LIBRARY OF CONGRESS, COPYRIGHT OFFICE. No registration (*m> f this book as a preliminary to copyright protec- tion has been found. Forwarded to Order Division _JMAR % W\Q (Apr. 5, 1901—5,000.) LUDLOW TEXTILE ARITHMETIC BY JOSEPH J. EATON, S. B. u Director, Ludlow Textile School, Ludlow, Mass. AND RICHARD BRADY i Overseer, Ludlow Manufacturing Associates, Ludlow, Mass. 1908 The C. R. Kaplinger Co. SPRINGFIELD, MASS. ^ f^eceivsd from Copyright Office, MAR i 191U V J0-SZ1O Preface This book is arranged for the use of the Ludlow Textile School. It shows by practical illustrations and examples how the different arithmetical processes are applied to mill work in this particular branch of the textile industry. • The arithmetic is not intended to take the place of a com- plete arithmetic but at the same time a complete set of abstract examples are arranged according to grade and the concrete examples are extremely practical. The general necessity for an early understanding of decimals and proportion caused the placing of those matters at an early part of the book. Fractions have been treated in detail although they are used to a limited extent in the mills. In many cases the language used is that of the mill, consequently some words, usually explained in the notes, have other than the generally accepted meanings. Only the calculations in general use in the Ludlow Mills, which have been tested practically, are included in the chapter on General Mill Work. The explanations have been made as simple as possible Later, it is intended to publish a school description of the various processes of manufacture in vogue in the jute and hemp industry which will contain a number of more advanced problems. Some of the examples in this book have been taken from the following arithmetics, Bradbury and Eaton's, Thomp- son's, Smith's, and Wentworth's. Other authorities con- sulted were Carter, Sharp, and Leggatt. * As this is the first attempt to publish an arithmetic relating to this particular branch of the textile industry, any suggestion for improvement will gladly be received. Contents PAGE Vocabulary, Notation and Numeration, 1 Addition, ■ 2 Subtraction 8 Multiplication, . » 9 Division, 11 Miscellaneous Examples 14 Analysis, 17 Factors and Multiples, 19 Cancellation, . 23 United States Money, 24 Decimals, 26 Proportion, 38 Fractions, 42 Measurements; Weighing; Ludlow Yarn Measure ; Tables, . 57 Alligation, 71 Percentage, 73 Roots, 79 General Mill Work; Belting; Horse Power; R. P. M. ; Rollers; Pulleys; Surface Speeds; Drafts; Twists, 82 Short Processes, 102 Answers 103 Vocabulary Bagging. — A coarse fabric made from jute for covers of bales'of cotton. A product of the Ludlow Mills. Bale. — A large package prepared for transportation or storage. All the fibre arrives in Ludlow in bales. Baling. — An unfinished twine made in the Ludlow Mills and used chiefly for sewing bales. Baller. — -The operator of the balling machines. The latter are used for the purpose of winding the twine into balls. Batch. — A quantity of hemp, or jute placed together at one time, over each layer of which is spread a certain amount of liquor composed chiefly of oil. This batch is allowed to lay a certain length of time in order that the liquor may penetrate through all of the fibre. Breaker Card. — -This is a machine on which the fibre is first broken into tow. It is called breaker card to dis- tinguish it from a finer machine which is known as a finisher card. Card.- — A machine used to comb and cleanse the fibres and to form them into sliver. Carpet Yarn. — Used in the manufacture of carpets, one of the main products of the Ludlow Mills, made from hemp, flax, and jute fibres. Creel. — That part of the machine in which is placed the rove bobbins while they are being unwound in a spinning frame or other machine. Cut. — 300 yards of yarn. Same as lea. Cuts per Spindle. — The number of yards of spun yarn produced from each spinning spindle. Doff. — 1. To take the full bobbins from a machine and replace them with empty bobbins. 2. The number of full bobbins on a spinning or roving or twisting frame. viii VOCABULARY Doffer. — 1. Girls who replace the full bobbins with empty ones. 2. The roller which takes the fibre of! the card cylinder. Draft. — The amount that the material is drawn out between the receiving and delivery rollers. Feed Roller. — The roller which receives or "feeds" the material into a carding machine. Filling. — The weft of woven fabrics. Fluted Roller.- — Rollers with corrugated surfaces used as retaining or drawing rollers of wet spinning frames and on other textile machines. Flyer. — An inverted, two pronged fork, which, fastened to the top of a spindle, is used for putting the twist into yarn and for winding the latter upon a bobbin. Gill. — The brass stock into which the pins are driven which are used for combing the fibre on drawing and roving frames. Hackle. — Used for combing flax, hemp, jute, and other fibres. Hank. — Twelve cuts or 3600 yards of yarn. Hemp. — Many kinds are grown in different parts of the world. The tough and strong fibre obtained from these plants is largely used in the manufacture of yarns, twines, and ropes. The principal source of supply for American hemp is the Ohio valley. Hemp of other kinds is imported from Italy and from Russia. The coarser kind, sisal, is imported from Yucatan, Mexico. Manila hemp comes from the Philippines. Horse Power. — The power necessary to raise 33000 pounds one foot in one minute. Jute. — An annual Asiatic plant of the linden family. The fibre obtained from the inner bark of this plant, is used in Ludlow for the manufacture of yarns and twines. Jenny. — One of the machines used in Ludlow in the manufacture of binder twines. Lap. — A number of slivers from a breaker card wound around a stick made in the form of an axle. This is done for convenience in handling and weighing. Lbs. per Spyndle. — The number of pounds contained in 14,400 yards of yarn. VOCABULARY ix Lea. — 300 yards of yarn. Same as cut. The number of leas or cuts in one pound of yarn gives the name to the yarn, thus yarn which measures 1800 yards to the pound would be called 6 lea yarn. 1800 -=-300 = 6. Length op Bell. — Where it is necessary to measure the material in the process of manufacture it is usual to attach a bell mechanism to some part of the machine over which the material passes. This mechanism is arranged so as to ring a bell when a certain length has been delivered. The length of bell is the number of yards the machine de- livers while the wheel which rings the bell makes one revo- lution. Loom. — The machine in which the yarn is woven into fabric. Marline. — -Coarse twine made in Mills 5 and 6. Polished Twine. — Twine after it is put through a mixture of starch, wax, and so forth, and has passed over rollers which put a glossy surface on it. Production.- — The amount of work which comes off one or more machines in a specified time. Reel. — A frame to wind yarn, called a 54", 72", or 90" reel according to its circumference. Rove. — A slightly twisted sliver. Roving Frame. — The machine on which the sliver is first formed into a thread in the process of manufacturing yarns. Sett of Laps. — A number of slivers of the same length whose combined weight will total a fixed number of pounds. Sliver. — Textile fibres formed into a continuous strand. Softener. — A machine having a number of fluted rollers pressed together with heavy springs. The fibres are passed between these rollers to prepare them for spinning. Spindle. — The vertical rotating rods on a spinning machine bearing the bobbins. Spyndle. — 14,400 yards of yarn. Step.— The bearing on which the spindle runs. Stripper. — A fast driven roller in the carding machine. System. — A group of machines consisting of cards, drawings, and roving frames. Tape. — Cotton webbing used in Ludlow to drive spinning spindles. x VOCABULARY Turns per Inch. — The number of revolutions the flyer makes while the rollers of a machine are delivering one inch of yarn. It is by the turns per inch that the twist on yarn or twine is calculated. Warp. — The threads running lengthwise in any woven fabric. Weft. — The threads running crosswise in any woven fabric. Wharve. — That part of the spindle around which passes the tape which causes the spindle to revolve. Worker. — The slow moving roller in a carding machine. Ludlow Textile Arithmetic NOTATION AND NUMERATION. LESSON I.— Oral. A Unit is a single quantity, for example, a reel, a softener, a pound. (A unit may also be a definite group of things such as a system, although a system is made up of several single machines.) A Number is a collection of units, for example, six reels, two softeners, ten pounds. Figures are used to represent numbers, as 0123456789 zero one two three four five six seven eight nine (The zero is sometimes called naught and cipher.) Notation means the method of writing numbers. Numeration means the method of reading numbers. Reading Numbers. — Numbers are read from left to right like ordinary printing. A single figure expresses a certain number of units and is said to be in the unit column. A number of two figures has one figure in the unit column, and, as the other figure represents a certain number of tens, it is said to be in the tens column. Each column has a certain name as shown below. IvUDLOW TEXTILE ARITHMETIC O w Pi Pi 03 CO pi ■d *d *d 1 *d Pi w Pi CO pi pi +3 *d 0) i-H w a) 773 Pi CD CD £ Pi O 'd a ^d •p 5-H -d tfl Pi Pi pi 1 Pi pi Pi CD ■p Pi pi CD 4-3 '3 pi 397,492,608,125 The above number would be read, three hundred ninety- seven billion, four hundred ninety-two million, six hundred eight thousand, one hundred twenty-five. (In reading, the name of the unit figures is omitted. The names of other columns, to the left, of figures of higher denomination, are omitted in this work.) Periods. — Numbers are separated into groups of three, by commas, beginning at the right. These groups are called periods. (As ten of any order, or column, equal one of the next column to the left, this method of writing numbers, is called the decimal system. The word decimal is derived from a Latin word meaning ten.) Writing Numbers. — To write numbers, begin at the left and write the hundreds, tens, and units of each period, putting in zeros in all vacant places, and putting in commas between each period and the following period. Like Quantities, are quantities of the same kind or denomination, for example, dollars, boys, bobbins, twines. ADDITION. Addition is the process of finding a number which is equal to two or more numbers, and the result is called their sum. The sign +, read plus, indicates addition. (Only like quantities can be added.) LUDLOW TEXTILE ARITHMETIC LESSON II.— Written. Add: 1. 2. 3. 4. 5. 6. 7. 8. 438 146 400 1400 23,547 376,489 963,525 706,145 951 257 584 1276 16,938 976,432 37,675 90,676 276 389 224 1342 61,847 43,763 538,629 3,865 (Test the accuracy of the work by adding down as well as up.) 9. — Find the total number of pounds in the following; one bale of jute weighing four hundred pounds, one bale of American hemp weighing five hundred eighty-four pounds, one bale Italian hemp weighing six hundred twenty-four pounds, and one bale of flax tow weighing two hundred twenty-four pounds. 10. — What is the total number of pounds contained in the following shipment of polished twines: one bale of 18 B American containing one hundred forty-four pounds^ one circular bale of seaming cord containing five hundred four pounds, one bale of 18 D shoe twine weighing one hun- dred sixty-eight pounds, one barrel of 24 BC American twine weighing one hundred thirty pounds, and one bag of 36 B Italian weighing one hundred fifty-four pounds? 11. — On March 19, 1900, the jute yarn production of No. 34 Mill was as follows: 12 lbs. A, Six hundred thirty -nine pounds. 14 lbs. B, Fourteen hundred six pounds. 14^ lbs. C, Four thousand, two hundred eighty-four pounds. 15 lbs. D, Fourteen thousand, three hundred fifty-nine pounds. 16 lbs. E, Eleven hundred sixty-four pounds. What was the total production in pounds? 12. The following is the form used for the daily jute yarn production slips which are sent to the office daily at 5 P. M. Department No. 38, LUDLOW TEXTILE ARITHMETIC March 18, 1902. JUTE YARN PRODUCT ION. No. of Bags Sorts Lbs, 8 10 lbs. CA 2534 7 12 lbs. BB 2478 38 13 lbs. BB 14439 35 14 lbs. C taper 13294 11 14 lbs. CC 4269 15 15 lbs. BB 5873 5 16 lbs. 8 Ply 1809 Totals, Over Find the totals. Note: — In the above examples the figures preceding the letter used in regard to the twine denotes the Size, and the letter denotes the Grade, while the word denotes Material from which the twine is made. For example, 18 B Amer- ican, means that the twine is called 18 size, denoting its diameter, is B grade, and is made from American hemp, JUTE YARN PRODUCTION SHEETS. LESSON III.— Written. Make out daily jute yarn production sheets or slips similar to the one in example 12, Lesson II, using the follow- ing quantities; 1.— March. 20, 1901, the production for Dept. 67, #14 Mill, was, 3 bags of 12 lbs. B Special weighing 798 pounds, 5 bags of 14 lbs. B Special weighing 1726 pounds, 10 bags of 14^ lbs. C weighing 2604 pounds, and 62 bags of 15 C Special weighing 13829 pounds, and 3 bags of 16 lbs. C weighing 1096 pounds. Make out regular slip showing totals. 2.— The production sheet of Dept. 69, Mill #14, for March 31, 1901, was, 1 bag of 12 B Ex. S weighing 237 pounds, 4 bags 14 B Ex. weighing 1565 pounds, 10 bags 14 C weighing 3901 pounds, 3 bags 14 Taper weighing 1209 pounds, 10 IvUDLOW TEXTILE ARITHMETIC 5 bags 143^ C weighing 2557 pounds, 5 bags 15 C weighing 2148 pounds, 31 bags 15 S weighing 6662 pounds, 2 bags 16 C weighing 876 pounds. Find totals from slip in regular ,orm. 3.— The production sheet for Dept. 35, Mill #14, for April 3, 1901, was, 6 bags 12 BX weighing 1730 lbs., 5 bags 14 BX weighing 1724 lbs., 10 bags of 14^ C weighing 2872 lbs., 62 bags 15 C weighing 4988 lbs., 13 bags 16 C weighing 5078 lbs., 2 bags 19 C weighing 660 lbs., 1 bag 24 C weighing 484 lbs. Find totals on regular sheet. 4. — The following sheet shows the total production of jute yarns in six mills for one day. Find the totals. April 3, 1908. No. of Bags Sorts Lbs. 2 6 lbs. A 780 8 7 lbs. A 1074 2 12 lbs. A 540 3 10 lbs. B 589 8 12 lbs. B 2740 4 13 lbs. B 1104 17 14 lbs. B 4809 7 15 lbs. B 1987 14 10 lbs. C 4205 83 14 lbs. C 23721 70 15 lbs. C 23808 15 16 lbs. C 5420 2 24 lbs. C 575 3 30 lbs. C 894 Totals, Note: — Yarns tabulated like above are arranged con- secutively, that is, A, B, C, and so forth, 6, 7, 8, and so forth. In other words the yarns are arranged in the order of their diameters and in groups according to their grade. 5. — Jute yarn production sheet for the mills for April 2, 1900. Arrange in proper form, according to example 4, and find the totals. 5 bags of 6 lbs. AK weighing 2000 lbs., 3 bags 7 lbs. AK weighing 954 pounds, 2 bags of 8 lbs. AK weighing 604 pounds, 9 bags 12 lbs. AK weighing 3475 pounds, 6 LUDLOW TEXTILE ARITHMETIC 3 bags." 12 lbs. KX weighing 11 84 pounds, 10 bags 13 lbs. KX weighing 3275 lbs., 13 bags 14 lbs..;KX weighing ;5400 lbs., 2 bags 15 lbs. BX weighing 799 lbs.', 4 bags. 10 lbs. NX weigh- ing 1389 pounds, 21 bags 14 lbs. C weighing 8568 pounds, 2 bags- 15 lbs. C weighing 8057 pounds, 10 bags 16 lbs. C weighing 4056 pounds, 2 bags 19 lbs. C weighing 912 pounds, 5 b'ags 30 lbs. C weighing 2042 pounds. "6. — Arrange the daily total jute yarn production sheet and find totals for April 7, 1900. 5 bags 30 C weighing 1741 pounds, 10 bags 16 C weighing 3575 pounds, 18 bags 15 C weighing 5840 pounds, 14 bags 14 C weighing 6460 pounds, 15 bags 14 MX weighing 5760 pounds, 8 bags 13 MX weighing 2900 pounds, 4 bags 12 AX weighing 1500 pounds, 4 bags 10 CX weighing 1464 pounds, 4 bags 7 RX weighing 2380 pounds, 6 bags 6 RX weighing 1370 pounds. PREPARING ROOM PRODUCTION SLIPS. LESSON IV.— Written. 1. — Find the totals in the following Preparing Room Jute Yarn Production Slip. March 17, 1907. Department No. 75. No. of System Sorts Lbs. Laps No . of D. 1* B Italian Twine 3160 18 2 Queen Twine 3000 23 3 Queen Twine 2750 22 4 BC American Twine 2430 20 5 C Jute Yarn 2000 24 6 Springfield Baling 2060 18 7 Boston Baling 1890 17 8 C Jute Yarn 3350 24 9 BX Jute Yarn 3500 25 10 BX Jute Yarn 2750 26 11 Extra Baling 2620 16 Totals, 2. — The production sheet of Dept. # 152, for March 20, 1905,' shows that #1 system used twenty-two hundred fifty pounds of laps and turned off 16 doffs of A Jute, #2 LUDLOW TEXTILE ARITHMETIC 7 system had two thousand one hundred fifty pounds of laps and 15 doffs of A "Jute, #3 system running A Jute used twenty-four hundred pounds of laps and turned off twenty doffs, #4 system had two thousand five hundred pounds of lap! of A Jute and turned off fourteen doffs. The daily slip for this day should show the sorts on each system, num- ber of pounds of laps used, number of doffs on each system, and also should have the totals for the day, made out in the form shown in example. 3.-«-0n March 28, 1903, in mill #14, Dept. 60, the following quantities were produced: System 1 made C Jute using 2250 lbs. of laps and turning off 24 doffs, on system #2, 20 doffs of A Jute were turned off and 4000 lbs. of laps used, on system #3 running B Jute 3250 lbs. of laps were used and 24 doffs turned off, system #4 running C Jute used 3500 lbs. laps and turned off 23 doffs, system #5 running C Jute used 3750 lbs. laps and turned off 21 doffs, svstem #6 running C Jute used 2500 lbs. laps and turned off 24 doffs, #7 running C Jute used 3050 lbs. laps and turned off 18 doffs, #8 running C Jute used 2450 laps and turned off 18 doffs, #9 running C Jute used 3750 lbs. laps and turned off 22 doffs, and system #10 used 2000 lbs. laps and turned off -20 doffs of C Jute. Make out slip and find the totals. WEEKLY PRODUCTION SHEETS. LESSON V.— Written. 1. — If the daily total production of carpet yarn in two Ludlow Mills is as follows, find the total production for the week: Monday 112,475 Tuesday 130,784 Wednesday 107,685 Thursday 104,746 Friday " 109,859 Saturday 61,846 Check 360,944 297,451 Total, 8 LUDLOW TEXTILE ARITHMETIC 2. — Make out slips for other weeks using the figures below, and find totals. On Monday the total production of carpet yarn was 101,574 lbs., on Tuesday, 111,487 lbs., on Wednesday, 115,967 lbs., Thursday, 118,243 lbs., Friday, 113,479 lbs., and on Saturday, 61,165 lbs. Check results as shown in Example 1. 3. — The total production of carpet yarn on Monday was 103,746 lbs., Tuesday, 110,692 lbs., Wednesday, 115,764 lbs., Thursday, 118,765 lbs., Friday, 107,776 lbs., and on Saturday, 63,127 lbs. Make out a sheet in the regular form, finding total and checking result. SUBTRACTION. Subtraction is the process of finding the difference between two numbers and the result is called their differ- ence. The sign — , called minus, indicates subtraction. The Minuend is the greater number. The Subtrahend is the smaller number. (Only like quantities can be subtracted.) Proof. — An example in subtraction may be proved or tested for accuracy by adding the result, or difference, to the number which is subtracted or taken away, the sum equaling the minuend if the subtraction is performed correctly. Example. Subtract 347 from 863 and prove. 863 minuend 347 subtrahend 516 difference 863 proof LESSON VI.— Written. Find differences, and prove: 1. 2. 3. 4. 5. 6. 7. 849 321 8642 3084 23675 832,645 132,641 278 219 730 2427 19498 794,658 93,867 LUDLOW TEXTILE ARITHMETIC 9 8. — From a lot of Italian hemp containing twelve hundred eighty-six pounds there are produced eight hundred twenty- nine pounds of polished twines. How many pounds are lost in manufacturing? 9. — In 1904 the number of school children in Ludlow was five hundred four, in 1907 the number was six hundred ninety-six. What was the increase? 10. — According to the Assessors' report for the year 1907 there were, in Ludlow, one thousand eighty-three assessed polls, and from the Registrar of Voters' report for the same year, the number of male voters was four hundred eighteen. How many men paid a poll tax who were not voters? 11. — According to the State Census of 1895 the popu- lation of Ludlow was 2562. At the time of the United States Census in 1900, the population had reached 3536. What was the gain in population during these five years? (Explain census reports.) 12. — The total production of jute carpet yarn in the Mills on Wednesday, March 17, 1903, was 322,684 lbs. Of this #18 Mill produced 100,720 lbs., and #11 Mill 92,250 lbs. How many pounds did the other mill manu- facture ? 13. — If the gross weight of a barrel of twine is 162 lbs., and the empty barrel weighs 19 lbs., what is the net weight? (Explain net and gross weight.) 14. — It is estimated that, in 1910, when the next United States census is taken, the population of Springfield will have reached 90,000. According to the last report in 1900, the population was 62,059. What gain in population will that be in ten years? MULTIPLICATION. Multiplication is the process of taking a number a certain number of times or it really is a short process of finding the sum of a number a certain number of times, and the result is called the product. Multiplicand is the name of the number multiplied or the number taken a certain number of times. 10 LUDLOW TEXTILE ARITHMETIC Multiplier is the name of the number which . multiplies- or, which shows ^how many times the multiplicand is to be taken. • _"„• . " "'. ' , " ; Abstract Numbers are those which do -not specify- any particular thing, for example, 6, 7, 9. Concrete '"Numbers indicate or refer to some particular thing, for example, 6 spindles, 7 boys, 9 mills. (The multiplier must be an abstract number while the multiplicand and product may both be either abstract or concrete. In finding the product of two abstract numbers either may be the multiplicand or the multiplier.) LESSON VII.— Written. 1.— Multiply 267 by 2, by 3, and by 4, and then add the three products. 2. — Multiply 3401 by 8 and by 9 and then add the two products. 3.— Multiply 3425 by 132. 4.— Multiply 62,084 by 4007. 5. — If three hundred fifty pounds of Jute can be run over a system of preparing in one hour, how many pounds can be run over the same system in a week of 55 working hours ? 6. — If the average production of each loom is 850 yards of bagging per day of 10 working hours, what would be the total production of 112 looms for the same length of time? 7. — What would be the wages per week of 55 working hours of a boy who is paid at the rate of ten cents an hour? 8. — -If one system of preparing will supply six sides of spinning with rove, how many sides should 16 systems keep going? (A rove is a slightly twisted sliver or fleecy strand of fibre.) 9. — If the number of help required to operate a system of jute preparing is 7, and it requires the services of 10 girls in the spinning department to spin the product of this system, how many help would be required to operate both the pre- paring and spinning room in a plant running 54 systems? 10. — If one cut or lea of yarn contains 300 yards and 48 cuts make a spyndle, how many yards are contained in one spyndle ? LUDLOW TEXTILE ARITHMETIC 11 11. — -Find the "number of pounds of yarn in 1450 spyndles of 14 lbs. C yarn. . , (14 lbs. yarn means that 1 spyndle weighs 14 lbs.) 12. — Find the number of pounds of jute in a stock house containing 3750 bales, weighing 400 pounds each. (The standard weight of a bale of "jute is 400 pounds.) LESSON VIII.— Written! 1.— Make out a list showing the wages paid for a week's work of 55 hours for each cent difference in the rate from 5 cents per hour to 20 cents per hour. LESSON IX.— Written. 1. — Make a list showing the price per ton for work that is paid for at the rate of from 1 to 20 cents per 100 pounds. DIVISION. Division is the process of finding how many times one number is contained in another. The sign of division is h-, read, divided by; for example 9-^3, shows that 9 is to be divided by 3. The Dividend is the number divided. The Divisor is the number used for dividing the dividend. The Quotient is the result of the division. The Remainder is the part left over. Short Division is a method of dividing when the divisor is small or of such a nature that the work may be performed mentally. Long Division is a method of dividing when the work is fully written, but is otherwise the same as short division. (The quotient is written over the dividend, the first figure being written over the right hand figure of the partial dividend used in obtaining it.) LESSON X.— Oral. Example. Find the quotient of 6975-^3, by short division. 12 LUDLOW TEXTILE ARITHMETIC The divisor is written at the left of the dividend, and the 3) 6975 . quotient under the dividend. Three is contained in 6 twice, thus 2325 2 is the first figure in the quotient and is placed below the 6 of the dividend. Three is contained in 9 three times, in 7 two times with a remainder of 1. This 1 is equal to 10 of the next lower order and with the 5, the next figure of the divi- dend makes 15. Three is contained in 15 five times. This work would be stated, 3 in 6, 2; in 9, 3; in 7, 2; in 15, 5; 6975 divided by 3 equals 2325. 1.— 936 2.— 17869 3.-46785 4.-786,491 5.-369,472 6= ? 6.-56783 7= ? 7.— 1440 5= ? 8.— 30456 8= ? 9.-274568 5= ? 10.— 896532 100= ? 12= ? 11= ? 13= ? 15= ? (The work may be tested or checked by multiplying the quotient by the divisor and adding the remainder to the product. If the work is correctly performed the result should equal the dividend.) LONG DIVISION. Example. Divide 85323 by 23. 3709 As 23 is more than 8 it is necessary to take two figures 23) 85323 of the dividend for a partial divi- 69 dend. Eighty-five will contain 23, three times, but not four 163 times, therefore 3 is the first 161 figure in the quotient and is placed above the right hand figure of 223 the trial dividend, namely 8. 207 Subtracting 3 times 23 from 85 leaves 16. Annexing the next 16 remainder figure in the dividend, the next partial dividend becomes 163. This contains 23, seven times with a remainder of 2. The next figure of the dividend, 2, is brought down or annexed and it is found that this partial dividend, 22, will not con- LUDLOW TEXTILE ARITHMETIC 13 tain 23 so the next figure of the dividend, 3, is brought down and annexed. Twenty-three goes into 223 nine times with a remainder of 16. Thus the quotient is 3709 and the remainder 16, or 23 goes into 85323, 3709 times and 16 over. Note: — When it is necessary to bring down more than one figure of the dividend, a zero should be placed in the quotient for each extra figure brought down. LESSON XI.— Written. Divide: 1.-232,848 by 56. 2.— 94,576 by 345. 3.— 78,942 by 372. 4.-609,725 by 25. 5.-345,632 by 948. 6.— 76,865 by 3425. 7.— 89,210 by 8740. 8.-100,456 by 3700. 9.-372,658 by 7005. 10.— 84,736 by 9876. LESSON XII.— Written. 1. — One cut or lea of carpet yarn measures 10800 inches' How many yards will one lea measure? (12 inches make one foot, 3 feet make 1 yard.) 2. — A spyndle of yarn contains 14400 yards. How many cuts does it contain? 3. — If a thread wound once around a reel measures 90 inches, how many inches would a thread wound around 120 times measure? How many yards? 4. — Now many bales of jute (400 lbs.) can be loaded into a car designed to carry 60000 lbs.? 5. — The twist on yarns is calculated by the number of revolutions the flyer makes during the time the roller is delivering one inch of yarn. Under these conditions what would be the twist when the roller delivers two inches of yarn while the flyer makes 10 revolutions? (Express as so many turns per inch.) 6. — Without making allowances for waste, how many spyndles of 30 lbs. yarn can be made from 19 bales of jute? (30 lbs. yarn means that 1 spyndle weighs 30 lbs.) 14 LUDLOW TEXTILE ARITHMETIC 7. — Find the average cuts per spindle if the total pro- duction is 348,900 cuts and the number of spindles in oper- ation is 13956. 8. — The report of the Assessors on March 1, 1908, gives the total valuation of the property in the Town of Ludlow, subject to taxation, as $3,472,474.00. The last official census gives the population as 3881. From these figures find the per capita assessed valuation; or, in other words, find the average wealth of each individual in Ludlow. SIGNS. + plus, the sign of addition. — minus, the sign of subtraction. X times, the sign of multiplication. -r- divided by, the sign of division. > greater than. < less than. .'. therefore, or hence. MISCELLANEOUS EXAMPLES. LESSON XIIL— Written. 1. — How many pins would be required for a push bar first drawing frame, with two heads, each head having 30 faller bars with 8 rows of gills on each faller, and 2 rows of pins in each gill with 18 pins in a row? 2. — In a three headed 2d drawing frame there are 30 faller bars to the head having 6 rows of gills on each bar, with 2 rows of pins in each gill and 20 pins in a row. How many pins would be required to fill the gills of one of these drawing frames? 3. — Find the number of hackle pins required for a seven headed roving frame, with 29 faller bars in head, 8 rows of gills on each faller, 2 rows of pins in each gill with 13 pins in a row. 4. — Find the total number of pins it would require for this system of preparing machinery consisting of 1st and 2d drawing frames and roving. LUDLOW TEXTILE ARITHMETIC 15 5. — The number of watts is equal to the number of amperes multiplied by the number of volts. There are 746 watts in one horse-power. How many amperes are required to furnish power for a 50 H. P. motor, the number of volts being 110? 6. — On a twine measuring machine is an indicator which records the number of yards contained in one pound of twine. The indicator reads 182 yards at the start and 922 yards at the finish when measuring a quantity of 9 B Amer- ican. What were the total number of yards? Find the number of yards per pound in each sort from the following readings of the indicator: Indicator At St; 7.— 12 A Italian, 8. — 18 B American, 9. — 24 B American, 10.— 36 B Italian, 11.— 48 B Italian, 12. — Triangle Baling, 13. — Boston Baling, 14. — Medium Cord, 15. — A girl is paid 12^ per hundred pounds for winding a certain size yarn and earns $8.40 in a week; for another kind of yarn she is paid at the rate of 10^ per hundred pounds. How much more of this kind of yarn will she have to wind in a week to earn the same wages? 16. — For opening jute a man is paid at the rate of 15^ per bale of 400 pounds. How many pounds of jute will he have to handle to earn $15.00? 17. — The revolutions per minute of the main driving shaft in No. 12 Mill were as follows: 140, 142, 145, 141; 139, 137, 143. What were the average number of revolutions per minute? 18. — A spinning room driving shaft makes 320 R. P. M. How many revolutions will it make in 3 hours? (R. P. M. equals revolutions per minute.) 19. — A bailer is paid at the rate of 25^ per hundred pounds for one kind of work; 30^ per hundred pounds for another kind of work; and 35^ per hundred pounds for At Start At Fini 341 941 541 921 921 1205 1205 1405 1405 1555 1555 1985 1985 2365 2365 2573 16 LUDLOW TEXTILE ARITHMETIC another kind. During the week she balls 1200 pounds at the first rate, 1700 at the second, and 300 at the third. What will her wages for the week amount to ? 20. — If one side of a dry spinning frame turns off a certain size yarn at the rate of 50 pounds per hour, how long will it take two sides to spin 6,000 pounds? 21. — How long will it take six sides to get out the same order? 22. — The following is the weight in grains of eight 50- yard tests of 14 C jute yarn. Find the average weight of the yarn? 334, 327, 346, 353, 330, 347, 350, 333. 23.— The United States produced in 1890, 6,940,898 bales of cotton. Of this number Texas produced 1,594,305 bales. How many bales did the other states produce? 24. — A lea or cut of carpet yarn measures 300 yards. A spyndle of yarn is 14,400 yards. How many leas in a spyndle ? 25. — If 54 systems use, on an average, 432 bales of jute every ten working hours, find the average number of pounds run over each system per hour. 26. — In 15 tons of 30 pounds C yarn, how many spyndles? (30 pounds indicates that yarn weighs 30 pounds per spyndle.) 27. — If 7000 grains equal one pound avoirdupois, how many grains in one ounce? 28. — If the net weight of jute rove on a single rove bobbin weighs two pounds, how many pounds of jute rove will there be in twenty-five doffs of a roving frame, each doff containing fifty-six bobbins? 29. — A foot of lumber is 12 inches square and one inch thick. How many feet- of lumber are contained in a plank 12 feet long, 1 foot wide, and 2 inches thick? 30. — If five spinning bobbins contain two pounds of yarn, how many of these bobbins will have to be filled to make a bale of twine weighing one hundred fifty pounds? 31.— What would be the weight of 1,000,000 pins if 42 pins equalled one ounce? 32. — W'hat are the wages per hour of a boy who receives $8.25 for a week of 55 working hours? 33. — What wages would a man receive for opening 96 bales of jute in one week at the rate of 15^ per bale? LUDLOW TEXTILE ARITHMETIC 17 34. — Find the number of pins contained in one thousand pounds of card pins weighing fifty to the ounce. 35. — Out of every 100 pounds of American hemp there are produced about 60 pounds of finished twines. On this basis how many pounds of twine could be made from a lot of hemp containing 10 tons? (1 ton equals 2000 pounds.) 36. — How many pins would be required for the lags of a breaker card cylinder, each lag or stave having six rows of pins* with 42 pins in each row, and 150 lags being the usual number required to cover an ordinary card cylinder four feet in diameter? 37. — What would be the weight of this number of pins if 42 pins weigh one ounce? 38. — It is computed that about 12 H. P. is required to drive one system of preparing machinery and that about 6 H. P. is required to drive one side of a dry spinning frame. On this basis what would be the H. P. required to operate a room containing 4 systems of preparing and 20 sides of spinning machinery, allowing 10 additional H. P. to drive shafting and belts? 39. — In the year 1815, Catharine Woods of Dunmore,' near Ballynahinch, in the County of Down, Ireland, then about 13 years old, spun 12 cuts of linen yarn weighing 10 grains, each cut 120 threads, each thread 2}/% yards. How many yards would there be in one pound of this yarn? (7000 grains equal one pound.) 40. — The circumference of the earth's surface is about 25,000 miles. A spyndle of yarn measures 14,400 yards. How many spyndles of yarn would it take to go around the earth? 41. — How many gallons of batching oil would be required for a week of 55 working hours in a room where the average product is 30,000 pounds per 10 hours, and this oil is used at the rate of one gallon for each 1000 pounds of jute? ANALYSIS. Analysis is the process of reasoning from the given' number to one, "and then from one to the number required. 18 LUDLOW TEXTILE ARITHMETIC . Example. If a weaver receives $1.00 for weaving 4 rolls of bagging, how much will he receive for 96 rolls? If $1.00 or 100^ are re- ceived for weaving 4 rolls of lOOjzf-^- 4 = 25^ bagging then as many cents as 25^X96 = 2400^ 4 is contained times in 100 ^ = $24.00 or 25^ will be received for 1 roll, and 96 times 25 cf, or 2400^ or $24.00 will be received for 96 rolls. LESSON XIV.— Written. 1. — If a roving frame turns off 20 doffs in a day of 10 hours, how many doffs should the same frame turn off in a week of 55 hours? 2. — If a winding machine turns off 6,000 pounds of jute yarn in a day of 10 hours, how many pounds can be taken off this machine in 55 hours? 3. — A man receives 75^ for opening 5 bales of jute, how much is he paid for opening 15 bales? 4. — If 25 girls do a certain amount of work in 10 hours, how many girls will be required to do the same work in one hour ? 5.— If 50 card pins weigh one ounce, what will be the weight of 10,000 pins? 6. — If there are 2030 fallers in 10 seven-headed roving frames, how many fallers are contained in 16 eight-headed frames of the same make? 7. — How many bales of jute will be required to make 150 bags of yarn if 3 bales of jute make 2 bags? 8. — With an output of 100,000,000 pounds yearly, what would be the average number of pounds turned out weekly? 9. — What would the import duties amount to for 100,000,- 000 pounds if each pound was taxed one cent? 10. — Find the difference one eighth of a cent a pound would amount to in buying 100,000,000 pounds of stock. 11. — How many feet of card clothing, 2" wide, would be required to cover a doffer 72" wide and 14" in diameter? 12.— With jute selling at $112 per ton in London, what is it worth per pound in Ludlow after adding one cent per pound for cost of freight, insurance, and so forth? (An English ton equals 2210 pounds.) LUDLOW TEXTILE ARITHMETIC 19 LESSON XV.— Written. 1. — What is the cost per pound of hemp which sells, in the United States, for $245 per ton? (An American ton equals 2000 pounds.) 2. — What is the cost per ton for polishing and balling twine if the polisher is paid at the rate of 18^ per 100 pounds and the bailer at the rate of 25{zf per 100 pounds? 3. — If 4 men can open 56 bales of jute in one day of 10 working hours, how many men will it take to open 70 bales in 5 hours? 4. — Twenty-five bales of jute weigh 10,000 pounds. Find the weight of 17 bales. 5. — Three winders can wind 3600 pounds of jute yarn in 10 hours, how many girls would be required to turn off 12000 pounds of yarn in the same time? 6. — How many polishing machines will have to be put on a certain kind of work to get out 33,000 pounds of twine in 55 hours if one machine can polish 1200 pounds in 10 hours ? 7. — How many sides of spinning will have to be put on to produce 99,000 pounds of yarn in .2 weeks of 55 hours each, if one side turns off 450 pounds of yarn in a day of 10 hours? 8. — How many bales of jute can be purchased for $1200, if one pound of jute costs 6^? 9. — If one foot of belting costs $1.44, what would be the cost of a roll of belting containing 250 feet? 10. — If 3 boys make a tool chest in 12 days, how many boys will be required to make it in one day? 11. — If 7 boys do a piece of work in 8 days, how many boys will it take to do the same work in one day? 12. — If 3 pounds of jute cost 27^, what will a bale con- taining 400 pounds cost? FACTORS AND MULTIPLES. Factors of a number are those whole numbers, which multiplied together, will equal the original number. A Prime Number has no factors except itself and one, for example, 2, 3, 5, 7, 11. 20 LUDLOW TEXTILE ARITHMETIC Even Numbers can be divided by 2 with no remainder, as 2, 4, 6, 8, 10. Odd Numbers can not be exactly divided by 2, as 3, 5, 7, 9, 11. Factoring is the process of separating a number into its factors, for example, the factors of 15 are 3 and 5. A number can be divided by 2 if the right hand figure is even, as 246, 738. 3 if the sum of all the figures can be divided by 3, as 48, 264. 5 if the right hand figure is either zero or five, as 35, 470. LESSON XVI.— Oral. 1. — Name all the prime numbers from 1 to 51. 2. — Name the even numbers in the following list: 7 9 23 48 21 50 67 5 74 26 13 36 51 53 8 84 72 96 69 78 90 3. — Name all the odd numbers in the above list. 1 4. — Name the prime factors in each of the following: 27, 36, 48, 95, 37, 108- 232, 438, 625. '■'■ 5. — In example 2, name all the numbers that are divisible by 3; by 5. LESSOR XVII.— Written. Find the prime factors of the following: 1.— 60 6.- - 75 11.— 480 16.- -1086 2.-36 ?.- -210 12.— 640 17.- -1550 3.-39 8.- -400 13.— 720 18.- -3150 4.-42 9.- -810 14.— 700 19.- -3900 5.— 17 10.- -756 15.— 800 20.- -1485 GREATEST COMMON FACTOR. A Common Factor of two or more numbers is a factor of each of the numbers. The Greatest Common Factor of two or more numbers is the greatest factor contained in each of the numbers. (The greatest common factor is written G. C. F. and is sometimes called the G. C. D. or greatest common divisor.) LUDLOW TEXTILE ARITHMETIC 21 (Numbers that have no common factors are said to be prime to each or other, or are mutually prime.) Example. Find the G. C. F. of 18, 30, and 42. Separate the different numbers into their prime fac- Case I. 18 = 3X3X2 tors, and then take those fac- 30 = 3X2X5 tors which are common to all 42 = 3 X 2 X 7 three numbers and multiply 2X3 = 6 G. C. F. them together. The product is the G. C. F. In this case the common factors are 2 and 3, therefore 2X3 = 6 and 6 is the G. C. F. In a simple case of this sort the G. C. F. should be found by inspection. In this second method the numbers are arranged Case II. 3) 18 — 30 — 42 horizontally in a row, and 2) 6—10 — 14 are divided by a common ol c _ 7 factor. The quotients are in 9 wo = g q. p p turn divided by a common factor and the process con- tinued until the quotients last obtained have no factor in common. The product oj the divisors or common factors, in this example, 2X3 = 6, is the G. C. F. LESSON XVIIL— Written. Find the G. C. F. of: 1.— 60, 72, 108. ' 6.-35, 49, 21. 2.-54, 72, 90. 7.-75, 175, 225. 3.-55, 99, 110. 8.-28, 77, 98. 4.-84, 120. 9.-39, 52, 78. 5.— 80, 112, 160. 10.— 60, 180, 240. LEAST COMMON MULTIPLE. A Multiple of a number is a number which will exactly contain the first. A Common Multiple of two or more numbers is a number which will contain each of the numbers without remainder. 22 UJDLOW TEXTILE ARITHMETIC The Least Common Multiple of two or more numbers is the smallest number that exactly contains all of the given numbers. (Least common multiple is written L. C. M.) LESSON XIX.— Oral. Name the least common multiple of: 1.— 2 and 3. 2. — 5 and 6. 3.-7 and 8. 4.-9 and 4. 5.-2, 3, and 4. 6.-2, 4, and 5. 7.-3, 6, and 10. 8.-8, 6, and 12. 9.-2, 4, and 10. 10.— 9, 3, and 4. LESSON XX.— Written. Example. Find the L. C. M. of 54, 72, 90. Any multiple of 54, 72, 90, will contain all Case I. 54 = 2X3X3X3 . the factors of each of the 72=2X2X2X3X3 three numbers. There- 90=2X3X3X5 fore, to find the least 2X3X3X3X2X2X5 common multiple, it is = 1080 L. C. M. necessary to find the product of all the prime numbers, or factors, us- ing each factor the greatest number of times that it occurs in any one number. This second method is usually more conven- Case II. 2)54-72-90 ient. Arrange the num- bers in a line and divide by any factors that are , common to any two of 3-4-5 them. When the last 2X3X3X3X4XO quotients are prime to = 1080 L. C. M. each other, multiply them together and then multiply this product by each of the divisors. The final product of them all is the L. C, M. 2) 54- -72- -90 3) 27- -36- -45 3) 9- -12- -15 LUDLOW TEXTILE ARITHMETIC 23 LESSON XXI.— Written. Find the L. C. M. of 1.— 9, 36, 45. 2.-6, 33, 99r 3.-35, 70, 140. 4.— 15, 30, 45, 90. 5.— 12, 48, 36. 6.— 16, 48, 96. 7.-4, 12, 18. 8.-*9, 15, 45. 9.— 300, 400, 500. 10.— 20, 16, 8. CANCELLATION. Cancellation is a short process in division, whereby- equal factors may be removed from both divisor and dividend. This process is used in calculating drafts and twists in mill work, and in tracing speed, or R. P. M. from one part of a machine to another. 10^-5 means that 10 is to be divided by 5. This also may be expressed 10/5. By short division this would be worked as follows: 5) 10 2 Thus 10 over 5 or 10 divided by 5 equals 2. A shorter way of finding the quotient would be as follows : 2 5 is an equal factor of both 10 10 2 and 5 and goes into 10 twice and 5 — =— = 2 once. As 1 goes into 2 twice the 5 1 final result is 2. This process is 1 called cancellation. Example. Divide 9 X 18 X 24 by 3 X 6 X 12. 1 3 6,4 9X18X24 = 3X6X1 = =.18 3X 6X12 1X1X1 1 1 8 1 (By this method it is not necessary to multiply the differ- ent quantities together before dividing.) 24 LUDLOW TEXTILE ARITHMETIC LESSON XXII.— Written. Divide : 1.—7X6X16 by 3X8x14. 2.— 11X27X30 by 9X15X3. 3.-9 X 14 X39 by 13 X7 X 18. 4.— 1728X3X7 by 12X12X12. 5.— 84X13X5 by 91X4X15. UNITED STATES MONEY. In the Ludlow Mills the cost of material, and cost of labor and manufacture is reckoned in dollars, cents and mills. The relationship of these different units is shown in the following table: 10 mills — 1 cent (^) 100 cents = 1 dollar ($) (The dollar sign, $, is written before, or at the left of the figures. Thus twenty-four dollars would be written $24. Twenty-five cents would be written 25$/, the sign, 0, following the figures. When both dollars and cents are written to- gether, a period or point is used to separate one from the other. This period is called the decimal point. The dollar sign only, is used. Thus $24.25 means twenty-four dollars and twenty-five cents. Mills are written in the third decimal place. Thus $9,467 would be read nine dollars and forty- six cents, seven mills.) Day workers are those who are paid a stated sum per week. Their rate per hour is found by dividing 55 into their wages for a week. Piece workers are those who are paid a certain rate for performing a certain amount of work, as, for instance, a reeler may receive 6 cents for every reel of yarn she turns off her machine, or a man may receive 15 cents for every bale of jute he opens. In making up the wages of both day and piece workers, parts of a cent are used. These parts are called mills. LESSON XXIIL— Written. Write the following in figures: 1. — Ten thousand dollars and forty cents. LUDLOW TEXTILE ARITHMETIC 25 2. — Seven hundred eighty-four dollars and sixteen cents, four mills. 3. — Five thousand six hundred seventy-four dollars and nineteen cents, two mills. 4. — Three million dollars and three mills. 5. — Seven thousand dollars and three cents, three mills. 6. — Eight hundred seven dollars and ten cents. 7. — Five thousand twenty-three dollars and seventy-six cents, eight mills. 8.-»One dollar and two mills. 9. — Twenty-seven dollars and thirty cents, five mills. (Use ciphers to fill in vacant places.) TO REDUCE DOLLARS AND CENTS TO CENTS. Example. Reduce $1.27 to cents. Dollars and cents are reduced to cents by dropping the decimal $1.27 — 127^ point and substituting the sign for cents, i, for the dollar sign. (Dollars, cents, and mills may be reduced to mills in the same way.) LESSON XXIV.— Written. 1.— Reduce $22.46 to cents. 2. — Reduce $47 to cents. (If no cents are expressed annex two zeros.) 3 — Reduce $6,759 to mills. 4.— Reduce $32,476 to mills. 5 —Reduce $66,432 to mills. TO REDUCE CENTS TO DOLLARS. Example. Reduce 3768^ to dollars. Cents are reduced to dollars and cents by pointing off two 3768w many tons? 8. — What would be the weight of 10,000 pins if 78 pins weigh one ounce? 9. — What would be the length of a roll of wire weighing 50 lbs., if the weight of one foot was 87^ grains? 10. — What is the cost per pound of American hemp that costs $150 per ton? LUDLOW TEXTILE ARITHMETIC (59 ADDITIONAL TABLES. DRY MEASURE. 2 pints = 1 quart. 8 quarts = 1 peck. 4 pecks = 1 bushel. SHIPPING MEASURE. 40 cubic feet = 1 U. S. shipping ton 42 cubic feet = 1 British shipping ton. NUMBERS. 12 units = 1 dozen. 12 dozen = 1 gross. CIRCULAR MEASURE. 60 seconds (") — 1 minute. 60 minutes (') = 1 degree. 360 degrees (°) = 1 circumference. 90 degrees = 1 quadrant. TIME MEASURE. 60 seconds = 1 minute. 60 minutes = 1 hour 24 hours = 1 day. 7 days = 1 week. 365 days, 5 hours, 48 minutes, 48 seconds = 1 year. HORSE POWER. 33,000 foot pounds = 1 horse-power (H. P.), that is equals the work required to raise 33,000 pounds 1 foot high in 1 minute. 70 LUDLOW TEXTILE ARITHMETIC WEIGHTS AND MEASURES. WIRE GAUGES. Gauges. Birmingham. American or Brown and Sharp inch. inch. 0000000 000000 00000 0000 0.454 0.46 000 0.425 . 40964 00 0.38 . 3648 0.34 . 32486 1 0.3 0.2893 2 0.284 0.25763 3 0.259 . 22942 4 0.238 0.20431 5 0.22 0.18194 6 0.203 0.16202 7 0.18 0.14428 8 0.165 0.12849 9 0.148 0.11443 10 0.134 0.10189 11 0.12 0.09074 12 0.109 . 08081 13 0.095 0.07196 14 . 083 0.06408 15 0.072 . 05707 16 0.065 0.05082 17 0.058 0.04526 36 0.004 0.005 MISCELLANEOUS EXAMPLES. LESSON LXXVI.— Written. 1. — A bale of jute measures, approximately, 50" high, 20" wide, 20" thick. How many of these could be packed in a storehouse 100 feet long, 50 feet wide and 20 feet high, allowing a passageway of 2 ft. all the way around the inside of the building? 2. — How many systems of preparing machinery could be placed in a room 198' X88' allowing 1450 sq. ft. of floor space for each system? LUDLOW TEXTILE ARITHMETIC 71 3. — How many feet of telephone wire would be required for a single line between Ludlow and Worcester, a distance of about 48 miles? 4. — How many storehouses could be erected on a lot of 20 acres, allowing 100 square feet for each house? 5. — How many gallons would a cistern hold that is 31 feet high and 8 ft. square? 6. — How many feet between the Ludlow and Spring- field Post Offices, a distance of 7.8 miles? 7 rf — Which box is larger and how much, one 36" long, 28" deep and 27" wide the other 30" long, 30" deep and 26" inches wide? 8. — If a man empties 24 boxes of rove, each box contain- ing 56 roves, and each rove weighing 2 lbs., in 10 hours, how many pounds would he have to lift in 55 hours? 9. — How much floor space will be required for 30 spin- ning frames 27 ft., 6 in. long and 7 ft. wide allowing for alleyways 7 ft. wide? 10. — How many gallons of water will be contained in a catch-basin 2 ft., 9 in. high, 65 in. long and 23 in. wide? 11. — How many pounds pressure would be put on a re- taining roller by 39 weights each weighing 3 lbs., 8 ozs.? 12. — How many bobbins would be required to fill a box 2 feet, 6 inches long, 2 feet wide, and 2 feet, 4 inches deep, each bobbin measuring 6 inches in length and 3 inches in diameter ? 13. — If one boiler consumes 18 c. w. t. of coal in 24 hours, how much coal would 3 boilers consume in a week of 6 days? 14. — If a doffer lifts 2 lbs., 6 ozs. of yarn every time she doffs a spinning frame how much will she have lifted at the end of a week of 55 hours, doffing every 30 minutes? 15. — At a dollar a cubic foot, what would it cost to con- struct a building 500 feet long, 72 feet wide, 80 feet high. ALLIGATION. Alligation is useful in mill work for finding the average price per pound of any of the products which are made from fibres of different qualities having different cost prices. Alligation treats of mixing different substances at differ- ent prices producing a compound of some intermediate quality or price. 72 LUDLOW TEXTILE ARITHMETIC Example. What is the average cost per pound of a mix made up as follows: Five 400 lbs. bales of jute @ 4$/ per pound? Four 400 lbs. bales of jute @ 50 per pound? Two 400 lbs. bales of jute @ 11$/ per pound? Three 400 lbs. bales of jute @ 12$/ per pound? Solution: 5X400 = 2000 2000 X 4= 8000 4X400 = 1600 1600 X 5= 8000 2X400= 800 800X11= 8800 3X400 = 1200 1200X12 = 14400 5600 lbs. 39200$/ 39200 - — = 70 average price per pound. 5600 Rule. To find the average cost per pound of a mix when the proportion of the materials mixed and their prices are given, divide the total value of the material mixed by the sum of the amounts put in and the quotient will be the average price per pound. LESSON LXXVIL— Written. 1. — What is the price per pound of the following mix: 2/10 jute at 80 per pound, 1/10 hemp at 14$/ per pound, 4/10 tow at 3 1/2$/ per pound, and 3/10 flax at 90 per pound? 2. — Find the cost per pound of a mix made up as follows: 1/3 at 10$/ per lb., 1/9 at 14$/ per lb., and 5/9 at lip' per lb. ? 3. — What is the cost per pound of the following mixture: 1/8 tow at 70 per lb., 1/4 flax at 11$/ per lb., 1/2 hemp at 12$/ per lb., 1/8 flax at 16$/ per lb.? 4. — What would be the average price per pound of the following mixture: 1/4 tow @ 80 per lb., 1/2 hemp @ 12$/ per lb., 1/4 flax @ 16/ per lb.? 5. — Find cost per pound of a mix composed as follows: 1/3 flax @ 12$/, 1/3 hemp @ 10$/, 1/6 tow @ 9$/, and 1/6 jute @ ll$z. Alligation Alternate is the process of mixing quanti- ties at different prices so as to obtain a mixture of a required intermediate price. LUDLOW TEXTILE ARITHMETIC 73 Example. It is desired to mix jute costing 60 per pound with hemp costing 90 per pound so as to make a mix worth 70 per pound. What proportions of the different materials will have to be used? . 06 2 jute 0.07 . 09 1 hemp Write the prices of the jute and hemp in a vertical column as above with the price' of the desired mix at the left. Wrjte the difference between the desired price of the mix and the price of the jute against the hemp, in this case 1, and the difference between the price of the mixture and that of the hemp against the jute, in this case 2, and these figures will represent the proportions in which each of those quanti- ties enter into the mixture. LESSON LXXVIIL— Written. 1. — Mixing flax costing 15 cents a pound with hemp costing 10 cents a pound t what proportions of the materials will have to be used to form a mixture worth 12 cents a pound?" 2. — With jute at 5 cents a pound and hemp at 15 cents a pound, what proportions will go together to make a mixture . worth 10 cents a pound? 3. — What proportion of hemp at 8 cents a pound and flax at 14 cents a pound will go together to form a mixture worth 12 cents a pound? 4. — How will four kinds of fibre worth respectively 60, 80, 130, 160, a pound, be mixed to make a mixture worth 10^ a pound? 5. — With hemp at 10^, flax at 12^, jute at 70, and tow at 50, what proportions will go together to form a mixture worth 90 a pound? PERCENTAGE. Percentage is used in mill work for a variety of purposes. It is used to find the loss by waste in the different processes of manufacture. It is used again to find the relative cost of manufacture in the different departments through which the material passes, and it may be used to calculate the different speeds at which different parts of the machinery are running. 74 LUDLOW TEXTILE ARITHMETIC Per Cent, means per hundred. The sign % means per cent. The Per Cent, of a number is the result obtained by- taking a stated number of hundredths of it. 1% of a number is 1/100 of it. 2% of a number is 2/100 of it. 9% of a number is 9/100 of it The Percentage is the amount computed on the given number thus the percentage on $500 at 5% is $25. The Base of percentage is the number on which the per- centage is computed. Thus $500 is the base on which the percentage is com- puted in the foregoing example. When the per cent, is expressed by a decimal of more than two figures the figures after the second place must be taken as parts of 1%. Thus 0.125 means 12 5/10% or 12 1/2%. The Rate Per Cent, being a certain number of hun- dredths may be expressed decimally or by a common fraction as is shown in the following table : 1% = 0.01 = 1/100 2% = 0.02 = 2/100 = 1/50 10% =0.10= 10/100 = 1/10 50% = . 50 = 50/100 = 1/2 100%, = 1.00 = 100/100 = 1 125%, = 1.25 = 125/100 = 1 1/4 6|% =0.0625 = 61/100 = 1/16 124% =0.125 = 12J/100 = 1/8 TO EXPRESS A COMMON FRACTION AS A RATE PER CENT. Example. Express 4/5 as a rate per cent. 1 = 100/100 = 100% 4/5 = 4/5 X 100/100 = 80/100 = 80% Rule. Divide 100 by the denominator of the fraction and multiply the quotient by the numerator. IvTJDLOW TEXTILE ARITHMETIC 75 LESSON LXXIX.— Written. Express the following in rates per cent.: 1.— 1/4 2.— 1/3 3.— 1/2 4.— 1/6 5.— 1/8 6.— 1/10 7.— 5/16 8*— 5/12 9.-3/7 10.— 4/9 11.— 9/40 12.— 23/16 TO EXPRESS A DECIMAL AS A RATE PER CENT. Example. Express 0.3 as a rate per cent. 0.3 = 0.30 = 30% Example. Express 0.5625 as a rate per cent. 0.5625 = 0'. 56 25/100 = 56)<%. Rule. Write the decimal as hundredths and the num- ber expressing the hundredths is the rate per cent. If the decimal has more than two places the figures that follow the hundredths place signify a part of 1%. LESSON LXXX.— Written. Express as a rate per cent.: 1.— 0.46 2.— 0.38 3.— 0.07 4.— 0.9 5.— 0.26 6.-7.79 7.-6.48 8.— 0.0005 9.— 10.04 10.— 0.00625 11.— 0.125 12.— 0.0375 76 LUDLOW TEXTILE ARITHMETIC TO FIND THE PERCENTAGE HAVING THE BASE AND THE RATE GIVEN. Example. — Base = 250 and rate per cent. =4%, find the percentage. 4% of 250 = 250X0.04=10.00 = 10 or 4% of 250 = 250 .04 10.00 Rule. Multiply the base by the rate and point off two places to the left in the result. LESSON LXXXL— Written. 1.— What is 6% of 250 pounds of jute? 2.— Find 8% of 400 lbs. of jute. 3.— Find 12i^% of 500 lbs. of flax. 4. — 16K% of 1200 lbs. of hemp is equal to what? 5.— What is 25% of 5000 lbs. of jute? 6.— Find 33^% of 6600 lbs. of hemp. 7.— Find 40% of 7000 lbs. of jute. LESSON LXXXIL— Written. 1. — In a spinning room having 3600 spindles, making both hemp and jute yarns, 40% are spinning jute yarn. Find the number of spindles running on hemp. 2. — The waste from a certain jute finisher card is averag- ing 4%. How many pounds of waste would this card make in a day's work of 3250 pounds? 3. — From a certain grade of American hemp the loss by waste in the process of manufacture is as follows: Softening, 3% Breaker card, 16% Finisher card, 7% Drawing and roving, 3% Spinning and reeling, 3% What is the total loss in pounds per ton? 4. — In the manufacture of twines from Italian hemp the waste is computed to be about 20%. On this basis how many pounds of finished twines can be made from, a lot con- taining 37„4> tons? 5. — If 80 bagging looms are turning off 3860 yards of cloth in a week, and it is intended to increase the production by 20%, how many more looms will have to be installed to LUDLOW TEXTILE ARITHMETIC 77 get this production, and what will be the total number of yards of cloth produced weekly when these additional looms are running ? TO FIND THE RATE PER CENT., THE BASE AND PER CENTAGE BEING GIVEN. Example. $1 is what per cent, of $4? 1 X 100 = 100 100-^-4 = 25% Example. 8 is what % of 200? 8X100 = 800 800^-200 = 4% Rule. Multiply the Percentage by 100 and divide the product by the Base. LESSON LXXXIII.— Written. 1.— What per cent, of $350 is $19? 2.— What per cent, of $340 is $14? 3. — What per cent, of 250 pounds is 5 pounds? 4 — What per cent, of 270 pounds is 16 pounds? 5. — From a sample lot of American hemp containing 1,200 pounds, there is a loss by waste and so forth of 444 pounds. Find the per cent, of loss in manufacture. 6. — From a sample test of 1,000 pounds of Italian hemp, 770 pounds of yarn are produced. What is the per cent, of loss? 7. — The number of children attending the schools of Ludlow from the year 1896 to 1908 is as follows: Year. No. of children. Per cent, increase or decrease. 1896 315 1897 324 1898 325 1899 362 1900 448 1901 487 1902 523 1903 515 1904 470 1905 578 1906 607 1907 605 1908 723 Show the annual rate of increase or decrease. 78 LUDLOW TEXTILE ARITHMETIC TO FIND THE BASE WHEN THE PERCENTAGE AND RATE ARE GIVEN. Example. $6 is 3% of what? $6 X 100 = $600 -r-3 Rule. Multiply the percentage by 100 and divide the product by the rate. LESSON LXXXIV.— Oral. 1.— $9 is 4% of what? 2.— $37.50 is 3% of what? 3.— 12 is 3% of what? 4.— 37 y 2 is 6% of what? 5.— 33 is 3/8% of what? 6.-75 is 25% of what? 7.— 120 is 20% of what? LESSON LXXXV.— Written. 1. — From a sample of Italian hemp there are 330 pounds of waste, which are 23% of the whole. How many pounds were in the sample? 2. — From a sett of jute laps there are 10 pounds of flyings which are 4% of the whole. What is the weight of the sett of lapps ? 3. — A lot of American hemp loses 40% in being manu- factured into twine. The total amount of loss is 4,800 pounds. How many pounds did the lot contain? 4. — A boy is absent from work, in one week, 20% of the running time, and loses $1.20 in wages on that account. What are his wages per week? 5. — In 1904 the number of children attending school in Ludlow was 470. Since that time the number has increased about 50%. How many are attending the schools now? 6. — If a system of preparing machinery is turning off 19,350 pounds weekly, and it is decided to increase the pro- duction by 5%, how many pounds per week will be produced ? LESSON LXXXVL— Written. 1. — From a lot of jute there was made 1320 pounds of waste which was at the rate of 5.5%. How many 400 lb. bales did the lot contain? LUDLOW TEXTILE ARITHMETIC 79 2. — -On May 1, 1900, the assessed valuation of personal estate in the Town of Ludlow was $458,138. On May 1, 1907, the valuation was $1,602,769. Find the per cent, of increase during the seven years. 3. — According to the assessors' report for the Town of Ludlow in the year 1900, the valuation of the real estate subject to taxation was $1,175,021. From the same source it is found that in 1907, the valuation of real estate had in- creased to $1,869,705. Find the per cent, of increase. 4.— From the foregoing two examples find the per cent, of increase in the total valuation of both personal and real estate in Ludlow from 1900 to 1907. 5. — According to the state census of 1905 the population of Ludlow was 3,881. In 1895 the census gives the popu- lation of the town as 2,562. Find the per cent, of increase during the ten years. SUMMARY OF RULES. Percentage =baseXrate. Base = percentage -r- rate. Rate = percentage -4- base. Amount = base X (1 + rate) . Difference = base X (1-rate). Base = amount -f- (1 -1-rate). Base = difference -4- (1-rate). SQUARE ROOT. Square Root is used in mill work to find the proper degree of twist to be put in the different sizes of yarns and ropes. The square root of a number is one of the equal factors of the number. The sign \/ , called the radical sign indicates square root, thus \/81 means the square root of 81 or 9. (Square root may also be expressed by the fraction ^ written to the right and slightly above the number, 81). Examples of square roots. Numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 Square roots of the above 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 80 LUDLOW TEXTILE ARITHMETIC In these cases all of the square roots are whole numbers. It is obvious that the square root of any number between 1 and 4, or between 4 and 9, and so on, will be a mixed number whose value, in the first case will be between 1 and 2, in the second case between 2 and 3, and so forth. Thus a square root is not necessarily always a whole number. 27X27 = 729 or it may be stated 27 is the square root of 729 or, inversely V729 = 27 Now 27 = 20 + 7 then 27X27= (20 + 7) X (20 + 7) = 20X(20 + 7)+7x(20 + 7) = 20X20 + 20X7 + 20X7 + 7X7 = 20.X 20 + 2X20X7 + 7X7 Thus a number may be separated into two figures, one, the value of the tens, and the other, the value of the units and the product of the original number by itself (the square of the number) is equal to the square of the number repre- senting the tens plus twice the product of the tens and the units plus the square of the units. The extraction of the square root of a number, which is the reverse of this process, is based upon the above. It has been shown that the square root of any number from 1 to 99, inclusive, lies between 1 and 10. It can also be shown that the square root of any numbei from 100 to 9,999 lies between 10 and 100, and so on. Thus the square root of any number expressed by one or two figures will have but one figure; the square root of any number of three or four figures will have two figures, and so on. Stated in an- other way, the square root of any number will have one figure for each two of the original number and one more in case of there being an odd number of figures in the original number. For example, the square root of 3,349 will have two figures; the square root of 7,946,347 will have four figures. This rule also applies to decimals, the square root contain- ing half as many decimal places as the original figure, with one extra if there is an odd number of decimal places in the original number. For example, the square root of 4,729.4345 will have two figures in the whole number and two additional decimal places. LUDLOW TEXTILE ARITHMETIC 81 Advantage may be taken of the above when the square root of a number that is not a perfect square is being found for as many ciphers in the decimal places may be annexed as is desired in finding an approximate square root. In finding the square root of a fraction, the fraction may be reduced to a decimal or the square root of both numerator and denominator may be extracted. RULE FOR EXTRACTING THE SQUARE ROOT. Mark off the number into groups of two figures, com- mencing at the decimal point. Find the greatest square in the first group at the left and subtract it from the group. Place the square root of this square at the right as the first figure in the square root of the original number. Bring down the next group or pair and annex this to the remainder for a new dividend. Double the root obtained, annex a cipher, and use as a trial divisor. . The quotient, or a smaller number, will be the next figure in the root. Add this figure to the trial divisor for a complete divisor and multiply by the last figure. Subtract the product from the dividend; bring down the next pair of figures and annex them to the remainder. Continue in this way until all of the pairs of figures have been brought down. The result will be the square root. Example. Find the square root of 21,224,449. Marking off in pairs commencing at the right we have four. Four fig- 21 22 44 49 ( 4607 sq. rt. ures will be the root. 16 . The highest square in the 86 ) 5 22 first pair of figures at the 5 16 right is 16. Subtracting this from 21 the remain- 9207 ) 6 44 49 der is 5. The square root 6 44 49 of 16 is 4. This is the first figure in the answer. Bringing down the next group, 22, and annexing to the remainder, 5, the new dividend is 522. 82 LUDLOW TEXTILE ARITHMETIC Doubling the root 4 and annexing a cipher the trial divisor is 80. 80 will go in 522 6 times. Add 6 to the 80 making 86 for a complete divisor. Multiply by 6 and sub- tract this product from the dividend. The remainder will be 6. Bring down the next pair, 44. Double the root obtained, 46, and annex a cipher for a new trial divisor. This will give 920. As it is larger than the dividend 644, bring down the next pair, 49, and annex to the former dividend making the new dividend, 64,449. Place a cipher in the root as it was found that the trial divisor went into the dividend no times. Double the root 460, for a new trial divisor and annex a cipher. The trial divisor will be 9,200 and the dividend 64,449. The divisor will be contained about 7 times Annex the 7 to the root and add 7 to the "trial divisor. The com- plete divisor 9,207 will now he contained exactly 7 times. As there is no remainder, the square root of 21,224,449 is 4,607. This may be proved by squaring 4,607, the product equaling the original number if the work is correct. LESSSON LXXXVII.— Written. Find the square root of: 1.-123,201 8.-729 2.-8,046.09 3.-49,112,064 961 4.— 110.25 9.-225 5.-96,275,344 — 6.-43.267 4489 7.— 0.0346 10.— 532 769 Note — For practical examples see chapter on General Mill Work. GENERAL MILL WORK. TO FIND THE CIRCUMFERENCE, HAVING THE DIAMETER. Example. What is the circumference of a roller whose diameter is 4 inches? 4"X 3. 1416 = 12.5664 = 12.6" approximately Rule. Multiply the diameter by 3.1416. LUDLOW TEXTILE ARITHMETIC 83 TO FIND THE DIAMETER HAVING THE CIRCUMFERENCE Example. Find the diameter of a roller whose circum- ference is 12.5664 inches. 12.5664" -3. 1416 = 4" Rule 1. Divide the circumference by 3.1416. Another way to find the circumference is by proportion. This is an approximate method but the results are fairly accurate. The ratio of the diameter of any circle to its circumference is about 7 : 22. Therefore the proportion is 7 : 22 :: 4 : ? LESSON LXXXVIIL— Written. 1 — Find the circumference of a roller whose diameter is 2% inches. 2. — What is the diameter of a roller whose circumference is 22 inches? 3. — What is the circumference of a bicycle wheel whose diameter is 28 inches? 4. — Find the circumference of a 5 foot cylinder. 5. — Find the diameter, in inches, of a reel 23^ yards in circumference. 6. — What is the diameter (working) of a fluted roller whose outer circumference is 8 inches? Note — -The circumference of fluted rollers such as are used in the wet spinning department is found by multiplying the diameter by 3.4. TO FIND LENGTH OF BELL ON LAPPER, CARD, REEL, OR OTHER MACHINES. Example. What is the length of bell on a lapper whose callendar roller measures 22^ inches in diameter and whose bell wheel has 84 teeth? 22> finding the speed of any roller is to multiply the R. P. M. of the shaft by diameter of the driver and divide this result by the diameter of the follower, in order to trace the speed to any roller. In the rules given here, the speed of the shaft is not taken into consideration in finding the speed of either roller, consequently the same difference is kept proportionally between the two rollers. In- stead of using the circumferences, the diameters of both rollers are used, thus keeping the proportion the same. In- stead of multiplying and dividing as is usually the case in tracing the speed, the divisors and dividends are all collected together and worked out by cancellation as much as possible. As stated before draft is the ratio between the rates of feed and delivery. LUDLOW TEXTILE ARITHMETIC 87 TO FIND THE DRAFT ON A ROVING, DRAWING, OR SPINNING FRAME. Example. Find the draft on a roving frame geared as follows : Boss roller pinion 38 Draft wheel 36 Back shaft pinion 24 Stud wheel 70 Stud pinion 24 Back roller wheel 70 Back roller diameter 2" Boss roller diameter 2%" 38X24X24X2 =9 draft 36X70X70X2^ Rule. Multiply the product of all the drivers by diame- ter of receiving roller for a divisor; multiply the product of all the followers by the diameter of delivering roller for a dividend. Divide the latter by the former and the quotient will be the draft of the frame. By dividing the draft produced, 9 in this case, into 36, the number of teeth in the draft wheel, the constant number for draft of this frame is obtained, that is 9 into 36 = 4. To produce any draft on this machine multiply draft required by 4. The result will be the pinion required. TO FIND THE DRAFT ON A BREAKER CARD. Multiply the number of teeth in the pinion on the end of the delivery roller by the pinions driving the feed roller and this product multiplied by the diameter of the feed roller for a divisor. For a dividend, multiply the driven wheel between the delivery and the feed by the diameter of the delivering roller. Divide the former into the latter and the result will be the draft of the breaker. Example. Find draft of gearing in actual operation in some English manufactured carding machines. 64X 17X 24X10 =12.8 draft 58X120X120X 4 88 LUDLOW TEXTILE ARITHMETIC Example. Find wheel for 10 of a draft. 10X4 = 40 wheel required For a draft of 8. 8X4 = 32 wheel. Note: — The change pinion for draft on a roving frame is usually on the back shaft. The draft on a second drawing frame is found in the same way as for a roving frame. Example. Drawing Frame. Boss roller pinion, 53 Back shaft wheel, 34 Back shaft pinion, 25 Back stud pinion, 25 Back stud wheel, 68 Back roller wheel, 69 Back roller diameter, 2" Boss roller diameter, 2%" 53X25X25X2 - = 6 draft. 34X68X69X2^ Note: — The change pinion for the draft on our drawing frames is on the end of the drawing rollers. The draft of a push bar drawing is a little more difficult to work out, but it is calculated in exactly the same manner as the draft on any other machine. The difference between the number of inches taken in by the receiving roller and the number of inches delivered by the roller which delivers the fibre constitutes the draft. Example of the draft gearing on latest style of push bar drawings. Find draft. 64X18X24X40X2 = 5 60X80X22X42X2^ On another style of push bar frame the gearing is : 48X20X27X42X39X2 = 5.1 52X84X42X22X43X23^ On an English push bar frame the draft gearing now working is 48X18X28X2 = 5.1 42X54X43X2^ LUDLOW TEXTILE ARITHMETIC 89 TO FIND DRAFT OF A ROLLER CARD. The draft of a roller finisher card is found by multiplying the diameter of the feed roller by the product of all the driving wheels between the feed and the delivery for a divi- nend; for a divisor find the product of all the followers between the feed and the delivery and multiply by the diam- eter of the delivery roller for a divisor. By dividing the latter into the former the theoretical draft of the card is obtained. The actual draft of the card may »ot be quite as long as the figures show because the diameter of the feed roller is taken at the bottom of the pins while the fibres do not all go to the bottom of them. Example 1. Find draft of finisher card with draft gearing as follows: Feed roller wheel, 96 Draft pinion, Stud wheel, 24 96 Stud pinion, Second stud wheel, 32 104 Delivery roller pinion, Diameter feed roller, 75 4" Diameter delivery roller, 4" 96X96X104X4 = 16.6 draft 24X32X75X4 This means that the sliver delivered is 16.6 times as long as the sliver that is received. Example 1. Find by proportion the draft pinion re- quired to have this machine on a draft of 20. Example 2. And the draft pinion for a draft of 12. On a Fairbairn Knife Card the particulars of the draft will run as follows: Feed roller wheel, 120 Draft pinion, 44 Inside stud wheel, 72 Delivery roller pinion, 16 Diameter feed roller, 3" Diameter delivering roller, 4" Then 120X72X4 =16.36 draft. 44X16X3 90 LUDLOW TEXTILE ARITHMETIC Example. Find the draft with a 36 draft pinion on the card. 2. . What draft pinion would be required to produce a draft of 18 on this card? SPINNING. TO FIND THE DRAFT ON A SPINNING FRAME. Rule. Multiply the number of teeth in stud wheel by number of teeth in top roller wheel and multiply this product by diameter of drawing roller for a dividend. Mul- tiply the diameter of top roller by the number of teeth in draft pinion and then multiply this product by the number of teeth on inside stud pinion for a divisor. The number of times this divisor is contained in the dividend is the draft of the frame. Example 1. Find the draft on a Z}4," spinning frame geared as follows: Draft pinion, 36 teeth Stud wheel, 80 " Inside stud pinion, 30 " Top roller wheel, 75 " Diameter top roller, 23^" Diameter drawing roller, 4" 80X75X4 = 8.8 draft 36X30X2^ Example 2. With the draft gearing on a 4" as follows, find the draft. Draft pinion, ' 37 Stud wheel, 80 Inside stud pinion, 40 Top roller wheel, • - 75 Diameter top roller, 2^" Diameter drawing roller, 4" 80X75X4 — =6.4 draft 37X40X2^ LUDLOW TEXTILE ARITHMETIC 91 Example 3. Find the draft of a 4^" spinning frame with the following gears: Draft pinion, 38 teeth Stud wheel, 60 " Inside stud pinion, 40 " Top roller wheel, 80 " Diameter top roller, 2%" Diameter drawing roller, 4" 60X80X4 = 5.6 draft 38X40X23^ Example 4. On a 5" spinning frame the draft gearing .ild run at follows: Draft pinion, 56 teeth Stud wheel, 50 " Inside pinion, 45 " Top roller wheel, 80 " Diameter top roller, 2H» Diameter drawing roller, 5" Find the draft. 50X80X5 = = 3.17 draft 56X45X2^ TO FIND THE CONSTANT NUMBER FOR DRAFT IN A SPINNING FRAME. Rule. Multiply number of teeth in stud wheel by number of teeth in top roller wheel and by diameter of drawing roller for a dividend; then multiply number of teeth in stud pinion (inside) by diameter of receiving roller for a divisor, or, use the figures for finding the draft with the draft pinion omitted. Example 1. To find the constant number of the 33^" spinning frame geared as shown in the example for finding the draft. 80X75X4 = 320 = constant number 30X2^ Example 2. Find the constant number for draft on a 4" spinning frame. 80X75X4 = 240 constant number 40X2.5 92 LUDLOW TEXTILE ARITHMETIC Example 3. Find the constant number for draft on a 4/^" spinning frame. 60X80X4 = 216 constant number 40X2.5 Example 4. On a 5" spinning frame find the constant number for draft. 50X80X5 =177 constant number 45X2.5 TO FIND THE PINION FOR ANY DRAFT HAVING THE CONSTANT NUMBER. Rule. Divide the draft required into the constant number ; the result will be the number of teeth on the pinion to be put on the spinning frame to produce the required draft. Example. What pinion would be required to produce a draft of 8 on a 3^2" frame, the constant number being 320? 320 -f- 8 = 40 teeth on draft pinion TO FIND DRAFT A CERTAIN PINION IS PRODUC- ING BY HAVING CONSTANT NUMBER. Rule. Divide draft pinion into constant number. Example. What draft is on a certain frame whose constant number is 320 and which is running with a 40 draft pinion? 320-^-40 = 8 draft For additional examples in drafting, vary draft pinions on different machines. TWISTING. The twist in yarn is calculated by the turns per inch. (The turns per inch means the number of revolutions the Flyer makes while the rollers are delivering one inch of yarn.) To find the number of turns per inch which any machine is putting into the yarn, divide the R. P. M. of the spindles by the number of inches that the rollers deliver per minute; the result will be the turns per inch. LUDLOW TEXTILE ARITHMETIC 93 For example, the twist going in the yarn on a spinning frame, with the rollers delivering 564 inches per minute and the spindles running at the rate of 1920 R. P. M., would be 1920^-564 = 3.4 turns per inch. The twist of a frame can be found without having the R. P. M. of any part of the machine by getting all the divisors and dividends together and working out the problem as far as possible by cancellation. Rule. Multiply the number of teeth in roller whee by the number of teeth in stud wheel and this product by the diameter of cylinder, for a dividend; multiply the cir- cumference of drawing roller by diameter of wharves by twist pinion and by cylinder pinion for a divisor. Example. Find the twist of a spinning frame geared as follows: Roller wheel, 96 Stud wheel, 136 Diameter of cylinder, 10 Cylinder pinion, 24 Twist pinion, 56 Circumference of roller, 14. 1" Diameter wharves, 2" 96X136X10 24X56X14.1X2 5.2 turns per inch Example. Find the turns per inch of yarn spun on a 4" frame with the twist geared as follows: Roller wheel, 120 Stud wheel, 90 Diameter cylinder, Cylinder pinion, Twist pinion, Roller circumference, 10 28 35 12.56 Diameter wharve, 1.75 120X90X10 — = 5 turns per inch 28X35X12.56X1.75 94 LUDLOW TEXTILE ARITHMETIC Example 4. What would be the twist of a spinning frame geared like this? Roller wheel, 150 teeth Stud wheel, 112 " Diameter of cylinder, 10" Cylinder pinion, 44 teeth Twist pinion, 39 " Circumference of roller, 15. 7" Diameter of wharve, 2 . 25" 150X112X10 — =tt V / +nr-n c nar 1-n/^n 44X39X15.7X2.25 A constant number for the twist of a spinning frame is . found by leaving the twist pinion out of the foregoing calcu- lations. For example, the constant number for a 3^ frame with the same gearing and diameter of roller as is given in example 4, would be 120X90X10 197 = = turns per inch T.P.X29xl.5xl2.56 T.P. Therefore the constant number for this machine would be 197. Example 5. Find the constant number for twist of a 4" frame. 120X90X10 = 175 constant number 28X1.75X12.56 Example 6. Find the constant number for a 5" frame. 150X112X10 = 105 constant number 44X2.25X15.7 The average degree of twist required for flax, hemp, and jute yarns is found by extracting the square root of the leas per pound and multiplying this product by 2 for the number of turns per inch. Thus the number of turns per inch required for 4's lea yarn equal 2X\/4 = 2X2 = 4 turns per inch. And again the number of turns per inch required for 25's lea would be 2X\/25 = 2X5= 10 turns per inch. For yarns numbered on the pounds per spyndle basis, it is usually ' taken that 3 lb. yarn requires 8 turns per inch. The number of turns per inch required for any other size LUDLOW TEXTILE ARITHMETIC 95 of yarn may be obtained by multiplying the turns per inch for 3 lb. yarn by the \/3 and dividing by the \/ lbs. per spyndle of the yarn to be twisted. Thus, the twist required for yarn weighing 8 lbs. per spyndle at the rate of 8 turns per inch for 3 lb. yarn would be SXv/3 64X3 = — = a/24 = 4.9 turns per inch V8 8 The reason that the square root of the size of the yarn is introduced into the twist calculations is that the twist should vary inversely as the diameter of the thread and that the diameter of the thread varies inversely as the square root of the leas per pound or directly as the pounds per spyndle. From this reasoning we get a constant number for finding the turns per inch by multiplying 8, the turns per inch, for 3 lb. yarn, by 1.73, the square root of 3, which equals 13.8. Thus for example to find the turns per inch required for 8 lb. yarn, divide the square root of 8 into the constant number 13.8 which equals 13.8-^2.8 = 4.9 turns per inch. \/ = radical sign EXAMPLES IN TWISTING. LESSON XCIL— Written. 1. — What would be the average twist or turns per inch for yarn weighing 9 pounds per spyndle? 2. — Find the turns per inch for 6 lea yarn. 3. — What is the twist for 14 pound yarn? 4. — Make a sheet showing the twist required for each lea of yarn from l's to 10 's inclusive. 5. — Make a list showing the turns per inch required for each one pound difference in weight per spyndle from 1 to 6 lbs. Note. These examples are worked out on the average degree of twist that is required for ordinary yarns. In yarn wanted for special purposes, the twist will vary from this it may be 1.75 times the square root of the lea or it may be 2.25 to 2.5 times the square root of the leas per pound. 96 LUDLOW TEXTILE ARITHMETIC STRANDS. TO FIND PROPER DEGREE OF TWIST PER STRAND OF ROPE. The number of turns per foot twist required in a strand is found by dividing the number of the yarn used by the number of threads per strand, extracting the square root of this result and multiplying by 3.75. Thus the number of turns per foot twist required for a strand containing 40 threads of number 20's rope yarn would be V20/40 X 3 . 75 = 2 . 7 turns per inch Note : — This twist must be put in the rope in the opposite direction to that in which the yarn was spun. TO FIND NUMBER OF THREADS PER STRAND OF ANY SIZE ROPE. The number of threads per strand for a 3 strand of any diameter may be found by dividing the product of the square of the rope's diameter and the number of the yarn by 0.81 or 0.9 2 , 0.9 being the diameter of a rope 3 inches in circum- ference. The number of threads per strand for a 4 strand rope of any diameter is found by squaring the diameter of the rope and multiplying by the number of the yarn which is used and dividing the product thus obtained by 1.2. Thus the number of threads of 20's rope yarn required to form a strand for a three strand rope 2 inches in diameter would be 2 2 X 20 ^ 0.81 = 99. For a four strand rope 2 inches in diameter it would be 2 2 x20 = 67 1.2 CUTS PER SPINDLE. The yarn production is calculated by the average number of 300 yard cuts which each spinning spindle produces in a stated time. Rule: — Where the yarn is reeled, the number of reels multiplied by the number of cuts contained in each reel will give the total number of cuts, and this product divided by LUDLOW TEXTILE ARITHMETIC 97 the number of spinning spindles running while this yarn was being produced, will give the average cuts per spindle for that particular space of time. Example. What are the average cuts per spindle for 10 hours where the number of reels turned off is 600, each reel containing 100 cuts and the number of spindles in oper- ation, during that time was 3300. 600 X 100 = 18. 1 cuts per spindle per 10 hours 3300 Where the yarn is numbered on the pounds per spyndle basis the rule to find the cuts per spyndle is: Rule : — Divide the production by the pounds per spyndle, for the number of spyndles; multiply this quotient by 48 (the number of cuts in a spyndle) for the total number of cuts ; divide this product by the number of spinning spindles in operation during the time this yarn was being produced, and the result will be the average cuts per spindle. Example. The average cuts per spindle from 200 spin- ning spindles which produce 1875 pounds of 15 lb. yarn in a day of 10 hours would be 1875/15=125 125X48 = 6000 6000^200 = 30 30 cuts per spindle per 10 hours, or 3 cuts per spindle per hour which means that each spinning spindle in operation on that particular size yarn is spinning, on an average, 900 yards of yarn each hour. TO FIND THE CUTS PER SPINDLE PER HOUR. Multiply the number of spindles running by the number of hours that they have been in operation, the result will be the spindle hours and this product divided into the total number of cuts will give the cuts per spindle per hour. 98 LUDLOW TEXTILE ARITHMETIC Example. Find the cuts per spindle per hour fc following work : Kind Pounds Spvndles Cuts 15 lbs. 13,500 900 43,200 Spindles Hours Spindle Hours Monday, 300 10 3,000 Tuesday, 240 10 2,400 Wednesday, 300 9 2,700 Thursday, 210 10 2,100 Friday, 300 10 3,000 Saturday, 240 5 1,200 Total spindle hours ? 14,400 Total number of cuts = 43,200 Q O11 + T-.0-T o-r-*i-nrl1o n&r Spindle hours = 14,400 TO FIND THE NUMBER OF YARDS IN ANY WEIGHT OF YARN, THE POUNDS PER SPYNDLE BEING GIVEN. Rule. Divide pounds per spyndle into weight, multiply this product by 48 for cuts, and the number of cuts multi- plied by 300 gives the number of yards. Example. How many yards are contained in 12,000 lbs. of jute yarn which weighs 15 lbs. per spyndle. 12000/15 = 800 800X48 = 38,400 yards LEVERS. TO FIND THE PRESSURE ON DRAWING ROLLER. Rule. Multiply length of lever in inches by the weight suspended in pounds, and this product divided by the length of fulcrum will give the pounds pressure on the roller. Example. To find weight on a pressing roller with Length of lever, 12" Weight suspended,. 14 lbs. Fulcrum, IX" 14X12 -134U lbs. \v LUDLOW TEXTILE ARITHMETIC 99 On drawings and rovings where compound levers are used the pressure is found by multiplying the length ofthe two levers by the weight and dividing this by the product of the two fulcrums. Example. Second drawing fra: me. Top lever, 9" Bottom lever, 9" Top fulcrum, 2" Bottom fulcrum, 3y 2 " Weight, 15 lbs. 9X9X15 = 173 4/7 lbs. pressure 2X3.5 ROVING FRAME. COMPOUND LEVER. Top lever, 4" Bottom lever, 5" Top fulcrum, 1" Bottom fulcrum, 2" Weight, 12 lbs, 4X5X12 = 120 lbs. 2X1 Note: — The word fulcrum in the above examples is a mill term meaning the short arm of the lever. TO FIND POUNDS PER SPYNDLE OF ROVE FROM WEIGHT OF 100 YARDS. Example. What are the pounds per spyndle of rove which weighs 10 ounces per 100 yards? 10X9 = 90 lbs. per spyndle Explanation. The weight of 100 yards in ounces mul- tiplied by 9 is equal to the weight of a spyndle (14,400 yds.) divided by 16, the number of ounces contained in one pound. Thus 100 yards = 10 ounces in weight and 1 spyndle (14,400 yds.) = 1440 oz. in weight and 1440^16 = 90 lbs. Rule. Multiply weight of 100 yards of rove in ounces by 9; the result will be the pounds per spyndle. 100 LUDLOW TEXTILE ARITHMETIC LESSON XCIIL— Written. Find the pounds per spyndle of 100 yards of rove weighing 1. — 5 ounces. 2. — 12 ounces. 3. — 7 ounces. 4. — 16 ounces. 5. — 10 ounces. TO FIND DRAFT TO BE PUT ON SPINNING FRAME HAVING POUNDS PER SPYNDLE OF ROVE GIVEN. Example. Find draft required to make yarn weighing 14 lbs. per spyndle from rove weighing 81 lbs. per spyndle. 81^14 = 5.8 draft Rule. Divide the weight of spyndle of yarn required into weight per spyndle of rove. LESSON XCIV.— Written. Find draft required to spin 1.— 6 lbs. yarn from 63 lbs. rove. 2.— 5 3.— 15 4.— 14 5.— 30 6.— 18 7.-24 63 81 81 81 90 108 TO FIND SPINNING DRAFT FOR ANY SIZE YARN FROM WEIGHT OF ROVE. Example. What draft would be required to spin yarn weighing 10 pounds per spyndle, from rove weighing 9 ounces per 100 yards? 50 yards rove = 43^ ounces 4373^2 grains = 1 ounce therefore 437 . 5 X 4 . 5 = 1968 . 75 grains and 50 yards, 10 lb. yarn = 243 grains 1968. 75 -=-243 = 8.1 draft on spinning frame Rule. Divide weight of 50 yards of yarn required into the weight of 50 yards of rove. The quotient will be the draft to be put on the spinning frame. LUDLOW TEXTILE ARITHMETIC 101 LESSON XCV.— Written. Find draft required to make yarn from rove weighing 10 ounces per 100 yards. 1. — 50 yards yarn to equal 243 grains. 2.— 50 " " " " 316 " 3.— 50 " " " " 340 " 4.— 50 " " " " 365 " TO FIND DRAFT TO SPIN ANY LEAS PER POUND HAVING WEIGHT OF ROVE GIVEN. Example. What draft is necessary to spin 3 lea yarn from rove weighing 9 ounces per 100 yards? 9X3-5-5.3 = 5.1 spinning draft 100 yards of rove equals 9 ozs. 1 lea or 300 yds. =27 ozs. 3 leas or 900 yds. of rove = 81 ozs. and the yarn required is 3 lea or 900 yds. to the pound which equals 10 ozs. Therefore 81^16 = 5.1 Placing this in the form of a fraction 300X9X3 = 5.1 100X16 Rule. Multiply weight of 100 yards of rove in ounces by leas per pound of yarn required. This product divided by 5 . 3 will give draft to be put on spinning frame. Note: — While the weight of rove and leas per pound are variable the number of yards in a lea (300), the ounces in a pound (16), and the number of yards of rove weighed (100), are unchanged. Therefore from these figures the following is obtained: 100 X 16 h- 300 = 5 . 3 a constant number LESSON XCVL— Written. Find draft for spinning frame to spin 1. — 3 lea from rove weighing 9 ozs. per 100 yards. 2.-4 " " 9 " " 100 " 3.-6 " " 6 " " 100 " 4.-8 " " 5i " " 100 " 5.-2 " " 10 " " 100 " 6.— 1 " " 12 " " 100 " 102 LUDLOW TEXTILE ARITHMETIC To obtain 10 X 100 X 5X 25 X 125 X 33KX 12^ X 5% of 25% of 50% of 20% of 33^% of 66^% of 12K% of 16K% of 9X HX To divide by 10 100 5 25 125 33^ SHORT PROCESSES. Move decimal point one place to right. Move decimal point two places to right. Multiply by 10 and divide by 2. Multiply by 100 and divide by 4. Multiply by 1000 and divide by 8. Multiply by 100 and divide by 3. Multiply by 100 and divide by 8. Divide by 10 and divide quotient by 2. Divide by 4. Divide by 2. Divide by 5. Multiply by 1/3. ' Multiply by 2/3. Multiply by 1/8. Multiply by 1/6. Multiply by 10, subtract multiplicand. Multiply by 1Q and add multiplicand. Move decimal point one place to left. Move decimal point two places to left. Multiply by 2 and divide by 10. Multiply by 4 and divide by 100. Multiply by 8 and divide by 1000. Multiply by 3 and divide by 100. Answers ADDITION. Lesson 2. Page 3-4. 1.— 1665 7.-1,539,829 2.-792 8.-800,686 3.— 1208 9.-1,832 fbs. 4.— 4018 10.— 1100 lbs. 5.-102,332 11.— 21,852 lbs. 6.-1,396,684 12.-44,696 lbs. JUTE YARN PRODUCTION SHEETS. Lesson 3. Page 4-5. 1.-20,053 lbs. 4.-72,246 lbs. 2.-19,155 lbs. 5.-42,715 lbs. 3.-26,060 lbs. 6.-32,990 lbs. PREPARING ROOM SHEETS. Lesson 4. Page 6-7. 1.-29,510 lbs. 233 doffs 2.-9,300 lbs. 65 doffs 3.-30,500 lbs. 214 doffs WEEKLY PRODUCTION SHEETS. Lesson 5. Page 7-8. 1.-627,395 lbs. 2.-621,915 lbs. 3.-619,870 lbs. SUBTRACTION. Lesson 6. Page 8-9. 1.— 571 8.-457 lbs. lost 2.— 102 9.— 192 pupils increase 3.— 7912 10 —665 men 4.-657 11.— 974 people 5.— 4177 12.-29,714 lbs. 6.-37,987 13.— 143 lbs. 7.-38,774 14.-27,941 people gain 104 LUDLOW TEXTILE ARITHMETIC MULTIPLICATION. Lesson 7. Page 10-11. 1.- —2403 7.— $5.50 2.- -57,817 8. — 96 sides 3.- —452,100 9.— 918 help 4.- -248,777,588 10.-14,400 yds. 5.- -19,250 lbs. 11.-20,300 lbs. 6.- -95,200 yds. WAGE 12.-1,500,000 lbs, LIST. Li ESSON 8. Page 11. 1.- -$2 . 75 9.— $7.15 2.- -$3.30 10.— $7.70 3.- -$3 . 85 11.— $8.25 4.- -$4.40 12.— $8.80 5.- -$4.95 13.— $9.35 6.- -$5.50 14.— $9.90 7.- -$6.05 15.— $10.45 8.- -$6.60 PRICE 16.— $11.00 LIST. Lesson 9. Page 11. 1.- -$0.20 11.— $2.20 2- -$0.40 12.— $2.40 3.- -$0.60 13.— $2.60 4.- -$0.80 14.— $2 . 80 5.- -$1.00 15.— $3.00 6- -SI . 20 16.— $3.20 7.- -$1.40 17.— $3 . 40 8.- -$1.60 18.— $3.60 9.- -$1 . 80 19.— $3.80 10.- -$2 . 00 20.— $4.00 DIVISION. Lesson 10. Page 11-12. 1.- -156 6.-567 83/100 2.- -2,552 5/7 7.— 120 3.- -9357 8.-2768 8/11 4.- -98,311 3/8 9.-21,120 8/13 5.- -73,894 2/5 10.-59,768 12/15 LUDLOW TEXTILE ARITHMETIC 105 LONG DIVISION. Lesson 11. Page 13. 1.— 4158 2.-274 46/345 3.— 212 13/62 4.-24,389 5.-364 140/237 Lesson 12. 1.— 300 yards 2^—48 cuts 3.-10,800 ins. 300 yds. 4.— 150 bales 6.-22 303/685 7.— 10 181/874 8.-27 139/925 9.-53 1393/7005 10.— 8 1432/2469 Page 13-14. 5. — 5 turns per inch 6.-253 1/3 spyndles 7. — 25 cuts 8.— $894 2860/3881 MISCELLANEOUS EXAMPLES. Lesson 13. Page 1.-17,280 pins 2.-21,600 pins 3.-42,224 pins 4.-81,104 pins 5. — 339 1/11 amperes 6.— 740 yds. 7.— 600 yds. 8.— 380 yds. 9.-284 yds. 10.— 200 yds. 11.— 150 yds. 12.— 430 yds. 13.— 380 yds. 14.— 208 yds. 15.— 1400 lbs. 16.-40,000 lbs. 17. — 141 revs. 18.-57,600 revs. 19.— $9.15 20.— 60 hours 21.— 20 hours ANALYSIS, 14-15-16-17. 22.— 340 grains 23.-5,346,593 bales 24.-48 leas 25.— 320 lbs 26.— 1000 spyndles 27. — 437 1/2 grains 28.— 2800 pounds 29.-24 feet 30.— 375 bobbins 31.-23,809 11/21 ozs. 32.-15^ 33.— $14.40 34.-800,000 pins 35.-12,000 lbs. 36.-37,800 pins 37.-56 1/4 lbs. 38.— 178 H. P. 39.-2,520,000 yds. 40.— 3055 5/9 spyndles 41.— 165 gals. Lesson 14. 1.— 110 doffs 2.-33,000 lbs. 3.— $2.25 4.— 250 girls 5. — 200 ounces 6.— 3712 fallers Page 18. 7.-225 bales 8.-1,923,077 lbs. 9.— $1,000,000 10.— $125,000 11.— 132 ft. 12. — 6 cents 106 LUDLOW TEXTILE ARITHMETIC Lesson 15. Page 19. 1.- 2.- 3.- 4.- 5.- 6.- —12 1/4 cents -$8.60 —10 men -6800 lbs. — 10 winders — 5 machines 7.— 20' sides 8.— 50 bales 9.— $360 10. — 36 boys 11. — 56 boys 12.— $36 PRIME FACTORS. Lesson 17. Page 20. 1.— 2-2-3-5 11.— 2-2-2-2-2-3-5 2.-2-2-3-3 12.— 2-2-2-2-2-2-2-5 3.— 3-13 13.— 2-2-2-2-3-3-5 4.-2-3-7 14.— 2-2-5-5-7 • 5.— 17 15.— 2-2-2-2-2-5-5 6.-3-5-5 16.— 2-3-181 7.-2-3-5-7 17.— 2-5-5-31 8.-2-2-2-2-5-5 18.— 2-3-3-5-5-7 9.-2-3-3-3-3-5 19.— 2-2-3-5-5-13 10.— 2-2-3-3-3-7 20.— 3-3-3-5-11 GREATEST COMMON FACTOR. Lesson 18. Page 21. 1.— 12 6.-7 2.— 18 7.-25 3.— 11 8.-7 4.— 12 9.— 13 ' 5.— 16 10.— 60 LEAST COMMON MULTIPLE. Lesson 19. Page 22. 1.— 6 6.— 20 2.— 30 7.— 30 3.-56 8.-24 4.-36 9.— 20 5.— 12 10.— 36. Lesson 21. Page 23. 1.— 180 .6.-96 2.— 198 7.-36 3.— 140 8.-45 4.— 90 9.— 6000 5.— 144 10.— 80 LUDLOW TEXTILE ARITHMETIC 107 -2 -22 -3 CANCELLATION. Lesson 22. Page 24. 4.— 21 5.— 1 UNITED STATES MONEY. Lesson 23. Page 24-25. 1.- 2.- 3.- 4.- 5.- -$10,000.40 -$784,164 -$5674.192 -$3,000,000,003 -$7,000,033 6.— $807.10 7.— $5023 . 768 8.— $1,002 9.— $27 . 305 Lesson 24. Page 25. 1.- 2.- 3.- -2246^ -4700^f -67 ; 590 mills 4.-32,476 mills 5.-66,432 mills Lesson 25. Page 25-26. 1.- 2.- 3.- -$49.87 -$7,658 -$460.36 4.— $9000.145 5.— $10,049,015 Lesson 26. Page 26. 1.- 2.- 3.- 4.- -$4.75 -$5 . 00 -$5 . 25 -$5.75 5.— $6.00 6.— $6.25 7.— $7.00 DECIMALS. Lesson 28. Page 28-29. 1.- 2.- 3.- 4.- 5.- 6.- 7, 8.- -0.1 -0.27 -0.437 -608.2701 -941.32721 -3.41159 -19.083 -0.001 9.— 0.01763 10.— 0.03003 11.— 60.006 12.— 170.704 13.— 5.0856 14.— 89.009 15.— . 0378 16.— 5.9764 108 LUDLOW TEXTILE ARITHMETIC ADDITION OF DECIMALS. Lesson 29. Page 29-30. 1.— 169.03 8.— 149.14438 2.— 280.466 9.— 174.379779 3.-224.9586 10.— 70.3932725 4.— 189.28745 11.— 184.5376349 5.— 512.362229 12.— 119.349505 6.— 59.656075 13.— $1188.52 7.— 90.1066379 14.— $113.39 SUBTRACTION OF DECIMALS. Lesson 30. Page 30. 1.— 3.379 8.-24.9975 2.-6.2236 9.-99.99 3.-4.784 10.— 33.393 4.— 0.00594 11.— 1.6483 5.— 4.79075 12.— 16.85835 6.— 5.9501 13.— 0.002711 7.-29.87633 14.— 5.9926 MULTIPLICATION OF DECIMALS. Lesson 31. Page 32. 1.— 41.82 9.-6.29629 2.-85 10.— 0.20397 3.— 121.5 11.— 1500 balls 4.— 12.15 12.— 13. 3518 inches 5.— 1.215 13.— $694,500 6.— 182.8948 14.— $9.02 7.-28.2744 15.— $2.15 8.-10,991 . 5 16.— 838-8 inches DIVISION OF DECIMALS. Lesson 32. Page 34. 1.— 8.257 15.— 61.386 2.— 13.367 16.— 0.25 3.-4.448 17.— 8 4.-4.857 18.— 6.0024 5.— 22.105 19.— 0.1016 6.— 0.378 20.— 0.101419 7.— 1.443 21.-8.6^ 8.— 0.00962 22.-24.3^ 9.-2 . 603 23.-28 . 644 inches 10.— 0.0015 24.— 17. 2 ins. 22.918 ins. 11.— 2.004 25.— 3. 75 ins. LUDLOW TEXTILE ARITHMETIC 109 12.— 0.006 26,— 11. 46 ins. 13.— 298.853 27.— 39.37 min. 14.— 0.02608 28.— 120 revs. EDUCTION OF COMMON FRACTIONS TO DECIMALS. Lesson 33 . Page 35. 1.— 0.5 6.— 0.3 2.— 0.25 7.— 0.75 3.— 0.2 8,-0.5 4.^-0.1 9.— 0.125 5.— 0.4 10.— 0.1666 REDUCTION OF COMMON FRACTIONS TO DECIMALS. Lesson 34. Page 35-36, 1.— 0.375 11.— 1.1875 2.— 0.4375 12.— 5.75 3.— 0.5625 13.— 11.4062 4.— 0.625 14.— 40.4531 5.— 0.6875 15.— 18.4687 6.— 0.15625 16.— 9.9844 7.— 0.875 17.— 8.44 8.— 0.28125 18.— 2.3958 9.— 0.171875 19.— 0.8298 10.— 0.390625 20.— 0.9734 MISCELLANEOUS EXAMPLES IN DECIMALS. Lesson 35. Page 36-37-38. 1.— 700 pounds 16.— $7250 2.— . 625 17.— 1 1 cents 5 mills 3.-2638 . 944 feet 18,— 400 reels 4.— . 1964 19.— 316 bobbins 5.— $1 . 38 20.— 840 cuts 6.— 502 revs. 21.— 38620 . 8 spyndles 7.-48 lags 22.— 14510 . 4 spyndles 8.— $2 . 36 23.— 10012 . 8 spyndles 9.— $2 . 24 24.-5 . 5 pounds 10.— $1 . 36 25.-335 . 5 H. P. 11.— 78.25 lbs. 26.-39 H. P. -12.— $0.11 27.— 0.87 ounces - 13.— 243 grains 28.-2 . 75 cuts 14.— 675 lbs. 29.— 200 R. P. M. 15.— $4 . 68 30.— 5616 revs. 110 LUDLOW TEXTILE ARITHMETIC PROPORTION. Lesson 36. Page 39. 1.— 10 to 5 2.-5 to 7 3.-3 to 5 4.-3 to 4 5.— 100 Lesson 37. Page 40-41. 1.— 15 1/2 lbs. 2. — 131 grains 3.-32 draft wheel 4.— 31 draft wheel 5. — 43 draft wheel 6.— 2011 R. P. M. 7.— 140 4/10 bales Lesson 38. Page 41. 1.— 5012 4/5 lbs. 2. — 6 inch diam. 3.-24.3 gr. 4'.— 340.2 gr. 5.— 121.5 gr. 6.-243 gr. 7.— 364.5 gr. 8.-486 gr. 9.— 607.5 gr 10.— 729 gr. Example 1. 1.— 24.3'gr. 2.-49 gr. 3.-73 gr. 4.-96 gr. 5.— 121 gr. 6.— 146 gr. 7.— 170 gr. 8.— 194 gr. 9.— 219 gr. 10.— 243 gr. 11.— 267 gr. 12.— 292 gr. 13.— 316 gr. 14.— 340 gr. 15.— 364 gr. Lesson 39. Page 41-42. 16.— 389 gr, 17.— 413 gr. 18.— 437 gr. 19.— 462 gr. 20.— 486 gr, 21.— 510 gr. 22.-534 gr. 23.— 560 gr. 24.-583 gr. 25.— 607 gr 26.-632 gr, 27.-656 gr. 28.— 680 gr. 29.— 705 gr. 30.— 729 gr. LUDLOW TEXTILE ARITHMETIC 111 Exampl 3 2. 1.— 3.1416 7.— 21.9912 2.-6.2832 8.— 25.1328 3.-9.4248 9.-28.2744 4.— 12.5664 10.— 31.416 5.— 15.708 11.— 34.5576 6.— 18.8496 12.— 37.6992 Lesson 40. Page 42. Example 1. 1 lea weighs 1166 grains 2 ' 583 ' - 3 ' 389 ' 4 ' 292 ' 5 ' 233 ' 6 ' 194 ' 7 ' 174 ' 8 ' 146 ' 9 ' 130 ' 10 " 117 ' 2.-288 yds. 6.— 2814 R. P. M 3.-47 wheel 7.-8.4 draft 4.-20,000 bales 8. — 18.9 inches 5.— $5641 FRACTIONS. Lesson 41. Page 44. 1.— 7/3 6.-44/5 2.-7/2 7.-44/7 3.— 19/5 8.— 101/10 4.— 503/100 • 9.— 100/8 5.-47/8 10.— 100/9 Lesson 42. Page 44. 1.— 55/8 • 6.— 331/14 2.-32/9 7.-989/22 3.-38/3 8.— 200/3 4.— 192/11 9.— 175/2 5.— 119/6 10.— 6279/64 REDUCTION OF FRACTIONS. Lesson 43. Page 45. 1.— 1 1/4 7.-5 1/10 2.-2 1/6 8.-6 1/7 3.-5 2/3 9.-7 2/5 112 LUDLOW TEXTILE ARITHMETIC 4.-2 1/11 10.— 4 1/3 5.-3 9/13 11.— 3 1/5 6.-3 1/12 12.— 5 5/8 Lesson 44. Page 45. 1.— 3 1/12 10.— 11 46/47 2.-2 17/20 11.— 9 3.-5 12.— 1 4.-3 13.— 43 10/47 5.-2 1/19 14.— 10 69/100 6.-5 11/23 15.— 17 38/43 7.— 21 11/12 16.— 1 35/94 8.-7 48/53 17.— 3 9.-8 10/29 18.— 4 REDUCTION OF FRACTIONS TO GIVEN DENOMINATOR. —99/11 —60/12 —144/12 —60/4 —100/5 —48/3 —34/2 Lesson 45. Page 45. 9.-72/3 10.— 126/6 11.— 100/4 12.— 130/5 13.— 320/8 14.— 84/2 15.— 300/6 REDUCTION OF FRACTIONS TO LOWEST TERMS. Lesson 46. Page 46. 1.— 1/4 2.-8/9 3.-6/7 4.— 1/3 5.— 1/5 6.— 1/5 1.— 3/5 2.— 21/25 3.— 1/4 4.— 211/243 5.-37/77 7.- -2/5 8.- 9- 10.- 11.- 12.- -3/4 -15/16 -3/8 -3/4 -9/20 'agi s 46. 6.- 7.- 8.- -1/3 -6/11 -1/12 9.- 10.- -429/1190 -33/40 LUDLOW TEXTILE ARITHMETIC 113 REDUCTION OF FRACTIONS TO HIGHER TERMS. Lesson 48. Page 47. 1.— 4/8 7.— 90/100 2.— 8/12 8.-9/48 3.— 18/30 9.-72/99 4.-8/28 10.— 16/36 5.— 9/60 11.— 15/51 6.— 21/45 12.— 18/39 m Lesson 49. Page 47. 1.— 84/144 7.— 440/460 2.— 78/143 8.— 570/600 3.— 105/112 9.— 78/198 4.— 112/400 10.— 85/210 5.— 81/234 11.— 77/336 6.— 40/135 12.— 240/260 MULTIPLY A FRACTION BY A WHOLE Lesson 50. Page 48. 1.— 6/7 7.-6 2.-2 1/4 8.-2 1/2 3.-4 9.— 1 3/4 4.-2 1/3 10.— 12/17 5.-5 1/2 11.— 24 6.-4/7 12.— 10 Lesson 51. Page 48. 1.— 51 7.-48 2/5 2.-222 3/4 8.-364 3.— 405 9.— 145 2/3 4.-224 10.— 66 5.— 71 2/5 11.— 68 6.— 140 12.— 372 MULTIPLY A FRACTION BY A FRACTION. Lesson 52. Page 49. 1.— 1/3 2.-9/35 3.-6/55 4.— 5/14 5.— 12/77 6.— 1/6 7.— 7/20 8.— 18/35 9.— 8/11 10.— 1/30 11.— 3/4 12.— 1/2 114 LUDLOW TEXTILE ARITHMETIC MULTIPLICATION OF FRACTIONS. Lesson 53. Page 49. 1.— 2/9 7.— 1/28 2.— 1 1/3 8.— 1/3 3.— 63/380 9.— 7/15 4.-377/945 10.— 2/9 5.-4/45 11.— 1/8 6.— 17/282 12.— 3/34 RECIPROCALS. Lesson 54. Page 49. ■ 1.— 1/3 7.-9/7 2.— 1/6 8.— 11/5 3.— 1/7 9.-23/6 4.— 1/13 10.— 7/23 5.— 1/17 11.— 13/29 6.— 1/22 12.— 6/41 DIVISION OF FRACTIONS BY WHOLE NUMBEE Lesson 55. Page 50. 1.— 1/9 7.— 2/169 2.— 1/15 8.-2/45 3.— 1/10 9.— 1/30 4.— 1/14 10.— 1/27 5.— 1/12 11.— 1/18 6.— 1/33 12.— 1/64 DIVISION OF FRACTIONS BY FRACTIONS. Lesson 56. Page 50. 1,-1 1/5 8.— 1 2.— 1 17/108 9.— 1 3/4 3.-2 2/7 10.— 36/49 4.-9/25 11.— 507/1000 5.— 1 79/210 12.— 2 2/47 6.— 1 19/27 13.— 2 77/150 7.— 33/70 14.— 1 FRACTIONAL PART OF ANOTHER FRACTION, Lesson 57. Page 51. 1.— 1/5 7.— 1/4 2.— 1/6 8.— 1/6 3.— 1/5 9.— 1/8 4.— 1/6 10.— 3/8 5.— 1/3 11.— 1/16 6.— 2/11 12.— 1/40 LUDLOW TEXTILE ARITHMETIC 115 SIMILAR FRACTIONS. Lesson 58. Page 52. 1.- 2.- 3.- 4.- 5.- -3/6, 2/6 -3/12, 4/12 -8/40, 5/40 -4/6, 5/6 -7/8, 6/8 6.— 5/16, 10/16 7.— 18/63, 28/63 8.— 9/15, 7/15 9.— 9/30, 20/30 10.— 40/48, 42/48 Lesson 59. Page 52. 1.- 2.- 3.- 4.- 5.- -2/16, 12/16, -42/54, 45/54, -40/50, 9/50, -54/60, 4/60, -10/30, 15/30, 5/16 28/54 46/50 35/60 12/30 6.— 14/46, 23/46, 7.— 28/70, 45/70, 8.— 28/60, 20/60, 9.— 18/48, 32/48, 10.— 25/90, 72/90, 9/46 30/70 25/60 9/48 70/90 ADDITION OF FRACTIONS. Le :sson 60. Page 53. 1.- 2.- 3.- 4.- 5.- -3/8 -13/16 -11/16 -7/8 -9/16 6.— 15/16 7.— 1 5/8 8.-27/64 9.— 15/64 10.— 1 1/64 Lesson 61. Page 53. 1.- 2.- 3- 4.- 5.- -2 9/16 -37/50 -1 1/5 -2 7/24 -71 11/16 6.— 101 7/48 7.-34 37/45 8.-2 59/84 9.— 51 11/16 10.— 21 5/12 SUBTRACTION OF FRACTIONS. Lesson 62. Page 54. 1.- 2.- 3.- 4.- 5.- -1/4 -2/5 -3/8 -3/16 -7/24 6.— 5/16 7.-4 1/4 8.— 21 7/16 9.-26 3/8 10.— 35 53/64 Lesson 63. Page 54. 1.- 2.- 3.- 4.- 5.- -23 3/8 -22 1/4 -20 9/16 -15 1/8 -33 13/16 6.-9 57/64 7.-27/64 8.-2 7/100 9.-3 31/40 10.— 37/60 116 LUDLOW TEXTILE ARITHMETIC CHANGE A DECIMAL TO A COMMON FRACTION. Lesson 64. Page 55. 1.- -1/5 11.— 3 24/25 2.- -3/5 12.— 6 333/500 3.- -1/2 13.— 7 1/8 4.- -1/4 14.— 4 1/16 5.- -3/4 15.— 12 12/125 6.- -61/200 16.— 1/2000 7.- -3/50 17.— 14 9/1000 8.- -3/500 18.— 7 3/8 9.- -649/2000 19.— 1/4000 10.- -501/1250 20.— 10 2119/5000 MISCELLANEOUS EXAMPLES. Lesson 65. Page 55-56. 1.- -5/12 7. — 5/8" cents 2.- -$10 8.-85 1/4 bales 3.- -$23 9.-27 1/2 doffs 4.- —75 cents 10.— 59 3/8 cents 5.- -92 1/2 cents 11.— 46 balls 6.- -21 bales 12.— 358 balls Lesson 66. Page 56-57. 1.- —347 1/3 pounds 11.— 0.1964 inches 2.- -$918.75 12.— $0,761 3.- -$18,125 13.— $0.25 4.- -$37 . 33 14.— $0,814 5.- —1162 pounds 15.— $5.53 6.- -0.60416 ton 16.— $54.58 7.- -$8.21 17.— 106 balls 8.- -$112 18.— 24 7/8 cents 9.- -$18.24 19.— $6.00 10.- — 6 yards WEIGHTS ! AND MEASURES. YARN EXAMPLES. Lesson 67. Page 60-61. 1.- —486 gr. 10.— 280 gr. 2.- —4 lea 11.— 64.3 gr. 3.- —40 threads 12.— 450 yds. 4.- —292 gr. 13. — 6 ozs. 5.- —10 lbs. 14.— 6000 spyndles 6.- —3 lea 16 lbs. 15.— 5119 cuts LUDLOW TEXTILE ARITHMETIC 117 7.-583 gr. 8.— 30 lbs. 9.— 25's lea 19.— 1 lb. = 2 1bs.= 3 1bs.= 41bs.= 5 lbs. = 6 lbs. Tibs. 8 lbs. 9 lbs. 10 lbs. 49 gr. 98 gr, 146 gr. 194 gr 243 gr = 292 gr = 340 gr = 389 gr, = 437 gr, = 486 gr, 16.— 3840 cuts 17.— 8000 spyndles 18. — 36 spyndles 20.— l's = 1166 gr. 2's = 583gr. 3's = 389 gr. 4's = 292gr. 5's = 233gr. 6's = 194gr. 7's = 174 gr. 8's = 146gr. 9's = 130 gr. . 10's = 117 gr. LONG MEASURE. Lesson 68. Page 63. 1. — 231 inches 2.— 179 inches 3. — 12 yards 4.— 11 yds., ft., 1 in. 5.— 3 mi., 39 rds., 2 yds. 6. — 6 mi., 126 rds., 1 yd. 7.— 270 inches 2.5 ft. 2 ft. WEIGHTS AND MEASURES. Lesson 69. Page 64. 1.— 12 yds., 1 ft., 10 ins. 2.-49 yds., 1 ft., 3 ins. 3.-23 mi., 1623 yds. 4.— 20 mi., 1015 yds., 2 ft. 5.— 11 yds., 1 ft., 10 ins. 6.-8 yds., 2 ft., 8 ins. 7.-2 mi., 1366 yds., 2 ins. 8.— 6 mi., 1756 yds., 1 in. LONG MEASURE. Lesson 70. Page 65. 1.— 186 yds. 2.— 319 yds., 1 ft., 4 ins. 3.— 18 mi., 804 yds., 2 ft. 4.— 61 mi., 229 yds. 5.-22 yds., 3 7/8 ins. 6.— 19 yds., 2 ft., 7 3/4 ins. 118 LUDLOW TEXTILE ARITHMETIC Lesson 71. Page 65. 1.— 2352 feet 5.-98,181 . 8 miles 2.— 1450 2/3 yards 6.— 11 spyndles 3.— 6035 rails 7.— 108 lengths 4.-255,200 yards 8.— 500 yards SQUARE MEASURE. Lesson 72. Page 66. 1.-79,200 sq. ft. 6.-36 sq. yds. 2.-17,424 sq. ft. 7.— 2404 sq. yds. 3.-6453.33 sq. yds. 8.— 12 sq. yds. 4.-196,020 sq. ft. 9.— 108.8 sq. yds. 5. — 66 sq. yds. CUBIC MEASURE. Lesson 73. Page 66-67. 1.— 96 cu. ft. 6.— 210.54 63/108 cu. ft. 2.— 126 cu. ft. 7.-185,856 cu. ft. 3.— 1800 cu. ft. 8.-^6666 + bales 4.-278,784 cu. ft. 9.-72 yds. 5.— 802. 12 cu. ft. 10.-41,472 cu. ins. LIQUID MEASURE. Lesson 74. Page 67-68. 1.— 45 gals., 2 qts., 1 pt. 6.— 3715 gals., 1 qt. 2.-67 gals., 1 qt., 1 pt 3. — 5 gals., 3 qts., 1 pt. 8 4.-8 gals., 2 qts., 1 pt. 9 5.-46,080 pts. 10 69 gals. —107 gals., 2 qts., 1 pt. — 29 gals., qts., 1 pt. —27 gals., qts., 1 pt. AVOIRDUPOIS WEIGHT. Lesson 75. Page 68. 1.— 1312.5 gr. 6.-50,000 tons 2.-75,900 lbs., oz., 3 gr. 7.— 1280 tons 3.-248 lbs., 6684 gr. 8.— 128 ozs. 4.-98,000 gr. 9.— 4000 ft. 5.— 17 tons, 1918 lbs., 4 ozs. 10.— $0,075 MISCELLANEOUS. Lesson 76. Page 70-71. 1.— 6104 . 67 bales 9.-5985 ft. 2.— 11 . 32 systems 10.— 213 . 55 gals. 3.-253,440 feet 11.— 2184 lbs. 4. — 8712 storehouses 12. — 475 bobbins LUDLOW TEXTILE ARITHMETIC 119 5.-14,840 . 32 gals. 13.— 324 cwt. 6.-41,184 feet 14.— 261 lbs., 4 ozs. 7.— The first 3816 cu. ins. 15.— $2,880,000 8.-14,784 lbs. ALLIGATION. Lesson 77. Page 72. 1.— 7.1jzf 2.-11^ 3.— 11 5/8^ 4-— 12jzT 5.— 10 2/3^ Lesson 78. Page 73. 1. — 2 flax, 3 hemp 2. — Equal parts 3. — 2 flax, 1 hemp 4. — Infinite number of answers 5. — Infinite number of answers PERCENTAGE. A COMMON FRACTION AS A RATE PER CENT. Lesson 79. Page 75. 1.-25% 7.-31.25% 2.-33.33% 8.-41.65% 3.-50% 9.-42.86% 4.-16.66% 10.-44.44% 5.-12.5% 11.-22.5% 6.-10% 12.-143.75% A DECIMAL AS A RATE PER CENT. Lesson 80. Page 75. 1.-46% 7.-648% 2.-38% 8.— .05% 3.-7% 9.-1004% 4.-90% 10.-0.625% • 5.-26% 11.-12.5% 6.-779% 12.-3.75% FIND PERCENTAGE HAVING BASE AND RATE GIVEN. Lesson 81. Page 76. 1.— 15 lbs. 5.— 1250 lbs. 2.-32 lbs. 6.— 2200 lbs. 3.-62 . 5 lbs. 7.— 2800 lbs. 4.— 200 lbs. 120 LUDLOW TEXTILE ARITHMETIC PERCENTAGE. Lesson 82. Page 76. 1.— 2160 spindles 2.— 130 lbs. 3.— 640 lbs. 4.-60,000 lbs. 5.-96 looms, 4632 yards FIND RATE HAVING BASE AND PERCENTAGE Lesson 83. Page 77. 1.-5.43% 7.— 1897 3. 17% increase 2.-4.12% 1898 3.-2% 1899 11.38% increase 4._5.92% 1900 23 . 76% increase 5—37% 1901 8.70% increase 6. — 23% 1902 7.39% increase 1903 1 . 55% decrease 1904 9 . 57% decrease 1905 23.00% decrease 1906 5 . 02% increase 1907 . 33% increase 1908 19.50% increase TO FIND BASE. Lesson 84. Page 78. 1.— $225 5.— 8800 2.— $1250 6.— 300 3.— 400 7.— 600 4.-625 Lesson 85. Page 78. 1.— 1435 lbs. 4.— $6 2. — 250 lbs. 5. — 705 children 3.-12,000 lbs. 6.— 20317. 5 lbs.' PERCENTAGE. Lesson 86. Page 78-79. 1.— 60 bales 2. — 250% increase 3. — 59.12% increase 4. — 112.62% increase 5. — 51 . 5% increase 6.- -6.577 7.- -0 . 186 8.- -0.871 or 27/31 9.- -0 . 689 or 15/67 0.- -0.831 LUDLOW TEXTILE ARITHMETIC 121 SQUARE ROOT. Lesson 87. Page 82. 1.— 351 2.-89 . 7 3.— 7008 4.— 10.5 5.— 9812 GENERAL MILL WORK. - DIAMETERS AND CIRCUMFERENCES. Lesson 88. Page 83. 1.— 7 . 0686 inches 4.— 15 . 708 feet 2.-7 inches 5.-28 . 68 inches 3.— 88 inches ; 6.— 2.353 inches LENGTH OF BELL. Lesson 89. Page 84. 1. — 175 yards 2.— 140 yards 3.— 155 1/2 yards 4. — 176 yards TRACING SPEED. Lesson 90. Page 85. 1.— 437.5 R. P. M. 2.-233.3 R. P. M. 3.— 131.25 R. P. M. 4.— 70 R. P. M. SURFACE SPEED. Lesson 91. Page 86. 1. — 473 . 2 feet per minute 2. — 30 . 628 feet per minute 3. — 104.3 feet per minute 4. — 100 revolutions per minute 5. — 5024 feet per minute TWISTING. Lesson 92. Page 95. 1. — 4.6 turns per inch 2. — 4 . 9 turns per inch 3. — 3 . 7 turns per inch 122 LUDLOW TEXTILE ARITHMETIC .— l's = 2.0 turns per inch 2's = 2.8 < i 3's = 3.4 i i 4's = 4.0 << 5's = 4.5 ii 6's=4.9 " 7's = 5.3 " 8's = 5.8 (< 9's = 6.0 " 10's = 6.3 ii ii i . — 1 pound yarn - 13.8 turns per inc 2 pounds " 9.9 " " " 3 ii 8.0 " " " 4 ii 6.9 " " " 5 1 1 6.1 " " " 6 ii 5.6 POUNDS PER SPYNDLE. Lesson 93. Page 100. 1. — 45 pounds per spyndle 2. — 108 pounds per spyndle 3. — 63 pounds per spyndle 4. — 144 pounds per spyndle 5, — 90 pounds per spyndle DRAFT ON SPINNING FRAME. Lesson 94. Page 100. 1.- -10.5 draft 5.-2.7 2.- -12.6 6.— 5.0 3.- -5.4 7.-4.5 4.- -5.8 SPINNING DRAFT. Lesson 95. Page 101. 1.- —9 draft 2.- -6.9 3.- -6.4 4.- -6 DRAFT. Lesson 96. Page 101. 1.- -5.1 draft 4.-8.3 draft 2.- -6.8 5.-3.8 3.- -6.8 6.-2.3 OCT 131909