fifteen and 
 eight? Twenty-five and eight? Thirty-five and. eight? 
 
8BC. I.] ^ ARITHMETIC. 2S 
 
 Forty-five and eight? Fifty-five and eight? Sixty-five 
 and eight? Seventy-five and eight? Eighty-five and 
 eight? Ninety-five and eight? 
 
 31. How many are five and nine? Fifteen and nine? 
 Twenty-five and nine? Thirty-five and nine? Forty- 
 five and nine? Fifty-five and nine? Sixty-five and nine? 
 Seventy-five and nine? Eighty-five and nine? Ninety- 
 five and nine? 
 
 32. How many are four and seven ? Fourteen and 
 seven? Twenty-four and seven? Thirty-four and 
 seven ? Forty-four and seven ? Fifty-four and seven ? 
 Sixty-four and seven ? Seventy-four and seven ? 
 Eighty-four and seven ? Ninety-four and seven ? 
 
 33. How many are four and eight? Fourteen and 
 eight? Twenty-four and eight? Thirty-four and eight? 
 Forty-four and eight? Fifty-four and eight? Sixty-four 
 and eight? Seventy-four and eight? Eighty-four and 
 eight? Ninety-four and eight ? 
 
 34. How many are four and nine? Fourteen and 
 nine? Twenty-four and nine ? Thirty-four and nine? 
 Forty-four and nine ? Fifty-four and nine ? Sixty-four 
 and nine? Seventy-four and nine? Eighty- four and 
 nine? Ninety-four and nine? 
 
 35. How many are three and eight? Thirteen and 
 eight? Twenty-three and eight? Thirty-three and 
 eight? Forty-three and eight? Fifty-three and eight.'* 
 Sixty-three and eight? Seventy-three and eight? 
 Eighty-three and eight? Ninety-three and eight? 
 
 36. How many are three and nine? Thirteen and 
 nine? Twenty-three and nine? Thirty -three and nine? 
 Forty-three and nine? Fifty-three and nine? Sixty- 
 three and nine? Seventy-three and nine? Eighty- 
 three and nine? Ninety-three and nine? 
 
 37. How many are two and nine? Twelve and 
 nine? Twenty-two and nine? Thirty -two and nine? 
 Forty-two and nine ? Fifty-two and nine ? Sixty-two 
 and nine? Seventy-two and nine? Eighty-two and 
 nine ? Ninety-two and nine ? 
 
24 ARITHMETIC. [parti. 
 
 Or. i. A man bought a firkin of butter for nine 
 dollars, a keg of molasses for six dollars, and five 
 bushels of wheat for seven dollars ; how much did he 
 give for the whole ? 
 
 2. A boy gave some apples to his companions, to one 
 he gave seven, to another six, and to another eight; 
 how many did he give to all ? 
 
 3. A man bought a cow for seventeen dollars, a sheep 
 for nine, and a calf for seven ; how much did he give 
 for the whole ? 
 
 4. A drover bought sheep as follows : of one man 
 he bought twenty-seven, of another eight, of another 
 ten, and of another five; afterwards he sold nine of 
 them ; how many had he then ? 
 
 5. A lady bought a comb for thirty-seven cents, some 
 tape for eight cents, some pins for ten cents, some 
 needles for six cents, and some thread for six cents ; 
 she gave seventy-five cents ; how much change ought 
 she to receive back ? 
 
 6. Eight, and nine, and six, are how many ? 
 
 7. Five, and seven, and three, are how many? 
 
 8. Four, and three, less two, are how many ? 
 
 9. Seven, and five, less three, are how many ? 
 
 10. Sixteen, and nine, and three, are how many ? 
 
 11. Twenty-three and eight are how many ? 
 
 12. Twenty-seven and five are how many ? 
 
 13. Twenty-five, less eight, are how many ? 
 
 14. Thirty- two and seven, less nine, are how many ? 
 
 15. Thirty-eight, and six, and four, less seven, are 
 how many ? 
 
 16. Forty-four, and eight, and three, and seven, are 
 how many ? 
 
 17. Fifty-two, and six, and four, and fi,ve, and three, 
 are how many ? 
 
 18. Fifty-seven, and six, and three, and five, and 
 two, less eight, are how many ? 
 
 19. Sixty-three, and five, and four, and six, and two, 
 less seven, are how many ? 
 
8KC. I.] ARITHMETIC. 26 
 
 20. Seventy-five, and six, and eight, and three, and 
 se\en, and four, less nine, are how many? 
 
 21. Eighty-three, and six, and five, and two, and 
 seven, and nine, less four, are how many ? 
 
 22. Fifty-eight, and ten, and five, and seven, and 
 three, and six, and four, less nine, are how many ? 
 
 23. Sixty-seven, and five, and eight, and nine, and 
 seven, less six, are how many? 
 
 24. Seventy-four, and nine, and seven, and five, and 
 two, less six, are how many ? 
 
 25. Seventy-eight, and seven, and six, and two, and 
 five, and eight, less nine, are how many ? 
 
 26. Eighty-four, and seven, and six, and eight, and 
 five, less ten, are how many? 
 
 27. Forty-seven, and eight, and six, and two, and 
 four, and eight, and three, and seven, and ten, and nine, 
 less five, are how many ? 
 
 28. Thirty-five, and eight, and four, and ^ix, and 
 three, and four, less eleven, are how many ? 
 
 29. Seventy, and ten, and six, and nine, and seven, 
 and two, and five, and eight, and nine, less three, are 
 how many ? 
 
 H. 1. A man bought a cow for twenty-eight dollars, 
 and a sheep for four dollars, and a pig for seven dollars ; 
 how much did he give for the whole ? 
 
 2. James had twenty-seven cents; John gave him 
 four more, David seven, and George eleven, and he 
 bought nine cents worth of cake ; how many cents had 
 he left? 
 
 3. A man paid sixteen dollars to A, nine dollars to 
 B, seven dollars to 0, ten dollars to D, six dollars to E, 
 four dollars to F, and had eight dollars left ; how many 
 had he at first ? 
 
 4. From Brown's to Black's it is three miles ; from 
 Black's to Blue's, six miles ; from Blue's to Green's, 
 eleven miles ', from Green's to Violet's, four miles ; 
 from Violet's to Olive's, four miles ; from Olive'a 
 
26 ARITHMETIC. [part i. 
 
 to Purple's, nine miles; from Purple's to White's, 
 four miles; how many miles is it from Brown's to 
 White's ? 
 
 5. One boy had fifteen nuts; another boy gave him 
 seven; another, nine; and another gave him enough 
 to make his number forty; how many did the last boy 
 give him ? 
 
 6. A boy had thirty-seven apples ; he gave five to 
 one companion; and eight to another; and when he 
 had given some to another, he had six left; how many 
 did he give to the last ? 
 
 7. A man owed fifty-six dollars ; at one time he paid 
 seventeen dollars; at another, eight; at another, five; 
 at another, seven; at last he paid the rest of the debt, 
 wanting four dollars ; how uiuch was the last payment ? 
 
 8. Six men bought a horse for seventy dollars ; the 
 first gave twenty-three dollars ; the second, fifteen ; the 
 third, twelve ; the fourth, nine ; the fifth, seven ; how 
 much did the sixth give ? 
 
 9. A man bought a horse for forty-five dollars and 
 paid fifteen dollars for keeping him ; he let him enough 
 to receive twenty dollars ; and then sold him for forty- 
 three dollars; did he gain or lose by the bargain ? and 
 how much ? 
 
 SECTION II. 
 
 A. 1. What cost three yards of tape, at two cents a 
 yard ? * 
 
 2. What cost four apples, at two cents apiece ? ' 
 
 3. What cost five peaches, at three cents apiece? 
 
 4. What must you give for two oranges, at six cents 
 apiece ? 
 
 ♦The pupil should be mado to observe that three yards will cost 
 three times as much as one yard ; and say, if one yard costs two cents, 
 three yards will cost three times two cents. He should be made to 
 jfive this reason for the solution of each question, varying tlie nunibera 
 according to the question. 
 
SEC. II.] ARITHMETIC. 27 
 
 5. What would be the price of three barrels of cider, 
 at three dollars a barrel ? 
 
 6. If one orange is worth three apples, how many 
 apples are four oranges worth ? 
 
 7. What are two barrels of flour worth, at five dollars 
 a barrel ? 
 
 8. What cost three yards of cloth, at four dollars a 
 yard ? 
 
 9. What cost two pounds of raisins, at eight cents a 
 pound ? 
 
 10. What cost three lemons, at six cents apiece ? 
 
 11. If a man travels three miles in an hour, how many 
 miles will he travel in four hours ? 
 
 12. What will five pair of shoes come to at two dol- 
 lars a pair ? 
 
 13. What is the price of seven yards of cloth, at 
 three dollars a yard ? 
 
 14. What is the value of two pounds of beef, at 
 seven cents a pound ? 
 
 15. If there are three feet in one yard, how many 
 feet are there in four yards? 
 
 16. How many feet are there in seven yards ? 
 
 17. How many feet are there in six yards and two 
 feet ? 
 
 18. If a man earns seven dollars in one week, how 
 much would he earn in five weeks ? 
 
 19. What cost seven hundred weight of sugar, at nine 
 dollars a hundred weight? 
 
 20. What cost seven pounds of sugar, at ten cents a 
 pound ? 
 
 21. If one half yard of cloth costs three dollars, what 
 would three yards cost ? 
 
 22. If one quarter of a yard of cloth costs two dol- 
 lars, what is that a yard? 
 
 23. How many yards of cloth are there in seven 
 pieces, each piece containing ten yards ? 
 
 24. What will five barrels of flour cost, at six dol- 
 lars a barrel ? 
 
28 ARITHMETIC. [part i. 
 
 25. Jf a man can travel four miles in an hour, how 
 
 far can he travel in eight hours ? 
 
 26. If it takes four bushels of wheat to make a bar- 
 rel of flour, how many bushels will it take to make 
 seven barrels ? 
 
 B. 1. Two times one are how many?* 
 2. Two times two are how many? 
 8. Two times three are how many? 
 4. Two times four are how many ? 
 6. Two times five are how many ? 
 
 6. Two times six are how many ? 
 
 7. Two times seven are how many ? 
 
 8. Two times eight are how many ? 
 
 9. Two times nine are how man}' ? 
 
 10. Two times ten are how many ? 
 
 11. Three times one are how many? 
 
 12. Three times two are how many ? 
 
 13. Three times three are how many? 
 
 14. Three times four are how many ? 
 
 15. Three times five are how many ? 
 
 16. Three time's six are how many ? 
 
 17. Three times seven are how many? 
 
 18. Three times eight are how many ? 
 
 19. Three times nine are how many? 
 
 20. Three times ten are how many? 
 
 21. Four times one are how many ? 
 
 22. Four times two are how many ? 
 
 23. Four times three are how many? 
 
 24. Four times four are how many? 
 
 25. Four times five are how many ? 
 
 26. Four times six are how many ? . 
 
 27. Four times seven are how many ? 
 
 28. Four times eight are how many? 
 
 29. Four times nine are how many ? 
 80. Four times ten are how many ? 
 
 * See the Key. 
 
SEC. ii.J ARITHMETIC. 29 
 
 • 31. Five times one are bow many ? 
 
 32. Five times two are how many ? 
 
 33. Five times three are how many ? 
 
 34. Five times four are how many ? 
 
 35. Five times five are how many ? 
 
 36. Five times six are how many? 
 
 37. Five times seven are how many ? 
 
 38. Five times eight are how many ? 
 
 39. Five times nine are" how many ? 
 
 40. Five times ten are how many 't 
 
 41. Six times one are how many ? 
 
 42. Six times two arc how many ? 
 
 43. Six times three are how many ? 
 
 44. Six times four are how many ? 
 
 45. Six times five are how many ? 
 
 46. Six times six are how many ? 
 
 47. Six times seven are how many ? 
 
 48. Six times eight are how many ? 
 
 49. Six times nine are how many? 
 
 50. Six times ten are how many ? 
 
 51. Seven times one are how many ? 
 
 52. Seven times two are how many ? 
 
 53. Seven times three are how many? 
 
 54. Seven times four are how many ? 
 
 55. Seven times five are how many ? 
 
 56. Seven times six are how many ? 
 
 57. Seven times seven are how many ? 
 
 58. Seven times eight are how many ? 
 
 59. Seven times nine are how many ? 
 
 60. Seven times ten are how many ? • 
 
 61. Eight times one are how many? 
 
 62. Eight times two are how many ? 
 
 63. Eight times three arc how many? 
 
 64. Eight times four are how many? 
 
 65. Eight times five are how many ? 
 
 66. Eight times six are how many? 
 
 67. Eight times seven are how many ? 
 
 68. Eight times eight are how many ? 
 
80 ARITHMETIC. [pakt'i. 
 
 69. Eight times nine are how many ? g 
 
 70. Eight times ten are how many ? 
 
 71. Nine times one are how many? . 
 
 72. Nine times two are how many? < 
 
 73. Nine times three are how many? ; 
 
 74. Nine times four are how many? : 
 
 75. Nine times five are how many? 
 7G. Nine times six are how many? 
 
 77. Nine times seven are how many? 
 
 78. Nine times eight are how many? : 
 
 79. Nine times nine are how many? 
 
 80. Nine times ten are how many? 
 
 81. Ten times one are how many? 
 
 82. Ten times two are how many? 
 
 83. Ten times three are how many? 
 
 84. Ten times four are how many? 
 
 85. Ten times five are how many? 
 
 86. Ten times six are how many? 
 
 87. Ten times seven are how many? 
 
 88. Ten times eight are how many? 
 
 89. Ten times nine are how many? 
 
 90. Ten times ten are how many? 
 
 C. 1. Two times two are how many times one? "'• 
 
 2. Three times two are how many times one? 
 
 3. Four times two are how many times one? '* 
 
 4. Five times two are how many times one? 
 
 5. Seven times two are how many? 
 
 6. Nine times two are how many ? 
 
 7. Six times two are how many? 
 
 8. Eight times two are how many? 
 
 9. Ten times two are how many? 
 
 10. Two times three are how many? 
 
 11. Three times three are how many? 
 
 12. Four times three are how many? • 
 
 13. Five times three are how many? 
 
 14. Six times three are how many? 
 
 15. Eight times three are how many? 
 
SEC. II.] ARITHMETIC. 81 
 
 16. Seven times three are how many? 
 
 17. Ten times three are how many? 
 
 18. Nine times three are how many? 
 
 19. Two times four are how many? 
 
 20. Six times four are how many? 
 
 21. Four times four are how many? 
 
 22. Seven times four are how many? 
 « 28, Nine times four are how many? 
 
 24. Three times four are how many? 
 
 25. Five times four are how many? 
 
 26. Ten times four are how many? 
 
 27. .Eight times four are how many? 
 
 28. Two times five are how many? 
 
 29. Five times five are how many? 
 
 30. Three times five are how many? 
 
 31. Six times five are how many? 
 
 32. Two times six are how many? 
 
 33. Four times five are how many? 
 
 34. Seven times five are how many? 
 
 35. Three times six are how many? 
 
 36. Seven times six are how many? 
 
 37. Seven times seven are how many? 
 
 38. Four times eight are how many? 
 39,. Six times seven are how many? 
 
 40. Eight times nine are how many? 
 
 41. Six times eight are how many? 
 
 42. Three times seven are how many? 
 
 43. Four times nine are how many? 
 
 44. Three times eight are how many? 
 
 45. Six times six are how many? 
 
 46. Six times nine are how many? 
 
 47. Nine times five arc how many? 
 
 48. Four times six are how many? 
 
 49. Two times nine are how many? 
 
 50. Seven times nine are how many? 
 
 51. Nine times eight are how many? 
 62. Two times eight are how many? 
 53. Three times ten are how many? 
 
82 ARITHMETIC. [pabt i. 
 
 64. Eight times seven are how many? 
 
 55. Five times six are how many? 
 
 56. Five times eight arc how many? 
 
 57. Two times seven are how many? 
 
 58. Two times six arc how many? 
 
 59. Eight times six are how many? 
 
 60. Four times seven are how many? 
 
 61. Eight times eight are how many? * 
 
 62. Ten times five are how many? 
 
 63. Seven times ten arc how many? 
 
 64. Ten times ten are how many? 
 
 65. Nine times six are how many? 
 
 66. Five times nine are how many? 
 
 67. Three times nine are how many ? 
 
 68. Nine times seven are how many? 
 
 69. Five times ten are how many? 
 
 70. Seven times eight are how many? 
 
 71. Five times seven are how many? 
 
 72. Ten times eight are how many ? 
 
 73. Ten times seven are how many? 
 
 74. Nine times ten are how many? 
 
 75. Eight times five are how many? 
 
 76. Nine times nine are how many? 
 
 77. Four times ten are how many? , 
 
 78. Ten times six are how many? 
 
 79. Eight times ten are how many? 
 
 80. Ten times nine are how many? 
 
 D. 1. What cost three yards of cloth, at five dollars 
 a yard ? 
 
 2. What cost four oranges, at six cents apiece? 
 
 8. What cost seven barrels of cider, at three dollars 
 a barrel ? 
 
 4. How much do three barrels of beer come to, at 
 seven dollars a barrel? 
 
 5. What cost four firkins of butter, at eight dollars 
 a firkin ? 
 
 6. What do nine pounds of veal come to, at six 
 cents a pound ? 
 
8E0. II.] ARITHMETIC. 83 
 
 7. What cost six reams of paper, at five dollars per 
 ream ? 
 
 S. What cost eight pairs of shoes, at three dollars a 
 pairi* 
 
 9. What is the value of nine yards of cloth, at six 
 dollars a yard ? 
 
 10. If a man travels five miles an hour, how many 
 miles will he travel in nine hours? 
 
 11. There is an orchard consistini; of ten rows of 
 trees, and nine trees in each row; how many trees are 
 there in the orchard? 
 
 12. On a chess-board there are eight rows of squares, 
 and eight squares in each row; how many squares are 
 there on the board ? 
 
 13. In one penny there are four farthings; how 
 many farthings are there in six pence? 
 
 1-1. How many farthings are there in eight pence? 
 
 15. IIow many farthings are there in nine pence ? 
 
 16. How many farthings are there in ten pence? 
 
 17. In one siiilling there are twelve pence; how 
 many farthings are there in a shilling? 
 
 IS. In one pint, there are four gills; how many gills 
 are there in five pints? 
 
 19. In one quart, there are two pints; how many 
 pints are there in six quarts? 
 
 20. How many pints are there in three quarts? 
 
 21. How many gills are there in six pints? 
 
 22. How many gills are there in one quart? 
 
 23. How many gills are there in three quarts? 
 
 24. In one gallon there are four quarts; how many 
 quarts are there in three gallons ? 
 
 25. How many quarts are there in five gallons? 
 
 26. How many quarts are there in seven gallons ? 
 
 27. How many pints are there in one gallon? 
 
 28. How many pints are there in three gallons? 
 
 29. How many gills are there in one gallon ? 
 
 30. How many gills are there in five quarts ? 
 
 31. How many gills are there in two gallons ? 
 
 2 
 
84 ARITHMETIC. [pakt i. 
 
 32. A person bought two oranges, at six cents 
 apiece; and seven lemons, at four cents apiece; and 
 five pears, at two cents apiece; how much did the 
 whole come to ? 
 
 33. If one pint of oil costs eight cents; what will one 
 quart cost? 
 
 3-4. If one gill of alcohol costs four cents, what will 
 one quart cost? 
 
 35. If one gill of beer costs two cents, what will one 
 gallon cost ? 
 
 36. If a stage runs seven miles in an hour^ how far 
 will it run in nine hours? 
 
 37. Two men start from the same place and travel 
 different ways ; one travels two miles in an hour ; the 
 other travels three miles in an hour ; how far apart 
 will they be at the end of one hour ? How far at the 
 end of two hours? How far at the end of three 
 hours ? How far at the end of four hours? 
 
 38. Two men start from the same place and travel 
 the same way; one travels at the rate of two miles 
 in an hour; the other, four; how far apart will they 
 be in one hour? How far in two hours ? How far in 
 four hours? 
 
 39. If three men can do a piece of work in two 
 days, how many days would it take one man to do it? 
 
 40. If four men can do a piece of work in five days, 
 how many days would it take one man to do it ? 
 
 41. If six men can do a piece of work in seven 
 days, how many men would it take to do it in one 
 day? 
 
 42. If a quantity of provisions will serve three men 
 five days, how many men would it serve one day ? 
 
 43. If a quantity of provisions will serve five men 
 seven days, how many days would it serve one man ? 
 
 44. If fifteen dollars' worth of provisions will serve 
 eight men five days, how many days will it serve one 
 man ? 
 
 45. A man had a piece of work to perform which 
 
SEC. III.] ARITHMETIC. 35 
 
 seven men could do in nine days, but it was necessary 
 that the whole should be completed in one day; how 
 many men must he employ? 
 
 4G. If the interest of one dollar is six cents a year, 
 what would be the interest of tea dollars for the same 
 time ? 
 
 47. If the interest of one dollar is six cents for one 
 year, what would be the interest of it for two years? 
 for three years? for seven years? 
 
 48. If a man can earn seven shillings in a day, how 
 many shillings will he earn in six days? 
 
 49. If a man can earn eight dollars in a month, how 
 much can he earn in six months? 
 
 50. At five dollars a week, what will nine weeks' 
 board come to? 
 
 51. A lady bought three yards of cambric at two 
 dollars a yard, seven yards of silk for three dollars a 
 yard, five yards of ribbon for four dollars, and some 
 crape for two dollars; she paid four ten-dollar bills; 
 how much must she receive back again ? 
 
 SECTION III. 
 
 A. 1. How many apples, at one cent apiece, can 
 you buy for four cents? 
 
 2. How many pears, at two cents apiece, can you 
 buy for four cents? 
 
 o. How many peaches, at three cents apiece, can 
 you buy for six cents ? 
 
 4. How many apples, at two cents apiece, can you 
 buy for six cents? How many for eight cents? 
 
 5. How many pears, at three cents apiece, can you 
 buy for nine cents ? How many for twelve cents ? 
 
 6. If you have eight apples to give to four boys, 
 how many can you give to each ? 
 
 7 If a man travels six miles in two hours, how many 
 miles does he travel in an hour? 
 
 8. If a man travels three miles in an hour, how many 
 bours will it take him to travel nine miles ? 
 
36 ARITHMETIC. [part i. 
 
 9. How many yards of cloth, at three dollars a yard, 
 can you buy for fifteen dollars ? 
 
 10. If you had sixteen cents, how many cakes could 
 you buy at four cents apiece ? 
 
 11. If you had ten dollars, how much cloth could 
 you buy at five dollars a yard ? 
 
 12. If you had twelve apples to give to six of your 
 companions, how many would you give them apiece ? 
 
 13. If a man can travel six miles in an hour, how 
 long will it take him to travel eighteen miles ? 
 
 14. If a man can travel five miles in an hour, how 
 long will it take him to travel twenty miles ? 
 
 15. In a certain orchard there are twenty-four trees 
 standing in rows, and there are three trees in each 
 row ; how many rows are there ? 
 
 16. In an orchard there are twenty-one trees, and 
 there are seven trees in each row; how many rows are 
 there ? 
 
 17. A man paid twenty-seven dollars for some sheep, 
 and he gave nine dollars apiece for them; how many 
 sheep did he buy? 
 
 18. A man paid twenty -eight dollars for seven bar- 
 rels of cider; how much did he give a barrel ? 
 
 19. At five ceuts a piece, how many oranges can 
 you buy for thirty cents? 
 
 20. Twenty-five are how maYiy times five ? 
 
 21. Thirty-two are how many times four ? How 
 many times eight ? 
 
 22. Thirty-five are how many times seven ? How 
 many times five? 
 
 23. Thirty-six are how many times six? How many 
 times nine ? How many times four ? • 
 
 B. Remarh. When any thing, or any number, is 
 divided into two equal parts, one of the parts is called 
 the half of the thing or the number. 
 
 1. If an apple is worth two cents, what is one half 
 of it worth ? 
 
SKC. Ill] ARlTiniETIC. '37 
 
 2. AVhat is one half .of two cents? 
 Ans. One cent. 
 
 ^Q. Why? 
 
 A. Because, if you divide two cents into two equal 
 parts, one of the parts is one oent. 
 
 3. If you can buy a cake for two cents, how much 
 can you buy for one cent ? 
 
 4. One is whht part of two? 
 Ans. One is the half part of two. 
 
 5. Two times one are how many times two? 
 
 G. If you can buy one pear for two cents, how many 
 can you buy for three cents? 
 
 7. Three are how many times two? 
 Ans. Once two and" one kalf of two. 
 
 8. Four arc how many times two? 
 
 9. If two shillinfTS will buy one yard of cotton cloth, 
 how many yards will five shillings buy ? 
 
 10. Five are how many times two ? 
 Ans, Two times two and half of two. 
 
 11. Six are how many times two? 
 
 12. If two dollars will buy a yard of cloth, how 
 many yards will seven dollars buy ? 
 
 lo. How many halves make a whole one? 
 
 14. Eight arc how many times two ? 
 
 15. Nine arc how many times two? 
 
 16. Ten are how many times two? 
 
 Remarks When any thing, or any number, is 
 divided into three equal parts, one of those parts is 
 called the third part of the thing or number. When 
 it is divided into four equal parts, one part is called the 
 fourth part, and so on. 
 
 17. If a yard of cloth be worth three dollars, and 
 it be cut into three equal pieces, what will one of the 
 pieces be worth ? that is, what will one third of a yard 
 be worth I 
 
 18. What is a third of three? 
 
 19. Suppose the yard of cloth to be cut as before, 
 what will two pieces of it cost ? that is, what will two 
 thirds of a yard cost ? 
 
38 ARITHMETIC. [part i. 
 
 20. What is two thirds of three ? 
 
 21. If three shillings will buy one bushel of corn, 
 what part of a bushel will one shilling buy? What 
 part of a bushel will two shillings buy? 
 
 22. One is what part of three ? 
 
 Ans. One is the third part of three; or one third 
 of three. 
 
 23. Two is what part of tlu'ee ? 
 
 A71S. Two is two times the third part of three, or 
 two thirds of three.* 
 
 24. Three times one are how many times three ? 
 
 25. If you can buy a barrel of cider for three dol- 
 lars, how much can you buy for four dollars ? How 
 much for five dollars ? 
 
 26. How many thirds make a whole one? 
 
 27. Five are how many times three ? 
 Ans. One three, and two thirds of three. 
 
 28. Six are how many times three ? 
 
 29. If you can buy a barrel of fish for three dollars, 
 how much can you buy for seven dollars ? How much 
 for eight dollars ? 
 
 30. What do you understand by a third, and by 
 two thirds of any thing ? 
 
 For the answer, see remark after example 16th. 
 
 31. Eight are how many times three? 
 
 32. Nine are how many times three ? 
 
 33. Ten are how many times three? 
 
 34. Eleven are how many times three? 
 
 35. Twelve are how many times three? 
 
 36. If a yard of cloth be worth four dollars, and it 
 be cut into four equal parts, what will one of the parts 
 be worth? that is, what is one fourth of it worth? 
 What are two fourths of it worth? What are three 
 fourths of it worth? 
 
 37. If you can buy a barrel of cider for four dol- 
 
 * The instructor may use either ofthe.se expressions; the former 
 will be most easily understood by the pupil : it would be well, there- 
 fore, for the instructor to use tJie "former frequently, though the latter 
 is used for the most part in this treatise. 
 
SEC. III.] ARITHMETIC. 39 
 
 lars, how mucli can you buy for one dollar? How 
 much for two dollars? How much for three dollars? 
 
 38. What part of four is one? 
 Ans. One is the fourth part of fgur. 
 
 39. What part of four is two ? 
 Alls. Two fourths of four. 
 
 40. What part of four is three? 
 Ans. Three fourths of four. 
 
 41. How many fourths make a whole one? 
 
 42. If you can buy a bushel of corn for four shil- 
 lings, how much can you buy for five shillings? 
 How much for six shillings? How much for seven 
 shillings ? 
 
 43. Five are how many times four ? 
 Ans. Once four, and one fourth of four. 
 
 44. Six are how many times four ? 
 
 Ans. Once four, and two fourths of four. 
 
 45. Seven are how many times four? 
 Ans. Once four, and three fourths of four. 
 
 46. Eight are how many times four ? 
 
 47. If four bushels of corn will buy one yard of 
 cloth, how many yards will nine bushels buy ? How 
 many yards will ten bushels buy ? How many yards 
 will eleven bushels buy ? 
 
 48. What do you understand by one fourth, two 
 fourths, or three fourths of any thing ? 
 
 See remark after example 16th. 
 
 49. Ten are how many times four ? 
 
 50. Eleven are how many times four ? 
 
 51. Twelve are how many times four? 
 
 52. -Thirteen are how many times four ? 
 
 53. Fourteen are how many times four? 
 
 54. Fifteen are how many times four ? 
 
 55. Sixteen are how many times four ? 
 
 56. If a barrel of flour be worth five dollars, and it 
 be divided equally among five men, what will one man's 
 share be worth ? that is, what is one. fifth of a barrel 
 worth ? What are two fifths of it worth ? What are 
 
40 ARITHMETIC. [part t. 
 
 three fifths of it worth ? What are four fifths of it 
 worth? 
 
 57. If five dollars will buy one box of butter, what 
 part of a box will one dollar buy ? What part will two 
 dollars buy ? What part will three dollars buy ? What 
 part will four dollars buy ? 
 
 58. What part of five is one ? 
 Alls. One is the fifth part of five. 
 
 59. Two is what part of five ? 
 
 A71S. Two fifths of five. * 
 
 60. Three is what part of five ? 
 Ans. Three fifths of five. 
 
 61. Four is what part of five ? 
 
 6'2. How many fifths make a whole one ? 
 
 63. If cherries are five cents a quart, how many quarts 
 can you buy for six cents ? How many for seveu cents ? 
 How many for eight cents ? How many for nine cents ?• 
 How many for eleven cents ? How many for thirteen 
 cents ? 
 
 64. What do you understand by one fifth, two fifths, 
 etc., of any thing? 
 
 See remark after example 16th. 
 
 65. Seven are how many times five ? 
 A71S. Once five and two fifths of five. 
 
 66. Eight are how many times five ? 
 
 67. Nine are how many times five? 
 
 68. Ten are how many times five ? • 
 
 69. Eleven are how many times five ? 
 
 70. Twelve are how many times five 'f 
 
 71. Thirteen are how many times five ? 
 
 72. Fourteen are how many times five ? 
 78.. Fifteen are how many times five ? 
 
 74. If a barrel of beef cost six dollars, and it were 
 divided into six equal parts, what would one of the 
 parts be worth ? that is, what is one sixth of it worth ? 
 What are two sixths of it worth ? What are three 
 sixths of it worth? Four sixths? Five sixths? 
 
 75. If fish is Worth six dollars a barrel, what part of 
 
SEC. III.] ARITHMETIC. 41 
 
 a barrel will one dollar buy ? What part of a barrel 
 will two dollars buy? Three dollars? Four dollars? 
 Five dollars ? 
 
 76. What part of six is one ? 
 Ans. One is the sixth part of six. 
 
 77. Wliat part of six is two ? 
 Ans. Two sixths of six. 
 
 78. Three is what part of six ? 
 Ans. Three sixths of six. 
 
 79. Four is what part of six? 
 
 80. How many sixths make a whole one ? 
 
 81. How mucli rye at six shillings a bushel can you 
 buy for sevea shillings ? How much for eight shil- 
 lings ? Nine shillings ? Ten shillings ? Eleven shil- 
 lings ? Twelve shillings? Thirteen shillings? Fif- 
 teen shillings? Seventeen shillings? 
 
 82. What do j-ou understand by one sixth, two sixths, 
 etc.? 
 
 83. Eight are how many times six ? 
 Ans. One time six and tw(3 sixths of six ? 
 
 84. Nine are how many times six ? 
 
 85. Ten are how many times six ? 
 
 86. Eleven are how many tijues six ? 
 
 87. Twelve are how many times six? 
 
 88. Thirteen are how many times six ? 
 
 89. Fourteen arc how many times six ? 
 00. Fifteen are how many times six ? 
 
 91. If coal is wqi'th seven dollars a chaldron, what 
 is one seventh of a chaldron worth? What are two 
 sevenths of a chaldron worth ? Three sevenths ? Four 
 sevenths ? Five sevenths'? Six sevenths ? 
 
 92. At the rate of seven dollars a yard, how much 
 broadcloth can you buy for one dollar ? How much for 
 two dollars ? How much for three dollars ? How much 
 for four dollars ? How much for five dollars ? How 
 much for six dollars? How much for eight dollars? 
 How much for ten dollars ? How much for twelve dol- 
 lars ? How much for fifteen dollars ? 
 
42 ARITHMETIC. [parti. 
 
 93. "What part of seven is one ? 
 Ans. One is one seventh of seven. 
 
 94. What part of seven is two? 
 Ans. Two se-venths of seven. . 
 
 95. What part of seven is three? 
 
 96. Four is what part of seven? 
 
 97. Five is what part of seven ? 
 
 98. What do you understand by one seventh, two 
 sevenths, etc., of any thing ? 
 
 99. How many sevenths make a whole one ? 
 
 100. Nine are how many times seyen ? 
 
 101. Ten are how many times seven ? 
 
 102. Eleven are how many times seven ? 
 
 103. Twelve are how many times seven ? 
 
 104. Thirteen are how many times seven ? 
 
 105. Fourteen are how many times seven ? 
 
 106. Fifteen are how many times seven ? 
 
 107. Sixteen are how many times seven? 
 
 108. When wheat is eight shillings a bushel, what is 
 one eighth of a bushel worth ? What are two eighths 
 of a bushel worth ? What are three eighths of a bushel 
 worth ? What are four eighths o.f a bushel worth ? 
 Five eighths ? Six eighths ? Seven eighths ? 
 
 109. When wood is eight dollars a cord, what part 
 of a cord can you buy for a dollar ? What part of a 
 cord can you buy for two dollars ? What part for three 
 dollars ? What part for four dollars ? What part for 
 five dollars ? What part for six dollars ? What part 
 for seven dollars? How much can you buy for nine 
 dollars ? How much for ten dollars ? How much for 
 eleven dollars ? How much for thirteen dollars ? How 
 much for fifteen dollars ? How much for nineteen dol- 
 lars? 
 
 110. What part of eight is one? 
 
 111. What part of eight is two? 
 
 112. Three is what part of eight? 
 
 113. Four is what part of eight? 
 
 114. Five is what part of eight? 
 
SEC.iii.] ARITHMETIC. 43 
 
 115. What do jou understand by one eighth, two 
 eighths, etc., of any number ? 
 
 IIG. Seven is what part of eight? 
 
 117. How many eighths make a whole one? 
 
 118. Ten are how many times eight? 
 
 119. Eleven are how many times eight? 
 
 120. Twelve are how many times ei^ht? 
 
 121. Thirteen are how many times eight? '' 
 
 122. Fourteen are how many times eight? 
 
 123. When sugar is nine dollars a hundred weight, 
 what IS one ninth of a hundred weight worth? What 
 are two ninths of a hundred weight worth? Three 
 ninths?. Four ninths ? Five ninths ? Six ninths^ 
 Seventh ninths? Eight ninths? 
 
 ^124. When rye is nine shillings a bushel, what part 
 ot a bushel can you buy for one shilling? What part 
 for two shillings? What part for three shillinog? For 
 four shillings ? For five shillings ? For six shillino-s f 
 Seven shillings? Eight shillings? How much l-or 
 ten shillings? For thirteen shillings? For fourteen 
 shuhngs?^ Sixteen shillings? Twenty shillings? 
 
 12.5. What do you understand by one ninth, two 
 ninths, three ninths, etc., of any number? 
 
 126. Three is what part of nine? 
 
 127. Four is what part of nine ? 
 
 128. Five is what part of nine ? 
 
 129. Seven is what part of nine? 
 
 130. How many ninths make a whole one? 
 
 131. Thirteen are how many times nine? 
 
 132. Fifteen are how many times nine? 
 
 133. Seventeen are how many times nine? 
 
 134. When hay is ten dollars a ton, what is one tenth 
 of a ton worth ? What are two tenths of a ton worth ? 
 What are three tenths of a ton worth ?" Four tenths? 
 Five tenths? Six tenths? Seven tenths? Eight 
 tenths? Nine tenths? 
 
 135. When sugar is ten dollars a hundred weight, 
 what part of a hundred weight can you buy for one 
 
U ARITHMETIC. [parti. 
 
 dollar? What part for two dollars? What part for 
 three dollars? What part for four dollars? What 
 part for five dollars? Six dollars? Seven dollars? 
 Eight dollars? Nine dollars? IIow much can you 
 buy for eleven dollars? Thirteen dollars? Fifteen 
 dollars? Seventeen dollars? 
 
 136. What do you understand by one tenth, two 
 tenths, three tenths, etc., of any thing?* 
 
 137. How many tenths make a whole one?* 
 
 C. Instead of writing the names of numbers, it is 
 usual to express them by particular characters, called 
 figures. 
 
 One is written 1 
 
 Two is written. 2 
 
 Three is written 3 
 
 Four is written 4 
 
 Five is written 5 
 
 Six is written 6 
 
 Seven is written 7 
 
 Eight is written 8 
 
 Nine is written 9 
 
 Ten is written 10 
 
 1. Eleven times one are how many times 2? 
 
 2. Twelve are how many times 2? 3 ? 4? 
 
 3. Fourteen are how many times 2? 4? 3? 
 
 4. If you had fifteen cents, how many cakes could 
 you buy at 4 cents apiece? How many at 2 cents 
 apiece? How many at 3 cents apiece? How many at 
 5 cents apiece? 
 
 6. Fifteen are how many times 4? 2? 3? 5? 
 
 6. Sixteen are how many times 5? 3? 6? 2? 7? 4? 
 
 7. Seventeen are how many times 6? 2? 7? 3? 
 5? 4? 
 
 8. Eighteen are how many times 4? 7? 9? 6? 3? 
 2? 6? 8? 
 
 * These questions should frequently b© put to the learners. 
 
SEC. III.] ARITHMETIC. 45 
 
 9. Nineteen are how many times 3? 7? 4:? 5? 8? 
 6? 9? 2? 10? 
 
 10. Twenty are how many times 6 ? 2 ? 8 ? 3 ? 9 ? 
 4? 10? 6? 7? 
 
 11. Twenty-one^are how many times 7? 3? 8? 2? 
 4? 6? 9? 5? 10? 
 
 12. Twenty-two are how many times 3? 8? 5? 4? 
 9? 6? 7? 10? 2? 
 
 13. If you had twenty-seven dollars, how much 
 cloth could you buy at 9 dollars a yard ? How much 
 at 6 dollars a yard? How much at 4 dollars a yard? 
 How much at 3 dollars a yard ? How much at 7 dollars 
 a yard? How much at 8 dollars a yard? How much 
 at 5 dollars a yard? How much at 10 dollars a yard? 
 
 14. Twentvscven are how many times 9? 6? 4? 
 3? 7? 8? 5"? 10? 
 
 15. Twenty-four are how many times 6? 8? 7? 
 6? 2? 10? 3? 4? 9? 
 
 16. Twenty-nine are how many times 3? 7? 5? 
 9? 6? 8? 4? 10? 
 
 17. Twenty-three are how many times 4? 2? 7? 
 8? 3? 9? 6? 5? 10? 
 
 18. Twenty-five are how many times 3? 7? 2? 6? 
 9? 4? 8? 5? 10? 
 
 19. Thirty are how many times 10? 2? 3? 7? 9? 
 6.? 5? 4? 8? 
 
 20: Thirty-three are how many times 6? 8? 7? 
 4? 9? 5? 10? 3? 
 
 21. Twenty-six are how many times 9? 4? 7? 3? 
 8? 5? 6? 10? 
 
 22. Thirty-five are how many times 5? 6? 3? 7? 
 9? 10? 4? 8? 
 
 23. Thirty-eight are how many times 8? 6? 3? 
 9? 5? 4? 7? 10? 
 
 24. Thirty-four are how many times 7? 3? 9? 10? 
 6? 8? 4? 5? 
 
 25. Thirty-six are how many times 8? 9? 4? 5? 
 3? 6? 7? 10? 
 
46 ARITHMETIC. [paet i. 
 
 26. Forty are how many times 8? 10? 6? 4? 3? 
 9? 5? I'i 
 
 27. For forty-seven cents, how many pounds of 
 meat can be bought at 6 cents a pound? How many 
 pounds at 8 cents ? How many at 9 cents ? How 
 many at 3 cents ? How many at 5 cents ? How many 
 at 4 cents ? How many at 7 cents ? How many at 10 
 cents^a pound? 
 
 28. Forty-seven are how many times 6? 8? 9? 3? 
 5? 4? 7? 10? 
 
 29. Forty-three are how many times 9? 8? 7? 6? 
 4? 3? 5? 10? 
 
 30. Forty-five are how many times 10? 8? 3? 6? 
 4? 7? 5? 9? 
 
 31. Forty-nine are how many times 6 ? 10? 6? 9? 
 4? 8? 7? 
 
 32. Fifty-three are how many times 8? 5? 6? 4? 
 7? 9? 10? 
 
 33. Fifty-seven are how many times 9 ? 7? 10? 6? 
 5? 8? 4? 
 
 34. Fifty-five are how many times 0? 4? 8? 10? 
 9? 7? 5? 
 
 35. Forty-eight are how many times 7? 5? 9? 4? 
 6? 8? 10? 
 
 36. Fifty-four are how many times 5? 9? 6? 4? 
 7? 10? 8? 
 
 37. Forty-four are how many times 4? 6? 9? 7? 
 5? 8? 10? 
 
 38. Fifty-eight are how many times 7 ? 6? 8? 4? 
 9? 5? 10? 
 
 39. Forty six are how many times 8? 10? 4? 6? 
 9? 7? 5? ^ 
 
 40. Fifty arc how many times 9? 5? 4? 10? 8? 
 6? 7? 
 
 41. Fifty-nine are how many times 4? 8? 7? 6? 
 10? 9? 5? 
 
 42. Sixty-four are how many times 7? 5? 8? 10? 
 6? 9? 
 
SEC. III.] ARITHMETIC. 47 
 
 43. Sixty-eight are how many times 6? 8? 9? 7? 
 10? 5? 
 
 44. Fifty-two arc how many times 4? 6? 8? 10? 
 6? 7? 9? 
 
 45. Sixty-three are how. many times 5? 4? 6? 10? 
 9? 7? 8? 
 
 46. Sixty-two are how many times 4? 10? 9? 7? 
 8? 5? 6? 
 
 47. Seventy-three are how many times 10 ? 7 ? 8 ? 
 6? 5? 9? 
 
 48. Seventy-five arc how many times 7? 8? 10? 5? 
 6? 9? 
 
 49. If you had sixty-seven dollars, how many barrels 
 of flour could you buy at 5 dollars a barrel ? How 
 many at seven dollars a barrel? How many at 6 dol- 
 lars a barrel ? How many at 8 dollars a barrel ? How 
 many at 10 dollars a barrel ? How many at 9 dollars a 
 barrel ? 
 
 50. Sixty-seven are how many times 5? 7? 6? 8? 
 10? 9? 
 
 51. Seventy-four are how many times 10? 7? 8? 6? 
 5? 9? 
 
 52. Sixty are how many times 9? 10? 6? 4? 7? 
 6? 8? 
 
 53. Seventy-two are how many times 5? 7? 6? 8? 
 10? 9? 
 
 54. Sixty-five are how many times 5? 10? 8? 6? 
 7? 9? 
 
 55. Sixty-one are how many times 4? 5? 7? 6? 8? 
 10? 9? 
 
 56. Seventy-nine are how many times 10? 9? 8? 
 7? 6? 5? 
 
 57. Seventy are how many times 9? 5? 6? 8? 
 7? 10? 
 
 58. Eighty-two are how many times 10? 7? 8? 6? 
 9? 
 
 59. Sixty-six are how many times 9? 5? 6? 7? 
 10? 8? 
 
48 ARITHMETIC. [part i. 
 
 60. Eighty are how many times 10? 7? 6? 8? 9? 
 
 61. Sixty-nine are how many times 9 '/ 5 ? 7 ? 10 ? 
 8? 6? 
 
 62. Eiehty-one are how many times 10? 6? 8? 7? 
 9? 
 
 63. Seventy -six are how many times 9 ? 5? 10? 6? 
 7? 8? 
 
 64. Eighty-three are how many times 10 ? 6 ? 7 ? 
 9? 8? 
 
 65. Seventy-one are how many times 9? 5? 7? 6? 
 8? 10? 
 
 66. Eighty-four are how many times 10 ? 6 ? 8 ? 
 9? 7? 
 
 67. Seventy-seven are how many times 9? 7? 5? 
 10? 8? 6? 
 
 68. Eighty-five are how many times 10 ? 8 ? 7 ? 
 6? 9? 
 
 69. Ninety are how many times 9? 10? 6? 7? 8? 
 
 70. Eighty-six are how many times 10 ? 9 ? 6 ? 7 ? 
 8? 
 
 71. Ninety-four are how many times 9 ? 10 ? 8 ? 
 6? 7? 
 
 72. Eighty-seven are how many times 10? 9? 7? 
 6? 8? 
 
 73. Ninety-two are how many times 9? 10? 6? 7? 
 8? ^ 
 
 74. Eighty-eight are how many times 10 ? 9 ? 8 ? 
 6? 7? 
 
 75. Ninety-five are how many times 9? 10? 6? 8? 
 7? 
 
 76. Eighty-nine are how many times 10 ? 9 ? 6 ? 
 7? 8? 
 
 77. Ninety-eight are how many times 10 ? 9 ? 8 ? 
 6? 7? 
 
 78. Ninety-four are how many times 9 ? 10 ? 6 ? 
 8? 7? 
 
 79. One hundred are how many times 6 ? 10 ? 8 ? 
 9? 7? 
 
SEC. in.] ARITHMETIC. 49 
 
 80. Ninety-three are how many times 10 ? 6 ? 9 ? 
 8? 7? 
 
 81. Ninety-nine are how many times 7? 10? 8? 
 6? 9? 
 
 82. Ninety-six are how many times 9? 8? 7? 6? 
 10? 
 
 83. Ninety-seven are how many times 10 ? 9 ? 6 ? 
 7? 8? 
 
 I). 1. If an orange is worth 3 apples, how many 
 oranges are fiftccen apples worth ? 
 
 2. In 8 pints how many quarts? 
 
 3. In 8 gills how many pints ? 
 
 4. If you divide twelve apples equally among throe 
 boys, how many would you give thorn apiece ? 
 
 5. How many hours Would it take you to travel 10 
 miles, if you travel three milt^s in an- hour ? 
 
 6. How many pence are there in eight farthings? 
 
 7. How many pence are there in twelve farthings ? 
 
 8. How many pence are there in seventeen farthings ? 
 
 9. How many gallons are there in ten quarts ? 
 
 10. How much broadcloth, at six dollars a yard, can 
 you buy for seventeen dollars ? 
 
 11. How many pounds of raisins, at 8 cents a pound, 
 can you buy for twenty-five cents ? 
 
 , '12. In twenty-eight shillings how many dollars ?* 
 
 13. In twenty-eight farthings how many pence ? 
 
 14. How many barrels of flour, at 7 dollars a barrel, 
 can you buy for thirty-four dollars ? 
 
 15. How many reams of paper, at five dollars a ream, 
 can you buy for thirty-seven dollars ? 
 
 16. In thirty-four gills how many pints? 
 
 17. In twenty-seven quarts how many gallons ? 
 
 18. If an orange is worth six apples, how many 
 oranges can you buy for forty apples ? 
 
 19. Thirty-six shillings are how many dollars ? 
 
 *Iu some States 8 shillings is equal to one dollar. 
 
50 ARITHMETIC. [part i. 
 
 20. A man bought thirty apples at the rate of 3 for 
 a cent, hew many cents did they come to ? 
 
 21. A laborer engaged to work 8 months for ninety- 
 six dollars; how much did he receive for a month? how 
 much a week, allowing 4 weeks to the month ? how 
 many shillings a day, allowing 6 working days to the 
 week ? 
 
 22. If wine is worth twenty cents a pint, what is 1 
 gill worth ? 
 
 23. If you buy a bushel* of apples for forty cents, 
 what is the price of a peck ? 
 
 24. If you buy a bushel of pears for forty-eight cents, 
 what will be the price of half a peck, at the same rate? 
 
 25. Four men bought a horse for forty-eight dollars ; 
 what did each man pay? 
 
 26. Five men bought a horse for seventy-five dollars, 
 and sold him again for forty dollars ; what did each 
 man lose by the bargain ? 
 
 27. A man gave sixty-three cents for a horse to ride 
 nine miles ; what was the price of one mile's ride ? 
 
 28. A man hired a horse to ride, and agreed to give 
 8 cents a mile ; he paid fifty-six cents : how many miles 
 did he ride ? 
 
 29. A man had forty-two dollars, which he paid for 
 wood at 7 dollars a cord ; how many cords did he buy ? 
 
 80. Two boys are forty-eight rods apart, and both 
 running the same way; but the hindmost boy gains 
 upon the other 3 rods in a minute; in how many 
 minutes will he overtake the foremost boy? 
 
 31. There is a vessel containing sixty-three gallons 
 of wine; it has a pipe which discharges 7 gallons in 
 an hour; how many hours will it take to empty the 
 vessel? 
 
 22. There is a vessel containing eighty-seven gal- 
 lons, and by a cock ten gallons will run into it in an 
 hour; in how many hours will the vessel be filled? 
 
 S3. If one man can do a piece of work in thirty 
 
 *1 bushel is 4 pecks. 
 
SEC. IV.] ARITHMETIC. 51 
 
 days, in how many days can 3 men do it? in how many 
 days can 5 men do it? 
 
 84. If you wish to put sixty-four pounds of butter 
 into 8 boxes, how many pounds would you put into 
 each box? 
 
 35. If you had seventy-two pounds of butter, which 
 you wished to put into boxes containing 8 pounds 
 each, how many boxes would it take? 
 
 36. If a man can perform a journey in thirty-six 
 hours, how many days will it take him to do it when 
 the days are nine hours long? 
 
 37. If a man can do a piece of work in forty-eight 
 hours, how many days would it take him to do it, if he 
 works twelve hours in a day ? 
 
 SECTION IV. 
 
 A. 1. At two cents a yard, what will three yards 
 and one half of a yard of tape cost? 
 
 2. 3 times 2 and one half of 2 are how many? 
 
 3. At three dollars a yard, what will 4 yards and one 
 third of a yard of cloth cost? 
 
 4. 4 times 3 and one third of 3 are how many? 
 
 5. At three dollars a barrel, what will 3 barrels and 
 2 thirds of a barrel of cider cost? 
 
 6. 3 times 3 and two thirds of 3 are how.many? 
 
 7. If a man earn 4 dollars in a week, how many 
 dollars will he earn in 3 weeks and 1 fourth of a week? 
 
 8. 3 times 4 and 1 fourth of 4 are how many? _ 
 
 9. tf a yard of cloth cost 4 dollars, what will 5 
 yards and 3 fourths of a yard cost? 
 
 10. 5 times 4 and 3 fourths of 4 are how many? 
 
 11. If a man spend five dollars in a week, how many 
 dollars will he spend in 3 weeks and 1 fifth of a^\:eek? 
 How much in 5 weeks and 2 fifths of a week? 
 
 12. 3 times 5 and 1 fifth of 5 are how many? 
 
 13. 5 times 5 and 2 fifths of 5 are how many? 
 
62 ARITHMETIC. [part i. 
 
 14. 6 times 5 and 8 fifths of 5 are liow many? 
 
 15. If beer is worth six dollars a barrel, what would 
 4 barrels and 1 sixth of a barrel cost? How much 
 would 7 barrels and 5 sixths of a barrel cost? 
 
 IG. 4 times 6 and 1 sixth of six are how many? 
 
 17. 7 times six and 5 sixths of six are how 
 many ? 
 
 18. At 7 dollars a barrel, what will o barrels and 1 
 seventh of a barrel of flour ti'ost? What will 5 barrels 
 and 2 sevenths of a barrel cost? 
 
 19. o times 7 and 1 seventh of 7 are how many? 
 
 20. 5 times 7 and 2 sevenths of 7 are how many? 
 
 21. 8 times 5 and 4 fifths of 5 are how niany? 
 
 22. 8 times 6 and 3 sixths of 6 arc how ma-ny? 
 
 23. At 8 dollars a yard, what will 4 yards and 1 
 eighth of a yard of broadcloth cost? 
 
 24. 4 times 8 and 1 eighth of 8 are how many? 
 
 25. 2 times 7 and 3 sevenths of 7 are how m9,ny? 
 
 26. 8 times 7 and 4 sevenths of 7 are how many? 
 
 27. 9 times 7 and 6 sevenths of 7 are how many? 
 
 28. 3 times 8 and 5 eighths of 8 are how many? 
 
 29. 9 times 8 and 7 eighths of 8 are how many? 
 
 30. If a hundred weight of sugar cost 9 dollars, 
 what will 2 hundred weight and 1 ninth of a hundred 
 weight cost? What will 5 hundred weight and 2 
 ninths of a hundred weight cost? 
 
 31. 2 times 9 and 1 ninth of 9 are how many? 
 
 32. 5 times 9 and 2 ninths of 9 are how many? 
 
 33. 6 times 9 and 4 ninths of 9. are how many? . 
 
 34. 2 times 10 and 3 tenths of 10 are how maqy? 
 
 35. 7 times 9 and 7 ninths of 9 arc how many? 
 
 36. 5 times 10 and 4 tenths of 10 are how many? 
 
 37. 8 times 9 and 5 ninths of 9 are how many? 
 
 38. 4 times 10 and 7 tenths of 10 are how many? 
 
 39. 6 times 10 and 9 tenths of 10 are how many? 
 
 B. 1. A man bought 2 oranges at 6 cents apiece; 
 how many cents did they come to ? He paid for them 
 
SEC. IV.] ARITHMETIC. 63 
 
 with cherries at four cents a pint j how many pints did 
 it take? 
 
 2. 2 times 6 are how many times 4? 
 
 3. A man bought 3 yards of cloth at 4 dollars a 
 yard; how many dollars did it come to? How much 
 flour at 6 dollars a barrel would it take to pay for it? 
 
 4. 3 times 4 are how many times 6? 
 
 5. A man boui2;ht 4 peaches at 3 cents apiece; how 
 many cents did they come to? He paid for them with 
 pears at 2 cents a piece; how many pears did it take? 
 
 6. 4 times 3 are how many times 2? 
 
 7. Bought 2 hundred weight of sugar, at 9 dollars 
 a hundred weight, and paid for it with wood.' at 6 dol- 
 lars a cord; how many cords did it take? 
 
 8. 2 times 9 are how many times G? 
 
 9. Bought 3 barrels of flour at 8 dollars a barrel, 
 and paid for it with cider at 4 dollars a barrel; how 
 many barrels did it take? 
 
 10. 3 times 8 are how many times 4? 
 
 11. 12 times 3 are how many times 5? 
 
 12. 6 times 4 are how many times 8? 
 
 13. 3 times 10 are how many times ,6? 
 
 14. 4 times 9 arcliow many times 6? 
 
 15. How much flannel worth 4 shillings a yard 
 must be given for 3 yards of silk worth 5 shillings a 
 yard ? 
 
 16. 3 times 5 are how many times 4 ? 
 
 17. 2 times 7 are how many times 3? 5? 4? 
 
 18. 4 times 5 are how many times 3 ? 6? 7? 
 
 19. 3 times 7 are how many times 4 ? 5? 6? 8? 9? 
 
 20. Bought 2 kegs and 2 sevenths of a keg'of 
 tobacco at 7 dollars a keg, and paid for it with wood at 
 4 dollars a cord; how many cords did it take? How 
 much butter at 3 dollars a box would it take to pay 
 for it? 
 
 21. 2 times 7, and 2 sevenths of 7 are how many 
 times 4? 3? 5? 6? 8? 
 
 22. Bought 3 bushels and 3 fifths of a bushel of 
 
5i ARITHMETIC. [part i. 
 
 corn at 5 shillings a bushel, and paid for it with wheat 
 at 6 shillings a bushel; how many bushels of wheat 
 did it take? 
 
 28. 3 times 5 and 3 fifths of five are how many 
 times 6? 9? 4? 7? 3? 8? 
 
 24. How much sugar that is 8 dollars a hundred 
 weight, can be bought for 4 cords and 2 sevenths of a 
 cord of wood, at 7 dollars a cord ? 
 
 25. 4 times 7 and 2 sevenths of 7 are how many 
 times 6? 8? 5? 3? 9? 10'/ 
 
 26. 5 times 5 and 3 fifths of 5 are how many 
 times 4? 8? 9? 7? 10? 3? 6? 
 
 27. 6 times 7 and 3 sevenths of 7 are how many 
 times 9? 4? 5? 8? 10? 
 
 28. 5 times 8 and 3 eighths of 8 are how many 
 times 6? 9? 4? 7? 10? 
 
 29. 7 times 8 and 5 eighths of 8 are how many 
 times 9? 6? 10? 4? 5? 
 
 30. 5 times 9 and 4 ninths of 9 are how many 
 times 7? 8? 6? 10? 4? 
 
 31. 7 times 9 and 7 ninths of 9 are how many 
 times 6? 8? 10? 5? 4? 
 
 32. 6 times 10 and 3 tenths of 10 are how many 
 times 7? 5 4? 9? 8? 
 
 33. 8 times 10 and 4 tenths of 10 are how many 
 times 6 ? 7 ? 9 ? 
 
 34. 8 times 9 and 3 ninths of 9 are how many 
 times 6 ? 10 ? 7 ? 
 
 C. 1. Bought 4 bushels of apples, at 3 shillings a 
 bushel; how many dollars did they come to? 
 
 2. How many apples, at 2 cents apiece, must you 
 give for 2 lemons, at 4 cents apiece? 
 
 3. How many pears, at 3 cents apiece, must you 
 give for 3 oranges, at 5 cents apiece ? 
 
 4. How many barrels of cider, at 3 dollars a barrel, 
 must be given for 5 boxes of butter, at 4 dollars a 
 box? 
 
SEC. IV.] ARITHMETIC. 55 
 
 5. A man bought 4 yards of broadcloth, at 7 dollars 
 a yard, and paid for it with flour, at 5 dollars a barrel; 
 how many barrels did he give? 
 
 6. If 2 apples cost 4 cents, what would 4 apples 
 cost? 
 
 7. If 3 apples are worth 6 cents, how many apples 
 must you give for 8 pears, that are worth 3 cents 
 apiece ? 
 
 8. James had 8 oranges that were worth 5 cents 
 apiece, and George had 5 quarts of cherries that were 
 worth G cents a quart, which he gave to James for a part 
 of his oranges; how many oranges did he buy, and how 
 many had James left? 
 
 9. Bought 8 yards of cloth, at 9 shillings a yard; 
 how many dollars did it come to? 
 
 10. Bought 5 bushels and 3 sevenths of a bushel of 
 salt, at 7 shillings a bushel ; how many dollars did it 
 come to? 
 
 11. Bought 9 boxes and 2 thirds of a box of raisins 
 for 3 dollars a box, and paid for it with cider, at 4 dol- 
 lars a barrel; how many barrels did it take? 
 
 12. Bought 8' pounds and 4 sevenths of a pound of 
 opium, at 7 dollars a pound, and paid for it with cloth, 
 at 5 dollars a yard; how many yards did it take? 
 
 13. Bought (3 chaldrons and 4 ninths of a chaldron 
 of coal, at 9 dollars a chaldron, and paid for it with 
 oranges, at 5 dollars a box ; how many boxes did it take ? 
 
 14. Bought 7 cases and 5 sixths of a case of Flo- 
 rence oil, at 6 dollars a case, and paid for it with sheet 
 lead, at 7 dollars a hundred weight; how many hundred 
 weight did it take? 
 
 15. liow inany dozen of eggs, at 8 cents a dozen, 
 must' you give for 7 pounds of sugar, at 10 ocnts a 
 pound ? 
 
 16. How much barley, at 3 shillings a bushel, must 
 be niven for 8 bushels of wheat, at 7 shillings a bushel ? 
 
 17. How much cloth, at 4 shillings a yard, must be 
 given for a firkin of butter worth 8 dollars ? 
 
56 ARITHMETIC. [part i. 
 
 18. IIoTT much clotli, at 5 shillings a yard, can be 
 bought for 2 reams of paper, at 5 dollars a ream? 
 
 19. IIow much -wha'at, at 7 shillings a bushel, can 
 be bought for 2 barrels of cider, at 4 dollars and a 
 half a barrel? 
 
 20. flow long would it take a man to lay up 10 
 dollars, if he saves 4 shillings a day? 
 
 21. If a man earns 8 shillings a day, how many dol- 
 lars would he earn in 10 days ? 
 
 22. A man bought twenty pears at the rate of 2 
 for 3 cents; how much did tjiey come to? 
 
 23. How many eggs, at the rate of 3 for 5 cents, 
 can you buy for thirty cents ? 
 
 24. A man hired a laborer and agreed to give him 
 5 dollars for every 3 days' work ; how much did he 
 give him a week, there being G working days in a 
 week ? How much was it a month, allowing 4 weeks 
 to a month ? 
 
 25. If a man receives 5 dollars for 3 days' work, 
 how many shillingi? is that a day ? 
 
 26. 5 men bought a horse for sixty-three dollars, 
 and paid two dollars a week for keening him; at the 
 end of 8 weeks they sold him for fifty-four dollars; 
 how much did each man lose by the bargain ? 
 
 SECTION y. 
 
 A. 1. James had 4 apples, and John had half as 
 many; how many had John ? 
 
 2. If an orange cost 6 'cents, and an apple half as 
 much ; how much docs the apple cost ? 
 
 3. If you divide 8 apples equally betwijen two boys, 
 ^<'hat part of them must each have ? • 
 
 Ans. One half of them. 
 
 4. What is 1 half of 8 ? 
 
 5. If you divide 8 apples equally among 4 boys, 
 what part of them must each have ? 
 
 Ans. One fourth of them. 
 
SEC. v.] ARITHMETIC. 57 
 
 6. What is 1 fourth of 8 ? 
 
 7. If you divide 6 oranges equally among 3 boys, 
 what part of thciii must 1 boy have ? 
 
 8. What is 1 third of 6? 
 
 9. If 4 yards of cloth cost 8 dollars, vv^hat part of 
 8 dollars would 1 yard cost? What part of 8 dollars 
 would 2 yards cost? Vv'hat part of 8 dollars would o 
 yards cost ? 
 
 10. What is 1 fourth of 8 ? What is 2 fourths of 
 8? What is 3 fourths of 8? 
 
 11. If G yards of cloth will make 8 coats, what part 
 of G yards will make 1 coat ? What part of G 3^ards 
 will make two coats ? 
 
 12. What is one third of 6? What is two thirds 
 of G? 
 
 13. If 3 barrels of cider cost 9 dollars, w^hat part 
 of 9 dollars will 1 barrel cost ? What part of 9 dol- 
 lars will 2 barrels cost ? 
 
 14. What is 1 third of 9 ? What is 2 thirds of 9 ? 
 
 15. If 2 yards of cloth cost 10 dollars, what part of 
 10 dollars will 1 yard cost ? What part of 10 dollars 
 will 3 yards cost ? 
 
 IG. What is 1 half of 10? What is 3 halves of 
 10? 
 
 17. If 2 barrels of flour cost twelve dollars, what 
 part of twelve dollars will 1 barrel cost? What part 
 of twelve dollars will 3 barrels cost ? What part of 
 twelve dollars will 5 barrels cost ? 
 
 18. What is 1 half of twejve ? What is 3 halves 
 of twelve ? What is 5 halves of twelve ? 
 
 19. If 4 barrels of cider cost twelve dollars, what 
 part of twelve dollars will 1 barrel cost? What part 
 of twelve dulhirs will 3' barrels cost ? What part of 
 twelve dollars will 5 barrels cost? Wliat part of 
 twelve dollars will 7 barrels cost ? 
 
 20. What is 1 fourth of twelve? What is 2 fourths 
 of twelve ? 3 fourths ? 5 fourths ? 7 fourths ? 
 
 21. If three oranges cost twelve cents, what part of 
 
58 ARITHMETIC. [part i. 
 
 twelve cents will 1 orange cost ? What part of twelve 
 cents will 2 oranges cost ? What part of twelve cents 
 will .4 oranges cost? What part of twelve cents will 
 5 oranges cost ? 7 oranges ? 1.0 oranges ? 
 
 22. What is one third V twelve? 2 thirds? 4 thirds? 
 
 5 thirds? 7 thirds? 10 thirds? 
 
 23. If five bushels of wheat cost 10 dollars, what 
 part of 10 dollars will 1 bushel cost ? What part of 
 10 dollars will 2 bushels cost ? 3 bushels ? 4 bushels ? 
 
 6 bushels ? 7 bushels ? 
 
 24. What is 1 fifth of ten? 2 fifths? 3 fifths? 4 
 fifths ? 6 fifths ? 7 fifths ? 
 
 25. What is 1 half of fourteen ? 1 seventh ? 2 sev- 
 enths ? 3 sevenths ? 5 sevenths ? 
 
 26. What is 1 third of fifteen ? 2 thirds ? 1 fifth ? 
 
 2 fifths ? 3 fifths ? 4 fifths ? 
 
 27. What is 1 half of sixteen? 3 fourths? 1 eighth? 
 
 3 eii^hths? 5 eighths? 7 eighths? 1 sixteenth? 
 
 28. What is 1 half of eighteen ?. 2 thirds ? 1 sixth? 
 5 sixths ? 1 ninth ? 2 ninths? 4 ninths? 5 ninths? 7 
 ninths ? 8 ninths ? 1 eighteenth ? 5 eighteenths ? 
 
 29. What is 1 half of twenty ? 1 fourth ? 3 fourths ? 
 1 fifth ? 3 fifths ? 2 fifths ? 4 fifths ? 1 twentieth ? 3 
 twentieths ? 7 twentieths ? 
 
 30. What is 1 third of twenty-one ? 2 thirds ? 1 
 seventh ? 5 sevenths ? 3 sevenths ? 6 sevenths ? 4 
 sevenths ? 
 
 31. What is 1 half of twenty-two? 1 eleventh? 3 
 elevenths ? 5 elevenths ? 7 elevenths ? 10 elevenths ? 
 
 32. What is 1 half of twenty-four? 1 third? 2 
 thirds ? 1 fourth ? 3 fourths ? 1 sixth ? 5 sixths ? 1 
 eighth? 3 eighths? 7 eighths? 1 twelfth? 5 twelfths? 
 
 7 twelfths ? " 
 
 33. What is 1 half of twenty-six? 
 
 34. What are 2 thirds of twenty-seven? 2 ninths? 
 
 4 ninths? 5 ninths? 8 ninths? 7 ninths? 
 
 35. What is 1 half of twenty-eight? 3 fourths? 2 
 sevenths ? 5 sevenths ? 3 sevenths ? 
 
SEC. v.] ARITHMETIC. 59 
 
 36. What is 1 sixth of tliirty? 3 fifths? 7 tenths? 
 2 thirds? 1 half? 
 
 37. What is 1 half of thirty-two ? 3 fourths ? 5 
 eighths ? 1 sixteenth ? 
 
 38. What is 1 half of thirty-four? 
 
 39. Whatis 5 sixths of thirty-six? 2 thirds? 1 half ? 
 7 twelfths ? 4 ninths ? 3 fourths ? 
 
 40. Whatis 1 half of thirty-eight? 
 
 41. What is 1 third of thirty-nine ? 
 
 42. What is 3 fourths of forty? 7 eighths? 9 tenths? 
 1 half? 
 
 43. AYhat is 3 sevenths of forty-two? 5 sixths? 
 
 B. 1. A boy having twelve apples, kept 1 fourth of 
 them himself, and divided the other 3 fourths of them 
 equally among 4 of his companions; how many did he 
 give them apiece ? 
 
 2. 3 fourths of twelve are how many times 4 ? 
 
 3. A man having fourteen bushels of grain, divided 
 5 sevenths of it equally among 3 men ; how much did 
 he give them apiece ? 
 
 4. 5 sevenths of fourteen are how many times 3 ? 
 
 5. A man having fifteen shillings, gave away 4 fifths 
 of it, how many dollars did he give away? 
 
 6. 4 fifths of nineteen are how many times 6? 
 
 7. A man having twenty-one cents, paid away 6 sev- 
 enths of them for oranges at 5 cents apiece; how many 
 oranges did he buy ? 
 
 8. 6 sevenths of twenty-one are how many times 5? 
 
 9. A man bought a piece of cloth for twenty-four 
 shillings, and sold it again for 5 thirds of what he gave 
 for it; how many dollars did he sell it for? 
 
 10. 5 thirds of twenty-four are how many times 6? 
 
 11. 7 fifths of twenty are how many times 6 ? 
 
 12. 5 sevenths of thirty-five are how many times 8? 
 
 13. 7 sixths of thirty-six are how many times 5 ? 
 
 14. 9 sevenths of forty-two are how many times 8 ? 
 
 15. 7 ninths of forty -five are how many times 6 ? 
 
GO ARITHMETIC. [i'art i. 
 
 16. 5 sixths of forty-eight are how many times 7 ? 
 
 17. 6 fifths of fifty are how many times 9 ? 
 
 18. 8 ninths of fifty-four are how many times 5 ? 
 
 19. 9 sevenths of fifty-six are how many times 10 ? 
 
 20. 7 sixths of sixty are how many times 8 ? 
 
 21. 9 sevenths of sixty-three are how many times 7 ? 
 
 22. 10 eighths of sixty-four are how many times 9 ? 
 
 23. 6 eighths of seventy-two are how many times 7? 
 
 24. 4 sevenths af eighty-four arc how many times 9? 
 
 25. 7 tenths of ninety are how many times 8 ? 
 
 G. 1. Charles had 6 apples, and gave 1 third of them 
 to John ; how many did he give him ? 
 
 2. Albert had 9 cents, and spent 2 thirds of them; 
 how many had he left ? 
 
 8. James had 10 pears, and gave 1 half of them to 
 one of his companions, and 2 fifths of them to another ; 
 how many did he give away? 
 
 4. If 3 yards of cloth cost 6 dollars, what is that a 
 yard ? 
 
 5. If 4 yards of cloth cost twelve dollars, what will 
 2 yards cost? 
 
 6. If 9 apples cost eighteen cents, what will 3 apples 
 cost ? 
 
 7. If 3 oranges cost 18 cents, what will 2 cost ? 
 
 8. James had twenty-five cents, and he gave 4 fifths 
 of them for 10 apples; how much did he give for all 
 the apples ? how much apiece ? 
 
 9. A man had thirty dollars, and gave 5 sixths of 
 them for 8 yards of cloth ; how much did he give a 
 yard ? 
 
 10. A man had forty yards of cloth, and sold 3 fifths 
 of it for twenty-four dollars ; what was that a 
 yard ? 
 
 11. A man had forty-two barrels of flour, and sold 2 
 sevenths of it for 6 dollars a barrel ; how much did it 
 come to. 
 
 12. A boy had sixty-three nuts, and divided 4 sev- 
 
SEC. v.] ARITHMETIC. 61 
 
 entlis of them equally among six of his companions ', 
 how many did he give them apiece ? 
 
 13. If 4 yards of cloth cost twelve dollars, what will 
 three yards cost? 
 
 14. If five oranges cost twenty-five cents, what will 
 3 cost ? 
 
 15. If 3 oranges cost fifteen cents, what will 7 cost? 
 
 16. If three barrels of cider cost twelve dollars, what 
 will 10 barrels cost ? 
 
 17. If 7 pounds of flour cost thirty-five cents, what 
 will nine pounds cost ? 
 
 18. If 5 firkins of butter cost forty dollars, what 
 will three firkins cost? 
 
 19. If 2 men can do a piece of work in six days, 
 how long would it take 4 men to do the same w^ork ? 
 
 20. If 6 men can do a piece of work in twelve days, 
 in how many days will 3 men do the same work ? 
 
 21. If 3 men can do a piece of work in twelve days, 
 in how many days will 4 men do the same work ? 
 
 22. If 2 cocks of a aertain size will empty a cistern 
 in 6 hours, in how long a time will 3 cocks of the same 
 size enjpty it ? 
 
 23. Three men, setting out on a journey, purchased 
 5 loaves of bread apiece, but before they had eaten any 
 of it, two other men joined them, and they agreed to 
 share the bread equally among the whole ; how many 
 loaves did they have apiece? 
 
 24. If 4 barrels of flour cost twenty-four dollars, 
 what would 7 barrels cost? IIow much cider at 3 dol- 
 lars a barrel would 7 barrels of flour buy ? 
 
 25. A man bought a quantity of flour for fifty-four 
 dollars ; and another man gave him 9 yards of 
 cloth for 5 sixths of it; what was the cloth worth a 
 yard? 
 
 26. If 9 yards of cloth cost fifty-four dollars, how 
 many boxes of butter, at 4 dollars a box, would 5 yards 
 of the same cloth buy ? 
 
 27. Bought 8 firkins of butter for seventy-two doU 
 
62 ARITHMETIC. [paet i 
 
 lars, and gave 6 of them for 7 yards of cloth ; what 
 was a yard of the cloth worth '/ 
 
 28. A man bought 6 barrels of flour for sixty dol- 
 lars, and gave 4 barrels of it for cider at 5 five dollars 
 a barrel ; how many barrels did he buy? 
 
 Note. — The manner of writing numbers with figures 
 has been explained as far as ten. The numbers from 
 ten to one hundred are written as follows : 
 
 Ten is written 10 
 
 Eleven 11 
 
 Twelve 12 
 
 Thirteen 13 
 
 Fourteen 14 
 
 Fifteen 15 
 
 Sixteen 16 
 
 Seventeen 17 
 
 Eighteen 18 ^ 
 
 Nineteen 19 
 
 Twenty 20 
 
 Twenty-one '. — 21 
 
 Twenty-two : 22 
 
 Twenty-three 23 
 
 Twenty-four 24' 
 
 Twenty-five 25 
 
 T wen ty-six ." 26 
 
 Twenty-seven 27 
 
 Twenty-eight 28 
 
 Twenty-nine 29 
 
 Thirty 30 
 
 Thirty-one, etc 31 
 
 Forty 40 
 
 Fifty 50 
 
 Sixty 60 
 
 Seventy 70 
 
 Eighty 80 
 
 Ninety 90 
 
 One hundred 100 
 
SEC. VI.] ARITHMETIC. 63 
 
 SECTION VI. 
 
 A. 1. Bought 1 half of a yard of cloth for I shil- 
 ling, what would be the price of a yard at the same 
 rate ? 
 
 2. If 1 half pint of cherries cost 2 cents, what will 
 a pint cost ? 
 
 3. If 1 fourth of a barrel of flour cost 2 dollars, what 
 would a barrel cost ? 
 
 4. 2 is 1 half of what number ? 1 fourth of what 
 number ? 
 
 5. If 1 third of a yard of cloth cost 2 dollars, what 
 is that a yard ? 
 
 6. 2 is 1 third of what number ? 
 
 7. If 1 tbird of a yard of cloth cost 3 dollars, what 
 is that a yard ? 
 
 8. 3 is 1 third of what number ? 
 
 9. If 1 fourth of a firkin of butter cost 3 dollars, 
 what is that a firkin ? 
 
 10. 3 is 1 fourth of what number? 
 
 11. A man bought 1 third of a bushel of wheat for 
 4 shillings, what would a bushel cost at that rate? 
 
 12. 4 is 1 third of what number? 
 
 13. If a man can ride 2 miles in 1 fifth of an hour, 
 how far can he ride in an hour ? 
 
 14. 2 is 1 fifth of what number? 
 
 15. A man, being asked the age of his eldest son, 
 answered that his youngest son, who was 3 years old, 
 was just I fifth of the age of his eldest son; how old 
 was the eldest son ? 
 
 16. 3 is 1 fifth of what number? « 
 
 17. A inan bought I sixth part of a hundred weight 
 of sugar for 2 dollars, what would a hundred weight 
 cost at the same rate? 
 
 18.2 is 1 sixth of what number ? 
 
 19. Bought 1 fifth of a pound of starch for 5 cents, 
 what was that a pound ? 
 
 20. 5 is 1 fifth of what number? 
 
64 AllITHMETIC. [part i. 
 
 21. Bought 1 fourth of a pound of aquafortis for 6 
 cents, what was that a pound ? 
 
 22. 6 is 1 fourth of what number? 
 
 23. Bouiiht 1 seventh of a hundred weight of cocoa 
 for 4 dollars, what would 1 hundred weight cost at the 
 same rate ? 
 
 24. 4 is 1 seventh of what number? 
 
 25. 7 is 1 fifth of what number ? 
 
 26. 5 is 1 third of what number? 
 
 27. 4 is 1 eio-hth of what number? 
 
 28. 6 is one sixth of what number? 
 
 29. 8 is 1 third of what number? • 
 80. 9 is 1 fourth of what number ? 
 
 31. 7 is 1 sixth of what number? 
 
 32. 8 is 1 seventh of what number? 
 
 33. 9 is 1 eighth of what number? 
 
 34. 8 is 1 tenth of what number? 
 
 35. 7 is 1 ninth of what number? 
 
 36. 6 is 1 fifth of what number? 
 
 37. 10 is 1 seventh of what number? 
 
 B. 1. A man bought some linen and some cotton 
 cloth ; for the linen he gave 4 shillings a yard, which 
 was twice as much as he gave for the cofton; what did 
 he give a yard for the corton ? 
 
 2. 4 is 2 times what number? 
 
 3. If 2 thirds of a yard of cloth cost 6 dollars, what 
 would 1 third cost? 
 
 4. 6 is 2 times what number? 
 
 5. If 3 fourths of a barrel of flour cost 6 dollars, 
 what wfll 1 fourth of a barrel cost ? 
 
 6. 6 is 3 times what number? 
 
 7. If 2 fifths of a pound of chocolate cost 8 cents, 
 what would 1 fifth of a pound cost? 
 
 8. 8 is 2 times what number? 
 
 9. If 3 fifths of a pound of candles cost 9 cents, 
 what will 1 fifth of a pound cost? 
 
 10. 9 is 3 times what number? 
 
SEC. VI.] ARITHMETIC. 65 
 
 11. If 2 sevenths of a pound of spermaceti candles 
 cost 10 cents, what will 1 seventh of a pound cost? 
 
 12. 10 is 2 times what number? 
 
 13. If 5 ei<jfhths of a pound of cotton cost 10 cents, 
 what will 1 eighth cost? 
 
 14. 10 is 5 times what number? 
 
 15. If 2 thirds of a yard of cloth cost 4 dollars, 
 what will 1 third cost? If 1 third of a yard Cost 2 
 dollars, what will a yard cost?* 
 
 16. If 4 is 2 thirds of some number, what is 1 third 
 of the same number? 2 is 1 third of what number?* 
 Then 4 is 2 thirds of what? 
 
 17. If 2 thirds of a barrel of flour cost 6 dollars, 
 what will 1 third of a barrel cost? If 1 third of a 
 barrel cost 3 dollars, what will a barrel cost?* 
 
 18. If 6 is 2 thirds of some number, what is 1 third 
 of the same number? 3 is 1 third of what number? 
 Then 6 is 2 thirds of what? 
 
 19. If 3 fourths of a bushel of wheat cost 6 shil- 
 lings, what will 1 fourth of a bushel cost? If 1 
 fourth of a bushel cost 2 shillings, what will a bushel 
 cost? 
 
 20. If 6 is 3 fourths of some number, what is 1 
 fourth of the same number? 2 is 1 fourth of what 
 number? Then 6 is 3 fourths of -what? 
 
 21. If 2 fifths of a gallon of wine cost 4 shillings, 
 what will 1 fifth of a gallon cost? If 1 fifth of a gal- 
 lon cost 2 shillings, what will a gallon cost? 
 
 22. If 4 is 2 fifths of some number, what is 1 fifth 
 of the same number? 2 is 1 fifth of what number? 
 Then 4 is 2 fifths of what? ' 
 
 23. If 3 sevenths of a pound of tobacco cost 6 
 cents, what will 1 seventh of a pound cost? If 1 
 seventh of a pound cost 2 cents, what will a pound 
 cost ? 
 
 24. If 6 is 3 sevenths of some number, what is 1 
 
 * See this section, article A. 
 
66 ARITHMETIC. [part i. 
 
 seventh of the same number? 2 is 1 seventh of what 
 number? Then 6 is 3 sevenths of what? 
 
 25. If 2 sevenths of a barrel of fish cost 4 dollars, 
 what will 1 seventh of a barrel cost? What will a 
 barrel cost? 
 
 26. 4 is 2 sevenths of what number ? 
 
 27. If 3 eighths of a pound of chocolate cost 6 
 cents,'' what will 1 eighth of a pound cost? What will 
 a pound cost? 
 
 28. 6 is 3 eights of what number? 
 
 29. If 8 cents will buy 2 fifths of a pound of aqua- 
 fortis, how many cents will buy a pound ? 
 
 ' 30. 8 is 2 fifths of what number? 
 
 31. A .man bought 3 fourths, of a hundred weight 
 of yellow ochre for 9 dollars; what was that a hundred 
 weight? 
 
 32. 9 is 3 fourths of what number? 
 
 33. 8 is 4 ninths of what number? 
 
 34. 9 is 3 tenths of what number? 
 
 35. 10 is 5 sevenths of what number? 
 
 36. 12 is 3 fifths of what number? 
 
 37. 12 is 4 ninths of what number? 
 
 38. 10 is 2 sevenths of what number? 
 
 39. 14 is 7 fifths of what number? 
 
 40. 15 is 3 elevenths of what number? 
 
 41. 16 is 2 fifths of what number? 
 
 42. 18 is 6 tenths of what number? 
 
 43. 20 is 5 ninths of what number? 
 
 44. 21 is 3 ninths of what number? 
 
 45. 25 is 8 ninths of what number? 
 
 0. 1. If 5 eighths of a cask of claret wine cost 15 
 dotlars, what is that a cask ? How much cider at 6 
 dollars a barrel would it take to pay for a cask of the wine? 
 
 2. 15 is 5 eighths of how many times 4? 
 
 3. If 2 thirds of a pound of coifee cost 18 cents, 
 how much would a pound cost? How many ox'anges 
 at 5 cents apiece, might be bought for a pound? 
 
SEC. VI.] ARITHMETIC. G7 
 
 4. 18 is 2 thirds of how many times 5? 
 
 5. A man bought 4 sevenths of a hundred wcii^ht 
 of sugar for 20 shillings, how many dollars would a 
 hundred weight come to at the same rate? 
 
 6. 20 is 4 sevenths of how many times 6? 
 
 7. A man sold a cow for 21 dollars, which was only 
 7 tenths of what she cost him ; how much did she cost 
 him ? When he bought her, he paid for her with cloth 
 at 8 dollars a yard ; how many yards of cloth did he 
 give ? 
 
 8. 21 is seven tenths of how many times 8? 
 
 9. A mau being asked the age of his youngest son, 
 answered, that the age of his eldest son was 24 years, 
 which was three fifths of his own age; and that his own 
 age was 10 times as much as that of his youngest son; 
 what was his age y and what was the age of his young- 
 est son ? 
 
 10. 24 is 3 fifths of how many times 10? 
 
 11. 27 is 3 fifths of how many times 7? 
 
 12. 28 is 7 tenths of how many times 9 ? ' 
 
 13. 30 is 5 eighths of how many times 7? 
 
 14. 32 is 4 sevenths of how many times 6 ? 
 
 15. 36 is 9 eighths of how many times 5 ? 
 
 16. 40 is 8 ninths of how many times 8 ? 
 
 17. 42 is 6 fifths of how many times 4? 
 
 . 18. 45 is 9 eighths of how many times 6 ? 
 
 19. 48 is 8 ninths of how many times 7 ? 
 
 20. * 50 is 5 sevenths of how many times 8 ? 
 
 21. 54 is 9 sixths of how many times 7 ? 
 
 22. 56 is 7 ninths of how many times 10? 
 
 23. 60 is 10 sevenths of how many times 4 ? 
 
 24. 63 is 9 eighths of how many times 5 ? 
 
 25. 64 is 8 ninths of how many times 7 ? 
 26.- 70 is 10 sevenths of how many times 8 ? 
 
 27. 72 is 9 fifths of how many times 6 ? 
 
 28. 80 is 10 thirds of how many times 4? 
 
 29. 80 is 8 fifths of how many times 6 ? 
 
68 ARITHMETIC. [parti. 
 
 D. 1. A boy gave away 4 cents, which was 1 third 
 of all he had ; how many had he at J5rst ? 
 
 2. A boy gave 5 apples to one of his companions, 
 which was 1 fourth of whatjie had; how many had he? 
 
 3. A man paid away 4 dollars, which was 2 thirds of 
 all the money he had; how much had he? 
 
 4. A man sold a watch for 18 dollars, which was 3 
 fourths of what it cost him ; how much did is cost ? 
 
 5. A man sold a cow for 15 dollars, which was 3 fifths 
 of what the cow cost ; how much did he lose by his 
 bargain ? 
 
 6. A man bought 12 yards of cloth, and sold it for 
 54 dollars, which was 9 eighths of what it cost him; 
 what did it cost him a yard ? and how much did he gain 
 by his bargain ? 
 
 7. There is a pole standing in the water, so that 10 
 feet of it is above the water, which is 2 thirds of the 
 whole length of the pole ; how long is the pole ? 
 
 8. There is a pole 2 thirds under water, and 4 feet 
 out; how long is the pole? 
 
 9 There is a pole 2 fifths under water, and 6 feet 
 out of the water ; how long is the pole ? 
 
 10. There is an orchard, in which 3 sevenths of the 
 trees bear cherries, and 2 sevenths bear pe'aches, and 
 10 trees bear plums ; how many trees are there in the 
 orchard ? and how many of each sort ? 
 
 11. There is a school, in which 2 ninths of the boys 
 learn arithmetic, 3 ninths learn grammar, 1 ninth learn 
 geography, 1 ninth learn geometry, and 12 learn to 
 write ; how many are there in the school ? and how 
 many attending to each study ? 
 
 12. A man sold a watch for 63 dollars, which was 
 7 fifths of what it cost him; how much did he gain by 
 the bargain ? 
 
SEC. VI.] ARITHMETIC. 69 
 
 MISCELLANEOUS EXAMPLES. 
 
 1. If 1 yard of cloth cost 4 dollars, what will 5 yards 
 cost? 
 
 2. A man bought 3 pounds of raisins, at 7 cents a 
 pound, and 16 oranges, at 4 cents apiece, and 1 pound 
 of candles for 16 cents; what did they all come to? 
 
 ' 3. A boy had 37 apples ; he kept 5 himself, and 
 divided the rest equally among four companions ; how 
 many did he give them apiece ? 
 
 4. Two men are 40 miles apart, and both travelling 
 the same way ; the hindermost man gains upon the other 
 5 miles each day ; in how many days will he overtake 
 him ? 
 
 5. Two men are travelling the same way, one travels 
 at the rate of 38 miles a day, the other, who is behind 
 the former, travels 44 miles ; how much does he gain 
 of the first dach day? and in how many days would he 
 gain 60 miles ? 
 
 6. A fox is 80 rods before a greyhound, and is run- 
 ning at the rate of 27 rods in a minute; the greyhound 
 is following at the rate of 31 rods in a minute; in 
 how many minutes will the greyhound overtake the fox? 
 
 7. If 2 yards of cloth cost 6 dollars, what would 4 
 yards cost ? what would 12 yards cost ? 
 
 8. If 8 sheep cost 24 dollars, what would B cost ? 
 
 9. If 4 tons of hay will keep 3 horses through the 
 winter, how many tons will keep 30 horses the same 
 time ? 
 
 10. If a man spends 8 shillings a day, how many dol- 
 lars will he spend in a week ? 
 
 11. Bought 10 pieces of cloth, each containing 5 
 yards, for 100 dollars; what was it a piece? and how 
 much a yard ? 
 
 12. If the wages of 12 weeks come to 60 dollars, 
 what is that a month ? and how much for 5 weeks ? 
 
 13. If 7 horses eat 14 bushels of oats in 1 week, how 
 many bushels would 15 horses eat in the same time? 
 
70 ARITHMETIC. [part i. 
 
 14. If 3 horses eat 8 bushels of oats in 2 weeks, 
 how long would it take them to eat 40 bushels ? 
 
 15. If 1 horse eat 1 bushel of oats in 3 days, in how 
 many days would 4 horses eat 36 bushels ? 
 
 16. If 2 men spend 12 dollars in 1 week, how many 
 dollars would 3 men, at the same rate, spend in 5 
 weeks ? 
 
 17. If a staff 3 feet long cast a shadow of 2 feet at 
 12 o'clock, what is the length of a pole that casts a 
 shadow eighteen feet at the same time of day? 
 
 18. If 47 gallons of water, in 1 hour, run into a 
 cistern containing 108 gallons, and by a pipe 38 gallons 
 run out in an hour, how much remains in the cistern 
 in an hour? and in how many hours will the cistern be 
 filled ? 
 
 19. If 4 men can do a piece of work in 8 days, how 
 many men would it take to do the same work in 4 days? 
 
 20. If 6 men can do a piece of work 'in .9 days, in 
 how many days would 2 men do it ? 
 
 21. If 8 men can do a piece of work in 5 days, in 
 how many days would they do a piece of work 4 times 
 as large ? 
 
 22. If 7 men can do a piece of work in 3 days, how 
 many men would it take to do a piece of work 5 times 
 as large in the same time ? 
 
 23. If 8 men can do a piece of work in 4 days, in 
 how many days would 2 men do a piece of work one 
 half as large ? 
 
 24. A man bought a cask of wine containing 63 gal- 
 lons, 3 sevenths of which leaked out; and he sold the 
 remainder for 36 dollars; how much per gallon did he 
 sell it for? 
 
 25. If a cask of wine cost 48 dollars, what is 5 
 eighths of it worth ? 
 
 26. A man bought 7 oranges for 6 cents apiece, and 
 sold them all for 54 cents, how much did he gain by 
 the bargain ? 
 
 27. A man bought 8 yards of cloth for 56 dollars. 
 
«EC. VII.] ARITHMETIC. 71 
 
 and sold it a<rain for 9 dollars a 3^ard ; how much did 
 he gain by the bargain ? 
 
 28. A man bought 8 barrels of flour for 40 dollars; 
 how much must he sell it at per barrel to gain 16 dol- 
 lars ? 
 
 29. A man bought 5 firkins of butter at 7 dollars a 
 firkin ; how much must he sell it per firkin to gain 10 
 dollars ? 
 
 30. A man gave 35 cents for his breakfast, which 
 was 5 eighths of what he gave for his dinner; what did 
 he give for his dinner ? 
 
 31. A ship's crew of 6 men have provision for 3 
 months ; how many months would it last one man ? 
 
 32. A ship's crew have provision sufficient to last 1 
 man 27 months ; how long would it last 9 men ? 
 
 33. A ship's crew have provision sufficient to last 3 
 men 10 months ; how long would it last 5 men ? 
 
 34. A man built 40 rods of wall in a certain time ; 
 another man can build 9 rods while the first builds 5; 
 how much would he build in the same time ? 
 
 SECTION VII. 
 
 If tlie comhinations in this section should he found 
 too difficulty they may he omitted until reviewing the 
 hooJc. 
 
 A. 1. A MAN being asked the age of his eldest son, 
 answered, that his youngest son was 6 years old, and 
 that 2 thirds of the youngest son's age was just 1 
 fifth of the eldest son's age. Required the age of the 
 eldest son ? 
 
 2. 2 thirds of 6 is 1 fifth of what number? 
 
 3. A man being asked how many sheep he had, said 
 that he had them in two pastures; in one pasture he 
 had 8; and that 3 fourths of these was just 1 third of 
 
72 * ARITHMETIC. [parti. 
 
 what he had in the other. How many were there in the 
 other ? 
 
 4. 3 fourths of 8 is 1 third of what number ? 
 
 5. Two boys talking' of their ages, one said he was 9 
 years old. Well, said the other, 2 thirds of your age 
 is exactly 3 fourths of my age, now, if you will tell me 
 how old I am, I will give you as many apples as I am 
 years old. What was his age ? 
 
 6. 2 thirds of 9 is three fourths of what number? 
 
 7. Two boys counting their money, one said he had 
 10 cents. The other says, 4 fifths of your money is 
 exactly 2 sevenths of mine ; now, if you will tell how 
 many I have, I will give you 1 half of them. How 
 many had he ? 
 
 8. 4 fifths of 10 is 2 sevenths of what number? 
 
 9. 5 sixths of 12 is 2 thirds of what number ? 
 
 10. 6 sevenths of 14 is 4 ninths of what number? 
 
 11. 6 fifths of 15 is 2 thirds of what number ? 
 
 12. 7 ninths of 18 is 2 fifths of what number? 
 
 13. 4 fifths of 20 is 8 sevenths of what number ? 
 
 14. 8 sevenths of 21 is 6 tenths of what number ? 
 
 15. 5 sixths of 24 is 10 sevenths of how many times 
 5? 
 
 16. 3 sevenths of 28 is 2 eighths of how many times 
 7? 
 
 17. 4 fifths of 30 is 6 sevenths of how many times 
 8? 
 
 18. 6 eighths of 32 is 8 ninths of how many times 5? 
 
 19. 4 ninths of 36 is 8 tenths of how many times 6? 
 
 20. 3 fourths of 40 is 5 sevenths of how many times 
 8? 
 
 21. 6 ninths of 45 is 3 fifths of how many times 7 ? 
 
 22. 5 sixths of 48 is 10 sevenths of how many times 
 3? 
 
 23. 4 sevenths of 63 is 6 fifths of how many times 
 8? 
 
 24. 5 ninths of 72 is 4 sevenths of how many times 
 9? 
 
SEC. VII ] ARITHMETIC. 78 
 
 B. 1. 4 fifths of 15 is 6 tenths of how many thirds 
 of 21? 
 
 2. 4 thirds of 18 is 8 ninths of how many sevenths 
 of 35 ? 
 
 3. 6 sevenths of 21 is 2 thirds of how many thirds of 
 24 i ^ 
 
 4. 5 fourths of 24 is 10 sevenths of how many fifths 
 of 40 ? 
 
 5. 5 eighths of 32 is 2 fifths of how many fifths 
 of 35? 
 
 6. 4 sevenths of 63 is 6 eighths of how many ninths 
 of 45 ? 
 
 7. 3 sevenths of 56 is 4 ninths of how many fourths 
 of 28 ? 
 
 8. 3 eighths of 64 is 6 tenths of how many sixths 
 of 30 ? 
 
 9. 2 eighths of 72 is 3 tenths of how many fifths of 
 40? 
 
 C. 1. Two times eleven are how many? 
 
 2. Two times twelve are how many ? 
 
 3. Two times thirteen are how many? 
 
 4. Two times fourteen are how many ? 
 
 5. Two times fifteen are how many? 
 
 6. Two times sixteen are how many? 
 
 7. Two times seventeen are how many? 
 
 8. Two times eighteen are how many ? 
 
 9. Two- times nineteen are how many? 
 
 10. Two times twenty are how many? 
 
 11. Three times eleven are how many? 
 
 12. Three times twelve are how many ? 
 
 13. Three times thirteen are how many? 
 
 14. Three times fourteen are how many? 
 
 15. Three times fifteen are how many? 
 
 16. Three times sixteen are how many? 
 
 17. Three times seventeen are how many? 
 
 18. Three times eighteen are how many? 
 
 19. Three times nineteen are how many? 
 
ARITHMETIC. [part i. 
 
 20. Three times twenty are how many? 
 
 21. Four times eleven are how many ? 
 
 22. Four times twelve are how many ? 
 
 23. Four times thirteen are how many ? 
 
 24. Four times fourteen are how many ? 
 
 25. Four times fifteen are how many ? 
 
 26. Four times sixteen are how many ? 
 
 27. Four times seventeen are how many? 
 
 28. Four times eighteen are how many ? 
 
 29. Four times nineteen are how many ? 
 
 30. Four times twenty are how many ? 
 
 31. Five times eleven are how many? 
 
 32. Five times twelve are how many? 
 
 33. Five times thirteen are how many? 
 
 34. Five times fourteen are how many? 
 
 35. Five times fifteen are how many? 
 
 36. Five times sixteen are how many ? 
 
 37. Five times seventeen are how many ? 
 
 38. Five times eighteen are how many? 
 
 39. Five times nineteen are how many? .• , 
 
 40. Five times twenty are how many? 
 
 41. ^ix times eleven are how many? 
 
 42. Six times twelve are how many? 
 
 43. Six times thirteen are how many? 
 
 44. Six times fourteen are how many? 
 
 45. Six times fifteen are how many? 
 
 46. Six times sixteen are how many? 
 
 47. Six times seventeen are how many ? 
 
 48. Six times eighteen are how many? 
 
 49. Six times nineteen are how many ? 
 
 50. Six times twenty are how many? 
 
 51. Seven times eleven are how many? 
 
 52. Seven times twelve are how many ? 
 
 53. Seven times thirteen are how many? 
 
 54. Seven times fourteen are how many? 
 
 55. Seven times fifteen are how many ? 
 
 56. Seven times sixteen are how many ? 
 
 57. Seven times seventeen are how many? 
 
SEC. VII.] ARITHMETIC. 75 
 
 58. Seven times eighteen are how many? 
 
 59. Seven times nineteen are how many? 
 
 60. Seven times twenty are how many ? 
 
 61. Eight times eleven are how many? 
 
 62. Eight times twelve are how many? 
 
 63. Eight times thirteen are how many? 
 
 64. Eight times fourteen are how many? 
 
 65. Eight times fifteen are how many? 
 
 66. Eight times sixteen are how many? 
 
 67. Eight times seventeen are how many? 
 
 68. Eight times eighteen are how many? 
 
 69. Eight times nineteen are how many? 
 
 70. Eight times twenty are how many? 
 
 71. Nine times eleven are how many ? 
 
 72. Nine times twelve are how many ? 
 
 73. Nine times thirteen are how many ? 
 
 74. Nine times fourteen are how many? 
 
 75. Nine times fifteen are how many? 
 
 76. Nine times sixteen are how many? 
 
 77. Nine times seventeen are how many? 
 
 78. Nine times eighteen are how many? 
 
 79. Nine times nineteen are how many? 
 
 80. Nine times twenty are how many? 
 
 81. Ten times eleven are how many? 
 
 82. Ten times twelve are how many? 
 
 83. Ten times thirteen are how many? 
 
 84. Ten times fourteen are how many? 
 
 85. Ten times fifteen are how many? 
 
 86. Ten times sixteen are how many? 
 
 87. Ten times seventeen are how many? 
 
 88. Ten times eighteen are how many? 
 
 89. Ten times nineteen are how many? 
 
 90. Ten times twenty are how many? 
 
76 AlUTHiMETlC. [parti. 
 
 ' SECTION YIII. 
 
 A. 1. If you cut an apple into two equal parts, what 
 is one of those parts called ?* 
 
 2. How many halves of an apple will make the 
 whole apple? 
 
 3. If you cut an apple into 3 equal parts, what is 
 1 of those parts called? what are 2 of the parts 
 called ? 
 
 4. How many thirds of an apple -will make the 
 whole apple? 
 
 5. If you cut an apple into 4 equal parts, what is 1 
 of those parts called ? what are 2 of those parts called ? 
 what are three of them called? 
 
 6. How many fourths of an apple make the whole 
 apple ? 
 
 7. If an apple be cut into 5 equal parts, what is 1 
 of the parts called ? what are 2 of the parts called ? 
 what are 3 of the parts called? what are 4 of the 
 parts called ? 
 
 8. How many fifths of an apple make the whole 
 apple ? 
 
 9. If an apple be cut into 6 equal parts, what is 1 
 of the parts called ? what are 2 of the parts called ? 
 what 3 ? what 4 ? what 5 ? 
 
 10. How many sixths of an apple make the whole 
 apple ? 
 
 11. If an apple be cut into 7 equal parts, what is 1 
 of the parts called ? what are 2 of the parts called ? 
 what 3 ? what 4 ? what 5 ? what 6 ? 
 
 Let the instructor ask the pupils the divisions of a unit in this man- 
 ner as far as the division into 10 parts. It would be well to ask them 
 further. Then let him begin again, and suppose an orange instead of 
 an apple. After applying the division to several different things, a 
 line may be drawn on the blackboard and divided. It is well often to 
 require the pupil to make the requisite division of a line on the board. 
 
 12. A man had a bushel of corn, and wished to 
 * See Section III, article B, remark before question 1 and 17. 
 
SEC. VIII.] ARITHMETIC. 77 
 
 give 1 half of a bushel apiece to some laborers; how 
 many could he give it to? 
 
 13. How many halves are there in 1 ? 
 
 14. A man divided 2 barrels of flour among his 
 laborers, giving them 1 half of a barrel apieoe; how 
 many men did he give it to? 
 
 15. How many halves are there in 2?* 
 
 16. In 3 bushels of corn, how many half bushels? 
 
 17. How many halves are there in 3? 
 
 18. A boy divided 4 oranges among his companions, 
 giving them 1 half of an orange apiece; how many boys 
 did he give them to? 
 
 19. How many halves are there in 4? 
 
 20. A man having some laborers, gave them 1 half 
 a dollar apiece; it took 3 dollars and 1 half a dollar to 
 pay them; how many laborers were there? 
 
 21. How many halves are there in 3 and 1 half? 
 
 22. How many halves are there in 5? 
 
 23. How many halves arc there in 7 and 1 half? 
 
 24. How can you tell how many halves there are in 
 any number? 
 
 A71S. 3ince there are 2 halves in one, there must be 
 twice as many halves as there are whole ones. 
 
 '25. If you had 1 orange, and should divide it among 
 your companions, giving them 1 third apiece, how 
 many could you give it to? 
 
 26. How many thirds are there in 1? 
 
 27. If you cut 2 oranges each into 3 pieces, how 
 many pieces would they make ? 
 
 28. If you cut 3 oranges into 3 pieces each, how 
 many pieces would they make? 
 
 29. If you cut 4 apples each into 3 pieces, how 
 many pieces would they make ? 
 
 30. How many thirds are there in 2? in 3? in 4? 
 in 5? 
 
 31. How can you tell how many thirds there are in 
 any number ? 
 
 * See the Key. 
 
78 ARITHMETIC. [pAitT i. 
 
 Ans. Since there are 3 thirds in one, there will be 3 
 times as many thirds as there are whole ones. 
 
 32. If you had 2 bushels and 1 third of a bushel of 
 corn to give to some poor persons, how many could you 
 give it to if you should give them 1 third of a bushel 
 apiece? 
 
 33. How many thirds are there in 2 and 1 third? 
 
 34. If a horse can eat 1 third of a bushel of oats 
 in 1 day, how many days would it take him to eat 3 
 bushels and 2 thirds of a bushel? 
 
 35. How many thirds are there in 3 and 2 thirds? 
 
 36. If 1 horse can eat 1 third of a bushel of oats 
 in a day, how many horses will it take to eat 5 bushels 
 and 2 thirds of a bushel in the same time? 
 
 37. In 5 and 2 thirds how many thirds? 
 
 38. In 7 and 1 third how many thirds? 
 
 39. If a horse will eat 1 fourth of a ton of hay in 
 1 month, how many horses will eat a ton in the same 
 time? How many will eat 2 tons? How many will 
 eat 3 tons? 4 tons? 5 tons? 6 tons? 
 
 40. How many fourths are there in 1 ? in 2 ? in 3 ? 
 in -4? in 5? in 6? 
 
 41. How can you tell how many fourths there are in 
 any number. ' """ 
 
 42. How many fourths are there in 4 and 1 fourth? 
 
 43. In 5 tons of hay and 3 fourths of a ton, how 
 many fourths of a ton ? 
 
 44. In 5 and 3 fourths how many fourths ? 
 
 45. In 7 and 1 fourth how many fourths? 
 
 46. In 9 and 3 fourths how many fourths? 
 
 47. If a horse eat 1 fifth of a ton of hay in a 
 month, how many horses will eat a ton in the same 
 time? 2 tons? 3 tons? 4 tons? 5 tons? 
 
 48. How many fifths are there in 1? in 2? in 3? 
 in 4? in 5? in 7? in 9? 
 
 49. How can you tell how many fifths there are in 
 any number? 
 
 50. In 2 and 1 fifth how many fifths? 
 
BBC. vm.] ARITHMETIC. 79 
 
 51. In 3 dollars and 2 fifths of a dollar, how many 
 fifths of a dollar. 
 
 52. In 3 and 2 fifths how many fifths? 
 
 53. In 5 and 3 fifths how many fifths? 
 
 54. In G and 4 fifths how many fifths? 
 
 55. How many sixths are there in 1 ? in 2? In 3 ? 
 in 4? in 5? in 7? in 8? 
 
 56. In 2 and 2 sixths how many sixths? 
 
 57. In 3 and 4 sixths how many sixths ? 
 
 58. In 6 and 5 sixths how many sixths ? 
 
 59. How many sevenths are there in 1? in 2? in 3 ? 
 in4J in 6? in 9? in 10? 
 
 60. In 3 and 1 seventh how many sevenths? 
 
 61. In 5 and 3 sevenths how many sevenths? 
 
 62. In 7 and 5 sevenths how many sevenths? 
 
 63. How many eighths are there in 1? in 3 ? in 5? 
 in 8? 
 
 64. In 2 and 3 eighths how many eighths? 
 
 65. In 3 and 5 eighths how many eighths ? 
 
 66. In 5 and 7 eighths how many eighths ? 
 
 67. How many ninths are there in 1 ? in 2 ? in 7? 
 in 5? 
 
 68. In 2 and 2 ninths how many ninths? 
 
 69. In 4 and 3 ninths how many ninths? 
 
 70. In 6 and 4 ninths how many ninths ? 
 
 71. In 8 and 7 ninths how many ninths ? 
 
 72. How many tenths in 1 ? in 2 ? in 5 ? in 8 ? 
 
 73. In 3 and 3 tenths how many tenths? 
 
 74. In 4 and 7 tenths how many tenths? 
 
 75. In 8 and 9 tenths how many tenths? 
 
 76. In 7 and 4 tenths how many tenths? 
 
 77. In 9 and 8 tenths how many tenths? 
 
 78. In 7 and 4 sevenths how many sevenths? 
 
 79. In 9 and 2 thirds how many thirds? 
 
 80. In 10 and 3 fourths how many fourths? 
 
 81. In 8 and 4 fifths how many fifths? 
 
 82. In 7 and 5 ninths how many ninths? 
 
80 ARITHMETIC. [part i 
 
 B, 1. If you give 4 men 1 half of a barrel of flour 
 apiece, how many barrels will it take? 
 
 2. In 4 halves how many times 1 ? 
 
 3. If you give 3 boys 1 half of an orange apiece, 
 how many oranges will it take ? 
 
 4. In 3 halves how many times 1 ? 
 
 5. If you give five men 1 half of a dollar apiece, \ 
 how many dollars will it take? k 
 
 6. In 5 halves how many times 1 ? 
 
 7. In 6 halves how many times 1 ? 
 
 8. In 7 halves how many times 1 ? 
 
 9. How can you tell how many whole ones there are 
 in any number of halves ? 
 
 10. A man divided some corn among 6 persons, 
 giving them 1 third of a bushel apiece; how many 
 bushels did it take ? 
 
 11. In 6 thirds how many times 1 ? 
 
 12. In 5 thirds how many times 1 ? 
 
 13. A man gave 8 paupei:s 1 third of a dollar apiece, 
 how many dollars did it take ? 
 
 14. In 8 thirds how many times 1 ? 
 
 15. In ten thirds how many times 1 ? 
 
 16. How can you tell how many whole ones there 
 are in any number of thirds ? 
 
 17. If a man spends 1 fourth of a dollar in 1 day, 
 how many dollars will he spend in 8 days? How many 
 in 7 days? How many in 11 days? 
 
 18. In 8 fourths how many times 1 ? 
 
 19. In 7 fourths how many times 1 ? 
 
 20. In 11 fourths how many times 1? 
 
 21. In 13 fourths how many times 1 ? 
 
 22. In 18 fourths how many times 1 ? 
 
 23. How can you tell how many whole ones there are 
 in any number of fourths ? 
 
 24. If 1 fifth of a barrel of flour will last a family 
 1 day, how many barrels will last thorn 10 days ? How 
 many 8 days? 11 days? 17 days? 
 
 25. In 10 fifths how many times 1 ? 
 
SEC. IX. 1 ARITHMETIC. ^81 
 
 26. In 8 fifths how many times 1 ? 
 
 27. In 11 fifths how many times 1 ? 
 
 28. In 17 fifths how many times 1 ? 
 
 29. In 18 sixths how many times 1 ? 
 
 30. In 23 fifths how many times 1 ? 
 
 31. In 21 sevenths how many times 1? 
 
 32. In 24ei2;hths how many times 1? 
 
 33. In 36 ninths how many times 1 ? 
 
 34. In 30 tenths how many times 1 ? 
 
 35. In 35 fourths how many times 1 ? 
 
 36. In 37 eighths how many times 1 '( 
 
 37. In 43 fifths how many times 1 ? 
 
 38. In 48 ninths how many times 1 ? 
 
 39. In 53 tenths how many times 1 ? 
 
 40. In 57 eighths how many times 1 ? 
 
 41. In 76 tenths how many times 1 ? 
 
 42. In 78 ninths how many times 1 ? 
 
 SECTION IX. 
 
 A. 1. If a breakfast for 1 man costs 1 third of a dol- 
 lar, what would a breakfast for 2 men cost ? 
 
 2. How much is 2 times 1 third ? 
 
 3. If it takes you 1 third of an hour to travel 1 mile, 
 how long will it take you to travel 3 miles? 
 
 4. How much is 3 times 1 third? 
 
 5. If 1 man can eat 1 third of a pound of meat at a 
 meal, how much can 5 men eat ? 
 
 6^How much is 7 times 1 third? 
 
 7v If 1 man can eat 2 thirds of a pound of meat 
 for dinner, how many thirds of a pound would 3 men 
 eat? 
 
 8. How much is 2 times 2 thirds ? 
 
 9. A man gave to 4 paupers 2 thirds of a dollar 
 apiece, how many thirds of a dollar did he give them ? 
 how many dollars? 
 
82 ARITHMETIC. [part i. 
 
 10.5 times 2 thirds are how many thirds ? how many 
 times 1 y 
 
 11. If you give 3 men 1 fourth of a dollar apiece, 
 how many fourths of a dollar will it take ? 
 
 12. 3 times 1 fourth are how many fourths? 
 
 13* If you give 3 men 3 fourths of a bushel of corn 
 apiece, how many fourths of a bushel will it take ? how 
 many bushels ? 
 
 14. 5 times 3. fourths are how many fourths ? how 
 many times 1 ? 
 
 15. If 1 horse eats 1 fifth of a bushel of oats in a day, 
 how much will 4 horses eat in the same time ? 
 
 16. 3 times 1 fifth are how many fifths ? 
 
 17. If 1 man can earn 3 fifths of a dollar in a day, 
 how much can he earn in 4 days ? 
 
 18. 7 times 3 fifths are how many fifths? how many 
 times 1 ? 
 
 19. If a family consume 2 sevenths of a barrel of 
 flour in a week, how much would they consume in 5 
 weeks? 
 
 20. 6 times 2 sevenths are how many sevenths ? how 
 many times 1 ? 
 
 21. 5 times 3 eighths are how many eighths? how 
 many times 1 ? 
 
 22. How much is 6 times 3 fifths ? 
 
 23. How much is 7 times 5 sixths ? 
 
 24. How much is 5 times 4 ninths ? 
 
 25. How much is 6 times 8 ninths? 
 
 26. How much is 7 times 9 tenths? 
 
 27. How much is 5 times 7 tenths ? 
 
 28. How much is 6 times 7 eighths? ^ 
 
 29. How much is 9 times 5 eighths ? ^ 
 
 30. How much is 8 times 5 sevenths ? 
 
 31. How much is 7 times 5 sixths? 
 
 32. How much is 8 times 7 fourths ? 
 
 33. How much is 7 times 4 fifths ? 
 
 34. How much is 5 times 3 eighths? 
 
SEC. IX.] ARITHMETIC. 83 
 
 B. 1. If 1 bushel of wheat costs a dollar and 1 half, 
 what will 2 bushels cost ? 
 
 2. How much is 2 times 1 and 1 half?* 
 
 3. If a barrel of cider costs 2 dollars and a half, what 
 will 3 barrels cost ? 
 
 4. How much is 4 times 2 and 1 halff 
 
 5. If a barrel of beer costs 3 dollars and a half, what 
 will 2 barrels cost ? 
 
 6. How much is 5 times 3 and 1 half? 
 
 7. How much is 6 times 3 and 1 lutlf? 
 
 8. If a box of butter costs 2 dollars and 1 third of a 
 dollar,- what will 3 boxes cost 1 
 
 9. How much is 4 times 2 and 1 third? 
 
 10. If you give to 2 persons 3 bushels and a third 
 of a bushel of wheat apiece, how many bushels will it 
 take ? 
 
 11. How much is 5 times 3 and 1 third? 
 
 12. If you give to 4 persons each 2 oranges and 1 
 fourth of an orange, how many oranges will it take? 
 
 13. How much is 5 times 2 and 1 fourth? 
 
 14/ If it takes 3 yards and 2 thirds of a yard of cloth 
 to make a suit of clothes, how many yards will it take 
 to make 2 suits ? 
 
 15. How much is 4 times 3 and 2 thirds? 
 
 16. If a family consume 2 bushels and 2 thirds of a 
 bushel of malt in 1 month, how much will they con- 
 sume in 3 months ? 
 
 17. How much is 5 times 2 and 2 thirds ? 
 
 18. How much is 4 times 3 and 3 fourths? 
 
 19. How much is 2 times 3 and 1 fourth ? 
 
 20. How much is 3 times 3 and 3 fourths ? 
 
 21. How much is 3 times 5 and 1 fourth? 
 
 22. If a horse eats 3 tons and 1 fifth of a ton of hay 
 in a year, how much will 2 horses eat in the same 
 time? 
 
 *This is to be understood 2 times 1 and 2 times 1 half, and to be 
 answered thus : 2 times 1 are 2, and two times 1 half are 2 halves or 1, 
 which, added to 2, makes 3. 
 
84 ARITHMETIC. [parti. 
 
 23. How much is 4 times 3 and 1 fifth ? 
 
 24. If a man can travel 4 miles and 2 fifths of a 
 mile in 1 hour, how far will he travel in 3 hours ? 
 
 25. How much is 5 times 4 and 2 fifths ? 
 
 26. How much is 3 times 5 and Z fifths ? 
 
 27. How much is 4 times 6 and 2 fifths? 
 
 28. How much is 3 times 2 and 1 sixth? 
 
 29. How much is 2 times 5 and 3 sixths? 
 
 30. If a yard of cloth costs 4 dollars and 5 sixths 
 of a dollar, what will 4 yards cost? 
 
 31. How much is 7 times 4 and 5 sixths? 
 
 32. How much is 2 times 3 and 3 sevenths? 
 
 33. How much is 3 times 4 and 3 sevenths? 
 
 34. If a bushel of wheat costs 7 shillings and 3 
 eighths, what will 5 bushels cost at that rate ? 
 
 35. How much is 3 times 4 and 5 eigldhs? 
 
 36. How much is 4 times 3 and 7 eighths? 
 
 37. A man bought 8 yards of cloth, at 9 dollars and 
 3 tenths a yard; how much did it come to? 
 
 38. How much is 6 times 2 and 2 ninths? 
 
 39. How much is 4 times 5 and 3 ninths? 
 
 40. A man bought 10 barrels of cider at 3 dollars 
 and 5 sixths a barrel ; how much did it come to? 
 
 41. How much is 3 times 7 and 2 tenths? 
 
 42. What cost 8 barrels of beef at 9 dollars and 3 
 sevenths a barrel? 
 
 C. 1. A boy wished to give 5 other boys 1 half pint 
 of chestnuts apiece; how many pints would it take? 
 
 2. A boy wished to give 3 other boys 3 fourths of 
 an orange apiece; how many oranges would it take? 
 
 3. A man gave to 10 persons 1 fifth of a bushel of 
 corn apiece; how many bushels did it take? 
 
 4. A man gave 7 men 3 fourths T)f a gallon of beer 
 apiece; how many gallons of beer did it take? 
 
 5. If it takes 1 yard and 1 fourth of a yard of cloth 
 to make a pair of pantaloons, how many yards woul(^ 
 it take to make 8 pairs ? 
 
SEC. X.] ARITHMETIC. 85 
 
 6. If a family consume 2 bushels and 3 fifths of a 
 bushel of grain in 1 week, how many bushels would 
 they consume in 4 weeks? 
 
 7. If a horse eats 3 bushels and 4 sevenths of a 
 bushel of oats in 1 week, how many bushels would he 
 eat in 8 weeks? 
 
 8. If a horse eats 5 loa<ls and 3 eights of a load of 
 hay in 1 year, how many loads would 6 horses eat? 
 
 9. If a man travels 4 miles and 5 ninths in an hour, 
 how many miles would he travel in 8 hours ? 
 
 10. If in an orchard of 10 trees, each tree bears 8 
 bushels and 3 sevenths, how many bushels will the 
 whole orchard bear? 
 
 11. If a man can build 5 rods and 7 eighths of a 
 rod of wall in 1 day, how many rods can he build in 8 
 days ? 
 
 12. If 3 men can build a piece of wall in 4 days 
 and 3 fifths of a day, how many days would it take 1 
 man to build it? 
 
 13. If 1 man can build 7 rods and 2 fifths of wall 
 in a day, how many rods would 10 men build? 
 
 • 14. If 1 man can build 3 rods and 2 ninths of wall 
 in 1 day, how many rods would 3 men build in 4 
 days? 
 
 15. If it takes 1 yard and 3 sevenths of a yard of 
 cloth to make 1 pair of pantaloons, and 2 yards and 4 
 sevenths for a coat ; how many yards would it take to 
 make 3 pairs of pantaloons and 3 coats? 
 
 SECTION X. 
 
 A. 1. If a yard of cloth costs 3 dollars, what will 1 
 half of a yard cost? 
 
 2. What is 1 half of 3 ?* 
 
 3. If a barrel of beer costs 5 dollars, what will 1 
 half of a barrel cost ? 
 
 * See Section V., article A. 
 
86 ARITHMETIC. [part i. 
 
 4. What is 1 half of 5? 
 
 5. If 2 barrels of cider cost 7 dollars, what is that a 
 barrel ? 
 
 6. What is 1 half of 7 ? 
 
 7. What is 1 half of 9 ? 
 
 8. What is 1 half of 11? 
 
 9. What is 1 half of 13? • 
 
 10. What is 1 half of 15? 
 
 11. If you divide 1 bushel of wheat equally among 
 3 persons, what part of a bushel will you give them 
 apiece? 
 
 12. If 3 yards of cloth cost 1 dollar, what part of a 
 dollar will 1 yard cost? 
 
 13. What is 1 third of 1? 
 
 14. How could you divide 2 oranges into 3 equal 
 parts? that is, how can you find 1 third of 2 oranges?* 
 
 15. One third of 2 oranges will be the same as how 
 many thirds of 1 orange? 
 
 16. If you divide 2 bushels of wheat equally among 
 3 persons, what part of a bushel will you give them 
 apiece ? 
 
 17. If 3 bushels of corn cost 2 dollars, what part' 
 of a dollar will 1 bushel cost? 
 
 Note. — One third of two things is twice as much as one third of one 
 thing. One third of one is one third, and consequently one third 
 of two things is two thirds. In the same manner, one tJiird of four 
 things is four thirds of one thing. If four oranges be cut each into 
 three parts, and then one part of each be talven, it will make four 
 pieces, each of which is one third of one orange. Plence one third 
 of four oranges is four thirds of one orange, that is, one whole one 
 and one third. 
 
 18. If 3 bushels of wheat cost 4 dollars, how much 
 is that a bushel? 
 
 19. What is 1 third of 2? of 4? 
 
 20. If 3 gallons of wine cost 5 dollars, what is that 
 a gallon ? 
 
 21. What is 1 third of 5? of 7? of 8? of 10? of 
 11? 
 
 * Divide each orange into three parts, and then take one part from 
 each. 
 
SEC. X.] ARITHMETIC. 87 
 
 22. If a bushel of apples be divided equally among 
 
 4 persons, what part of a bushel will they have apiece? 
 What would they have apiece if 2 bushels were divided 
 among them? What if 3 bushels? W^hat iff) bush- 
 els? What if 6 bushels? 
 
 23. W^hat is 1 fourth of 1? of 2? of 3? of 5? of 
 6? of 7? of 9? of 10? 
 
 2-4. If a bushel of malt will serve 5 persons 1 month, 
 how much will serve 1 person the same time? 
 
 25. If 2 barrels of cider will serve 5 persons 1 
 month, how much will serve one person the same time? 
 
 26. If 3 barrels of flour be divided among 5 men, 
 how much will each have ? If 4 barrels were divided, 
 what would each have ? What if 6 barrels were 
 divided? What if 7 barrels were divided ? 
 
 27. What is 1 fifth of 1? of 2? of 3? of 4? of 6? 
 of 7? 
 
 28. What is 1 sixth of 1? of 2? of 3? of 4? of 
 5? of 7? of 8? of 9? of 10? 
 
 29. What is 1 seventh of 1? of 2? of 3 ? of 4? 
 of 6? of 6? 
 
 30. What is 1 eighth of 1? of 2 ? of 3? of 4? of 
 5? of G? of 7? of 8? of 9? of 10? 
 
 31. What is 1 ninth of 1? of 2? of 3? of 4? of 
 5? of 6? of 7? of 8? of 9? of 10? of 11? 
 
 32. What is 1 tenth of 1 ? of 2 ? of 3 ? of 4 ? of 
 
 5 ? of 6 ? of 7 ? of 8 ? of 9 ? of 10 ? of 11 ? of 12 ? 
 of 13? 
 
 33. If 3 yards of cloth cost 2 dollars, what will 1 
 yard cost ? What will 2 yards cost ? 
 
 34. If 1 bushel of wheat costs 2 dollars, what will 1 
 third of a bushel cost? What will 2 thirds of a 
 bushel cost? 
 
 35. What is 1 third of 2? 
 
 36. What is 2 thirds of 2? 
 
 37. If a load of wood costs 5 dollars, what will 1 
 third of a load cost? What will 2 thirds of a load 
 cost? 
 
88 ARITHMETIC. [part i. 
 
 38. What is 1 third of 5? 
 
 39. What is 2 thirds of 5? 
 
 40. What is 2 thirds of 7 ? 
 
 41. If 4 bushels of salt cost 3 dollars, what will 1 
 bushel cost ? What will 3 bushels cost? 
 
 42. What is 1 fourth of 3? 
 
 43. What is 3 fourths of 3? 
 
 44. If a barrel of cider costs 2 dollars, what will 1 
 fifth of a barrel cost? What will 2 fifths of a barrel 
 cost ? 
 
 45. What is 1 fifth of 2? 
 
 46. What is 2 fifths of 2? 
 
 47. What is 1 fifth of 7? 
 
 48. What is 3 fifths of 7? 
 
 49. If 7 gallons of wine cost 5 dollars, what will a 
 gallon cost ? What will 4 gallons cost ? 
 
 50. What is 1 seventh of 5? 
 
 51. What is 4 sevenths of 5? 
 
 52. What is 1 sixth of 4? 
 
 53. What is 5 sixths of 4? . . 
 
 54. If you divide 7 dollars among 8 men, what part 
 of a dollar will you give them apiece? What would 
 3 of them have? 
 
 55. What is 1 eighth of 7? 
 
 56. What is 3 eii^hths of 7? 
 
 57. What is 1 ninth of 10? 
 
 58. What is 7 ninths of 10 ? 
 
 59. What is 1 tenth of 14? 
 
 60. What is 4 tenths of 14? 
 
 61. If 5 yards of cloth cost 17 dollars, what is that 
 a yard? What would 3 yards cost? What would 8 
 yards cost? 
 
 62. What is 3 fifths of 17? 
 
 63. What is 8 fifths of 17? 
 
 64. What is 5 sevenths of 20? 
 
 65. What is 2 ninths of 22? 
 
 66. What is 3 eighths of 27 ? 
 
 67. What is 2 thirds of 28? 
 
SEC. X.] 
 
 68. 
 
 69. 
 load? 
 cost? 
 
 70. 
 
 71. 
 
 72. 
 
 73. 
 
 74. 
 
 75. 
 
 76. 
 
 77. 
 
 78. 
 
 79. 
 
 80. 
 
 ARITHiMETIC. 
 
 89 
 
 What is 3 fourths of 31? 
 
 If 5 loads of hay cost 47 dollars, what is that a 
 
 What will 2 loads cost? What will 7 loads 
 
 AVhat will 12 loads cost? 
 
 -What 
 What 
 What 
 What 
 What 
 What 
 What 
 What 
 What 
 What 
 What 
 
 s 2 fifths of 47 ? 
 s 7 fifths of 47? 
 s 12 fifths of 47? 
 s 4 sevenths of 48? 
 s 4 ninths of 50? 
 s 2 sevenths of 58? 
 s 3 eighths of 61? 
 s 4 tenths of 73 ? 
 s 8 ninths of 6^? 
 s 9 tenths of 78? 
 s 7 eighths of 70? 
 
 B. 1. If you divide 7 apples equally between 2 boys, 
 how many would you give them apiece? 
 
 2. How can you divide 5 oranges eijually among 3 
 persons? 
 
 3. Divide 3 bushels of corn equally among 5 men, 
 how much would you give them apiece? 
 
 4. A boy had 7 pears, and gave away 3 fifths of 
 them; how many did he give away? and how many 
 had he left ? 
 
 5. If 2 yards of cloth cost 3 dollars, what is that a 
 yard ? 
 
 6. If4yardsof cloth cost 2 dollars, what is that a yard? 
 [Let the answers be given in dollars and cents, or in 
 
 shillings. 1 dollar is 100 cents.] 
 
 7. If 5 bushels of corn cost 7 dollars, what is that 
 a bushel? 
 
 8. If a man receives 8 dollars for 6 days' work, what 
 is that per day ? 
 
 9. If 3 bushels of wheat cost 8 dollars, what will 2 
 bushels cost ? 
 
 10. A man had 30 dollars, and gave away 3 sevenths 
 of it; how much did he give away? 
 
90 ARITHMETIC. [part i. 
 
 11. If 4 yards of cloth cost 10 dollars, what will 3 
 yards cost? 
 
 12. If 3 barrels of cider cost 8 dolkrs, what will 10 
 barrels cost? 
 
 13. If 7 pounds of flour cost 40 cents, what will 10 
 pounds cost? 
 
 14. If 4 firkins of butter cost 26 dollars, what will 
 7 firkins cost? 
 
 15. If 3 men can do a piece of work in 7 days, how 
 long will it take 1 man to do it? How long would it 
 take 4 men? 
 
 16. If 2 cocks will empty a cistern in 3 hours, in 
 how long a time would 1 empty it? In how long a 
 time would 7 cocks empty it? 
 
 SECTION XL 
 
 A. 1. Bought 1 half of a yard of cloth for 1 dol- 
 lar and 1 half ; what was that a yard ? 
 
 2. 1 and 1 half is the half of what number?* 
 
 3. If 1 third of a yard of cloth costs 1 dollar and 3 
 fourths of a dollar, how much does a yard cost ? 
 
 4. 2 and 3 fourths is 1 third of what number? 
 
 5. If 1 half of a barrel of beer costs 2 dollars and 1 
 fourth of a dollar, how much will a barrel cost ? 
 
 6. 2 and 3 fourths is 1 half of what number? 
 
 7. If 1 fourth of a box of lemons costs 3 dollars and 
 2 thirds of a dollar, what will a box cost? 
 
 8. 2 and 2 thirds is 1 fourth of what number ? 
 
 9. 3 and 1 half is 1 fourth of what number? 
 
 10. If 1 third of a barrel of pork costs 4 dollars and 
 1 third of a dollar, what will a barrel cost ? 
 
 11. 4 and 2 thirds is 1 third of what number? 
 
 12. If 1 sixth of a barrel of fish costs 2 dollars and 
 1 fifth of a dollar, what will a barrel cost? 
 
 * See Section VI. article A., and Section IX. article B. 
 
SEC. XI ] ARITHMETIC. 91 
 
 13. 3 and 2 fifths is 1 sixth of what number? 
 
 14. If 1 fifth of a barrel of salmon costs 3 dollars 
 and 2 sevenths of a dollar, what is that a barrel? 
 
 15. 4 and 3 sevenths is 1 fifth of what number? 
 
 16. If a man can travel 4 miles and o sevenths of a 
 mile in 1 ninth of a day, how far will he travel in a 
 whole day? 
 
 17. 5 and 3 sevenths is 1 ninth of what number? 
 
 18. 2 and 4 fifths is-l seventh of what number? 
 
 19. 6 and 3 eighths is 1 eijihth of what number? 
 
 20. 7 and 5 sixths is 1 third of what number? 
 
 21. 8 and 3 sevenths is 1 fourth of what number? 
 
 22. 5 and 3 tenths is 1 seventh of what number? 
 
 23. 8 and 5 ninths is 1 fifth of what number? 
 
 24. 9 and 4 fifths is 1 eighth of what number? 
 
 25. 6 and 4 tenths is 1 tenth of what number? 
 
 26. 7 and 4 ninths is 1 fifth of what number? 
 
 27. 8 and 7 tenths is 1 seventh of what number ? 
 
 B. 1. If 2 thirds of a barrel of beef cost 3 dollars, 
 what does 1 third of a barrel cost ?* 
 
 2. 3 is 2 times what number? 
 
 Ans. 3 is 2 times the half of 3 ; but 1 half of 3 is 
 3 halves, or 1 and 1 half,! therefore 3 is 2 times 1 and 
 1 half. 
 
 3. If 3 fifths of a yard of cloth cost 4 dollars, what 
 will 1 fifth of a yard cost ? 
 
 4. 5 is 3 times what number ? 
 
 5. If 3 sevenths of a barrel of pork cost 2 dollars, 
 .what will 1 seventh of a barrel cost? 
 
 6. 2 is three times what number? 
 
 7. If 4 thirds of a bunch of shingles cost 5 dollars, 
 what does 1 third of a bunch cost? 
 
 8. 7 is 4 times what number? 
 
 9. 2 is 4 times what number ? 
 
 10. A man bought 4 ninths of a barrel of flour for 
 
 * See Section VI., article B, 
 t See Section X., article A. 
 
92 ARITHMETIC. [parti. 
 
 3 dollars ; what would be the price of 1 ninth of a bar- 
 rel at the same rate ? 
 
 11. 5 is 4 times what number ? 
 
 12. A man bought 5 eighths of a hundred weight of 
 sugar for six dollars ; what would 1 eighth of a hun- 
 dred weight cost at the same rate ? 
 
 13. 7 is 5 times what number ? 
 
 14. 3 is 5 times what number ? 
 
 15. 8 is 5 times what number? 
 
 16. 9 is 4 times what number? 
 
 17. 11 is 6 times what number ? 
 
 18. 13 is 7 times what number ? 
 
 19. 14 is 8 times what number ? 
 
 20. 17 is 5 times what number ? 
 
 21. 18 is 8 times what number ? 
 
 22. 17 is 9 times what number ? 
 
 23. 15 is ten times what number ? 
 
 24. 20 is 9 times what number ? 
 
 25. 22 is 10 times what number ? 
 
 26. 24 is 7 times what number ? 
 
 27. If 2 thirds of a barrel of beef cost 8 dollars, 
 what will 1 third of a barrel cost ? What will the whole 
 barrel cost ? 
 
 28. *If 5 is 2 thirds of some number, what is 1 third 
 of the same number? 2 and 1 half is 1 third of what 
 number ? Then 5 is 2 thirds of what number ? 
 
 29. If 3 fourths of a barrel of flour cost 5 dollars, 
 what will 1 fourth of a barrel cost? What will the whole 
 barrel cost ? 
 
 30. If 8 is 3 fourths of some number, what is 1 fourth 
 of the same number? 2 and 2 thirds is 1 fourth of what 
 number ? Then 8 is 3 fourths of what number ? 
 
 31. A man boucjht 2 sevenths of a barrel of oil for 
 5 dollars j how much will 1 seventh cost at the same 
 rate ? how much would a barrel cost ? 
 
 32. If 9 is 2 sevenths of some number, what is 1 
 
 * See Sec. VI., Art. B., Examples 8 and following. 
 
sec: XI.] ARITHMETIC. 93 
 
 seventh of the same number 't 4 and 1 half is 1 seventh 
 of what number ? Then 9 is 2 sevenths of what num- 
 ber ? 
 
 33. Boutxht 5 eighths of a chaldron of coal for 7 
 dollars ; what is the price of 1 eighth, at the same rate ? 
 What is the price of the whole chaldron 't 
 
 34. If 12 is 5 eighths of some number, what is 1 
 eighth of the same number? 2 and 2 fifths is 1 eighth 
 of what number? Then 12 is 5 eighths of what num- 
 ber ? 
 
 35. If a man ejin do 7 ninths of a piece of work in 
 4 days, how long would it take him to do 1 ninth of it ? 
 how long would it take him to do the whole ? 
 
 3(). If 3 is 7 ninths of some number, what is 1 ninth 
 of the same number ? 3 sevenths is 1 ninth of what 
 number? Then 3 is 7 ninths of what number? 
 
 37. If 3 fifths of a chaldron of coal cost 8 dollars, 
 what is the whole chaldron worth ? 
 
 38. 7 is 3 fifths of what number ? 
 
 39. A man bought 6 sevenths of a cask of raisins 
 for 5 dollars ; what was the whole cask worth ? 
 
 40. 8 is 6 sevenths of what number ? 
 
 41. A man had 4 sixths of a week's board for 3 dol- 
 lars ; how much is that for a whole week ? 
 
 42. 3 is 5 sixths of what number ? 
 
 43. 9 is 5 eighths of what number? 
 
 44. 10 is 7 thirds of what number? 
 
 45. 11 is 6 fifths of what number ? 
 
 46. 12 is 7 fifths of what number ? 
 
 47. 15 is 2 sevenths of what number? 
 
 48. 17 is 3 tenths of what number? 
 
 49. A man bought 5 eighths of a cask of wine for 
 19 dollars, what would the whole cask cost ? 
 
 50. 19 is 3 eighths of what number ? 
 
 51. 21 is 4 fifths of what number ? 
 
 52. Bought 5 ninths of a ton of logwood for 23 dol- 
 lars ; what would a ton cogt at that rate ? 
 
 53. 23 is 7 ninths of what number? 
 
94 ARITHMETIC [part i. 
 
 54. 21 is 3 tenths of what number? 
 
 55. 29 is 4 sevenths of what number ? 
 
 56. 31 is 5 eighths of what number ? 
 
 57. 33 is 6 sevenths of what number ? 
 
 58. 38 is 4 ninths of what number ? 
 
 59. A man bought 1 barrel and 1 seventh of a bar- 
 rel of wine for 41 dollars; what was that a barrel ? 
 
 Note. — 1 barrel and 1 seventh of a barrel is the 
 salne as 8 sevenths of a barrel. If 8 sevenths of a 
 barrel cost 41 dollars, what does a barrel cost ? 
 
 60. 35 is 8 sevenths of what number ? 
 
 61. A man bought 1 ton and 4 fifths of a ton (that 
 is, 9 fifths of a ton) of fustic for 43 dollars, what was 
 that a ton ? 
 
 62. 52 is 9 fifths of what number? 
 
 63. Bought 1 ton and 2 sevenths of a ton of logwood 
 for 48 dollars, what was that a ton ? 
 
 64. 67 is 9 sevenths of what number ? 
 
 65. 53 is 7 tenths of what number ? 
 
 66. 58 is 9 eighths of what number ? 
 
 67. 61 is 10 ninths of what number ? 
 
 68. Bought 2 barrels and 1 fifth of a barrel of oil 
 (that is, 11 fifths of a barrel) for 65 dollars; what was 
 that a barrel ? 
 
 69. 65 is 9 fifths of what number? 
 
 70. 71 is 8 elevenths of what number ? 
 
 C. 1. A boy gave away 2 apples and 1 half, which 
 was 1 fourth of all he had ; how many had he ? 
 
 2. A man gave away 3 dollars, which was. 2 fifths of 
 all the money he had ; how much had he ? 
 
 3. A man sold a cow for 15 dollars, which was 4 
 fifths of what she cost him ; how much did he lose by 
 the bargain ? 
 
 4. A man sold a piece of cloth for 37 dollars, which 
 was 9 eighths of what it cost him; how much did he 
 gain by the bargain ? 
 
SEC. XI.] ARITHMETIC. 95 
 
 5. There is a pole 3 fifths under water, and 7 feet 
 out of the water ; how Ions: is the pole ? 
 
 6. A man sold a piece of cloth for 47 dollars, by 
 which bargain he lost 2 ninths of what the cloth cost 
 him ; how much did it cost him, and how much did he 
 lose ? 
 
 MISCELLANEOUS EXAMPLES. 
 
 1. If a staff 5 feet lonor casts a shadow 4 feet at 12 
 o'clock, what is the length of a pole that casts a shadow 
 67 feet at the same time ? 
 
 2. If 58 gallons of water, in 1 hour run into a cistern 
 containing 97 gallons, and 44 gallons run out in an hour, 
 in what time will it be filled '/ 
 
 3. A mafi bought a cask of wine containing 75 gal- 
 lons ; 2 sevenths of it leaked out, and he sold the 
 remainder for 1 dollar a gallon; how much did he sell 
 it for ? 
 
 4. A cask of wine cost 67 dollars; what is 5 eighths 
 of it worth? 
 
 5. A man bought 9 oranges for 6 cents and 2 sev- 
 enths apiec?e, and sold them for 67 cents; what did he 
 gain by the bargain ? 
 
 6. A man bought 10 yards of broadcloth for 70 
 dollars; how must he sell it per yard in order to gain 
 14 dollars? 
 
 7. If, when the days are 12 hours long, a man per- 
 forms a journey in 3 days; how many hours is he in 
 performing it? 
 
 8. If a man performs a journey in 36 hours, how 
 many days would he be in performing it when the days 
 are 9 hours long ? 
 
 9. If, when the days are II hours long, a man can 
 perform a journey in 5 days, in how many hours will 
 he perform it ? In how many days, when the days are 
 9 hours long ? , 
 
96 ARITHMETIC. [part i. 
 
 10. What number added to 2 fifths of 33 will make 
 the number 17 ? 
 
 11. How many yards of cloth that is 1 quarter of 
 a yard wide, will line 10 yards that is 3 quarters 
 wide ? 
 
 12. 8 yards of cloth that is 1 quarter wide, are 
 equal to how many yards that is 4 quarters wide ? 
 
 13. How many yards of cloth that is 3 quarters 
 wide, are equal to 7 yards that is 5 quarters wide ? 
 
 14. How many yards of cloth that is 6 quarters 
 wide, are equal to 37 that is 4 quarters wide ? 
 
 15. If a piece of cloth, 5 quarters wide, is worth 
 37 dollars, what is a piece of the same length, 3 quar- 
 ters wide, worth ? 
 
 16. If cloth 4 quarters wide is worth 8 dollars a 
 yard, what is 1 yard of the same kind of cloth, that is 
 5 quarters wide, worth? 
 
 'SECTION XII. . 
 
 Parts of one are called fractions. Fractions may 
 be expressed by figures, as well as whole numbers. It 
 requires two numbers to express a fraction ; one to 
 show into how many parts one is divided, and the other 
 to show how many of those parts are used. For exam- 
 ple, if we wish to express one half, (which means that 
 one is divided into two equal parts, and that one part 
 is used,) we must use the figure 2 to express that one 
 is divided into two equal parts and the figure 1 to show 
 that one part is used. And these must be written in 
 such a manner that we may always know what each 
 of them is intended to express. 
 
 One half is usually written thus, I; one number 
 above a line, and the other below it. The number 
 below the line shows into how many parts one is 
 divided, and the number above the line shows how 
 many parts are used. 
 
BEC. XII.] ARITHMETIC. 97 
 
 One third is written } 
 
 Two thirds I 
 
 One fourth J 
 
 Three fourths f 
 
 Two fifths I 
 
 Example. | of an apple signifies that the apple is 
 to be cut into 7 equal parts, and that 3 parts are to be 
 used. 
 
 Illustrate bj a line, divided into 7 equal parts, and 
 three of the parts taken. In the same way illustrate 
 the meaning of the fractions |, |, j^^. 
 
 We may observe, that, when one is divided into 3 
 parts, the parts are called fhit^ds ; when one is divided 
 into 4 parts, the parts arc called fourOu, etc.; that is, 
 the fraction takes its name from the number of parts 
 into which one is divided. The number under the 
 line is called the denominator, because it gives name 
 to the fraction ; and the number above the line is called 
 the numerator, because it shows the number of parts 
 used. Thus y^, 10 is the denominatoTy and 3 the 
 numerator. 
 
 N. B. — The pupil must be made familiar with this 
 mode of expressing fractions, and must be able to 
 apply it to any familiar objects; as apples, oranges, 
 etc.;* or by blackboard, before he is allowed to pro- 
 ceed any farther. Particular care must be takfen to 
 make him understand what the denominator signifies, 
 and what the numerator, as explained above. The 
 denominator should always be explained first. 
 
 The following examples are a recapitulation of some 
 of the foregoing sections, for the purpose of showing 
 the application of the above method of writing frac- 
 tions. Having analyzed the question, the pupil may 
 write the required fraction on the blackboard. 
 
 * When the numerator is larger than the denominator, the fraction 
 is called an improper fraction. See Key. 
 4 
 
98 ARITHMETIC. [part i. 
 
 -' « 
 
 See Section VIII. A. 
 
 A. 1. In 2 liow many times J ?* 
 
 2. In 3 how many times ^ t 
 
 3. In 2 how many times i ? 
 
 4. In 4 how many times ^ ? 
 
 5. In 6 how many times \ ? 
 
 6. In 7 how many times i ? 
 
 7. In 8 how many times i ? 
 
 8. In 25^"!* how many times ^ ? 
 
 9. In 3i how many time \ ? 
 
 10. Reduce 4i to an improper fraction. J 
 
 11. Reduce 3^ to an improper fraction. 
 
 12. Reduce 5f to an improper fraction. 
 
 13. Reduce 6| to an improper fraction. 
 
 ' 14. Reduce 8j^^ to an improper fraction. 
 15. Reduce 9| to an improper fraction. 
 
 B. 1. I are how many times 1? 
 
 2. ^ are how many times 1 ? 
 
 3. 1 are how many times 1? 
 
 4. I are how many times 1 ? 
 
 5. I are how many times 1 ? 
 
 6. '^if are how many times 1 ? 
 
 7. ^-^ are how many times 1 ? 
 
 8. ^^^ are how many times 1 ? 
 .9. ^^ are how many times 1? 
 10. f J are how many times 1? 
 
 See Section IX. 
 
 A. 1. How much is 3 times ^ ? 
 2, How much is 4 times i? 
 
 * When the numerator is lare;er than the denominator, the fraction 
 ift called an wiproper fraction. See Key. 
 
 + 2}/^ is read 2 and 1 half. It is called a mixed number. 
 
 I That is, to find how many fifths there are in 4 and 1 fifth, first, and 
 how many fifths there are in 4. 
 
SEC. XII.] 
 
 3. 
 4. 
 5. 
 6. 
 
 7. 
 
 8. 
 
 9. 
 
 10. 
 
 How much 
 How much 
 How much 
 How much 
 How much 
 How much 
 How much 
 How much 
 
 ARITHMETIC. 
 
 s 3 times | ? 
 s 4 times f ? 
 s 5 times |? 
 s 6 times |? 
 s 8 times j\? 
 s 9 times ^? 
 s 10 times J ? 
 is 9 times |? 
 
 99 
 
 B. 1. How much 
 
 2. How much 
 
 3. How much 
 
 4. How much 
 
 5. How much 
 
 6. How much 
 
 7. How much 
 
 8. How much 
 
 9. How much 
 10. How much 
 
 s 3 times 2^? 
 s 4 times 3|? 
 s 5 times 6| ? 
 s 6 times 4| ? 
 s 7 times 5'|? 
 s 8 times 6| ? 
 is 4 times 1(1 §? 
 s 9 times 7| : 
 s 8 times 9^ ? 
 s 10 times 7|? 
 
 t^^-e Sections Y. & X. 
 
 1. 
 
 3. 
 
 5. 
 
 7. 
 
 9. 
 11. 
 13. 
 15. 
 17. 
 19. 
 
 What is 
 What is 
 What is 
 What is 
 What is 
 What is 
 What is 
 What is 
 What is 
 What is 
 
 of 6? 
 
 df 8?. 
 
 of 9? 
 
 of 14? 
 
 of 5? 
 
 of 7? 
 I of 17? 
 I of 27? 
 f of 47 ? 
 I of 65? 
 
 2. 
 
 4. 
 
 6. 
 
 8. 
 10. 
 12. 
 14. 
 16. 
 18. 
 20. 
 
 What is I 
 What is I 
 What is J 
 What is I 
 W^hat i* ^ 
 
 _3 
 7 
 3 
 7 
 4 
 15 
 
 What is 
 What is 
 What is 
 What is 
 What is 
 
 of 6? 
 of 9? 
 of 10? 
 of 5? 
 of 7? 
 of 35? 
 of 26? 
 of 37? 
 of 42? 
 of 75 ? 
 
 See Sections VI. & XI. 
 
 A. 1. 2 is i of what number? 
 2. 4 is i of what number ? 
 
100 ARITHMETIC. [part i. 
 
 3. 8 is i of what number ? 
 
 4. 1 J is i of what number ? 
 
 5. 2| is i of what number ? 
 
 6. 4| is J of what number ? 
 
 7. 6| is i: of what number? 
 
 8. 7| is i of what number ? 
 
 9. 8| is i of what number ? 
 10. 9y^Q is J(j of what number ? 
 
 B. 1. 4 is f of what number? 
 
 2. 6 is f of what number ? 
 
 3. 8 is I of what number? 
 
 4. 12 is % of what number ? 
 
 5. 15 is I of, what number ? 
 
 6. 18 is ^ of what number ? 
 
 7. 20 is I of what number ? 
 
 8. 24 is I of what number ? 
 
 9. 28 is J of what number ? 
 
 10. 30 is y^5 of what number ? 
 
 11. 3 is f of what number ? 
 
 12. 4 is I of what number? 
 
 13. 5 is I of what number ? 
 
 14. 8 is I of what number? 
 
 15. 9 is I of what number? 
 
 16. 17 is J of what number ? 
 
 17. 25 is I of what number? 
 
 18. 38 is f of what number? 
 
 19. 48 is I of what number ? 
 
 20. 54 is y^ of what number ? 
 
 MISCELLANEOUS EXAMPLES. 
 
 1. A man sold 8 yards of cloth for 3| dollars a 
 yard; what did it come to? 
 
 2. A man sold a horse for 76 dollars, which was | of 
 what it cost him ; how much did it cost him ? 
 
SEC. XIII.] ARITHMETIC. 101 
 
 3. A man sold | of a gallon of wine for 40 cents; 
 what was that a gallon ? 
 
 4. If it will take li yards of cloth to make a coat, 
 how many yards will it take to make 7 coats ? 
 
 5. If 1 horse consumes o^ bushels of oats in 2 days, 
 how much would 2 horses consume in 5 days 't 
 
 6. If, when the days are Of hours long, a man per- 
 forms a journey in 10 days, in how many days would he 
 perform it when the days are 12 hours long ? 
 
 7. A man sold 8 j'ards of cloth for 7| dollars a yard, 
 and received 8 firkins of butter at G| dollars a firkin ; 
 how much was then due to him ? 
 
 8. Two men are o8 miles apart, and are travelling 
 towards each other, one at the rate of 3 miles an hour, 
 the other 2 miles; how much do they approach each 
 other in an hour ? How much in 2 hours? In how 
 many hours will they meet? At what distance from 
 each place from which they set out? 
 
 SECTION XIII. 
 
 A. 1. If you give ^ of an orange to oi>e boy, and k 
 to another, how much more do you give the first than 
 the second ? 
 
 2. i of an orange is how many $ of an orange ? 
 
 3. If you give ^ of an orange to one boy, and I to 
 another, how many i would you give away? How 
 many i would you have left ? 
 
 4. i and J are how many 1 ? 
 
 5. A man gave to 1 laborer ^ of a bushel of wheat, 
 and f to another; how many i of 'a bushel did he give 
 to both ? How many bushels ? 
 
 6. »} and f are how many i ? How many times 1 ? 
 
 7. A man gave ^ of a barrel of flour to 1 man, and 
 § of a barrel to another; to which did he give the 
 most ? 
 
 8. J is how many i ? 
 
102 ARITHMETIC. [parti. 
 
 9. A man bought J of a bushel of wheat at one time, 
 and I of a bushel at another; at which time did he 
 buy the most ? 
 
 10. i is how many | ? 
 
 11. A man bought f of a yard of cloth at one time, 
 and I of a yard at another ; at which time did he buy 
 the most ? 
 
 121 t are how many i ? 
 
 13. A man wished to give i of a bushel of wheat to 
 one man, and i of a bushel to another; but he could 
 not tell how to divide it. Another man standing by 
 advised him to divide the whole bushel into six equal 
 parts first, and then take i of them for one, and i of 
 them for the other. How many parts did he give to 
 each ? How many to both ? How many had he left ? 
 
 14. 2 is how many I? ^ is how many i ? J and i 
 are how many i ? 
 
 15. A man paying some money to his laborers, gave 
 each man | of a dollar, and each boy ^ of a dollar; 
 how much more did he give to a man than to a boy? 
 
 16. AVhat is the difi'erence between J and i ? 
 
 17. If a man can earn | of a dollar in a day, and a 
 boy ^ of a dollar, how much does the man earn more 
 than the boy ? 
 
 18. What is the difference between | and i? 
 
 19. A boy distributing some nuts among his com- 
 panions, gave i^ of a quart to one, and ^ of a quart to 
 another ; how much more did he give to one than to 
 the other ? 
 
 Note. — Change them to sixths. 
 
 20. What is the difference between i and ^ ? 
 
 21. A man having 2 bushels of grain to distribute 
 among his laborers, wished to give ^ of a bushel to one, 
 and f of a bushel to another, and the rest to a, third ; 
 but was at a loss to tell how to divide it ; at last he con- 
 cluded to divide each bushel into six equal parts, or 
 sixths, and then to distribute those parts. How many 
 sixths did he give to each ? 
 
SEC. XIII.] ARITHMETIC. 103 
 
 22. I is how many i ? 
 
 23. A man had a horse, and a cow, and a sheep. 
 The horse would eat f of a -load of hay in the win- 
 ter, the cow ^, and the sheep. i. How many i of a 
 load would each eat ? How many i would they all eat? 
 How many loads ? 
 
 24. A boy having a quart of nuts, wished to divide 
 them,, so as to give one companion i, another It. and a 
 third i of them ; but in order to make a proper division, 
 he first divided the whole into eight equal parts, and 
 then he was able to divide them as he wished. How 
 many eighths did he give to each ? How many eighths 
 had he left for himself? 
 
 25. i is how many i ? J is how many i ? ^ and \ 
 and i are how many i ? 
 
 26. A man gave f of a barrel of flour to one man, 
 and I of a barrel to another ; to which did he give the 
 most ? How much ? 
 
 27. Which is the largest, f or | ? How much the 
 largest ? 
 
 28. A boy having a pound of almonds, said he in- 
 tended giving i of them to his sister, and I to his 
 brother, and the rest to his mamma. His mamma, 
 smiling, said she did not think he could divide them so. 
 
 yes I can, said he, I will first divide them into twelve 
 equal parts, and then I can divide them well enough? 
 Pray how many twelfths did he give to each ? 
 
 29. i is how many J^? i is how many J^^ o and 
 i are how many y^y ? 
 
 30. Mr. Goodman having a pound of raisins, said he 
 would give Sarah J, and Mary 1, and James I of them, 
 and he tol4 Charles he should have the rest, if he could 
 tell how to divide them. Well, said Charles, I would 
 first divide the whole into twelve equal parts, and then 
 
 1 could take ^ and J and i of them. How many 
 twelfths would each have ? 
 
 31. i and J and ^ are how many yV ? 
 
 32. George bought a pine apple, and said he would 
 
104 ARITHMETIC. [parti. 
 
 give i of it to his papa, and | to his mammaj and j^^ 
 to his brother James, if he could divide it. James 
 took it and cut it into twenty equal pieces, and then 
 distributed them as George had desired. How many 
 twentieths did he give to each? 
 
 33. i is how many ^^ ? J is how many -^j^ ? § is how 
 many J^ ? j^^ is how many -^j^ 't 
 
 34. i is how many J^? 
 
 35. ^ is how many jL ? 
 
 36. ^ is how many -i 't 
 
 37. 3 is how many -^^ ? 
 
 38. f are how many ^ ? 
 
 39. f are how many -^j? 
 
 40. i is how many y^ r 
 
 41. I are how many jK? 
 
 42. I are how many j^^ ? 
 
 43. 4 ^^6 how many -Jj? 
 
 44. I are how many -^^^ ? 
 
 45. I are how many g^g ? 
 
 46. I are how many -3'jj ? 
 
 47. YO ^^^^ ^^^ many ^^^ ? 
 
 48. lieduce ^ to sixths and i to sixths. 
 
 49. § and | are how many i ? 
 
 50. Reduce 2 ^i^id 1 to eighths. 
 
 51. i and ^ are how many i? 
 
 52. i- an<i i are how many I ? 
 
 53. f and I are how many } ? 
 
 54. i and' I are how many i? 
 
 55. i and | are how many A ? 
 
 56. J and | are how many y^j ? 
 
 57. '} and ^ and i are how many i ? 
 
 58. '} and | and Jg are how many y^^j ? 
 
 59. f and i- are how many y^^ ^ 
 
 60. f and i and 1 are how many j^^? 
 
 61. I and y^ and i are. how many ^^ ? 
 
 62. k and i and f and i and y^^j ai^e how many J^ ? 
 
 63. i and -| are how many j\ ? 
 
 64. I and 1 are how many J^? 
 
SEC. XIII.] ARITHMETIC. 105 
 
 65. f and | are how many J^ ? * 
 
 66. i less i are how many i ? 
 
 67. t and f , less y^^j, are how many J^ ? 
 
 68. I less I are how many jL? 
 
 69. I less I are how many ^ j ? 
 
 70. I less I are how many J^? 
 
 71. I, and f, and J, and -f^, less |, are how many 
 
 72. i, and ^, and |, and -j'y, and 2*0, less i, are l^ow 
 ■many-^i^? 
 
 73. 4 ^"d 6 ^^® ^^^ many ^^^ ^ 
 
 74. f and | are how many g^^? 
 
 75. I and | are how many g^. 
 
 9 
 
 63 ' 
 
 When the denominators in two or more fractions are 
 the same, the fractions are said to have a common de- 
 nominator. Thus I and | have a common denomina- 
 tor. We have seen /hat, when two or more fractions 
 have a common denominator, they may be added and 
 subtracted as well as whole numbers. We add or sub- 
 tract the numerators, andwrite their sum or difference 
 over the common denominator. The first part of the 
 process in the above examples was to reduce them to a 
 common denominator. 
 
 76. lleduce f and f to a common denominator. 
 Note. — They may be" reduced to twelfths. 
 
 If it cannot be immediately seen what number m^st 
 be the common denominator, it may be found by mul- 
 tiplying all the denominators together; for that will 
 always produce a number divisible by all the denomi- 
 nators. 
 
 77. Reduce | and ^ to a common denominator. 
 
 78. Reduce f and f and | to a common denomi- 
 nator. 
 
 79. Reduce ^ and | to a common denominator. 
 
 80. Reduce | and | to a common denominator. 
 
106 ARITHMETIC. [part i. 
 
 81. Reduce } and f and f to a common denominator. 
 
 82. Add toorether | and ^. 
 
 83. Add together | and y^^. 
 
 84. Add together | and fg. 
 
 85. Add together | and | and y^^. 
 
 86. Subtract i from ^. 
 
 87. Subtract j\ from |. • 
 
 88. Subtract 'j from f . 
 
 89. Subtract | from |. 
 
 B. 1. Mr. F. said he would give ^ of a pine apple 
 to Fanny, and | to George, and the rest to the one 
 that could tell how to divide it, and how much there 
 would be left. But neither of them could tell; so he 
 kept it himself. Could you have told if you had been 
 there? How would you divide it? How much would 
 be left? 
 
 2. A man sold li bushel of wheat to one man, 4f 
 bushels to another; how many bushels did he sell to 
 both? 
 
 3. A man bought 6 J bushels of wheat at one time, 
 and 2 J at another; how much did he buy in the 
 whole? 
 
 4. A man bought 7t yards of one kind of cloth, 
 and 6f yards of another kind; how many yards in the 
 whole ? 
 
 5. A man bought | of a barrel of flour at one time, 
 2^ barrels at another, and 6| at another; how much 
 did he buy in the whole? 
 
 . 6. A man bought one sheep for 4| dollars, and 
 another for 5| dollars; how much did he give for both? 
 
 7. There is a pole standing, so that ^ of it is in the 
 mud, and | of it in the water, and the rest out of the 
 water; how much of it is out of the water? 
 
 8. A man having undertaken to do a piece of work, 
 did i of it the first day, i of it the second day, and i 
 of it the third day; how much of it did he do in three 
 days? 
 
SBC. xin ] ARITHMETIC. 107 
 
 9. A man having a piece of work to do, hired two 
 men and a boy to do it. The first man could do ^ of 
 the work in a day, and the other i of it, and the boy 
 I of it; how much of it would they all do in a day? 
 
 Note. — By dividing a line into halves, and then into 
 fourths, it will be seen that | is the same as ^j a line 
 divided into halves and then into sixths, will show that 
 I is the same as i, and | as |j f> i> can therefore be 
 reduced to J, and ^ to f , This is called reducing frac- 
 tions to their lowest terms. It is done by dividing the 
 greatest number that will divide it without a remainder. 
 
 1. Reduce | to its lowest terms.* Aiis. f . 
 
 2. Reduce -^^ to its lowest terms. 
 
 3. Reduce | to its lowest terms, 
 
 4. Reduce y*^ to its lowest terms. 
 
 5. Reduce jj to its lowest terms. 
 
 6. Reduce f^ to its lowest terms. 
 
 7. Reduce ^q to its lowest terms. 
 
 8. Reduce ^| to its lowest terms. 
 
 9. Reduce J 4 to its lowest terms. 
 
 10. Reduce ^^^ to its lowest terms. 
 
 11. Reduce || to its lowest terms. 
 
 12. Reduce |i to its lowest terms. 
 
 13. Reduce || to its lowest terms. 
 
 14. Reduce |j to its lowest terms. 
 
 Note. — It will be seen by the above section that if 
 both the numerator and denominator be multiplied by 
 the same number, the value of the fraction will not be 
 altered; or if they can both be divided by the same 
 number without a remainder, the fraction will not be 
 altered. 
 
 * If this article should be found too difficult for the pupil, he may 
 omit it till after the next section. 
 
108 ARITHMETIC. [part i. 
 
 SECTION XIV. 
 
 A. 1. A BOY having i of an orange gave away^ of 
 that, what part of the whole orange did he give away? 
 
 2. What is i of i? 
 
 3. If you €ut an apple into three pieces, and then 
 cut each of those pieces into two pieces, how many 
 pieces will the whole apple be cut into? What part 
 of the whole apple will one of the pieces be ? 
 
 4. What is J of i? 
 
 5. A boy had J of a pine apple, and cut that half 
 into three pieces, in order to give away i of it. What 
 part of the whole apple did he give away? 
 
 6. What is i of ^? ^ 
 
 7. If an orange be cut into 4 parts, and then each 
 of the parts be cut in two, how many pieces will the 
 whole be cut into? 
 
 8. What is -J of i? 
 
 9. A man having } of a barrpl of flour, sold i of 
 that; how much did he sell? 
 
 10. What is i of *? 
 
 11. If an orange be cut into 4 equal parts, and each" 
 of those parts be cut into 3 equal parts; how many 
 parts will the whole orange be cut into ? 
 
 12. What is iof i? 
 
 13. A boy having i of a quart of chestnuts, gave 
 away I of what he had ; what part of the whole quart 
 did he give away? 
 
 14. What is i of i? 
 
 15. What is Jof J? • 
 
 16. A man owning ^ of a ship, sold J of his share; 
 what part of the ship did he sell, and what part did 
 he then own ? 
 
 n. What is i of 1? 
 
 18. What is J of i ? 
 
 19. What is i of i? 
 
 20. What is i of ^? 
 
 21. What is i of ^? 
 
SEC. XIV.] 
 
 ARITHMETIC. 
 
 109 
 
 ^ofi? 
 
 s i of 4 ? 
 siof i? 
 s J of i? 
 s i of \? 
 s i of J? 
 s^of Vr' 
 
 22. What 
 
 23. What 
 
 24. What 
 
 25. What 
 
 26. What 
 
 27. What 
 
 28. What 
 
 29. A boy having f of an orange, (that is, 2 pieces,) 
 gave his sister ^ of what he had ; how many thirds 
 did he give her ? 
 
 30. What is iof t? ' 
 
 31. A boy having f of a pine apple, said he would 
 give one half of what he had to his sister, if she could 
 tell him how to divide it. His sister says, you. have 
 I, or three pieces, if you cut them all in two, you 
 can give me ^ of them. But J of J is i, therefore I 
 shall have | of the whole pine apple. 
 
 32. What is I of f ? 
 
 33. A man owning f of a share in the Planters' 
 Bank, sold i of his part ; what part of a share did 
 he sell? 
 
 34. What is i of f ? 
 
 35. A man owning | of a ship, sold k of his share; 
 what part of the whole ship did he sell? What part 
 had he left? 
 
 36. 
 
 What ] 
 
 IS i of 1? 
 
 37. 
 
 What 
 
 IS * of 1? 
 
 38. 
 
 What 
 
 IS i of 4 ? 
 
 39. 
 
 What 
 
 IS i of 9 ? 
 
 40. 
 
 What 
 
 Is \ of V ? 
 
 41. 
 
 What] 
 
 S ^ of 3 ? 
 
 42. A man owning § of a share in the Planters' 
 Bank, sold i of his part; what part of a whole share 
 did he sell ? 
 
 43. What is J of I? ' • 
 
 44. What is i of f 
 
 45. A boy having | of a watermelon, wished to 
 divide his part equally between his sister, his brother, 
 
110 
 
 ARITHMETIC. 
 
 [part I. 
 
 and himself, but was at a loss to know how to do it ; 
 but his sister advised him to cut each of the fifths into 
 3 equal parts. How many pieces did each have ? and 
 what part of the whole melon was each piece? 
 
 46 
 47. 
 
 48. 
 
 51. 
 52. 
 53. 
 54. 
 55. 
 56. 
 
 What 
 What 
 What 
 
 49. What 
 
 50. What 
 What 
 What 
 What 
 What 
 What 
 What 
 
 57. What 
 
 58. What 
 What 
 What 
 What 
 What 
 What 
 What 
 What 
 What 
 What 
 What 
 
 59. 
 60. 
 61. 
 62. 
 63. 
 64. 
 65. 
 66. 
 67. 
 68. 
 
 iof I 
 lof f? 
 iof 1? 
 iof f? 
 T^uOf I? 
 
 s ^i of i ? 
 s I of i? 
 si of I? 
 sf of I? 
 s i of § ? 
 s I of I ? 
 si of I? 
 slof I?' 
 s |of f? 
 s f of I ? 
 sf of I? 
 s^of f? 
 s|of.|? 
 
 s 1 of t? 
 s y Q or 8 : 
 
 S JO 01 8 i 
 S I of y3_? 
 
 s ^ of ^ ^ 
 sf of f? 
 
 69. If a yard of cloth cost 2j dollars, what will i 
 of a yard cost? 
 
 70. What is ^ of 2i? 
 
 71. A boy had 2^ oranges, and wished to give i of 
 them to his sister, and 2' to his brother, but he did not 
 know how to divide them equally. His brother told 
 him to cut the whole into halves, and then cut each of 
 the halves into 3 pieces. What part of a whole orange 
 did each have? 
 
 72. What is i of 2i? 
 
fiEC. XIV.] ARITHMETIC. Ill 
 
 73. A man bought 4 bushels of corn for 3f dollars; 
 what part of a dollar did 1 bushel cost? 
 
 Change the 3f to thirds, and then Jind k of ^ as 
 above, 
 
 74. What is iof 5§? 
 
 75. If 5 bushels of wheat cost 71 dollars, what is 
 that a bushel ? 
 
 76. What is -Jof 7f? 
 
 77. A man bought 6 gallons of alcohol for 8| dol- 
 lars; what was that a gallon? 
 
 78. What is i of 8§ ? 
 
 79. A man bought 7 gallons of wine for 8| dollars; 
 how much was that a gallon? 
 
 80. What is 4 of 8|? 
 
 81. A man bought 10 pieces of nankin for 6| dol- 
 lars; how much was it a piece? 
 
 82. What is J^of 6|? 
 
 83. If 9 bushels of rye cost 7| dollars, what is that 
 a bushel? 
 
 84. What is 1 of 7f? 
 
 85. What is ^ of 5|? 
 
 86. What is i of 8|? 
 
 87. What is \ of 6-^-3^? 
 
 88. What is i of 94? 
 
 89. A man bought 7 yards of cloth for 18| dollars; 
 what was that a yard? What would 3 yards cost at 
 that rate? 
 
 90. What is 4 of 18f.? What is -| of 18 j? 
 
 91. A man bought 5 barrels of cider for 27| dollars; 
 what was it a barrel? What would 7 barrels cost at 
 that rate? 
 
 92. What is \ of 271 What is f of 27|? 
 
 93. If 6 barrels of flour cost 38| dollars, what would 
 10 barrels cost at that rate ? 
 
 94. What is i^o of 38|? 
 
 3. 1. A man bought a piece of cloth for 42| dollars, 
 
112 ARITHMETIC. [paet i. 
 
 and was obliged to sell it for | of wliat it cost Mm; 
 how much did he lose? 
 
 2. A man bought a quantity of flour for 53f dol- 
 lars, and sold it for | of what it cost him; how much 
 did he gain? 
 
 3. If 7 men can do a piece of work in 4| days, 
 how long will it take 1 man to do it? How long will 
 it take 3 men to do it ? 
 
 4. If 4 men can do a piece of work in 9| days, 
 how long would it take to do it if 7 men were em- 
 ployed ? 
 
 5. There is a pole standing so that | of it is in the 
 water, and | as much in the mud; how much is in the 
 mud? 
 
 6. If a man can travel 13| miles in 3 hours, how 
 many miles can he travel in 8 hours? 
 
 7. If 5 horses will eat 26f loads of hay in a year, 
 what will 8 horses eat in the same time? 
 
 8. If 4 cocks will empty a cistern in 6| hours, how 
 long will it take 7 cocks of the same size to empty it? 
 
 SECTION XV. 
 
 A. 1. A BOY having 2 oranges wished to give } of 
 an orange apiece to his playmates; how many could he 
 give them to? If he had given f of an orange apiece, 
 how many could he have given them to? 
 
 2. How many times i are there in 2? How many 
 times f are there in 2? 
 
 3. A man having 3 bushels of corn, distributed it 
 among some poor persons, giving them | of a bushel 
 each; to how many did he give it? 
 
 Note. — First Jind Jwiv many lie, would have given it 
 to, if he had given \ of a, bushel to each. 
 
 4. In 3 are how many times i ? How many times f ? 
 
 5. If I of a barrel of flour will last a family one 
 
SEC. XV.] ARITHMETIC. 113 
 
 month, how long will 4 barrels last the same family? 
 How long will 6 barrels last? How long will 10 bar- 
 rels last? 
 
 6. How many times is | contained in 4? How 
 many times in G ? How many times in 10 ? 
 
 7. If 4 of a bushel of wheat will last a family one 
 week, how many Weeks will 6f bushels last the same 
 family? 
 
 8. How many times is f contaiacd in 6|? 
 
 9. There is a cistern havingr a cock which will fill 
 it in I of an hour; how mariy times would the cock 
 fill the cistern in 3| hours? 
 
 10. How many times is | contained in 3|? 
 
 11. How much cloth at 1? dollars (that is | dollars) 
 a yard, can be -bought for 4 dollars ? 
 
 12. How many times is 1? or | contained in 4? 
 
 13. A man distributed 8^ bushels of wheat among 
 some poor persons, giving 1^ bushels to each; how 
 many did^he give it to? 
 
 14. How many times is 1^ contained in 8^? 
 
 15. If a soldier is allowed 1^^ pound (that is | of a 
 pound) of meat in a day, to how many soldiers would 
 6| pounds be allowed? 
 
 16. How many times is IJ contained in 6|? 
 
 17. If If tons of hay will keep a horse through the 
 winter, how many horses will 10 tons keep? 
 
 18. How many times is If contained in 10? 
 
 19. At 2i dollars a box, how many boxes of raisins 
 can be bought for 10 dollars? 
 
 20. How many times is 2^ contained in 10? 
 
 21. At 1| dollars a pound, how many pounds of 
 indigo can be bought for 9| dollars? 
 
 22. How many times is 1| contained in 9|? 
 
 23. At 1| dollar a barrel, how many barrels of 
 raisins can be bought for 9| dollars ? 
 
 24. How many times is 14 contained in 9|? 
 
 25. At I of a dollar apiece, how many pieces of 
 nankin can be bought for 8| dollars? 
 
114 ARITHMETIC. [paet i. 
 
 26. How many times is i contained in 8|? 
 
 27. At ^ of a dollar a pound, how many pounds of 
 tea can be bought for 7| dollars? 
 
 28. How many times is | contained in 7| ? 
 
 29. How many times is 3^ contained in 7f ? 
 80. How many times is 5 J contained in 17 ? 
 
 31. How many times is 4i contained in 9|? 
 
 32. How many times is 3| contained in 12^? 
 
 B. 1. At J^ of a dollar a pound, how many pounds 
 of meat can be bought for i^ of a dollar? 
 Note. — Change ^ to tenths. 
 
 2. How many times is ^^ contained in J? 
 
 3. A man having f of a barrel of flour, distributed 
 it among some poor persons, giving them i of a barrel 
 apiece; how many did he give it to? 
 
 Note. — Change both fractions to twelfths; that is, 
 reduce them to a common denominator. 
 
 4. How many tim.es is i contained in f ? , 
 
 5. If a pound of almonds cost i of a dollar, how 
 many pounds can be bought for f of a dollar ? 
 
 Note. — Reduce the fractions to a common denomi- 
 nator. 
 
 6. How many times is ^ contained in f ? 
 
 7. If a piece of nankin cost | of a dollar, how many 
 pieces can be bought for 4f dollars? that is, ^9 dollars? 
 
 8. How many times is | contained in 4f ? 
 
 9. If a bushel of barley cost | of a dollar, how 
 many bushels can be bought for i of a dollar? How 
 many for If of a dollar. 
 
 10. How many times is | contained in |^? How 
 many times in 1| ? 
 
 11. How many times is | contained in | ? 
 
 12. How many times is | contained in | ? 
 
PARTLJ ARITHMETIC. 115 
 
 TABLES OF COINS, WEIGHTS, AND MEASURES. 
 Table 1. — Federal Money. 
 
 Federal Money is the currency of the Confederate States, and 
 of the United States. 
 
 10 mills (m.) make 1 cent, marked cf. 
 
 10 cents " 1 dime, " d. 
 
 10 dimes " 1 dollar, " doll, or $. 
 
 10 dollars " 1 Eagle, " E. 
 
 Table II. — Sterling Monet. 
 
 Sterling Money is the currency of England. 
 
 4 farthings (qr. or far.) make 1 penny, marked d 
 
 12 pence " 1 shilling, " s. 
 
 20 shillings " 1 pound, or a sovereign, " £,. 
 
 21 shillings " 1 guinea. 
 
 Table III— Troy Weight. 
 
 Used in weighing Gold, Silver, and some Liquids. 
 
 24 grains (gr.) make 1 pennj-weight, marked dwt. 
 20 pennyweights " 1 ounce, " oz. 
 12 ounces " 1 pound. " lb. 
 
 Table /,F.— Avoirdupois Weight. 
 
 Used in weighing Groceries, Hay, etc., and all the Metals, 
 except Gold and Silver. It is now the custom to allow 100 
 lbs. for a hundred weight, instead of 112 as formerly. 
 
 16 drams (dr.) make 1 ounce, marked oz. 
 
 16 ounces " 1 pound, ' " lb. 
 
 25 pounds " 1 quarter, " qr. 
 4 quarters " 1 hundred weight, " cwt 
 
 20 hundred weight " 1 ton. " T. 
 
 Table V. — Cloth Measure. 
 Used in measuring Woollens and other Cloths. 
 
 214 inches (in.) make 1 nail, marked na. 
 
 4 nails, or 9 in. « 1 quarter of a yard, " qr. 
 
 4 quarters « 1 yard, " yd. 
 
 3 quarters « 1 Flemish ell, « Fl. e. 
 
 5 quarters " 1 English ell, " E. e. 
 
 6 quarters « 1 French ell. " F. e. 
 
116 ARITHMETIC. [pakt i. 
 
 Table VL — Wine Measure. 
 Used in measuring "Wines, etc. 
 
 4 giUs (gi.) make 1 pint, marked pt. 
 
 2 pints " 1 quart, "■ qt. 
 
 4 quarts " 1 gallon, " gal. 
 
 5VA gallons " 1 barrel, " bar. or bbl. 
 
 42 gallons " 1 tierce, " tier. 
 
 63 gallons, or 2 barrels " 1 hogshead, " hhd. 
 
 2 hogsheads " 1 pipe, or butt, " pi. 
 
 2 pipes " 1 tun. " tun. 
 
 Table VII. — Dry Measure. 
 
 Used in measuring Grains, Salt, Oysters, etc. 
 
 2 pints make 1 quart, marked qt. 
 
 8 quarts " 1 peek, " pk. 
 
 4 pecks " 1 bushel. " bu 
 
 Table VIII. — Measure of Time. 
 
 60 seconds (sec.) make 1 minute, marked min. 
 
 60 minutes " 1 hour, " hr. 
 
 24 hours " 1 day, " d;/. 
 
 7 days " 1 week, " wk. 
 
 4 weeks " 1 month, " mo. 
 
 13 months. 1 day and 6 hours, I « ^ j. « ,^ 
 
 or 12 calendar months, J 
 
 For convenience of reckoning, it is usual in calen- 
 dars to call the year 365 days for 3 successive years, 
 and every fourth year 366, (for in 4 years, the six 
 hours overplus amount to a day,) which is called 
 bissextile, or leap year. This day is added to February. 
 
 The common year is divided into twelve months, 
 which are sometimes called calendar months, because 
 they are the months used in calendars. 
 
 The names of the months, and the number of days 
 in each, are as follows : 
 
 Names. Number of days. 
 
 January 31 
 
 February 28, in leap year 29. 
 
 March 31 
 
 April 30 
 
 May 31 
 
PART I] ARITHMETIC. 117 
 
 Names. Number of days. 
 
 June 30 
 
 July 31 
 
 August 31 
 
 September 30 
 
 October 3l 
 
 November 30 
 
 December 31 
 
 MISCELLANEOUS EXAMPLES. 
 
 1. In 2 pounds how many ounces? 
 
 2. In 8 yards how many quarters? 
 
 3. In 3 quarters of a yard, how many nails ? 
 
 4. -A of a dollar is how many cents ? 
 
 5. How many farthings is | of a penny? 
 
 6. How many pence is | of a shilling? 
 
 7. I of a yard is how many quarters and nails? 
 
 8. In £| how many shillings ? 
 
 9. How much is | of a shilling ? 
 
 10. How much is | of a bushel of wheat ? 
 
 11. How much would | of a barrel of wine cost, at 
 one dollar a gallon ? 
 
 12. How much would i cwt. of sugar cost, at 8 cents 
 a pound ? 
 
 13. How much is f of a day? 
 
 14. How much is | of a day ? 
 
 15. How much is | of a week ? 
 
 16. How much is f of an hour ? 
 
 17. How much would | of a hogshead of wine cost, 
 at 2 dollars a gallon ? 
 
 18. If a man%pends 28 dollars in a month, what is 
 that a week ? How much a day ? 
 
 19. If a man spends 16 dollars a week, what is that 
 a day ? 
 
 20. If a man buys 4 bushels of grain for 5 dollars, 
 how much is that a bushel ? 
 
118 ARITHMETIC. [part i. 
 
 21. If wine is 2 dollars a gallon, how much is that a 
 pint ? 
 
 22. If you give 5 cents a gill for wine, what is that 
 a pint y What is it a quart ? What is it a gallon ? 
 
 28. If wine is worth 20 cents a pint, what is that a 
 gill ? What is it a quart ? What is it a gallon ? 
 
 24. If a yard of cloth is worth 7 dollars, what are 
 2 1 'yards worth ? , 
 
 2o. If a man earns 11 dollars a week, what is that a 
 
 day? What for 3 days 't What for 4^ days ? 
 
 2o. If a man earns 2| dollars in a day, what will ho 
 earn in a week ? 
 
 27. What is | of a hogshead of wine? 
 
 2^. 1 farthing is what part of a penny ? 
 
 29. 2 fjirthings is what part of a pennyf 
 
 30. 3 farthings is what part of a penny? 
 
 31. 1 penny is what part of a shilling ? 
 
 32. 2' pence is what part of a shilling ? 
 
 33. 3 pence is what part of a shilling? 
 
 34. 5 pence is what part of a shilling ? 
 85. 6 pence is what part of a shilling? 
 
 36. 7 pence is what part of a shilling ? 
 
 37. 8 pence is what* part of a shilling? 
 
 38. 9 pence is what part of a shilling? 
 
 39. 10 pence is what part of a shilling? 
 
 40. 11 pence is what part of a shilling? 
 
 41. 1 shilling is what part of a pound? 
 
 42. 2 shillings is what part of a pound? 
 
 43. 3 shillings is what part of a pound? 
 
 44. 4 shillings is what part of a pound? 
 
 45. 5 shillings is what part of a pound ? 
 
 46. What part of a pound is 6 shillings? 7 shil- 
 lings? 8 shillings? 9 shillings? 10 shillings? 11 shil- 
 lings? 12 shillings? 13 shillings? 14 shillings? 15 
 shillings? 16 shillings? 17 shillings? 18 shillings? 
 19 shillings? 
 
 47. How many farthings are there in a shilling ? 
 
 48. One farthing i% what part of a shilling? 
 
PART I.] ARITHMETIC. 119 
 
 49. 2 farthings is what part of a shilling? 3 far- 
 things? 4 farthings? 5 farthings? 6 farthings? 7 
 farthings? 8 fafthmgs? 9 farthings? 10 farthings? 
 
 50. How many pence are there in a pound? . 
 
 51. One penny is what part of a pound? 
 
 52. What part of a pound is 2 pence? o pence? 
 4 pence? 5 pence ? 6 pence ? 7 pence ? 8 pence? 11 
 pence? 15 pence? 27 pence ? 35 pence? 
 
 53. How many pence are there in 1 shilling and 6 
 pence? 
 
 54. In 2 shillings and 4 pence, how many pence ? 
 
 55. In 4 shillings and 5 pence, how many pence? 
 " 56. In 5 shillings and 8 pence, how many pence? 
 
 57. In 9 shillings and 11 pence, how many pence ? 
 
 58. What part of £1 is 2s. 6d. ? 
 
 59. 3s. 5d. is what part of £1 ? 
 
 Note. — Reduce the icJiole to pence. 
 
 60. 7s. 8d. is what part of £1 ? 
 
 61. What is the price of 2 yards of cloth, at 3s. 
 4d, a yard ? 
 
 62. What will 8 yards of cloth cost, at 2s. 8d. a 
 yard ? 
 
 68. What will 4 bushels of wheat cost, at 5s. 9d. a 
 bushel ? * 
 
 64. What must you give for 4 barrels of cider, at 
 2} dollars a barrel ? 
 
 65. If 3 bushels of wheat be divided between 2 
 men, how much would they have apiece ? 
 
 66. If 4 bushels of corn be divided amSng 5 men, 
 how much would they have apiece ? 
 
 67. If 3 bushels of corn be divided among 7 men, 
 how much would they have apiece ? 
 
 68. How many nails are there in 1 yard? 
 
 69. How many nails are there in 4 yards ? 
 
 70. How many nails are there in 5 yards and 2 
 nails ? 
 
 71. In 7 yards and 3 quarters how many quarters? 
 
120 ARITHMETIC. [part i. 
 
 72. In 4 yardSj 2 quarters,.and 3 nails, how many- 
 nails ? 
 
 73. 1 nail is what part of a quarter? 
 
 74. 3 nails is what part of a quarter ? 
 
 75. 1 nail is what part of a yard ? 
 
 76. What part of 1 yard is 3 nails ? 5 nails ? 7 
 nails ? 10 nails ? 15 nails ? 
 
 77. In B quarters of a yard how many yards ? 
 
 78. In 12 quarters of a yard how mauy yards ? 
 
 79. In 10 quarters of a yard how many yards? 
 
 80. In 15 quarters of a yard how many yards? 
 
 81. In 12 nails how many quarters of a yard ? 
 
 82. In 16 nails how many quarters of a yard ? 
 How many yards ? 
 
 83. In 24 nails how many quarters of a yard? 
 How many yards ? 
 
 84. In 35 quarters of a yard how many yards ? 
 ' 85. In 45 nails how many yards? 
 
 86. In 63 nails how many yards ? 
 
 87. At 2 cents a nail what would 4 yards of cloth 
 cost? 
 
 88. At 2f dollars for 1 quarter of a yard, what 
 would 2 yards cost? 
 
 89. 1 oz. is what part of a lb. ? 
 
 90. What fkrt of a lb. is 2 oz.? 3 oz.? 4 oz. ? 5 
 oz. ? 7 oz.? 10 oz.? 15 oz.? 
 
 91. What part of a qr. of 1 cwt. is 1 lb. ? 2 lbs.? 
 3 lbs.? 4 1bs..i' 7 lbs.? 9 lbs.? 14 lbs.? 18 lbs.? 23 
 lbs.? 
 
 92. At 3 cents for 1 oz. what would 1 lb. cost? 
 
 93. At 3 cents for 2 oz. what would 1 lb. cost ? 
 
 94. At 3 cents for 8 oz. what would 1 lb. cost? 
 
 95. At 5 cents for 10 pz. what would 1 lb. cost? 
 
 96. At 8 shillings for 4 lbs. what would 10 lbs. 
 cost? 
 
 97. If a man consumes 1 lb. and 3 oz. of meat in a 
 day, how much would he consume in a week? 
 
PARTI.] ARITHMETIC. 121 
 
 98. If a man spends 2| dollars in a day, how much 
 would he spend in a wcekj? 
 
 99. If a man travels 3| miles in an hour, how far 
 would he travel in 3 hours ? How far in 7 hours if 
 How far in 12 hours? 
 
 100. If 2 men start from the same place, and travel 
 in opposite directions ; one at the rate of o| miles in 
 an hour, and the other 4?- miles; how far will they be 
 apart at the end of 1 hour? How far at the end of 2 
 hours ? How far at the end of 3 hours ? How far at 
 the end of 7 hours ? 
 
 101. Two men start from the same place, and travel 
 the same way; one at the rate of 4\ miles #n an hour, 
 the other at the rate of 4| miles in an hour; how far 
 will they be apart at the end of 1 hour ? How far in 
 
 2 hours ? How far in 5 hours? How far in 10 hours? 
 How far in o days, if they travel 10 hours in a day? 
 
 102. How many yards of cloth, at 5 dollars a yard, 
 must be given for 8 barrels of flour, at 7 dollars per 
 barrel ? 
 
 103. What part of a month is 1 week ? 2 weeks ? 
 
 3 weeks ? 
 
 104. What part of a year is 1 month? 2 months? 
 3 months? 4 months? 5 months? 6 months? 7 
 months? 8 months? 9 months? 10 months? 11 
 months ? 
 
 105. What part of 1 month is 1 day ? 2 days ? 3 
 days? 7 days? 8 days? 11 days? 15 days? 18 
 days ? 20 days ? 24 days ? 27 days ? 
 
 106. If 5 bushels of oats will keep 7 horses throui!;h 
 the winter, how many bushels will it take to keep 12 
 horses the same time ? 
 
 107. If you give 7 men 2i bushels of corn apiece, 
 how many bushels would it take for the whole? 
 
 108. A man, failing in trade, was able to pay his 
 creditors only 4 shillings on a dollar; how much would 
 he pay on 2 dollars ? How much on 3 dollars ? How 
 much on 7 dollars ? How much on 10 dollars ? 
 
122 ARITHMETIC. [paet i. 
 
 109. A man, failing in trade, is able to pay only 9 
 shillings on a pound; how. much would he pay on a 
 debt of 2 pounds ? How much on 3 pounds ? How 
 much on 12 pounds ? 
 
 110. A man, failing in trade^ is able to pay only 4 
 shillings and 7 pence on a dollar ; how much would he 
 pay- on a debt of 7 dollars? 
 
 111. If 6 dollars' worth of provisions will serve 3 
 men 5 days, how many days will it serve 1 man ? How 
 many days will it serve 2 men ? How many days will 
 it serve 8 men ? 
 
 112. If 10 dollars' worth of provisions will serve 7 
 men 4 daysk, how many days will it serve 9 men ? 
 
 113. If 12 dollars' worth of provisions will serve 7 
 men 3 days, how many men would it serve 1 day ? 
 How many 2 days ? How many 8 days ? 
 
 114. If 11 dollars' worth of provisions will serve 6 
 men 8 days, how many men will it serve 5 days ? 
 
 115. If 8 dollars' worth of provisions will serve 7 
 men 5 days, how many days would 16 dollars' worth 
 of provisions last 4 men ? 
 
 116. If 1 peck of wheat affords 12 ten-penny loaves, 
 how many penny loaves may be obtained from it? 
 How many two-penny loaves? How many -three- 
 penny loaves ? How many seven-penny loaves ? 
 
 117. If 1 peck of wheat affords 11 eight-penny 
 loaves, how many ten-penny loaves will it afford ? 
 
 118. A man having hired some men and some boys, 
 agreed to give each man. 3 shillings, and each boy 2 
 shillings; how much would it take to pay a man and a 
 boy? How much 2 men and 2 boys? How much 7 
 men and 7 boys ? 
 
 119. A man having 18 shillings to pay among his 
 laborers, would give to every man 2 shillings, and to 
 every boy 1 shilling; the number of men and boys 
 was equal ; how many were there of each ? 
 
 120. A gentleman having 50 shillings to pay among 
 his laborers, would give to every man 8 pence, and to 
 
PART I.] ARITHMETIC. 123 
 
 every boy 4 pence; the number of men and boys Was 
 equal; how mftny were there of each ? 
 
 121. Two men bought a bushel of corn, one gave 
 1 shilling, the other 2 shillings ; what part of ^e 
 whole did each pay ? What part of the corn must 
 each have ? 
 
 122. Two men 1>ought a barrel of flour for 8 dollars ; 
 one gave 3 dollars, the other 5 dollars ; what part (^id 
 ^ach pay ? and what part must each have ? 
 
 128. Three men, A, B, and C, rented a garden ;**A 
 paid 6 dollars, B 5 dollars, and G 9 dollars ; how much 
 did they all pay? What part of the whole did each 
 pay? They sold the produce for 40 dollars; what 
 part of it must each have ? What did each one's 
 share amount to ? 
 
 124. Three men bought a lottery ticket for 10 dol- 
 lars ; the first gave 3 dollars, the second 5 dollars, and 
 the third 2 dollars; they drew a prize of 120 dollars; 
 what was each man's share ? 
 
 125. Two men rented a pasture for 58 dollars; one 
 put in 7 horses, and the other 3 horses; what ought 
 each to pay ? 
 
 126. Three men commenced trade together; they 
 put in money in the following proportion ; the first 3 
 dollars, as often as the second put in 4, and as often as 
 the third put in 5; they gained 87 dollars; what was 
 each man's share of the gain? * 
 
 127. Two men rented a pasture for 32 dollars ; the 
 first put in 3 sheep for 4 months, the second put in 4 
 sheep for 5 months ; how much ought each to pay ? 
 
 Note. — Three sheep for 4 months is the same as 12 
 sheep for 1 month, 4 sheep for 5 months is the same 
 as 20 sheep for 1 month. This question is, therefore, 
 the same as if 1 man put in 12 sheep, and the other 
 20 sheep. 
 
 128. Two men, A and B, traded in company : A 
 put in 1 dollar for 4 months, and B 2 dollars for 3 
 
124 ARITHMETIC. [part i. 
 
 months, and they gained ninety cents ; how many 
 cents must each have ? 
 
 129. Three men, A, B, and C, traded in company; 
 and put in money in the following proportions . A put 
 in 4 dollars as often as B put in 3, and as often as C 
 put in 2 ; A's money was in 2 months, B's 3 months, 
 and C's 4 months, and they gained 100 dollars ; what 
 was each one's share ? 
 
 130. Two men, A and B, traded in company; A 
 put in 2 dollars as often as B put in 3 ; A's money was 
 employed 7 months, and B's money 5 months ; they 
 gained 58 dollars; what was each man's share of the 
 gain ? 
 
 131. Three men, A, B, and C, traded in company, 
 and put in money in the following proportions : A put 
 in 2 dollars as often .as B put in 4, and as often as C 
 put in 6. B's money was in twice as long as C's, and 
 A's two times as long as B's; they gained 88 dollars; 
 what was each one's share of the gain ? 
 
 Note. — Interest is a reward or premium allowed by a 
 debtor to a creditor for the use of money. The interest for 
 1 year, as established by law in some of the States, is 6 
 cents on a dollar, 6 dollars on a hundred dollars, or in fine 
 ■j§iy of the sum whatever be the denomination. It is called 
 6 per cent., that is, 6 on the hundred, because it is always 
 reckoned by the hundred. So 3 per cent., 4 per cent., etc., 
 signify j§^, y^^^, etc., or so much on the hundred. The 
 teacher can vary the examples to illustrate the different 
 rates per cent. 
 
 132. The interest of 1 dollar being 6 cents for 1 
 year, what is the interest of 7 dollars for the same 
 time? What is the interest of 10 dollars? of 15 dol- 
 lars? of 20 dollars? of 30 dollars? of 50 dollars? 
 of 75 dollars ? of 100 dollars ? of 118 dollars ? 
 
 133. If the interest of 1 dollar is 6 cents for 1 year, 
 what would it be for 2 years ? What would be the 
 interest of 8 dollars for 2 years ? of 17 dollars ? of 
 43 dollars ? 
 
PART I.] ARITHMETIC. 125 
 
 134. If the interest of 100 dollars is 6 dollars for a 
 year, what would be the interest of 50 dollars for the 
 same time ? of 2 hundred ? of 3 hundred ? of 4 hun- 
 dred ? of 1 hundred and 50 ? of 2 hundred and 50 ? 
 
 135. If the interest of 100 dollars is 6 dollars for 1 
 year, what would be the interest of it for G mouths ? 
 for 3 months? for 4 months? for 8 months? for 9 
 months? for 1 month? for 2 months? for 5 months? 
 for 7 months? for 10 months ? for 11 months? 
 
 136. What is the- interest of* 100 and 32 dollars for 
 2 years, at 6 per cent.? 
 
 137. What is the interest of 300 dollars for 1 year 
 and 6 months, at 6 per cent. ? 
 
 138. What is the interest of 1 dollar for 6 months, 
 at 6 per cent. ? what for 2 months ? what for 1 month ? 
 what for 3 months ? 4 months ? 5 months ? 7 months? 
 9 months? 11 months? 
 
 139. What is the 'interest of 57 dollars for 1 year 
 and 7 months, at 6 per cent. ? 
 
 140. What is the interest of 200 and 67 dollars for 
 
 1 year and 4 months, at 6 per cent. ? 
 
 141. If the interest of 1 year is 6 per cent., what 
 would be the per cent for 2 years ? for 3 years ? for 6 
 months ? for 2 months ? for 1 month ? for 4 months ? 
 for 5 months ? for 7 months ? for 8 months ? for 9 
 fiionths ? 
 
 142. If the interest of 2 months, or 60 days, is 1 per 
 cent,, what would be the per cent for 20 days? what 
 for 40 days ? what for 15 days ? what for 45 days ? 
 what for 12 days? what for 10 days? what for 5 
 days ? 
 
 143. What is the interest of 100 and 37 dollars for 
 
 2 years 3 months and 20 days ? 
 
 144. A can do a piece of work in 2 days ; how 
 much of it can he do in 1 day ? 
 
 145. B can do a piece of work in 4 days; how 
 much of it can he do in 1 day ? 
 
 146. If A can do J of a piece of work in 1 day, 
 
126 ARITHMETIC. [parti. 
 
 and B can do i of it in 1 day, how much would both 
 do in a day ? How long would it take them both 
 together to do the whole ? 
 
 147. If 1 man can do a piece of work in 2 days, 
 and another in 3 days, how much of it would each do 
 in a day? How much would both together do? How 
 long would it take them both to do the whole ? 
 
 148. A cistern has 2 cocks; the first will fill it in 3 
 hours, the second in 6 hours ; how much of it would 
 each fill in an hour ? How much would both together 
 fill? How long would it take them both to fill it ? 
 
 149. A man and his wife found by experience, that, 
 when they were both together, a bushel of meal would 
 last them only 2 weeks ; but when the man was gone, 
 it would last his wife 5 weeks; how much of it did both 
 together consume in 1 we<ek ? What part did the woman 
 alone consume in 1 week ? What part did the man 
 alone consume in 1 week ? How*long would it last the 
 man alone? 
 
 150. If 1 man could build a piece of wall in 5 days, 
 and another man could do it in 7 days, how much of 
 it would each do in 1 day ? How many days would it 
 take them both to do it ? 
 
 151. A cistern has 3 cocks; the first would fill it in 
 3 hours; the second in 6 hours; the third in 4 hours ; 
 whrft part of the whole would each fill in 1 hour ? and 
 how long would it take them all to fill it, if they were 
 all running at once? 
 
 152. A and B together can build a boat in 8 days, 
 and with the assistance of C they can do it in 5 days ; 
 how much of it can A and B build in 1 day ? How 
 much of it can A, B, and C, build in 1 day ? How 
 much of it can C build alone in 1 day? How long 
 would it take C to build it alone ? 
 
 153. Suppose I would line 8 yards of broadcloth that 
 is 1^ yards wide, with shalloon that is f of a yard wide; 
 how many yards of the shalloon will line 1 yard of the 
 broadcloth ? How many yards will line the whole ? 
 
PAKT I.] ARITHMETIC. 127 
 
 154. If 7 yards of cloth cost 13 dollars, what will 
 10 yards cost ? 
 
 155. If the wages of 25 weeks come to 75 dollars, 
 what will be the wa<res of 7 weeks? 
 
 156. If S.tons of hay will keep 7 horses three months, 
 how much will keep 12 horses the same time ? 
 
 157. If a staff 4 feet long casts a shadow 6 feet long, 
 what is the length of a pole that casts a shadow 58 feet 
 at the same time of day ? 
 
 158. If a stick 8 feet long casts a shadow 2 feet in 
 length, what is the height of a tree which casts a shadow 
 42 feet at the same time of day ? 
 
 159. x\t six dollars per week, how many months 
 board can I have for 100 dollars ? 
 
 160. A ship has sailed 24 miles in 4 hours; how 
 long will it take her to sail 150 miles at the same rate? 
 
 161. 30 men can perform a piece of work in 20 days ; 
 how many men will it take to perform the same work 
 in 8 days '/ 
 
 162. 17 men can perform a piece of work in 25 
 days ; in how many days would 5 men perform the same 
 work ? 
 
 163. A hare has 76 rods the start of a greyhound, 
 but the greyhound runs 15 rods to 10 of the hare; how 
 many rods must the greyhound run to overtake the 
 hare ? 
 
 164. A garrison has provision for 8 months, at the 
 rate of 15 ounces per day; how much must be allowed 
 per day, in order that the provision may last 11 months? 
 
 165. If 8 men can build a wall 15 rods in length in 
 10 days, how many men will it take to build a wall 45 
 rods in length in 5 days ? 
 
 166. If a quarter of wheat affords 60 ten-penny 
 loaves, how many eight-penny loaves may be obtained 
 from it ? 
 
 167. Said Harry to Dick, My p\irse and money 
 together are worth 16 dollars, but the money is worth 7 
 
128 ARITHMETIC. [part i. 
 
 times as much as the purse; how much money was there 
 in the purse ? and what is the value of the purse ? , 
 
 168. A man being asked the price of his horse, 
 answered, that his horse andsaddle together were worth 
 100 dollars, but the horse was worth 9 times as much 
 as the saddle; what was each worth ? 
 
 169. A man having a horse, a cow, and a sheep, was 
 asked what was the value of each. He answered, that 
 the cow was worth twice as much as the sheep, and the 
 horse 8 times as much as the sheep, and that all together 
 were worth 60 dollars ; what was the value of each ? 
 
 170. A man bought an apple, an orange and a melon, 
 for 21 cents ; for the orange he gave twice as much as 
 for the apple, and for the melon he gave twice as much 
 as for the orange; how much did he give for each ? 
 
 171. If 80 dollars' worth of provisions will serve 20 
 men 24 days, how many days will 100 dollars' worth 
 of provisions serve 80 men 1 
 
 172. There is a pole \ and \ under water, and 10 
 feet out ; how long is the pole ? 
 
 173. In an orchard of fruit trees, \ of them bear 
 apples, \ of them bear plums, \ of them pears, 7 of 
 them peaches, and 3 of them cherries ; how many trees 
 are there in the whole, and how many of each sort? 
 
 174. A farmer being asked how many sheep he had, 
 answered, that he had them in 4 pastures ; in the first 
 he had \ of his flock ; in the second i; in the third i; 
 and in the fourth 15; how many sheep had he ? 
 
 175. A man driving his geese to market, was met by 
 another, who said. Good-morrow, master, with your 
 hundred geese. Says he, I have not a hundred; but if 
 I had half as many more as I now have, and two geese 
 and a half, I should have a hundred; how many had 
 he? 
 
 176. What number is that, to which if its half be 
 added the sum will be 60 ? 
 
 177; What number is that, to which if its third be 
 added the sum will be 48 ? 
 
PART I.] ARITHMETIC. 129 
 
 178. What number is that, to which if its fifth be 
 added the sum will be 54 ? 
 
 179. What number is that, to which ifits half and its 
 third be added the sum will be 55 ? 
 
 ^ 180. A man being asked his age, answered, that if 
 Its half and its third were added to it, the sum would be 
 77 ; what was his age ? 
 
 181. What Dumber is that which being increased by 
 Its half, its fourth, and eighteen more, will be doubled? 
 
 182. A boy being asked his age, answered, that if 
 ^ and i of his age, and 20 more were added to his 
 age, the sum wo«ld be 3 times his age. What was his 
 
 age / 
 
 18o. A man being asked how many sheep he had, 
 answered, that if he had as many more, ^ as many more, 
 and 2 J sheep, he should have 100. How many had he*'' 
 
ARITHMETIC. 
 
 PART II. 
 
 KEY. 
 
 The Key contains remarks on the principles employed, 
 and illustrations of the manner of solving the examples in 
 each section. All the most difficult of the practical exam- 
 ples are solved in such a manner as to show the principles 
 by which they are performed. Care has been taken to 
 select examples for solution, tliat will explain those which 
 are not solved.. Many remarks with refjard to the manner 
 of illustrating the })rinciples to the pupils, are inserted in 
 their proper places, , 
 
 Instructors who may never have attended to fractions, 
 need not be afraid to undertake to teach this " book. The 
 author flatters hihiself that the principles are so illustrated, 
 and the processes are made so simple, that any one, who 
 shall undertake to teach it, will find himself familiar "with 
 fractions before he is aware of it, although he knew nothing 
 of them before ; and that every one will acquire a facility 
 in solving questions, which he never before possessed. 
 
 The reasoning used in performing these small examples, 
 is precisely the same as that used upon large, ones. And 
 when any one finds a difficulty in solving a question, he 
 will remove it much sooner, and much more effectually, 
 by taking a very small example of the same kind, and 
 observing how he does it, than by recurring to a rule. 
 
 The practical examples at the commencement of each 
 section and article, are generally such as to show the pupil 
 w^hat the com])ination is, and how he is to perform it. 
 This will teach the pupil gradually to reason upon abstract 
 numbers. In each combination, there are a few abstract 
 examples without practical ones, to exercise the learnei* in 
 the combinations, after he knows what these combinations 
 are. It would be an excellent exercise for the pupil to put 
 
 (131J 
 
132 KEY. [PAET II. 
 
 these into a practical form when he is reciting. For instance, 
 ■when the question is, how many are 5 and 3 ? Let him 
 make a question in this way : if an orange cost five cents, 
 and an apple 3 cents, what would they both come to ? This 
 may be done in all cases. 
 
 The examples are often so arranged, that several depend 
 on each other, so that the preceding explains the following 
 one. Sometimes also, in the same example, there are sev- 
 eral questions asked, so as to lead the pupil gradually from 
 the simple to the more difficult. It would be well for the 
 pupil to acquire the habit of doing this for himself, when 
 difficult questions occur. 
 
 The operations can be illustrated by counters, or marks 
 on the blackboard, according to the necessity of the pupils. 
 These illustrations will be less necessary as the pupils 
 advance in the work; but a frequent reference to them 
 throughout most of the book will be useful, in fixing more 
 clearly in mind the principles involved in the operations. 
 
 The book may be used in classes where it is convenient. 
 The pupil may answer the questions with the book before 
 him or not, as the instructor thinks proper. A very useful 
 mode of recitation is for the instructor to read the example 
 to the whole class, and then, allowing sufficient time for 
 them to perform the question, call upon some one to answer 
 it. In this manner, all the pupils will be obliged to perform 
 the example, because they do not know who is to answer it. 
 In this way it will be best for them to answer without the 
 book. 
 
 It will often be well to let the elder pupils hear the 
 younger. This will be a useful exercise for them, and an 
 assistance to the instructor. 
 
 SECTION I. 
 
 A. This section contains addition and subtraction. The 
 first example may be solved by means of beans, peas, etc., 
 or by means of the blackboard. The former method is 
 preferable, if the pupil be very young, not only by the 
 examples in the first part of this section, but by the first 
 examples in all the sections. 
 
 The pupil will probably solve the first examples without 
 any instruction. 
 
SEC. II.] KEY. 133 
 
 B & C. The articles B and C contain the common addi- 
 tion table as far as the first 10 numbers. In the first, the 
 numbers are placed in order; and in the second, out of 
 order. 
 
 The pupil should study these until he can find the 
 answers readily, and then he should commit the answers 
 to memory. > 
 
 D. In this article the numV)er8 are laro;er than in the 
 preceding; and, in some instances, three or more numbers 
 are added togetlier. In the abstract examples, the numbers 
 from one to ten are to be added to the numbers from ten to 
 twenty. 
 
 E. This article contains subtraction. 
 
 F. This article is intended to make the pupil familiar 
 with adding the first nine numbers to all others. The pupil 
 should study it until he can answer the questions very 
 readily. 
 
 G. In this article all the preceding arc combined together, 
 and the numbers from 1 to 10* are added to all numbers 
 from 20 to 100, and subtracted in the same manner. 
 
 18. 57 and G are G3, and 3 are 66, and 5- are 71, and 2 
 are 73, less 8 are 05. 
 
 H. This article contains practical questions which show 
 the application of all the preceding articles. 
 
 6. 37 less 5 are 32, less 8 are 24, less 6 (which he kept 
 hiinself ) are 18; consequently he gave 18 to the third boy. 
 
 SECTION II. 
 
 This section contains multiplication. The pupil will see 
 no difi'erence between this and addition. It is best that he 
 should not at first, though it may be well to explain it to 
 him after a while. 
 
 A. This article contains practical questions, which the 
 
 pupil will readily answer. 
 
 * Figures are used in the key. because the instructor is supposed 
 to be acquainted with tliem. They are not used in the first part of the 
 book, because the pupil would not understand them so well as he will 
 the words. 
 
134 KEY. - [part n. 
 
 I. Three yards will cost 3 times as much as 1 yard. 
 
 N. B. — Be careful to make the pupil give a similar reason 
 for multiplication, both in this article and elsewhere. 
 
 II. A man will travel 4 times as far in 4 hours as he will 
 in 1 hour. 
 
 15. There are 4 times as many feet in 4 yards as in 1 
 yard, or 4 times 3- feet. 
 
 B. This article contains the common multiplication table, 
 as far as the product of the first ten numbers. The pupil 
 should find the answers once, or twice through, until he 
 can find them readily, and then let him commit them to 
 memory. 
 
 C. This article is the same as the preceding, except in 
 this, the numbers are out of their naturaLprder. 
 
 D. In this article, multiplication is applied to practical 
 examples. They are of the same kind as those in article A 
 of this section. 
 
 12. There are 8 times as many squares in 8 rows, as in 1 
 row. 8 times 8 are 64. • ^ 
 
 13. There are 6 times as many farthings in 6 pence, as 
 in 1 penny. 6 times 4 are 24. 
 
 17. 12 times 4 are 48. 
 
 23. There are 3 times as many pints in 3 quarts as in 1 
 quart, 3 times 2 are 6. And in 6 pints there are 6 times 
 4 gills, or 24 gills. 
 
 28. In 3 gallons there are 12 quarts, and in 12 quarts 
 there are 24 pints. 
 
 31. In 2 gallons are 8 quarts ; in 8 quarts 16 pints ; in 16 
 pints 64 gills. 16 times 4 are 64. 
 
 35. In 1 gallon are 32 gills ; and 32 times 2 cents are 64 
 cents. Or, 1 pint will cost 8 cents, and there are 8 pints 
 in a gallon. 8 times 8 are 64. 
 
 38. They will be 2 miles apart in 1 hour, 4 miles in 2 
 hours, etc. 
 
 SECTION III. 
 
 A. This section contains division. The pupil will 
 scarcely distinguish it from multiplication. It is not im- 
 portant that he should at first. 
 
SEC. III.] KEY. 135 
 
 The pupil will be able to answer these questions by the 
 multiplication table, if he has committed it to memory 
 thoroughly. 
 
 B. In this article the pupil obtains the first ideas of 
 fractions, and learns the most important of the terms which 
 arc applied to fractions.* The pupil has already been 
 accustomed to look upon a collection of units, as formiufj; a 
 number, or as being itself a part of another number. He 
 knows, therefore, that one is a part of every number, and 
 that ever}^ number is a part of every numljor larger than 
 itself As every number may have Ji variety of parts, it is 
 necessary to give names to the different parts, in order to 
 distinguish tliem from each other. The parts receive their 
 names, according to the numl»er of parts into which any 
 numljcr is divided. If the number is divided into two' equal 
 parts, the parts arc called halves ; if it is divided into three 
 equal parts, they are called thirds ; if into four parts, 
 fourths, etc. ; and having divided a number into parts, we 
 can take as many of the parts as we choose. If a number 
 is divided into five equal parts, and three of the parts are 
 taken, the fraction is called three Jifths of the number. 
 The name shows at once into how many parts the number 
 is to be divided, and how many parts are taken. 
 
 The examples in this book are so arranged that the 
 names will usuall}^ show the pupil how the operation is to 
 be performed. In this section, although the pupil is taught 
 to divide numbers into vjlrious parts, he is not taught to 
 notice any fractions, except those where the numbers are 
 divided into their simple units, which is the most simple 
 kind. 
 
 It will be best to use beans, pebbles, etc., first. 
 
 20. Since 1 is 1 third of 3, 2 must be 2 thirds of 3. 
 
 34. Illustrate by grouping the marks or counters by 
 threes. 
 
 Pi'oceed in the same manner with the other divisions. 
 
 This being one of the most useful combinations, and one 
 but very little understood by most people, especially when 
 applied to large numbers*, the pupil must be made perfectly 
 familiar with it. Ask qjaestions like those in the book for 
 large numbers, and- also some like the following: What 
 part of 7 is 18? the answer will be V. 
 
 * As soon as the terms applied to fractions are fully comppohended, 
 the operations on them are as simple as those on whole numbers. 
 
136 KEY. [part ii. 
 
 C. The first ten figures are here explained. They are 
 used as an abridged method of writing numbers, and not 
 with any reference to their use in calculating. 
 
 This article is only a continuation of the last. All the 
 numbers from 1 to 100 are introduced into the two articles, 
 and are divided by all the numbers from 1 to 10 ; except 
 that some of the largest are not divided by some of the 
 smallest. 
 
 2. The pupil answers first, how many times 2 is con- 
 tained in 12, then how many times 3. 
 
 D. These examples, which are similar to those in article 
 A of this section, are solved in the same manner. 
 
 5. It would take as many hours as 3 miles are contained 
 in 10 miles. 3 hours and I of an hour. 
 
 20. They cost as many cents as there are 3 apples in 30 
 apples ; that is, 10 cents. 
 
 21. 12 dollars a month ; and 12 dollars a month is 3 dol- 
 lars a week ; that is, 18 shillings a week, which is 3 shil- 
 lings a day. 
 
 26. The whole loss was 35 dollars, which was 7 dollars 
 apiece. 
 
 SECTION IV. 
 
 A. This article contains multiplication simply. It is 
 repeating a number a certain number of times and a part 
 of another time. 
 
 14. 6 times 5 are 30, and | of 5 are 3, which added to 
 30 make 33. 
 
 B. In this article the pupil is taught to change a certain 
 number of twos into threes, threes into fives, etc. This 
 article combines all the preceding operations. 
 
 24. 4 cords of wood will cost 28 dollars, and f of a cord 
 will cost 2 dollars, which makes 30 dollars. 30 dollars will 
 buy 3 hundred weight of sugar and f of another hundred 
 weight. 
 
 29. 7 times 8 are 56, and f of 8 are 5, which added to 
 56 make 61 ; 61 are 6 times 9, and ^ of 9. 
 
 C. 1. 4 bushels of apples, at 3 shillings a bushel, come 
 to 12 shillings ; and 12 shillings are 2 dollars. 
 
SEC. v.] KEY. 137 
 
 2. The 2 lemons come to 8 cents, and 8 cents will buy 4 
 apples, at 2 cents apiece. 
 
 This is usually called Barter. The {j^eneral principle is 
 to find what the article will come to, whose price and quan- 
 tity are given, and then to find how much of the other 
 arlicle that money will buy. 
 
 G. If 2 npples cost 4 cents, 1 will cost 2 cents, and 4 will 
 cost 8 cents. Or 4 apples will cost 2 times as much as 2 
 apples. 
 
 22. Find how many times 2 pears arc contained in 20 
 poars, which is 10 times. 10 times 3 cents are 30 cents. 
 Or first find what 20 pears would come to at 3 cents apiece ; 
 and since it is 2 for 3 cents, instead of 1 for 3 cents, the 
 price will be half as much. 
 
 23. See how many times you. can have 5 cents in 30 
 cents, and you can buy so many times 3 eggs. 30 is 6 
 times 5, and 6 times 3 are 18. 18 eggs. 
 
 24. 10 dollars a week, and 40 dollars a month. 
 
 25. 5 dollars are 30 shillings, which is 10 shillings a day. 
 6 shillings is" equal to 1 dollar in some of the States. 
 
 26. 5 dollars apiece. 
 
 SECTION V. 
 
 In this section the principle effractions is applied to 
 larger numbers, btrt such as are divisible, into the parts 
 proposed to be taken. The pupil, who is familiar with 
 what precedes, will easily understand the examples in this 
 section. They require nothing but dinsion and multipli- 
 cation. 
 
 A. Let the pupil explain each example in the following 
 manner: What is 1 sixth of 18? Aus. 3. Why? Be- 
 cause 6 times 3 are 18; therefore if you divide 18 into 6 
 equal parts, one of the parts will be 3. 
 
 The pupil will be very likely to say 3 is the 6th part of 
 18, because 3 times 6 arc 18. Be careful to make him say 
 it the other way, viz., 6 times 3 are 18. 
 
 14. 1 third of 9 is 3 ; | is 2 times as much as ^, there- 
 fore I of 9 is 6. 
 
 19. 1 barrel will cost { part of 12 dollars; 3 barrels will 
 cost f of 12 dollars ; 7 barrels will cost | of 12 dollars. 
 
 37. What is f of 32? ^ of 32 is 4, f are 5 times 4, or 20. 
 
138 KEY. [part ii. 
 
 B. 11. I of 20 is 4; I are 7 times 4, of 28; and 28 is 4 
 times 6 and f of 6. 
 
 C. 3. 1 half of 10 is 5; f of 10 are 4; 5 and 4 are 9. He 
 gave away 9, and had 1 left. 
 
 4. 1 yard will cost ^,of what 3 yards cost. ^ of 6 dol- 
 lars is 2 dollars. 
 
 5. 2 yards will cost 1 half of what 4 cost; or 6 dollars. 
 
 6. 3 apples will cost ^ of what 9 cost, or 6 cents. 
 
 7. 2 is f of 3 ; therefore 2 oranges will cost | of what 3 
 cost, f of 18 cents are 12 cents. 
 
 ■ 8. f of 25" are 20. The 10 apples cost 20 cents, which 
 was 2 cents apiece. 
 
 11. f of 42 are 12, and 6 times 12 are 72. 72 dollars. 
 
 13. 3 is f of 4; f of 12 dollars are 9 dollars; or 4 yards 
 at 12 dollars is 3 dollars a yard, and 9 dollars for 3 yards. 
 
 14. Solved like the 13. Ans. 15 cents. 
 
 15. Since 1 is ^ of 3, 7 is ^ of 3: | of 15^ cents are 35 
 cents ; or 3 oranges at 15 cents, is 5 cents apiece. 7 times 
 5 cents are 35 cents. 
 
 Note. — In questions of this kind, it is generally the sim- 
 plest way to find what 1 article will cost, then it may easily 
 "be told how much any number will cost. 
 
 19. 4 men Avould do it in 1 half the time that 2 would do it. 
 Or, you may say, if 2 men would do it in 6 days, 1 man would 
 do it 12 days, and 4 men in I of that time, or 3 days. 
 
 SECTION VI. 
 
 A. 4. 2 halves of any number ;nake the whole number. 
 Therefore 2 is 1 half of 2 times 2 ; or 4. It is i of 4 times 
 2, or 8. 
 
 Let the pupil answer these questions in the following 
 manner: 4 is } of 3 times 4; 3 times 4 are 12. 5 is if of 
 7 times 5 ; 7 times 5 are 35. 
 
 B. 2. 4 is 2 times2. 
 4. 6 is 2 times 3. 
 
 16. 2 thirds of any number is twice as much as ^ of the 
 same number. If 4 is f of some number, then 1 half of 4 
 or 2 is i of that number ; 2 is J of 6 ; therefore 4 is f of 6. 
 
SEC. VI.] KEY. 139 
 
 20. If 6 is f of a number, |^ of 6 or 2 is J of the same 
 number ; 2 is ^ of 8 ; therefore 6 is f of 8. 
 
 23. It is evident that ^ of a pound will cost only J of 
 what ^ will cost. If ^ co.st 6 cents, ^ will cost 2 cents, 
 and the whole pound 14 cents. 
 
 26*. It will probably be perceived by this time, that f of 
 a number being given, it is necessary to find \, and then 
 the number is easily found ; 4 being f , 2 is i,, and 2 is | 
 of 14. 
 
 45. 24 being |, i of 24 or 3 will be ^ ; 3 is ^ of 27. 
 
 C. 6. 20 being f , 5 is j, and 5 is 4 of 35 ; and 35 is 5 
 times 6, and f of 6. 
 
 D. 4. 18 is three times 6, and 6 is 1 of 4 times 6, or 24. 
 A71S. 24 dollars. 
 
 6. 54 is I of 48 ; 12 yards at 48 dollars i^ 4 dollars a yard. 
 He cainod 6 dollars. 
 
 7." 10 feet is | of 15 feet. 
 
 8.. If I are under water, there must be ^ out of the water. 
 4 is i of 12. 
 
 9. Iff arc underwater, there must be f out of the water. 
 6 is I of 10. 
 
 10. f and f ai'c |. f bear cherries and peaches ; conse- 
 quently, the 10 which bear plums must be the other f . 10 
 is f of 35. 10 bear peaches, and 15 bear cherries. 
 
 11. |. and |, and |, and ^, are | ; therefore 12 must be 
 the other f of the whole. The whole number is 54. 
 
 MISCELLANEOUS EXAMPLES. 
 
 6. The greyhound gains upon the fox 4 rods in a minute. 
 It will take him 20 minutes to gain 80 rods. 
 
 8. I of 24. Or you may say, 1 sheep would cost 3 dollars, 
 and 3 sheep 9 dollars. 
 
 9. 30 horses will eat 10 times as much as 3 horses. 
 
 11. 10 dollars apiece, and 2 dollars a yard. 
 
 12. 5 dollars for 1 week, 20 dollars^ for a month, and 25 
 dollars for 5 weeks. 
 
 14. It would take them 5 times as long to eat 40 bushels, 
 as it would to eat 8 bushels. 
 
 15. 4 horses would eat 4 bushels in 3 days, and it would 
 take them 9 times as long to eat 36 bushels. Ans. 27 days. 
 
140 . KEY. [part II. 
 
 16. If 2 men spend 12 dollars in 1 week, 1 man will 
 spend 6 dollars in 1 week, and 30 dollars in 5 weeks, and 
 3 men would spend 3 times as much, or 90 dollars. 
 
 17. The shadow of the staff is f of the length of the staff, 
 therefore the shadow of the pole must be | the length of the 
 pole. 18 feet is | of 27 feet. 
 
 20. It would take 2 men 3 times as long to do it as it 
 would 6 men. 
 
 23. 8 men would do a piece of work 1 half as large in 2 
 days, and it would take 2 men 4 times as long to do it, or 8 
 days. 
 
 28. He must sell it for 56 dollars in order to gain 16 dol- 
 lars. 56 dollars is 7 dollars per barrel. 
 
 29. It cost him 36 dollars, and he must sell it for 45 to 
 gain 10 dollars ; 45 dollars is 9 dollars a firkin. 
 
 30. Ans. 56 cents. See Section VI. 
 
 33- If it would last 3 men 10 months, it would last 1 man 
 30 months, and 5 men 6 months. 
 
 34. There are 8 times 5 in 40; and since the other would 
 build as many times 9 as the first does 5, he would build 6 
 times 9, or 72 rods. 
 
 SECTION VII. 
 
 A. 13. i of 20 is 4, f are 16 ; 16 being ^, 2 is | ; 2 is j 
 of 14, and 6 is f of 14. 
 
 16. f of 28 are 12 ; 12 is 2 times 6, and 6 is i of 48, (12 
 is I of 48, ) and 48 is 6 times seven and f of 7. 
 
 B. 1. f of 15 are 12 ; 12 is 6 times 2 ; 2 is yV of 20 ; (12 
 is A of 20 ;) i of 21 is 7 ; 20 is 2 times 7 and f of 7. 
 
 2. I of 18 are 24 ; 24 is f of 27 ; + of 35 is 5 ; 27 is 5 
 times 5 and | of 5. 
 
 C. This article contains the multiplication table, in which 
 the numbers from 10 to 20 are multiplied by the ten first 
 numbers. 
 
 SECTION VIII. 
 
 A. 1. In one there are 2 halves ; in 2 there are twice as 
 many halves, that is 4 halves or f . 
 
SEC. IX.] KEr. 141 
 
 2. In one there are 2 halves, in 3 there are three times 
 two halves, that is, 6 halves, or f . 
 
 3. In one there are 3 thirds, in 2 there are twice 3 thirds, 
 that is, 6 thirds, or f. 
 
 15. Draw two lines on the board thus ■ . Each line 
 
 call one ; divide the upper line into two equal parts, each 
 part is one half of one, or one half; divide the lower line 
 in the same way, and then call for the answer. The form 
 of the question may be varied by asking how many half 
 apples there are in 2 apples. 
 
 37. In 5 and 2 thirds, how many thirds? Draw five 
 parallel equal lines, divide them into thirds, and draw 
 another parallel with the others and two thirds as long, let 
 the scholar count the thirds. In all these cases of illustra- 
 tion oh the boaixi, it is presumed that the scholar will soon 
 take the hint, and devise illustrations for himself at his seat. 
 
 SECTION IX. 
 
 A. 2. ^ signifies that 1 thing is divided into 3 equal 
 parts, and 1 part taken. Therefore 2 times 1 third is 2 parts, 
 or f . _ 
 
 6. 7 times ^ is ^, or 2}. 
 
 B. 4. 4 times 2 are 8, and 4 times 1 half are 4 halves, or 
 2, which added to 8 make 10. 
 
 18. 4 times 3 are 12, and 4 times f are ^ or three whole 
 ones, which added to 12 make 15. 
 
 32. 2 timfes 3 are 6, and 2 times ^ are f , which added to 
 6 make 6f . 
 
 40. 10 barrels at 3 dollars and f a barrel ; 10 barrels at 
 3 dollars, would be 30 dollars, then 10 times f is *^, or 8 
 and f of a dollar. Ans. 38f dollars. 
 
 C. 2. I to each would be 3 times f , or f , which are 2J 
 oranges. 
 
 3. V* or 2 bushels. 
 
 4. 7 times f are ^^ , or 5^ gallons. 
 
 5. 8 yards and f or 2 yards, that is 10 yards. 
 
 6. 4 times 2 are 8, and 4 times f are ^, or 2f, which 
 added to 8 make lOf bushels. 
 
142 KEY. [part ii. 
 
 12. It would take 1 man 3 times as long as it would 3 
 men. Ans. 13f da}-,?. 
 
 14. 3 men would build 3 times as much as 1 man ; and 
 in 4 days they would build 4 times as much as in 1 day. 
 A)is. 38f rods. 
 
 15. A7is» 12 yards. 
 
 SECTION X. 
 
 A. 21. |- of 1 is i-. i of 2 is 2 times as much, or |. ^ 
 of 4 is f , or U. ^ of 5 is f , or If. J of 6 is f , or 2. l 
 of 7 is I, or 2i 
 
 27. J of 1 is ^. ^ of 2 is f . ;^ of 3 is f . ^ of 7 is ?, 
 or If. 
 
 This manner of reasoning may be applied to any num- 
 ber. To find j of 38: it is 3/, for j of 38 is 38 times as 
 much as ^ of 1, and | of 1 is |, consequently j of 38 is V, 
 and ^'^ is 5f. 
 
 40. To find |- of a number, -J must be found first, and 
 then I will be 2 times as much. :J of 7 is f , and 2 times 
 I are y, or 4f. 
 
 74. ^ of 50 is ^gO, or 5f ; | is 4 times as much; 4 times 
 5 are 20, 4 times f are ^JJ, or 2f, which added to 20 make 
 22f. 
 
 Note. — The manner employed in example 40th is best 
 f a- small numbers, and that in the 74th for large numbers. 
 
 B. 2. Ans. If apiece. 
 
 3. ;J^ of 3 is f ; f of a bushel apiece. 
 
 4. f of 7 is 4|; he gave away 4^, and kept 2f. 
 G. 1 half dollar a yard, or 50 cents. 
 
 7. i of 7 is I, or If ; | of a dollar is f of 100 cents,- 
 which is 40 cents. Ans. 1 dollar and 40 cents a bushel. 
 
 8. i of_8 is If; f of 100 is 33|. Ans. 1 dollar and 33| 
 cents, or it is 1 dollar and 2 shillings. 
 
 9. If 3 bushels cost 8 dollars, 1 bushel will cost 2 dollars 
 and f, and 2 bushels will cost 5J dollar's. Ans. 5 dollars 
 and 2 shillings, or 33| cents. 
 
 13. If 7 pounds cost 40 cents, 1 will cost 5f cents; 10 
 pounds will cost 57^^ cents. 
 
 16, 1 cock would empty it in 6 hours, and 7 cocks would 
 empty it in ^ of 6 hours, or f of 1 hour, which is f of 60 
 minutes; f of 60 minutes is 51§ minutes. 
 
SEC. XI.] KEY. 143 
 
 SECTION XL 
 
 A. 2. 2 halves of a number make the number, conse- 
 quently 1 and 1 half is the half of 2 times 1 and 1 half, 
 which is 3. 
 
 15, 4f is j; of 5 times 4 and f , which is 22|. 
 17. 4f is i of 9 times 4^, which is 39f . 
 
 B. 4, 5 is 3 times ^ of 5, which is f , or If. 
 
 30. If 8 is I of some number, ^ of 8 is ^ of the same 
 number. ^ of 8 is 2f, 2| is 1 of 4 times 2f which is lOf; 
 therefore 8 is f of lOf. 
 
 40. If 8 is f, h of 8 is ,^; i of 8 is |, f is ^ of V, or 9| ; 
 therefore 8 is f of 9|. 
 
 52. If f of a ton cost 23 dollars, ^ of a ton must be | of 
 23, that is 4f dollars, and the whole would cost 9 times as 
 much, that is 41?, 
 
 09. ^ of 65 is 7f ; 7| is i of 5 times 7f , which is 36^. 
 C5 is f of 36^. 
 
 C. 4. 37 is f of 32f, which taken from 37 leaves 4|. 
 Ans. 4^ dollars. 
 
 5. 7 feet must be | of the whole pole. 
 
 6. If he lost f, he must have sold it for | of what it cost. 
 47 is I of 60f . Ans, 60 dollars and 42f cents. 
 
 Miscellaneous Examples. 
 
 1. The shadow of the staff is f of the length of the staff; 
 therefore the shadow of the pole is f of the length of the 
 pole. 67 is f of 83 f. _ Ans. 83 f feeL 
 
 2. 9 gallons remain in the cistern in 1 hour. It Avill be 
 filled in 10 hours iand ^ ; ^ of 60 minutes are 46| minutes 
 and I ; | of 60 seconds are 40 seconds. Aris. 10 hours, 46 
 minutes, 40 seconds. 
 
 10. Find f of 33, and subtract it from 17. Ans. 3f . 
 
 11. It will take 3 times 10 yards. 
 
 13. 5 is f of 3 ; it will take f as much. Or, 7 yards, 5 
 quarters wide, are equal to 35 yards 1 quarter wide, which 
 is equal to llf yards that is 3 quarters wide. 
 
 15. f of 37 dollars. 
 
 16. f as much. 
 
144 KEY. [part 11. 
 
 SECTION XII. 
 
 The examples in this section are performed in precisely 
 the same manner as those in the sections to M^hich they 
 refer. All the difficulty consists in comprehending that 
 fractions expressed in figures signify the same thing as 
 when expressed in words. Make the pupil express them in 
 words, and all the difficulty will vanish. Let particular 
 attention be paid to the explanations of fractions given in 
 this section. 
 
 VIII. A. 6. In 7 how many ^? expressed in words, is 
 "in 7 how many sixths? Ans. */. 
 
 14. Reduce 8/^ to an improper fraction; that is, in 8 
 and 3 tenths how many tenths? Ans. f§. 
 
 B. 8. ^yp are how many times 1 ? That is, in 23 sev- 
 enths how many whole ones ? A7is. 3f . 
 
 IX. B. 3. How much is 5 times 6f ? That is, how much 
 is 5 times 6 and 4 sevenths? Ans. 32f. 
 
 V. & X. 15. What is | of 27? That is, what is 5 eighths 
 of 27? Ans. 16|. 
 
 VI. & XI. A. 8. 7f is i of what number? That is, 7 
 and 6 sevenths is 1 eighth of what number? Aois. 62f . 
 
 B. 4. 12 is f of what number? That is, 12 is 3 sevenths 
 of what number? Ans. 28-. 
 
 12. 4 is f of what number? That is, 4 is 3 fifths of 
 what number? Ans. 6| 
 
 SECTION XIII. 
 
 The operations in this section are the reducing of frac- 
 tions to a common denominator, and the addition and 
 subtraction of fractions. The examples will generally 
 show what is to be done, and how it is to be done. 
 
 4. It will readily be seen that ^ and ^ are f . 
 
 25. In the fourth square of the second row, it will be 
 seen that 1 half is | ; and in the second square of the 
 fourth row, j is |, both together make | and ^ make ■^. 
 
 27. f is the same as f . 
 
 When these questions are performed in the mind, the 
 pupil will explain them as follows. He will probably do 
 it Tvithout assistance. Twenty twentieths make one whole 
 
SEC. XIII.] KEY. 145 
 
 one. i of 20 is 5, and f of 20 is 8, and ^s of 20 is 2; 
 therefore J is /j, f is ^%, and -j^^ is g^y. All the examples 
 should be explained in the same manner, 
 
 45.. One whole orte is f g, one eighth of f | is j^. | is 3 
 times as much, which is |^. 
 
 51. 1 half is |, and i is f, which added together make |. 
 
 ^1- f is 2V. 1^6 i« 2%, i is /ff, which added together 
 make ^^. 
 
 67. f is j8y, I is j%, which added together make jj; from 
 fl take /j, and there remains jf, or 1. 
 
 82. It will be easily perceived that these examples do 
 not dijQTer from those in the first part of this section, except 
 in the language used. They must be reduced to a common 
 denominator, and then they may be added and subtracted 
 as easily as v.hole numbers, f "is \^, and f is -j^j, and both 
 together make ]f, or 11. 
 
 80. ^ is I, and '} is §. If | be taken from f, there re- 
 mains i. 
 
 B. This article contains only a practical application of 
 the preceding. 
 
 3. This example and some of the following contain mixed 
 numbers, but they arc quite as easy as the others. The 
 whole numbers may be added separately, and the fractions 
 reduced to a common denominator, and then added as in 
 other cases, and afterwards joined to the whole numbers. 
 and 2 are 8; 1 half and J are |, making in the whole 8f 
 bushels. 
 
 ^ 5. 6 and 2 are 8; f, and ^, and f are f^, or 1^|, which 
 joined with 8 make 9^g. 
 
 C. It is difficult to find examples which will aptly illus- 
 trate this operation. It can be done more conveniently by 
 tlie instructor. Whenever a fraction occurs, which may be 
 reduced to lower terms, if it be suggested to the pupil^ he 
 will readily perceive it and do it." This may be done in 
 almost any part of the ■ book, but more especially after 
 studying the 13th section. Perhaps it would be as well to 
 omit this article the first time the pupil goes through the 
 book, and, after he has $^ea the use of the operation, let 
 him study it. 
 
146 KEY. [part II. 
 
 SECTION XIY. 
 
 A. This roetion contams the division of fractions "by 
 whole numhei's, and the multiplication of one fraction hy 
 another. Though these operations sometimes appear to be 
 division, and sometimes multiplication, yet there is actually 
 no difference in the operations. 
 
 The practical examples will generally show how the 
 operations are to he performed, hut it will be well to illus- 
 trate the operation for young pupils. 
 
 1 and 2. ^ of ^^ is i of the whole. 
 
 3 and 4. | of ^ is ^. 
 
 16 and 17. ^ of | is y^ of the whole. 
 
 33. Since | of a share signify 3 parts of a share, it is 
 evident that ^ of the three parts is 1 part, that is, |-. 
 
 39. f signify 9 pieces or parts, and it is evident that ^ 
 of 9 parts is 3 parts, that is f . 
 
 43. We cannot take | of 5 pieces, therefore we must take 
 i of i, which is j'g , and f is 5 times as much as i, therefore 
 i of f is j-V- 
 
 78. 8f is V-iof V'isl. 
 
 79. 8f is *V', I of I is J^, consequently, ^ of ^ is f §, or 
 111. 
 
 *86. We may say ^ of 8| is 2, and 2| over, then 2| is %^, 
 and i of y is f f , hence ^ of 8| is 2|f . 
 
 90. ^ of 18f is"2ff, and f is 3 times as much, or 7ff. 
 
 B. 4. It would take 1 man 4 times 9f , or 37f days, and 
 7 men would do it in | of that time, that is, in 5^f days. 
 
 SECTION XV. 
 
 A. This section contains the divisions of whole numbers 
 by fractions, and fractions by fractions. - 
 
 1. Since there are f in 2 it is evident that he could give 
 them to G boys if he gave them J apiece ; but, if he gave 
 them f apiece, he could give them to only one half as many, 
 or 3 boys. » 
 
 5, If -J^ of a barrel would last them one month, it is evi- 
 dent that 4 barrels would last 20 months ; but since it takes 
 I of a barrel, it will last them but one half as long, or 10 
 months. 
 
SEC. XV.] KEY. 147 
 
 7. 6| is y. If I of a bushel would last a week, 6| 
 bushels would last 27 Aveeks ; but, since it takes f it will last 
 only J of the time, or 9 weeks. 
 
 13. If ho had given j of a bushel apiece, he might have 
 given it to 17 persons ; but, since he gave 3 halves apiece, 
 he could give it to only ^ of that number, that is, to 5 per- 
 sons, and he would have 1 bushel left, which would be f of 
 enou2:h for another. 
 
 23! Of is V' {^n^ U is V- If it had been onljit of a 
 dolhir a barrel, he might have bought 66 barrels for 9f dol- 
 lars ; but, since it was V a barrel, he could buy only ^j of 
 that numlier, that is, 6 barrels. • 
 
 25 and 26. Ans. 9 4. 
 
 31. 4| is ^^ , and 9f is ^. Now ^ is contained in Y 
 48 times,- and V is* contained only jy P^^t as many times, 
 consequently only 25^ or 2f . 
 
 B. 1. J is /jj ; consequently 5 pounds can be bought for 
 J^ of a dollar. 
 
 3. f is -fr^, and \ is j\. If he had given only -^ apiece, 
 he could have given it to 9 persons; but since he gave -^^ 
 he could give it to only 1 half as many, or 4^ persons. 
 
 5. if is j^\, and f is |f . If a pound had cost j'y of a dol- 
 lar, 14 pounds could be bought for |f of a dollar ; but 
 since it costs /y, only | as many can be bought ; that is, 4f 
 pounds. 
 
 9. f is \l, and If is f§. If a bushel had cost 5^ of a 
 dollar, 65 bushels might have been bought ; but, since it 
 cost ^^, only ^'g^ part as much could be bought; that is 4^*5- 
 bushels. 
 
 12. I is j^g, and f is jf ; ^s i^ contained in |f 15 times, 
 but -^ is contained only \ as many times ; that is, 3| times. 
 
 MISCELLANEOUS EXAMPLES. 
 
 5. f of a penny is f of 4 ftirthings. Ans. 2f farthings. 
 
 6. f of 12 pence. Ans. 10 pence. 
 
 7. f of 4 quarters is 2 quarters and f of a quarter ; f 
 of a quarter is | of 4 nails, which is If nails. Ans. 2 
 quarters. If nails. 
 
 13. f of 24 hours is 15 hours. 
 
 14. I of 24 hours is 14 hours and f of an hour ; | of 60 
 minutes is 24 minutes. Ans. 14 hours, 24 minutes. 
 
148 KEY. PART II, 
 
 28. There "beinp; 4 farthings in a penny, 1 farthing is ^ 
 part of a penny. 
 
 30. 3 farthings is f of a penny. 
 
 31. 1 penny is -^^ of a shilling, because there are 12 
 pence in a shilling. 
 
 34. 5 pence is /^ of a shilling. 
 41. 1 shilling is ^V ^^ ^ pound. 
 43. 3 shillings is /^ of a pound. 
 
 48. 1 farthing is ^^ of one shilling. 
 
 49. 2 farthings is ^^g, or ^^ of a shilling. 5 farthings is 
 /^ of a shilling. 
 
 51. 1 penny is ^Ij^ of 1 pound. 7 pence is j^^ of i£l. 
 59. 3s. 5d. is 41 pence, which is ^^^ of £1. 
 75. 1 nail is j'g^ of a yard. 5 nails is /^ of a yard. 
 89. 1 oz. is tV of 1 lb. 15 oz. is |f of 1 lb. 
 91. 1 lb. is i^-g of 1 quarter. 9 lbs. is /^ of 1 quarter. 
 100. At the end of 1 hour they would be 7 and f miles 
 ■ apart. In 7 hours, 7 times 7f , Avhich is 54f miles. 
 
 121. This is the principle of fellowship ; 3 shillings were 
 paid ; one paid i, the other f . 
 
 122. One paid |, the other |. "" . 
 
 123. 20 dollars were paid in the whole; one paid ^^, 
 another /j^, and the third /(^. 
 
 126. 3, and 4, and 5 are 12. The first put in -^^ ; the 
 second j\ ; the third /g-* 
 
 129. 4 dollars for 2 months is the same as 8 dollars for 1 
 month ; 3 dollars for 3 months is the same as 9 dollars for 
 1 month ; and 2 dollars for 4 months is the same as 8 dol- 
 lars for 1 month. The question is the same as if A had 
 put in 8 dollars, B 9 dollars, and C 8 dollars. A must 
 have /j, B j^-g, and C j\ of 100 dollars. 
 
 131. A's money was in 4 times as long as C's. It is the 
 same as if A had put in 8 dollars for the same time, and B 
 8 dollars for the same time. A must have /j, B -2^, and C 
 2^ of 88 dollars. 
 
 The examples 127, 128, 129, 130, and 131, are double or 
 compound foUowship. 
 
 139. The interest of 50 dollars for 1 year and 6 months 
 is 4 dollars and 50 cents, and for 1 month it is 25 cents. 
 The interest of 7 dollars for 19 months (a dollar is J of a 
 cent a month) is 66j cents. The whole amounts to, 5 dol- 
 lars and 41 1 cents. 
 
 140. The interest of 200 dollars for 1^ year is 16 dol- 
 lars. The interest of 67 dollars is 67 cents for every 2 
 months, for 16 months it will be 8 times 67 cents, which 
 
SEC. XV.] KEY. 149 
 
 are 5 dollars and 36 cents. The whole interest is 21 dol- 
 lars and 36 cents, 
 
 143. The interest of 100 dollars for 2} years is 13 dollars 
 and 50 cents. The interest of 100 dollars for 60 days 
 would be 1 dollar, the interest for 20 days will be i of a 
 dollar, or 33^ cents. The interest of 1 dollar for 2;^ years 
 is 13k cents ; for 10 dollars the interest would be 1 dollar 
 and 35 centa, and for 30 dollars, 4 dollars and 5 cents. 
 The interest of 7 dollars for 21 years is 7 times 13^ cents, 
 or 94J cents. The interest of 3Y dollars for 60 days would 
 be 37 cents, and for 20 days i of 37 cents, or 12i cents. 
 The whole interest is 18 dollars and 95^ cents. 
 
 146. They would both together do | of the work in 1 
 day, and it would take them ^ of a day to do the other i. 
 Ans. 1^ day. 
 
 149. They both together consume J of a bushel in a 
 week, but the woman alone consumes only ^ of a bushel in 
 a week. That is, they both together consume /j in a week, 
 but the woman alone only j*jj ; consequently the man alone 
 would consume ^^^ ; and a bushel would last him 3 J weeks. 
 
 152. A and B can build i of it in 1 day ; A, B, and C, 
 can build | of it in 1 day ; 4he difference between ^ and ^ 
 is ^Q ; therefore C can build -^j^ of it in 1 day ; and it would 
 take him 13i days to build it alone. 
 
 164. Find how much they might eat in a day, in order to 
 make it last 1 month, and then it will be easy to find how 
 much they may eat in a day, to make it last 11 months. 
 
 167. The money is 7 parts of the whole, and the purse 
 1 part ; consequently the money is | and the purse i of 16. 
 
 170. He gave 1 part for the apple, 2 parts for the orange, 
 and 4 parts for the melon. These make 7 parts. The 
 apple 3 cents, the orange 6 cents, and the melon 12 cents. 
 
 175. If to a number half of itself be added, the sum is 
 I of that number ; hence subtract 2J from 100, and the 
 remainder is f of the number of geese that he had. 
 
 180. This must be reduced to 6ths. 1 half is f , and i 
 is f , and the number itself is f . If therefore to the whole 
 number its half and its third be added, the sum will be 
 y ; hence, 77 is y of the number. 
 
 181. i is f ; therefore if to a number h and i of itself 
 be added, the whole number will be I ; but when 18 more 
 is added to ^, the first number is doubled; that is, the 
 number is f of the first number ; therefore 18 is J of the 
 number. 
 
AEITHMETIC. 
 
 PART III. 
 
 WRITTEN ARITHMETIC 
 
 NOTATION AND NUMERATION. 
 
 1. Instead of writing the names of numbers, it is 
 ■usual to express them by particular characters, called 
 figures. 
 
 One is 
 
 written 
 
 1 
 
 Twenty-six is written 
 
 . 26 
 
 Two 
 
 li 
 
 2 
 
 Twenty-seven 
 
 a 
 
 27 
 
 Three 
 
 a 
 
 3 
 
 Twenty-eight 
 
 li 
 
 28 
 
 Four 
 
 (( 
 
 4 
 
 Twenty-nine 
 
 11 
 
 29 
 
 Five 
 
 (( 
 
 5 
 
 Thirty 
 
 .11 
 
 30 
 
 Six 
 
 a 
 
 6 
 
 Thirty-one 
 
 11 
 
 31 
 
 Seven 
 
 (( 
 
 7 
 
 Forty 
 
 11 
 
 40 
 
 Eight 
 
 it 
 
 8 
 
 Fifty 
 
 (( 
 
 50 
 
 Nine 
 
 a 
 
 9 
 
 Sixty 
 
 i< 
 
 60 
 
 Ten 
 
 a 
 
 10 
 
 Seventy 
 
 i( 
 
 70 
 
 Eleven 
 
 (( 
 
 11 
 
 Eighty 
 
 li 
 
 80 
 
 Twelve 
 
 i( 
 
 12 
 
 Ninety 
 
 (I 
 
 90 
 
 Thirteen 
 
 a 
 
 13 
 
 One hundred 
 
 li 
 
 100 
 
 Fourteen 
 
 i( 
 
 14 
 
 One hundred and 
 
 
 
 Fifteen 
 
 i( 
 
 15 
 
 one 
 
 11 
 
 101 
 
 Sixteen 
 
 li 
 
 16 
 
 One hundred and 
 
 
 
 Seventeen 
 
 li 
 
 17 
 
 two 
 
 li 
 
 102 
 
 Eighteen 
 
 a 
 
 18 
 
 One hundred and 
 
 
 
 Nineteeh 
 
 a 
 
 19 
 
 ten 
 
 u 
 
 110 
 
 Twenty 
 
 a 
 
 20 
 
 One hundred and 
 
 
 
 Twenty-one 
 
 11 
 
 21 
 
 twenty 
 
 11 
 
 120 
 
 Twenty-two 
 
 a 
 
 22 
 
 Two hundred 
 
 a 
 
 200 
 
 Twenty-three 
 
 a 
 
 23 
 
 Three hundred 
 
 11 
 
 300 
 
 Twenty-four 
 
 a 
 
 24 
 
 One thousand 
 
 11 
 
 1000 
 
 Twenty-five 
 
 iC 
 
 25 
 
 Ten thousand 
 
 'I 10,000 
 
PART III.] ARITHMETIC. 151 
 
 2, When several figures stand side by side, the 
 value of each figure depends on its place as counted 
 from the right towards the left. A figure standing in 
 the first place, signifies so many units, or ones; the 
 same figure, in the second phice, signifies so many fens; 
 in the third place, hundreds; in the fourth place, 
 thousands ; in the fifth place, fens of thousands ; and 
 in the sixth place, hundreds of thousands. Therefore 
 every time a figure is removed one place to the left, its 
 value is made tenfold, as may be seen in the following 
 table : 
 
 03 
 
 o-^ 
 
 '^ a '^ ' 
 
 "S a o « c-"^ 
 
 W^hwhp [hundred and eleven. 
 
 111^111, read One hundred and eleven thousand, one 
 4 " Four. 
 49 " Forty-nine. 
 496 " Four hundred and ninety-six. [five. 
 
 4,965 " Four thousand nine hundred and sixty- 
 49,652 ^' Forty-nine thousand six hundred and 
 fifty-two. ^ • 
 496,527 '^ Four hundred and ninety-six thousand 
 five hundred and twenty-seven. 
 S. Let the pupil read the following numbers: 
 
 1. 
 
 34 
 
 5. 
 
 142 
 
 9. 
 
 6,744 
 
 13. 
 
 11,504 
 
 2. 
 
 76 
 
 6. 
 
 279 
 
 10. 
 
 8,329 
 
 14. 
 
 50,603 
 
 3. 
 
 68 
 
 7. 
 
 758 
 
 11. 
 
 4,960 
 
 15. 
 
 124,769 
 
 4. 
 
 95 
 
 8. 
 
 423 
 
 12. 
 
 9,080 
 
 16. 
 
 784,984 
 
 4, Let the pupil express the following numbers in 
 figures : 
 
 1. Seventy-eight. 
 
 2. Ninety. 
 
152. ARITHMETIC. [part iii. 
 
 3. One hundred and fifty-six. 
 
 4. Three hundred and eight. 
 
 5. One thousand four hundred and sixty-nine. 
 
 6. Five thousand and two. 
 
 7. Seventy-seven thousand and eighty-four, 
 
 8. Forty thousand seven hundred and twenty-two. 
 
 9. Five hundred and sixty-six thousand one hundred 
 
 and fifty. 
 10. Six hundred thousand. 
 
 ADDITION. 
 
 5. 5 and 4 and 7 are how many ? This is called 
 a question in addition. We put together, or add the 
 numbers 5, 4 and seven, to find their sum or amount. 
 Thus, the sum of 5 and 4 and 7 is 16. 
 
 Ex. 1. What is the sum of 46 and 37? 
 gi It is the most convenient to write the numbers 
 
 ^;§ in order, units under units, tens under tens, etc.; 
 46 then besinnino; with the units, we find that 6 
 37 units and 7 units are 13 units, or 1 ten and 3 
 — units. We write down the 3 units under the 
 83 units and add the 1 ten to the other tens; 1 (ten) 
 and 4 (tens) and 3 (tens) make 8 (tens) which 
 written with the 3, makes 83. 
 
 Ex. What is the sum of 736 and 875 and 394? 
 .§ Adding the' units we find 6 and 5 are 11, and 
 
 ^ . ^ 4 are 15, or 1 ten and 5 units. Writing the 
 s g S 5 under the column of units, wc reserve the 1 
 ^^P ten, and add it to the tens of the next column; 
 i.-r thus, 1 and 3 are 4, and 7 are 11, and 9 are 20 
 7, La (tens) or 2 hundreds, and tens.' We write 
 
 2. the under the column of tens, and add the 
 
 200 5 2 (hundreds) to the column of hundreds; thus, 
 2 and 7 are 9, and 8 are 17, and 3 are 20 (hundreds,) 
 which being the sura of the last column we write down. 
 The answer, therefore, is 2005. 
 
 6. Add the following : 
 
PAKT III.] 
 
 
 ARITHMETIC. 
 
 
 
 163 
 
 3. 
 
 4 
 
 5. 
 
 6. 7. 
 
 8. 
 
 9. 
 
 10. 
 
 11. 
 
 15 
 
 33 
 
 8 
 
 7 19 
 
 28 
 
 45 
 
 56 
 
 69 
 
 4 
 
 5 
 
 9 
 
 5 5 
 
 7 
 
 8 
 
 7 
 
 5 
 
 12. 
 
 lo. 
 
 14. 
 
 15. 16. 
 
 17. 
 
 18. 
 
 19. 
 
 20. 
 
 85 
 
 17 
 
 75 
 
 35 62 
 
 84 
 
 93 
 
 60 
 
 80 
 
 42 
 
 50 
 
 
 26 79 
 
 45 
 
 58 
 
 30 
 
 50 
 
 21. 
 
 22. 
 
 23. 
 
 24. : 
 
 i5. 
 
 26. 
 
 27. 
 
 28. 
 
 27 
 
 60 
 
 47 
 
 234 311 
 
 435 
 
 731 
 
 478 
 
 32 
 
 88 
 
 55 
 
 362 5 
 
 25 
 
 924 
 
 311 
 
 298 
 
 56 
 
 99 
 
 72 
 
 403 293 
 
 808 
 
 579 
 
 527 
 
 29. 
 
 
 30. 
 
 31. 
 
 32. 
 
 3 
 
 3. 
 
 34. 
 
 500 
 
 7667 
 
 7543 
 
 1812 
 
 4648 
 
 789 
 
 314 
 
 1: 
 
 235 
 
 1858 
 
 1732 
 
 4e 
 
 120 
 
 2253 
 
 409 
 
 5555 
 
 1776 
 
 2712 
 
 8665 
 
 226 
 
 
 
 
 1492 
 
 37. 
 
 3333 
 
 452 
 
 45 
 
 
 36. 
 
 35. 
 
 38. 
 
 
 39. 
 
 990 
 
 
 8 
 
 6540 
 
 
 38,924 
 
 
 468 
 
 6209 
 
 
 295 
 
 6030 
 
 
 67,798 
 
 
 26,792 
 
 2316 
 
 
 17 
 
 2709 
 
 
 54,554 
 
 
 15,864 
 
 2201 
 
 40. 
 
 7897 
 
 5967 
 together 39, 
 
 67,1; 
 
 33,333 
 
 ^4, 51 
 
 80,112 
 
 7. 
 
 Add 
 
 12, 43, I 
 
 
 
 41. 
 
 Add together 27C 
 
 ), 489 
 
 , 126, 46.' 
 
 
 
 42. 
 
 Add together 2368, 23 
 
 5, 496, 
 
 1736. 
 
 
 
 43. 
 
 Add together 66,476, 8 
 
 01,400! 
 
 9,32, 
 
 127. 
 
 8. 44. A merchant bought 4 pieces of cloth ; for 
 the first piece he gave 53 dollars; for the second, 78 
 dollars; for the third, 68 dollars; and for the fourth, 
 67 dollars. How much did he give for the whole ? 
 
 45. In an orchard there are 22 peach trees, 95 apple 
 trees, 17 pear trees, and 56 plum trees. How many 
 trees in the orchard? 
 
 46. A man bought four horses; for the first he gave 
 84 dollars; for the second, 150 dollars; for the third, 
 
154 ARITHMETIC. [part iil 
 
 475 dollars; and for the fourth, 526 dollars. How 
 much did he give for all the horses ? 
 
 47. In Mr. Green's school there are 810 scholars ; 
 in Mr. White's, 697 scholars; in Mr. Brown's, 956 
 scholars; in Mr. Blue's, 776 scholars; in Mr. Black's, 
 572 scholars; and in Mr. Crimson's, 731 scholars. How 
 many scholars in these six schools ? 
 
 48. A gentleman, building a house, paid the mason 
 2,964 dollars, the carpenter 5,723 dollars, the plasterer 
 625 dollars, and the painter 354 dollars. How much 
 did the whole house cost ? 
 
 SUBTRACTION. 
 
 O. 9 less seven are how many ? This is called a 
 question in Subtraction. We take away, or subtract, 
 7 from 9 to find the difference or remainder, which is 2. 
 
 Ex. 1. From 476 take 245. 
 ^ It is most convenient to write the less num- 
 
 ^ . ^ ber under the greater, units under units, tens 
 s£a under tens, etc., as in addition. Beginning at the 
 '^^^ units, we say 5 from 6 leaves 1, which we write 
 ^ ^ " under the units ; then 4 (tens) from 7 (tens) 
 *^^^ leaves 3 (tens,) which we write under the tens; 
 and 2 (hundreds) from 4 (hundreds) leaves 2 
 ^ ^ -^ (hundreds,) which we write down, and the 
 answer is 231. 
 
 Ex. 2. From 939 take 458. 
 4 In this example we take 8 from 9 and there 
 
 ^ . . remains 1, which we write down ; but since we 
 cgS cannot take 5 (tens) from 3 (tens), we borrow 
 ^^^ 1 (hundred,) or 10 (tens) from the 9 (hundreds,) 
 9 3 9 which joined with the 3 (tens,) makes 13 (tens ;) 
 45 8 then 5 (tens) from 13 (tens) leaves S (tens,) 
 — — which we write down. As one of the 9 (hun- 
 481 drcds) has been put with the 3 (tens,) there 
 ^ remains but 8 (hundreds.) Therefore we say 4 
 
 (hundreds) from 8 (hundreds) leaves 4 (hundreds.) 
 This written down makes, the answer 481. 
 
PART II] 
 
 •] 
 
 
 AIIITIIMETIC. 
 
 
 
 155 
 
 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 fr 
 
 8. 
 
 9. 
 
 From 
 
 18 
 
 58 
 
 35 
 
 59 
 
 68 
 
 82 
 
 95 
 
 Take 
 
 4 
 
 6 
 
 12 
 
 33 
 
 34 
 
 51 
 
 75 
 
 Ans. 
 
 ll 
 
 
 
 26 
 
 
 
 20 
 
 
 10. 
 
 11. 
 
 12. 
 
 13. 
 
 
 14. 
 
 15. 
 
 From 
 
 285 
 
 467 
 
 9685 
 
 7856 
 
 
 8744 
 
 2607 
 
 Take 
 
 74 
 
 16. 
 
 46 
 
 17. 
 
 5473 
 
 5722 
 
 20. 
 
 4302 
 
 1405 
 
 
 18. 
 
 19. 
 
 21. 
 
 22. 
 
 From 
 
 12 
 
 13 
 
 19 
 
 16 
 
 24 
 
 56 
 
 92 
 
 Take 
 
 G 
 
 5 
 
 9 
 
 7 
 
 ,8 
 
 7 
 
 4 
 
 From 
 Take 
 
 30. 
 
 134 
 25 
 
 31. 
 
 244 
 73 
 
 32 
 
 252 
 171 
 
 33. 
 
 888 
 459 
 
 34. 
 
 2236 
 1145 
 
 35. 
 
 3456 
 2246 
 
 
 23. 
 
 24. 
 
 25. 
 
 26. 
 
 27. 
 
 28. 
 
 29. 
 
 From 
 
 30 
 
 53 
 
 46 
 
 33 
 
 64 
 
 92 
 
 84 
 
 Take 
 
 20 
 
 10 
 
 27 
 
 25 
 
 29 
 
 88 
 
 59 
 
 36. 
 
 9927 
 6090 
 
 37. From 96 take 58. 
 
 38. From 4785 take 3679. 
 
 39. From 8344 take 5324. 
 
 40. From 10982 take 6470^ 
 
 41. From 984 take 176. 
 
 42. From 4788 take 967. 
 
 43. From 7898 take 6898. 
 
 44. From 40816 take 5622. 
 
 10. 45. In a certain garden there were 86 rose 
 bushes, but the worms destroyed 33 of them. How 
 many were left ? 
 
 46. A man is 66 years of age, and his son 22. How 
 old was the man at the birth of his son ? 
 
 47. A farmer had 240 sheep -, but, one night, wolves 
 gained admittance to the fold, and carried off 121. How 
 many sheep were left? 
 
 48. Harry had 84 cents given him to buy some mar- 
 bles with, but on his way to the store he lost 25 cents. 
 How many cents had he left ? 
 
 49. A girl was carrying 72 eggs to market, but she 
 let the basket fall, and 39 eggs were broken. How 
 many had she left ? 
 
156 ARITHMETIC. [part hi. 
 
 50. In an army there were 14,642 men, but in a 
 battle 789 men were killed or wounded. How many 
 serviceable men were there left ? 
 
 MULTIPLICATION. 
 
 11. 7 times 5 are how mamy ? This is called a 
 question in multiplication. To obtain the answer we 
 repeat or multiply the number 5^ 7 times; thus, 7 times 
 5 are 35. 
 
 Ex. 1. Multiply 5G4 by 7. 
 ^ Writing the numbers thus, the 7 under the 
 
 I . ^. units, we say, 7 times 4 (units) are 28 (units,) 
 § 1 1 or 2 tens and 8 units. We write down the 8 
 ^^^ (units) and reserve the 2 (tens;) then 7 times 
 5 6 4 6 (tens) are 42 (tens) and the 2 (tens) which 
 7 we reserved are 44 (tens) or 4 hundreds and 
 4 tens. We write the 4 (tens) and reserve the 
 
 3 948 4 (hundreds;) then, 7 times 5 (hundreds) are 
 
 35 (hundreds,) which, with the 4 (hundreds) 
 
 we reserved, are 39 (hundreds) or 3 thousands and 9 
 
 hundreds. This we write down, and the answer is 3948. 
 
 Multiply 
 
 By 
 
 8 9 
 3 5 
 
 13 , : 
 
 2 
 
 23 
 3 
 
 43 51 
 6 9 
 
 65 
 
 7 
 
 Multiply 
 By 
 
 9. 10. 
 
 143 424 
 
 2 4 
 
 11. 
 511 
 
 7 
 
 
 12. 13. 
 
 821 410 
 6 5 
 
 14. 
 
 821 
 4 
 
 Multiply 
 By 
 
 15. 
 
 813 
 3 
 
 10. 
 
 962 
 
 8 
 
 
 17. 
 
 1243 
 2 
 
 18. 
 
 2104 
 4 
 
 Multiply 
 
 By 
 
 19. 
 
 5134 
 6 
 
 • 20. 
 7110 
 
 7 
 
 
 21. 
 8947 
 9 
 
 22. 
 4355 
 8 
 
PART III.] ARITHMETIC. 157 
 
 23. Multiply 129 by 35. 
 
 4 ^ Writing the numbers in order, units under 
 
 I £ . units, and tens under tens, we multiply, first, 
 
 ocS| by the 5 units, and write the result in its 
 
 hWh;:^ proper place, as when we multiplied by one 
 
 ;^ 2 9 figure. We multiply next by the 3 (tens) 
 
 3 5 and write the first figure of the result in the 
 
 . place of tens, because units multiplied by 
 
 (3 4 5 tens ought to produce tens. Then adding 
 
 3 8 7 together the results obtained by multiplying 
 by the 5 (units ) and 3 (tens,) their sum will 
 
 4 515 be 4515, which is the answer, 
 
 24. 25. 26. 27, 28. 29. 30. 
 
 Multiply 34 35 48 45 99 012 844 
 By 23 25 32 44 56 54 29 
 
 102 
 
 68 
 
 Ans. 782 
 
 31, 32. 33. 34. 35. 36. 37. 
 
 Multiply 676 434 467 548 1835 2972 5963 
 By 76 322 212 403 768 256 4678 
 
 38. Multiply 816 by 6, 142, Multiply 2523 by 47. 
 
 39. Multiply 363 by 48. 43. Multiply 682 by 746. 
 
 40. Multiply 846 by 65, 44. Multiply 8422 by 186. 
 
 41. Multiply 152 by 87. I 45. Multiply 3107 by 761. 
 
 46. How much will 48 barrels of flour cost at 8 dol- 
 lars a barrel ? 
 
 47. In a room there are 9 windows with 16 panes 
 of glass in each. How many panes of glass in all the 
 windows? 
 
 48, If a man receives 47 dollars a month, how much 
 will he receive in twelve months ? 
 
 49, In a certain regiment there are 13 companies, 
 and 126 men in each company. How many men in 
 the regiment ? 
 
153 ARITHMETIC. [paut hi. 
 
 50. There are 365 days in a j'car. How many days 
 in 19 years ? 
 
 DIVISION. 
 
 IS. 24 are liow many times 8? This is •cninu a 
 question in division , because we divide, or separate, 
 the larger number into as many equal parts as there are 
 uuits in the smaller number. 
 
 13. The number tp be divided is called the divi- 
 dend ; the number by which we divide, the divisor ; 
 and the result, the quotient. If anything is left after 
 the division, it is called the remainder. 
 
 14. Ex. 1. Divide 584 by 4. 
 
 Divisor, 4) 584 Dividend. We Write down the dividend, 
 
 584, and draw a line beneath 
 
 146 Quotient. it. Write the divisor, 4, at the 
 left of the dividend, and draw a 
 curved line between them. We first find how many times 
 4 is contained in 5, the first figure of the dividend ; it 
 is contained 1 time and there is 1 remainder. We write 
 the 1 under the 5, and suppose the remainder, 1, written 
 at the left of the next figure, 8, making the number 18. 
 4 is contained in 18, 4 times, with 2 remainder. The 
 4 we write below, and suppose the 2 written before the 
 next figure, 4, making it 24. 4 is contained in 24, 6 
 times, without remainder. We write the 6 underneath, 
 and the division is completed. 
 
 Division performed in this way is called slioi-t divi- 
 sion. 
 
 15. Perform in this way the following examples : 
 
 2. 3. 4. 5. ' 6. 7. 
 
 Divisor, 3)9 Dividend. 2)26 3)36 4)88 5(15 3)18 
 
 3 Quotient. 
 8. 9. 10. 11. 12. 13. 
 
 7(21 9)36 6)48 8)72 9)81 4)40 
 
PAKT III.] ARITHMETIC. 159 
 
 14. 15. 16. 17. 18. 19. 
 
 6)72 2)444 4)256 3)963 7)847 9)279 
 
 20. 21. 22. 23. 24. 
 
 5)6825 6)4650 9)9972 4)4484 8)1312 
 
 10. When the divisor consists of two or more 
 figures, it is more convenient to perform the operation 
 in the following manner : 
 
 25. Divide 6432 by 12. 
 Divisor. Dividend. Quotient. Write the dividend, and draw 
 12)6432(536 a curved line on both sides. We 
 6 first inquire how many times 12 
 
 is contained in 64, the first two 
 
 4 3 figures of the dividend. We find 
 
 3 6 that it is contained 5 times. We 
 
 place the 5 at the right of the 
 
 72 dividend, as the first figure of 
 
 7 2 the quotient, and then, multiply- 
 
 — ing the divisor, 12, by 5, we 
 
 obtain the number 60, which we write under the 64 of 
 the dividend, and subtract. To the remainder, 4, we 
 annex 3, the next figure of the dividend, and find how 
 many times the divisor, 12, is contained in 43. We 
 place the result, 3, at the right of the 5 in the quotient, 
 and, multiplying the divisor by 3, we obtain 36, which 
 we write under the 43, and subtract. To the remainder, 
 7, we annex the 2 of the dividend, and find how many 
 times the divisor is contained in 72, which is 6 times. 
 We write 6 in thQ quotient, and, as there is no re- 
 mainder, the division is complete. 
 
 Division performed in this way is callled long division. 
 17. Perform in this way the following examples : 
 
 26. 27. 28. 29. 
 
 13)286(22 15)235( 17)425( 13)585( 
 
 26 
 
 "26 
 26 
 
160 
 
 ARITHI\IETIC. 
 
 [part III. 
 
 30. 
 
 16)X168( 
 
 31. 32, 
 
 13)676( 12)1728( 
 
 S3. 
 
 15)2445( 
 
 13)36l6( 
 
 35. 36. 
 
 17)2057( 19)2994( 
 
 37. 
 
 15;4445( 
 
 38. Divide 144 by 12. 
 
 39. Divide 224 hv r>. 
 
 40. d; 
 
 bouclit 
 
 45. l.": 
 acres can yo 
 
 46. If a man c. 
 will it take liim to ;v 
 
 47. A man divided 
 
 41. Divide 5456 by 22. 
 
 42. Divide 1411 by 36. 
 
 43. Divide 8844 by 22. 
 
 'jts a pound can be 
 iars, how many 
 day, how long 
 
 ■"6 m;ies? 
 23,796 dollars among his 6 
 children. How much did each receive? 
 
 48. A railroad company employed 25 workmen, and 
 at the end of two months it took 1825 dollars to pay 
 them. How much did each man receive ? 
 
 THE END. 
 
JOSEPH ftUZtoM 
 
 800KBl«Ct«3 
 
 I Salt I MORE a. o '