THE SCIENCE 
 
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 MATHEMATICS LISRARY 
 
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THE SCIENCE 
 
 OF 
 
 0 mm 
 
 AND 
 Their * Practical * Application 
 FOR 
 
 TEACHERS AND PRIVATE LEARNERS 
 
 BY 
 
 MERRITT L. DAWKINS 
 
 DOWNING, MO. 
 PEERLESS PUBLISHING COMPANY. 
 1890. 
 
Entered according to act of Congress, in the year 1890, by 
 MERRITT L. DAWKINS, 
 In the Office of the Librarian of Congress at Washington. 
 
 ; PRINTED AT 
 THE RECORD PRINTING HOUSE, 
 DOWNING, MO. 
 
ISMy 44 Spoes evel. Quer, 
 
 513 
 paza MATHEMATICS LIBRARY 
 
 I NTRODUCTORY. 
 
 ———— 
 
 The design of the author in preparing this little 
 volume has been to embrace, within moderate space, a 
 practical and useful treatise on ARITHMETICAL SCIENCE, 
 and to place it within the reach of every home and every 
 business office throughout the land. 
 
 The author does not arrogate to himself the first 
 knowledge of the methods found in this work. He sim- 
 ply contends that no similar work issued previous to the 
 date of his copyright is so convenient, instructive and 
 satisfactory. 
 
 By the utmost brevity and precision the author 
 has been enabled to compress a vast number of 
 useful methods into a small compass and by eliminating 
 a few subjects which serve rather to perplex than to en- 
 Jighten, he has, in the opinion of those competent to 
 judge, produced a work that will prove valuable to the 
 business man, the farmer, the mechanic, and the large 
 class of people who have not had the opportunity or 
 ability to master the various technical terms of arithmetic, 
 and to whom many of its principles and processes have 
 heretofore been laborious and difficult. 
 
 The author has availed himself of many valuable hints 
 
 and suggestions from business men, practical teachers, 
 
 6022183 
 
4 INTRODUCTORY. 
 
 and educators, all of whom he desires to thank most 
 cordially for the aid they have rendered. 
 
 The greatest care has been taken to ensure accuracy, 
 but where imperfection is so general in works of this 
 class, it is too much to hope that errors have not crept 
 in. Should the reader note any such, the author will 
 deem it a favor to be informed of them, with a view to 
 their correction in subsequent editions. 
 
 Trusting that the work will in some measure supply 
 the popular demand for a Practican AritHmertic, the 
 author presents his work to the public. 
 
 M.: dae; 
 
 September 1, 1890. 
 
 NOTICE. 
 
 We solicit correspondence with teachers and others 
 who are open for a profitable engagement. 
 
 This is by universal consent the best selling book pub- 
 lished, and if you want to make Bia Money secure an 
 agency for this great work without delay. 
 
 Sample copy mailed on receipt of one dollar. 
 
 PEERLESS PUBLISHING Co. 
 
SGIENGE OF NUMBERS. 
 
 ~~ ——— 
 
 DEFINITIONS. 
 
 1. A Number is a unit or a collection of units. 
 
 2. An Even Number is one that is exactly divis- 
 
 ible by 2. 
 
 3. An Odd Number is one that is not exactly divis- 
 ible by 2. 
 
 4. A Prime Number is one that has no exact di- 
 visor besides itself and 1. 
 
 5. A Composite Number is one that has exact 
 divisors besides itself and 1. 
 
 6. An Abstract Number is one that denotes no 
 particular thing. 
 
 @. A Denominate Number is one which denotes 
 some particular thing; as two pounds, four boys. 
 
 $. An Integer is a whole number. 
 
 9. The Complement of a number is the difference 
 between it and a unit of the next higher order. 
 
 10. The Supplement of a number is the difference 
 between it and a unit of the next lower order. 
 
 11. The Reciprocal of a number is 1 divided by 
 that number. 
 
ADDITION. 
 
 >< 
 
 2. Addition is uniting two or more numbers into 
 one. 
 
 IS. The Addemnds are the numbers to be added. 
 if. The Sumi is the result obtained by adding. 
 
 15. The Sign of Addition is the perpendicular 
 cross, +, and is read Pius. 
 
 16. The Sign of Equality consists of two hori- 
 zontal parallels, = , and is read EQUALS, Or, IS EQUAL TO. 
 
 iv. WRapidity and acuracy in addition can be se- 
 cured only by frequent and careful practice. These two 
 acquirements are the most Sse qualifications of an 
 accountant. 
 
 18. Careful Practice will enable the student to 
 add several columns at once. This method cannot well 
 be extended to more than three columns with any prac- 
 tical advantage, unless the columns are incomplete. 
 
 19. The Work should be tested by adding in an 
 opposite direction from that in which the additions were 
 _first made. 
 
 20. When there are but Two Numbers to be 
 added, it is more convenient to begin at the left to add, 
 
 observing to carry ONE when the sum of the next lower 
 figures is more than 9. 
 
ADDITION. 7 
 
 21. Make Combinations of tens when possible 
 and think resutts only. Thus, instead of saying 3 and 
 “ai 7 and 8 are 15, think 8, 7, 15. 
 
 22. Numbers increasing by a common difference, 
 as 2,4, 6, 8, may be added by multiplying the first and 
 last by the number of addends. 
 
 23. Write Totals of each column under each other, 
 by one of the two methods shown below, and then com- 
 - bine them in one result. The first method is perhaps 
 the better, as it obviates the difficulty of carrying tens. 
 
 24. 1. A has 798 dollars, B 875 dollars, and C 769 
 
 dollars. How much have they together? $2442. 
 PROCESS. ANOTHER PROCESS. 
 22 798 2'2 
 22 875 2/4 
 22 769 24 
 2442 
 
 2. I owe one man $375, another. $280, a third $564, 
 a fourth $119, a fifth $75. How much do I owe? 
 
 7 $1413. 
 
 3. Paid for coffee $245, for tea $325, for sugar $196, 
 for flour $217, and for spices $273. What did all cost? 
 
 $1256. 
 
 4, <A drover paid $265 for one horse, $198 for another, 
 $237 for another, and $216 for another. What did he pay 
 for all? #916. 
 
 5. Four men formed a partnership. A furnished 
 $2467, B $1674, C $2312, and D $1893. What was their 
 capital? $8346. 
 
 6. A farmer has 328 sheep in one field, 675 in an- 
 other, 682 in another, 729 in another, and 2380 in another. 
 How many has he in the five fields? 4794, 
 7. How many strokes does a clock strike in 12 
 hours? 78, 
 
he) ADDITION. 
 
 8. A carpenter receives $246 for one job, for another 
 $327, for another $683, for another $456, and for another 
 $713. How much did he receive for all? $2425. 
 
 9. A merchant pays $320 a year for rent, $1230 to one 
 elerk, $832 to another, and $438 for sundry expenses. 
 What does his business cost him a year? $2820. 
 
 10. A landlord receives $329 rent for one house, $671 
 for another, $476 for another, $524 for another, $387 for 
 another. How much does he receive for all? $2387. 
 
 11. A gentleman bought a store for $625, and paid 
 $148 for repairs, and $175 for having it enlarged. For 
 how much must he sell it, in order to gain $160? 
 
 $1108. 
 12. A man owns six horses. The first is worth $384, 
 
 the second $488, the third $348, the fourth $843, the 
 fifth $483, and the last $834. What is the value of the 
 six horses? $3330. 
 13. A gentleman having lost $2380 finds that he has 
 $7289 remaining. How much would he have had, if in- 
 stead of losing, he had found $2380, and then earned 
 $2256 more? $14305. 
 14. Having a note due, I paid $153 at one time, $216 
 at another, $435 at another, $673 at another, $327 at an- 
 other, and there were $174 still unpaid. What was the 
 face of the note? $1978. 
 15. A gentleman making his will, left $3260 to his 
 wife,-$2875 to his oldest son, $1890 to his second son, 
 $1250 to his youngest son. What amount did he be- 
 queath in his will? $9275. 
 16. A man sold his piano for $409, his collection of 
 paintings for $541, his library for $718, his carpets for 
 $726, other furniture for $1738. How much money did 
 he obtain by the sale? $4182. 
 17. A drover bought horses for $628, and cows for 
 $382; on his horses he gained $324, and on his cows $125. 
 What should he have received for both to have gained 
 $216 more than he did? $1675. 
 
ADDITION. Q 
 
 18. What is the sum of all numbers from 1 to 24 in- 
 clusive? 300. 
 
 19. A boy agreed to work 30 days for 2 cents the 
 first day and an increase of 2 cents per day. What 
 
 did he receive? $9.30. 
 20. What is the sum of all the odd numbers from 1 
 to 49 inelusive? 625. 
 
 21. In one book there are 785 pages, in another 867 
 pages, in another 879 pages, and in another 684 pages. 
 How many pages in the four books? 3215. 
 
 22. A merchant pays $98 a month to his clerks, $60 
 to his bookkeeper, $37 to his janitor, $18. for fuel, $11 
 for gas, $36 for rent, and $70 for advertising. What 
 are his expenses per month? $330. 
 
 23. There are 31766 square miles in Maine, 9280 in 
 New Hampshire, 10212 in Vermont, 7800 in Massachu- 
 setts, 1306 in Rhode Island, and 4674 in Connecticut. 
 How many square miles in all of New England? 
 
 65038. 
 
 24. The population of the five principal cities of 
 Missouri in 1880 was as follows: St. Louis 350222, 
 Kansas City 55813, St. Joseph 32484, Hannibal 11074, 
 Sedalia 8202. What was their entire population at’ that 
 time? 457795. 
 
 25. Jackson, Greenslate & Co. ground 11936 pounds 
 of wheat on Monday, 10117 pounds on Tuesday, 8135 
 pounds on Wednesday, 9963 pounds on Thursday, 7215 
 pounds on Friday, 12634 pounds on Saturday. How 
 many pounds did they grind during the week? 
 
 60000. 
 
 26. A gentleman being asked his age, said he was 18 
 years old when he Jeft the common school, he spent 3 
 years in college, 4 years in a medical school, practiced 
 his profession 24 years, was a member of congress 8 
 years, and it was 13 years since he retired from business. 
 How.old was he? 70. 
 
SUBTRACTION. 
 
 a 
 
 25. Subtraction is taking one number from an- 
 other. 
 
 26. The Minuend is the number from which an- 
 other is to be taken. 
 
 26. The Subtrahend is the number to be sub- 
 tracted. 
 
 24@. The Remainder is ths result obtained by sub- 
 tracting. 
 
 28. The Sign of Subtraction is —, and is read 
 MINUS. 
 
 29. Only Like Numbers can be added or sub- 
 tracted. 
 
 30. It is Generally more convenient to write the 
 subtrahends under the minuend, but sometimes it is 
 better to reverse this position. The student should be 
 able to subtract either downward or upward as may be 
 required. 
 
 3i. Test the Work by adding the remainder to 
 
 the subtrahend. If their sum is equal to the minuend, 
 the work is correct. 
 
 32. Several Numbers may be subtracted from 
 another at a SINGLE OPERATION, as shown in the following 
 example. 
 
SUBTRACTION. 1] 
 
 33. 1. Ina regiment of 1280 men, 248 are killed in 
 battle, 324 are taken prisoners, 192 are drowned, and 273 
 join the enemy. How many remain? 243, 
 
 PROCESS. 
 1280 
 
 248 
 324 
 192 
 273 
 
 243 
 
 Expianation.— Add each column of subtrahends sep- 
 arately, and write in the remainder such figure as added 
 to the sum of each column will make its right hand figure 
 equal to the corresponding figure of the-minuend. The 
 sum of the first column is 17 and we perceive that 3 
 must be added to it to make 20, the next higher number 
 in which the unit figure in both terms are ALIKE, hence 
 we write 3 in the remainder and carry 2 to the next 
 column, the sum of which is 24 which, taken from 28 
 leaves 4 in the remainder; carry 2 and proceed as be- 
 fore. 
 
 2. A merchant has 3050 dollars. What sum will he 
 
 have after paying 2364 dollars? $686. 
 3. A farm that cost 8275 dollars was sold at a loss of 
 
 396. What was the selling price? $7879. 
 4. Bought a farm for 3280 dollars, paying 1820 dol- 
 
 lars down. How much remained unpaid? $2460. 
 
 5. J. Morgan & Co. owe me on account 8350 dollars. 
 If they pay 6428, how much remain unpaid? $1922. . 
 6. Boughta farm for 1820 dollars and sold it for 
 2460 dollars. How much did I make by the bargain? 
 $640. 
 7.° If a man’s income is 900 dollars a year and his 
 expenses 678 dollars, how much can he save in a year? 
 
 $222. 
 
ii SUBTRACTION. 
 
 8. From a tract of land containing 10000 acres the 
 owner sold to A 4750 acres, and to B 875 acres. How 
 many acres had he left? 4375, 
 
 9. If I have $2000 on deposit, and give a check for 
 #135 to one man, to another for $673.50, and to another 
 for $429.50, how much will remain? $762. 
 
 10. The captain of a ship having a cargo worth 16375 
 dollars, threw overboard in a storm 7532 dollars worth, 
 What was the value of the goods left? $8843. 
 
 11. Charles had 376 cents, James gave him 432 cents, 
 and Henry gave him enough to make his number 1000. 
 How many cents did Henry give him? 192. 
 
 12. A merchant had a quantity of flour on hand, for 
 which he asked 780 dollars, but for cash he sold it for 36 
 dollars less. How much did he receive for his flour? 
 
 $744. 
 
 13. A merchant having 248 pounds of coffee, bought 
 198 pounds, and then sold 73 pounds to one man, and 
 184 pounds to another. How many pounds had he left? 
 
 189. 
 
 14. A man set out on a journey of 1250 miles. Dur- 
 ing the first day he traveled 342 miles, and during the 
 second day he traveled 416 miles. How many miles had 
 he yet to travel? 492. 
 
 15. A merchant worth 20000 dollars, lost a store by 
 fire worth 2750 dollars and goods to the amount of 3650 
 dollars. How much had he left after receiving 3200 
 dollars insurance? $16800. 
 
 16. A man having 6789 dollars, invested 500 dollars 
 in bank stock, 480 dollars in a creamery 675 dollars in a 
 mining stock, and 1200 dollars in United States bonds. 
 How much had he left? $3934. 
 
 17. A man owns property valued at 6480 dollars, of 
 which 2316 dollars are in personal property, and 1673 
 dollars in real estate. The remainder is deposited in 
 bank. How much has he in bank? ; $2491. 
 
 x 
 
SUBTRACTION. 13 
 
 18. A gentleman had 4800 dollars to destribute among 
 his three sons. 4’o the eldest he gave 1834 dollars, to 
 the second 1683 dollars, and the remainder to the young- 
 est. How much did the youngest receive? $1283, 
 
 19. If a man’s annual income is 3600 dollars, and he 
 spends 363 dollars for house rent, 267 dollars for cloth- 
 ing, 684 dollars for provisions, 416 dollars for servants, 
 and 238 dollars for traveling, how much will he have left 
 at the end of the year? $1632. 
 
 20. On Monday morning a bank had on hand 1826 
 dollars. - During the day 2191 dollars were deposited 
 and 3412 drawn out, on Tuesday 3256 dollars were de- 
 posited and 2164 dollars drawn out. How many dollars 
 were on hand Wednesday? $1697. 
 
 21. At the end of the year 1890, [ found I had spent 
 2400 dollars. Of this amount 350 dollars were paid for 
 board, 125 dollars for clothing, 475 dollars for books, 
 150 dollars for incidentals, and remainder for real estate. 
 what was the cost of the real estate? $1300. 
 
 22. The following is my private account for two weeks: 
 First week, received 75 dollars as salary and spent 27 
 dollars for clothing, 6 dollars for board, 4 dollars for 
 washing, and 7 dollars for sundries. Second week, re- 
 ceived 75 as salary, loaned 25 dollars to A. D. Lewis, 
 paid 6 dollars for board, 3 dollars for washing, and 16 
 dollars for sundries. How much did I have at the end 
 of the two weeks? HHK. a, WY 
 
MULTIPLICATION. 
 
 $4. Multiplication is taking one number as many 
 times as there are units in another. 
 
 3. The Multiplicamd is the number to be multi- 
 plied. 
 
 36. The Multiplier is the number by which we 
 multiply. 
 
 3¢@. The Product is the result obtained by multi- 
 plying. 
 
 38. The Sign of Multiplication is the oblique 
 cross, X, and is read TIMES. 
 
 39. The multiplicand and multiplier are called the 
 Factors of the product. 
 
 49. Ordinary Multiplication is so well known 
 that it requires no explanation. The methods given in 
 this work are quite practical and very easily learned. 
 No one method of contraction is best for all cases, but it 
 is well for the pupil to be familiar with several and apply 
 the most appropriate in each particular case, or any one 
 that seems most simple may be adopted for general use. 
 
 441. A portion of the Multiplication Table is 
 inserted here for the benefit of those who are not 
 thoroughly acquainted with it. Now, dear reader, if you 
 
MULTIPLICATION. 15 
 
 belong to this class, do not rest until you have mastered 
 PERFECTLY that portion of the table here given. 
 
 TABLE. 
 
 5 6 7 8 91011 12 
 6| 8} 10) 12) 14| 16] 18) 20| 22) 241 
 
 we 
 ir 
 | abe 
 i) 
 
 ao w= 
 
 CO! Oo! > 
 Je) 
 a" 
 we) 
 jet 
 i | 
 —" 
 ie 2) 
 bo 
 —_ 
 ho 
 pis 
 bo 
 ~J 
 vs 
 = 
 1S) 
 wy) 
 Yo 
 oe 
 
 | 10) 15) 20) 25! 30) 35) 40) 45) 50) 55) 60 
 72 
 
 a" 
 Ts 
 bo 
 peed 
 bo 
 QO 
 do 
 or 
 bo 
 bo 
 > 
 Ne) 
 On 
 o> 
 Ss 
 oo 
 =~] 
 S 
 ~] 
 ~] 
 QO 
 ns 
 
 fo on eer ot pe 
 —" 
 iw) 
 — 
 CO 
 bo 
 a 
 Yo 
 er 
 — 
 bo 
 cm“ 
 io.6) 
 or 
 ae 
 or) 
 i=) 
 er) 
 er) 
 
 | 16 24 32) 40 48) 56 64 72 80 88) 96 
 | | 63, 72, 81) 90 99 108 
 110 120 
 121132 
 182)144 
 
 = 
 jas, 
 io 8) 
 bo 
 IN 
 vo 
 or) 
 > 
 ot 
 Or 
 1s 
 
 nd 
 —) 
 ho 
 =, 
 or 
 = 
 S 
 © 
 ~I 
 = 
 ee) 
 ss 
 c© 
 8) 
 fae 
 c 
 =) 
 
 et | bk 
 | = 
 bho | bo 
 H> |b 
 wo | o9 
 S| & 
 Hm | 
 Go | 
 l| on 
 Cle 
 ~1| D> 
 bo 1S 
 oo;~x 
 ml 
 co | 00 
 S| DW 
 am 
 Ole 
 HS! 
 —) 
 nh | — 
 Sn 
 
 NUMBERS LESS THAN 20. 
 
 43. The following method will be found valuable in 
 Multiplying Mentally any two numbers less than 20. 
 
 44. 1. Multiply 16 by 14. 224. 
 é PROCESS. 
 
 16 
 14 
 
 200 
 24 
 
 224 
 
 Expranation.—Add the unit figure of one number to 
 
16 MULTIPLICATION. 
 
 the other number, annex a cipher, and to the result add 
 the product of the unit figures. After a little practice 
 the operation can be performed mentally, thus extending 
 the multiplication table to the twenties. 
 
 2. How many miles will a man travel in 16 days, if 
 
 he travel 18 miles per day? 288. 
 
 3. If there are 17 yards of cloth in one piece, how 
 many yards in 13 pieces? 221. 
 
 4. Ifaman can dig 19 bushels of potatoes in one 
 day, how many bushel can he dig in 16 days? 304. 
 
 5. Ifa boat can sail 18 miles per hour, how many 
 miles can she sail in 14 hours? 252. 
 
 EN dS DIN Gra VWVils ee CP EL Bi: 
 
 45. To Multiply when there are ciphers at the 
 right of one or both factors, proceed as if there were no 
 ciphers, then annex the ciphers to the result. 
 
 46. 1. Multiply 3200 by 60. 192000. 
 PROCESS. 
 3200 
 60 
 192000 
 2. What will 42 cows cost at 40 dollars a head? 
 $1680. 
 3. What costs 56 yards of muslin at 30 cents a yard? 
 $16.80. 
 4. What will 21 horses cost at the rate of 80 dollars 
 per head? $1680. 
 5° What will be the cost of 70 acres of land at 24 
 dollars per acre? $1680. 
 
 6. Ifa ship sail 28 miles an hour, how many miles 
 will she sail in 60 hours? 1680. 
 
MULTIPLICATION. 17 
 
 SMALL NUMBERS. 
 
 47. The Method of multiplying by 11 and other 
 small numbers is explained below. 
 
 48. 1. Multiply 236 by 11. 2596. 
 PROCESS. 
 2596 
 
 ExpLanation :— Write the unit figure and then add the 
 digits, two at a time, beginning at the right, finally writ- 
 ing the left hand figure. 
 
 2. Multiply 423 by 16. 6768. 
 PROCESS, 
 
 423 
 16 
 
 6768 
 
 Expianation:—Multiply through by 6 and carry the 
 succeeding figure of the multiplicand each time, finally 
 writing the left hand figure of the multiplicand increased 
 by the units to carry from the next lower order. 
 
 3. Multiply 123 by 24. 2952. 
 PROCESS. 
 
 123 
 24 
 
 2952 
 ExpLaNaTion:—Proceed as above, but carry double 
 the succeeding figure each time, or double the multipli- 
 cand mentally and multiply by half the multiplier. 
 
 4. Ifa clerk receive 125 dollars a month, how much 
 will he receive in 16 months? $2000. 
 
18 MULTIPLICATION. 
 
 5. If it takes 132 laborers 18 months to build a rail- 
 road, how many months will it take 1 man to build it? 
 
 2376. 
 6. Allowing 365 days to the year, how many days has 
 a man lived who is 24 years old? 8760. 
 7. Sound moves 1142 feet in a second, how many feet 
 will it move in 27 seconds? 30834. 
 8. A cooper can make 127 barrels in a week. How 
 many can he make in 17 weeks? 2159. 
 
 9. Ifaman put 28 dollars in a savings-bank in a 
 month, how much will he dposit in 14 months? $392. 
 
 FACTORING. 
 
 49. The ordinary method of factoring the multiplier 
 when it is a Composite Number requires no ex- 
 planation. 
 
 50. To Multiply when one part of the multiplier 
 is a factor of another part, the following contraction will 
 
 be found valuable. 
 51. 1. Multiply 421 by 312. 131352. 
 
 PROCESS. 
 
 421 
 312 
 
 1263 
 3052 
 
 131352 
 Expianation.—Multiply first by the 3, and then this 
 result by 4, writing each partial product in its proper 
 order. 
 
 2. If an ocean steamer sail 287 miles per day, how 
 many miles will she sail in 84 days? 24108. 
 
MULTIPLICATION. 19 
 
 3. Sold my farm of 248 acres at 186 dollars per acre. 
 How much did I get for it? M6128. 
 
 4. How much will it cost to lay 325 miles of ocean 
 cable at an expense of 217 dollars per mile? = $70525. 
 
 5. What will be the cost of 239 bushels of potatoes at 
 84 cents per bushel? $200.76. 
 
 MULTIPLIER NEAR 100, 1000, ELC. 
 
 52. The method of Multiplying by a number a 
 little less than a unit of the next-highest order is illus- 
 trated below. 
 
 53. Ll Multiply 135 by 98. 13230. 
 
 PROCESS. 
 13500 
 270 
 
 13230 
 
 ExpLaNnation.—Annex two ciphers to the multiplicand, 
 thus multiplying by 100, and from the result subtract 
 the product of the multiplicand by the complement of 
 the multiplier. 
 
 2. What will be the cost of 216 bushels of corn at 
 
 97 cents per bushel? $209.52. 
 3. What is the value of 796 pounds of tea at 96 
 cents per pound? ) $764.16. 
 4, What will 175 acres of land cost at 97 dollars per 
 acre? $16975. 
 5. What is the value of 1234 bushels of wheat at 95 
 cents per bushel? $1172.30. 
 
 6. A nurseryman counted the trees in his orchard and 
 found that he had 127 rows, each row containing 980 
 trees. How many trees were in the orchard? 124460 
 
20 MULTIPLICATION. 
 
 ALCIQOUGT PARTS; 
 
 54. Much Time can be saved when the multiplier 
 is an aliquot part of some higher unit, by abbreviating 
 the ordinary method as here illustrated. 
 
 55. 1. How many trees in 327 rows containing 25 
 trees to the row? 8175. 
 
 PROCESS. 
 4)32700 
 S175 
 
 Expianation:—Multiply by 100 by annexing two 
 ciphers, and then divide by 4. 
 
 TABLE: 
 
 Of 10 Of 100 Of 1000 
 14 is, 4:64 1s 4y5)1624---is- ae 
 185 ris kt 8k is eye O68 Shige ss 
 y iss $y LOS RISP Wey tooe ess ae 
 24 is 4,123 is 4/125 is #32 
 3 is 4/162 is +4 | 1662 is + 
 5 is #¢/]20 is 4/250 is 4 
 62 is 2/25 is 4/ 3334 is 34 
 7% - is» 44334658 4 | 750 is 4 
 Bhar id 5S 00 isan ae | Os pe de ee 
 
 2. What will be the cost of 28 yards of cloth at 123 
 cents per yard? $3.50, 
 
MULTIPLICATION. 21 
 
 3. Find the cost of 256 yards of cloth at $1.064 per 
 
 yard. $272. 
 4. What will be the cost of 258 cords of wood at 
 $3.334 per cord? $860. 
 5. What will 176 bushels of corn cost at 374 cents per 
 bushel? $66. 
 6. What will 192 bushels of wheat cost at $1.163 per 
 bushel? $224. 
 7. What will be the cost of 48 yards of silk at $1.163 
 per yard? $56. 
 8. What is the value of 225 barrels of flour at $3.332 
 per barrel? $750. 
 9. What will 368 bushels of potatoes cost at 624 cents 
 per bushel? $230. 
 10. What will be the cost of 324 pounds of tea at 75 
 cents per pound? $243. 
 11. What is the value of 216 bushels of apples at 
 $1.374 per bushel? $297. 
 12. What will be the cost of 72 gallons of wine at 
 $1.125 per gallon? $81. 
 13. When butter is worth 334 cents per pound, what 
 will 786 pounds be worth? $262. 
 
 NUMBERS ENDING WITH 5. 
 
 54. The following method of multiplying any two 
 numbers whose Unit Figures are 5 will be found 
 valuable. 
 
 55. 1. Multiply 45 by 25. 1125. 
 PROCESS. 
 
 45 
 25 
 
 1125 
 ExpLanaTIon :—Multiply the figures in tens place to- 
 
22 MULTIPLICATION. 
 
 gether, increase this by half their sum, and to the result 
 annex 25, or if the sum of the figures in tens place is 
 odd, annex @5. 
 2. What cost 75 acres of land at 35 dollars per acre? 
 $2625. 
 3. How many bushels of corn can be raised on 65 
 acres of land at the rate of 35 bushels per acre? 
 
 2275. 
 4, If 35 men can build a wall in 25 days, how many 
 days will it take one man to build it? 875. 
 
 5. Ifit requires 125 tons of iron rail for one mile of 
 
 railroad, how many tons will be required for 45 miles? 
 | 5625, 
 
 6. A merchant bought 115 yards of cloth at 35 cents 
 per yard. What did it cost? $40.25. 
 
 7. What will be the cost of 85 pounds of butter at 
 
 45 cents per pound? $38.25. 
 
 NEWTON § MULTIPLICATION RULE. 
 
 38. This Valuable Method may be used in mul- 
 tiplying any two numbers in which the unit figures add 
 to TEN and the. other figures are ALIKE. 
 
 49. 1. Multiply 26 by 24. 624, 
 PROCESS. 
 26 
 24 
 624 
 
 ExpLANAtion:—We say 4 times 6 are 24, put down 
 both figures and carry 1 to the second figure of the mul- 
 tiplier. Then say 3 times 2 are 6. Always carry ONE 
 to the tens figure of the multiplier. The product of the 
 
MULTIPLICATION. 23 
 
 unit figures must occupy two places, hence, if their 
 product is less than TEN, a cipher should be written in 
 the product. 
 
 2. If aman travel 33 miles per day, how many miles 
 will he travel in 28 days? 924. 
 
 PROCESS. 
 
 33 
 28 
 
 924 
 
 Note:—The above rule is likewise applicable when. 
 the digits of the multiplicand are auixe and the digits of 
 the multiplier add to TEN, also when the complement of 
 the unit figure in the multiplier cross-multiplied by the 
 tens in the multiplicand is equal to the product of the 
 other figures cross-multiplied. 
 
 3. What will 84 bushels of wheat cost at 86 cents per 
 bushel? $72.24. 
 
 4. How many square rods in a field 147 rods long and 
 48 rods wide? 7056. 
 
 5. How man pounds of sugar are there in 126 packages, 
 each package weighing 86 pounds. 10836. 
 
 6. How many bushels of wheat can be raised on 39 
 acres of land, if one acre produces 24 bushels? 936. 
 
 7. What will be the cost of 77 bushels of corn at 37 
 cents per bushel? $28.49. 
 
 8. What cost 64 bushels, of apples at 38 cents a 
 bushel? $24.32. 
 
 9. A travels 30 miles a day, and B travels 38 miles a 
 day. How many miles will both travel in 36 days? 
 
 2448, 
 10. How many sheets of paper in 26 quires, if there 
 are 24 sheets in a quire? 624. 
 
 11. A mining company built 395 tenement houses at 
 
24 MULTIPLICATION. 
 
 an average cost of 395 dollars apiece. What did they all 
 
 cost? : $156025. 
 12. A drover bought 42 oxen at 384 dollars a head. 
 What did they cost? $1617. 
 13. What cost 29 pounds of coffee at 21 cents per 
 pound? $6.09. 
 13. What is the value of 1574 yards of cloth at 48 
 cents per yard? £75.60. 
 
 COMPEE MENU EMU DASE Ish ART Giarss 
 
 38. This rule is valuable when both multiplicand 
 and multiplier are a Little Less than a unit of the 
 next higher order. 
 
 39. 1. Multiply 96 by 94. 9024. 
 PROCESS. 
 96-4 
 94 -6 
 90 24 
 ExpLanation:—Multiply the complements of both fac- 
 tors together and write the result in the product. Then 
 cross-subtract; that is, take the complement of one num- 
 ber from the other number, and write the remainder at 
 the left of the first product. If the product of the com- 
 plements does not contain as many figures as the multi- 
 plier, insert cipher to make up the deficit. 
 
 2. At 97 dollars per acre, what will a farm of 96 acres 
 
 cost? $9312. 
 3. What will be the cost of 89 bushels of corn at 92 
 cents per bushel? $81.88. 
 
 4. Ifa planet travel 996 miles per minute, how far 
 will it travel in 996 minutes? 99216. 
 
MULTIPLICATION. 25 
 
 5. What will 92 horses cost at 97 dollars per head? 
 
 $8924. 
 6 How many square feet in a garden 93 feet long and 
 86 feet wide? 7998. 
 7. What will be the cost of 196 yards of cloth at 95 
 cents a yard? $186.20. 
 8. Ifa vessel sail 93 miles a day, how far will she 
 sail in 98 days? 8649. 
 9. What will be the cost of 92 bushels of rye at 95 
 cents a bushel? $8740. 
 10. A has 99 dollars, and B has 61 times as much. 
 How much has B? $6039. 
 11. How much will I receive for a farm of 95 acres at 
 48 dollars per acre? | $4560. 
 12. What must I pay for 990 town lots at an average 
 price of 994 dollars? #98460. 
 13. How many square rods in a square field which 
 measures 92 rods on each side? 8464. 
 
 14. <A farmer sold 94 bushels of wheat at 92 cents a 
 bushel. What did he get for it? $86.48. 
 
 15. A drover bought 96 head of cattle at 85 dollars 
 per head. What did they cost him? $8 160. 
 
 16. <A grocer packed 92 cases of eggs, each containing 
 8 dozen. How many eggs did he pack? 8832. 
 
 17. Mercury moves in its orbit at the rate of 975 miles 
 per minute. How many miles will it movein 975 miutes? 
 ’ 950625. 
 
 18. A travels 37 miles a day and B travels 42 miles a 
 day. How many miles will they both travel in 96 days? 
 iw 
 (584, 
 
 19. The earth is 95 million miles from the sun, and 
 Neptune is 42 times as far. How many million miles is 
 Neptune from the sun? 3990. 
 
26 MULTIPLICATION. 
 
 17. Two ships start from the same place and sail in 
 opposite directions, both at the rate of 48 miles a day. 
 How many miles apart will they be in 91 days? 8736. 
 
 GROSS VEU ie Tt bale ites erie 
 
 60. This Valuable Method now for the first time 
 presented to the public in its simplest form, has been 
 known perhaps as long as the ordinary method, but the 
 wise men of past ages guarded the beautiful system with 
 so much care and secrecy that it has been kept out of 
 school and college text books up to the present day, and 
 hence is not generally known. This rare and practical 
 method, soon to become the common property of man- 
 kind, is now given in its complete and unabridged form, 
 as a free offering to the world. 
 
 61. Cross Multiplication appears very complh- 
 cated, but is quite easy, and a little practice will enable 
 the student to multiply rapidly and accurately by a 
 number consisting of several figures. This method 
 should be adopted for general use by all who desire to 
 compute with rapidity. 
 
 62. The Ordinary Method may be used for test- 
 ing the work until the pupil becomes familiar with the 
 
 process. » 
 63. 1. Multiply 54 by 32. 1728. 
 PROCESS. 
 54 
 32 
 
 1728 
 
Nw 
 ~] 
 
 MULTIPLICATION. 
 ® 
 
 EXxpLaANATion.—-First. 2 times 4 are 8. 
 Second. 2 times 5 + 3 times 4 are 22: write 2 and 
 
 carry 2. 
 Third. 3 times 5 + 2 are 17. 
 2. Multiply 256 by &4. 21504. 
 PROCESS. 
 256 
 st 
 21504 
 
 ExpLaNnation:— First. + times 6 are 2-44: write 4 and 
 carry 2. 
 
 Second. - times 5 + 8 times 6 + 2 are 70; write O 
 and carry @. 
 
 Third. 4 times 2+ 8 times5 + 7 are 55:3 write 5 
 and carry 5. 
 
 Fourth. 8 times 2+ 5 are 21. 
 
 3. Multiply 649 by 325° 210925, 
 PROCESS. 
 649 
 325 
 210925 
 
 EXpLANATION:—First. 5 times 9 are 45; write 5 and 
 carry 4. 
 
 Second. 5 times 4+ 2 times 9 + 4 are 42: write 2 
 and carry 4. 
 
 Third. 5 times 6+ 3 times 9 + 2 times 44+ 4 are 
 69; write 9 and carry 6. 
 
 Fourth. 2 times 6 + 3 times 4 + 6 are 30; write O 
 and earry 3. 
 
 Fifth. $3 times 6 -! 3 are 21. 
 
28 MULTIPLICATION. 
 
 4, “Multiply 1357 by 2468. 3349076. 
 PROCESS. 
 1357 
 2468 
 3349076 
 
 Exp.Lanation:—First. 8 times 7 are 563 write 6 and 
 carry 3. 
 
 Second. 8 times 5 + 6 times 7 + 5 are 87: write @ 
 and carry 8. 
 
 Third. S& times 3 + 4 times 7 + 6 timesS + 8 are 
 90; write 0 and carry 9. 
 
 Fourth. Stimes i+ 2 times @ + 6 times 3+ 4 
 times 5 + 9 are 69; write 9 and carry 6. 
 
 Fifth. 6 times I + 2 times 5 + 4 times 3 + 6 are 
 34; write £4 and carry 3. 
 
 Sixth. 4 times Il + 2 times 3+ 3 are 13; write 3 
 and carry 1. 
 
 Seventh. 2 times i + I are 3. 
 
 5. What cost 169 yards of silk at $3.25 per yard? 
 $549.25. 
 6 What must I pay for a farm of 360 acres at 27 dollars 
 per acre? $9720. 
 7. At the rate of 53 miles per hour, how far will a 
 car run in 54 hours? 2862. 
 8. How much can a man earn in 64 months at a salary 
 of 56 dollars a month? $3584, 
 9. What will be the cost a tract of land containing 
 4528 acres at 27 dollars per acre? $122256. 
 10. A is worth 5482 dollars, and B has 134 times as 
 much. How much have they both? $740070. 
 11. One cord of wood. contains 128 cubic feet. How 
 many cubic feet are there in 75 cords? 9600 
 
 12. A travels 79 miles a day and B 40 miles. How 
 much farther does A travel in 31 days than B? — 1209. 
 
EE ————— ee ee 
 
 MULTIPLICATION. 29 
 
 13. A regiment of soldiers has provisions to last 718 
 men 32days. How many days would they last one man? 
 22816. 
 14. J. Morgan & Co. make 2528 handles per day. At 
 this rate how many handles can they make in a year of 
 312 days? 788736. 
 15. It requires 2176 pickets to fence a square lot. 
 How many pickets will be required to fence 1575 lots of 
 the same size and shape? 3427200. 
 16. A drover sold 260 sheep at $3.20 per head, and 
 425 hogs at an average price of $9.64 per head. How 
 much did he get for them? $4929, 
 17. A drover bought 1386 cows at 382 dollars each, 
 and the same number of horses at 123 dollars each. 
 \\ hat did he pay for both? $21080. 
 18. A farmer counted the trees in his orchard and 
 found that he had 324 rows, each row containing 536 
 trees. How many trees were there in the orchard? 
 173664. 
 19. Two vessels are 4320 miles apart and sail towards 
 each other, one at the rate of 26 miles per hour, the other 
 at the rate of 18 miles per hour. How far apart will 
 they be in 37 hours? 2692. 
 20. Mr. Chas. Hale sold a farm of 325 acres at $65.50 
 per acre, and received in payment 345 sheep at $3.25 per 
 head, a note for $2684.95, and the rest in cash. How 
 much cash did he receive? $17481.30. 
 21. Two ships sail from the same place, in 
 opposite directions, one at the rate of 127 miles per day, 
 and the other at the rate of 119 miles per day. How far 
 will they be apart in 49 days? 12054. 
 
DIVISION. 
 
 >< 
 
 64. Division is finding how many times one num- 
 ber contains another. 
 
 65. The Dividend is the number to be divided. 
 66. The Divisor is the number by which we divide. 
 
 67. The Quotient is the result obtained by divis- 
 ion. 
 
 68. The Remainder is the number which is some- 
 times left after dividing. 
 
 69. The Sign of Division is +, and is read DIvID-— 
 ED BY. Division is also indicated by writing the dividend 
 above or at the right of the divisor, with a line between 
 them. 
 
 70. Short Division is that in wich the steps in 
 the solution are performed mentally. 
 
 @i. Long Division is the method of dividing 
 when all the work is written. 
 
 @2. When a Remainder occurs at the end of divis- 
 ion, it may be written over the divisor in the form of a 
 fraction and annexed to the quotient. 
 
DIVISION. ; 31 
 
 73. When the Divisor is large, consider only the 
 
 first two or three figures as a trial divisor, and compare 
 them with the first two or three figures of dividend. 
 
 @4. To Test the work, multiply the divisor by the 
 quotient, and to the product add the remainder, if any. 
 If the work is correct, the result will equal the dividend. 
 
 FACTORING: 
 
 @5. Frequently, when the divisor can be resolved in- 
 to Factors, the work can be shortened by dividing by 
 each of the factors in succession. If there is more than 
 one remainder, the true remainder may be found by sub- 
 tracting the product of the divisor and quotient from the 
 dividend. 
 
 @6. 1. Find the quotient of 594 divided by 18. 
 33. 
 
 Expianation :— Divide the dividend by one factor of 
 the divisor, the quotient by another, and so on tillall the 
 factors are used. 
 
 2. In one hogshead there are 63 gallons. How many 
 
 hogsheads in 15435 gallons? 245. 
 
 3. Ifa boat sail 24 miles an hour, how many hours 
 will it be in sailing 1824 miles? 16, * 
 4. The product of two numbers is 26973; one of the 
 
 numbers is 81; what is the other? aon. 
 
 5. A farmer raised 8288 bushels of wheat, averaging 
 56 bushels to the acre. How many acres did he plant. 
 148. 
 
3:2 DIVISION. 
 
 6. A man bought 160 acres of land at 25 dollars an 
 acre, giving in payment a house valued at 928 dollars, 
 and horses at 96 dollars each. How many horses did he 
 cive? 32. 
 
 CR NG Es TIN Gee Deere 
 
 @¢. This method often renders it quite easy to apply 
 Short Divisiom to large numbers when there are 
 ciphers at the right of the divisor. 
 
 @%. 1. Divide 8765 by 600. 14; Rem. 365. 
 
 PROCESS. 
 6 00)87 | 65 
 14 - 385 
 
 ExpLanation:—Cut off the ciphers at the right of the 
 divisor, and the same number of places at the right of 
 the dividend. Divide the remaining part of the dividend 
 
 by the remaining part of the divisor, and prefix the re- 
 mainder to the figures cut off for the true remainder. 
 
 2. If 120 acres of land cost 4080 dollars, what will 
 
 one acre cost? $34. 
 3. A drover paid 28800 dollars for 360 horses. How 
 much was that per head? $80. 
 4, Aman gave 5600 dollars for 160 acres of land. 
 How much was that per acre? $35, 
 5. How many months musta person labor to earn 
 3430 dollars at 70 dollars per month? 49, 
 - 6. Into how many lots of 40 acres each can a tract of 
 land containing 6400 acres be divided? 160. 
 
 7. Two men start from the same place and travel in 
 opposite directions, one at the rate of 23 miles a day, and 
 the other at the rate 27 miles a day. When 4750 miles 
 apart, how many days had they traveled? 95. 
 
a 
 
 nf 
 
 DIVISION. 33 
 
 ALIQUOT PARTS. 
 
 79. This method of division is the reverse of Multi- 
 plication by aliquot parts, shown on page 20. 
 $0. 1. Divide 450 by 25. | 18. 
 PROCESS. 
 456 
 4 
 
 18(00 
 
 Expianation:—Multiply by 4 and then divide by 
 100, by cutting off two places. ' 
 
 2. At $1.25 per yard, how many yards of silk can be 
 
 bought for 15 dollars? 12. 
 3. A dealer bought 25 horses for 2650 dollars. How 
 much did he give a head? $106. 
 4. At 163 cents per yard, how many yards of muslin 
 can be bought for 8 dollars? 48, 
 5. At 123 dollars per barrel, how many barrels of flour 
 can be bought for 450 dollars? 36. 
 6. Ifaman can save 250 dollars per year, in how 
 many years can he save 2250 dollars? J, 
 If a locomotive run 25 miles an hour, how many 
 hours will it be in running 1625 miles? 65. 
 8. If it takes 163 yards of silk to make a dress, how 
 many dresses can be made from 450 yards? eer 
 9. How many weeks will it take a printer to earn 
 250 dollars, if he receives 162 dollars a week? 15. 
 
 10. A farmer received 14 dollars for a load of apples 
 
 at 334 cents a bushel. How many bushels did he have? 
 42. 
 
 11. A farmer bought a tract of land for 2000 dollars. 
 
 How many acres did he buy, if it cost him 123 dollars 
 per acre? 160. 
 
. 34 DIVISION. 
 
 12. If the distance across the Atlantic ocean is 3000 
 miles, how many days will a vessel, sailing 125 miles per 
 day, be in crossing? 24. 
 
 COMPLEMENT DIVISION. 
 
 $i. When the divisor is a Little Less than 100, 
 1000, ete., the following method will be found valuable. 
 
 $2. Divide 2138672 by 98. 21823; Rem. 18. 
 
 PROCESS. 
 
 21386 | 72 
 
 427 | 72 
 
 8 | 54 
 
 | 16 - 
 
 21823 | 14 
 
 brekie’ 
 
 is 
 
 Expianation:—Cut off from the right of the dividend, 
 by a vertical line, as many figures as the divisor contains. 
 Multiply the part on the left of the line by the comple- 
 ment of the divisor and set the product under the divi- 
 dend. Multiply the part of this product on the left of 
 the line by the same multiplier and set down as_ before. 
 Continue in like manner until no figure remains on the left 
 of the line. Add theseveral results and forevery 1 carried 
 across the line, add the number used as a multiplier. 
 The part on the left of the line will be the quotient, 
 and the part on the right, the remainder. Ifthe remain- 
 der is equal to or larger than the divisor, carry one to 
 quotient, and the excess will be the true remainder. 
 This rule may be advantageously applied to many other 
 numbers. To illustrate: If we wish to divide by 49 
 we may divide by 98 and multiply by 2, or if we wish 
 
er 
 
 DIVISION. 35 
 to divide by 198, we may divide first by 99 and then 
 that result by 2. 
 
 2. Divide 2326 by 32. 72; Rem. 22. 
 PROCESS. 
 
 23 | 26 
 92 
 
 ExpLanation:—In this case we divide by 96, or three 
 times the given divisor, which gives a quotient of 24. 
 This multiplied by 3 gives 72, which is the true quoti- 
 ent. 
 
 3. In one cask there are 94 gallons. How many casks. 
 
 in 4230 gallons? 45, 
 
 4. Ifaship sail 99 miles a day, in how many days 
 will it sail 12375 miles? 125. 
 
 5. In 240 quires of paper there are 5760 sheets. How 
 many sheets in a quire? 24. 
 
 6. In one day there are 24 hours; how many days 
 are there in 5200 hours? 216 da. 16 hr. 
 
 7. A man bought a drove of 95 horses for 4750 dollars. 
 How much did he give apiece? $50. 
 
 8. Amansolda farm of 480 acres for 15360 dol- 
 lars. How much did he get per acre? $32. 
 
 9. If aman can earn 998 dollars in a year, how many 
 years will it take him to earn 15968 dollars? 16. 
 
 10. A farmer having 6272 dollars bought land at 32 
 dollars per acre. How many acres did he buy? 196. 
 
 11. A man wishes to invest 1645 dollars in railroad 
 stock. How many shares can he buy at 47 dollars per 
 share? 35. 
 
CANCELLATION. 
 
 >< 
 
 $3. Cancellation is the method of shortening 
 computations by omitting common factors from both 
 dividend and divisor. 
 
 $4. Write the dividend on the right and the divisor - 
 on the left of a verticaL LINE. Cancel all common fac- 
 tors and divide the product ofthe remaining factors of 
 the dividend by the product of the remaining factors of 
 the divisor. 
 
 $5. <A Factor in one term will cancel Two or more 
 factors in the other term, when their product is equal to 
 the former. 
 
 $6. Cancellation is frequently rendered easier by an- 
 nexing Decimal Ciphers. For example: Suppose 
 we have every number on the left of the line canceled, . 
 except 4, it also will disappear by cancelling, if we annex 
 two decimal ciphers to some number on the right. 
 
 $@. In some cases it is better to reduce decimals to 
 Common Fractions. To illustrate: If we have $1.50 
 on the right of the line, perhaps it would be better to call 
 it of a dollar, writing the 8 on the right of the line 
 and the 2 on the left. 
 
 $8. To Test the work, write the statement as it was 
 
PROPERTIES. 37. 
 
 first written, with the result also on the left, and if all 
 cancel, the work is correct. 
 
 $9. 1. Divide 4 x 5 x 7 by 4x 7. D. 
 STATEMENT. 
 
 4,4 
 5 
 W\7 
 
 2. How many barrels of flour, at $8 a barrel, must be 
 given for 48 tons of coal, at 6 dollars a ton? 36. . 
 
 3. How many yards of muslin, at 35 cents per yard, 
 will pay for 20 yards of calico, at 14 cents per yard? 
 
 8, 
 
 4. How many cords of wood, at 4 dollars per cord, 
 must be given for 3 tons of hay at 12 dollars per ton? 
 
 9. 
 
 5. How many bushels of corn, at 63 cents a bushel, 
 must be given for 7 pieces of cloth, each containing 27 
 yards, at 40 cents a yard? 120. 
 
 6. A farmer bought 12 cows at 25 dollars a_ head, 
 and paid for them in hay at 15 dollars per ton. How 
 many tons of hay were required? 20. 
 
 7. A farmer buys 4 pieces of cloth, each containing 
 60 yards, at 15 cents per yard and pays for it with wheat 
 at $1.50 per bushel. How many bushels are required? 
 
 24. 
 
 8. Exchanged 15 pieces of calico, each containing 
 30 yards at 10 cents a yard, for 3 pieces of muslin, each 
 containing 50 yards. What was the muslin per yard? 
 
 $0.30. 
 
 9. I sold 70 kegs of nails of 100 pounds each at 5 cents 
 a pound, paying for them with pieces of calico of 35 
 yards each, at 8 cents a yard. How many pieces did I 
 give? 125. 
 
 10. A merchant bought 24 tubs of butter, each contain- 
 ing 32 pounds, at 40 cents a pound, paying for them with 
 
38 PROPERTIES. 
 
 4 patterns of silk of 48 yards each. How much was the 
 silk per yard? $1.60. 
 11. A miller bought 4 loads of wheat, each containing 
 18 sacks of 2 bushels each, worth $1.25 per bushel, and 
 paid for it in flour at 6 dollars per barrel. How many 
 barrels of flour were required? 30. 
 12. I bought 75 barrels of apples, of 4 bushels each, at 
 40 cents a bushel and paid for them with barrels of po- 
 tatoes, of 5 bushels each, at 50 cents per bushel. How 
 many barrels of potatoes did I give? 48, 
 13. <A dealer exchanged 480 bushels of wheat worth 
 70 cents a bushel, for an equal number of bushels of 
 barley, worth 84 cents per bushel, and corn worth 56 
 cents per bushel. How many bushels of each did he 
 receive? 240. 
 
 GREATEST COMMON DIVISOR. 
 
 90. The Greatest Common Divisor of two or 
 more numbers is the greatest number that will exactly 
 divide each of them. 
 
 91. 1. What is the greatest common divisor of 42, 
 84, and 126? 42. 
 
 PROCESS. 
 
 alas - $4 - 126 
 
 3/21 - 42 - 63 
 CPi tr dene tS | 
 CS eee og 
 
 2x3x7=42 
 
 ExpLANATION.—-Write the numbers one beside another, 
 
PROPERTIES. 39 
 
 with a vertical line at the left, and divide by any com- 
 mon factor of Att the numbers. Divide the quotient in 
 like manner, and so on until the quotients have no com- 
 mon factor; the product of all the divisors will be the 
 greatest common divisor. 
 
 2. A man bought three pieces of land containing 28, 
 36, and 44 acres respectively, which he wished to divide 
 into the largest possible fields, each having the same 
 number of acres. How many acres can he put ina field? 
 
 4. 
 
 3. A gentleman having a triangular piece of land, 
 the sides of which are 176 feet, 224 feet, and 320 feet, 
 wishes to enclose it with a fence having panels of the 
 greatest possible uniform length. What will be the 
 length of each panel? 16. 
 
 4, I have 3 pieces of carpet, the first containing 54, 
 the second 90, and the third 126 yards. If these pieces 
 were cut into the largest possible equal pieces, what will 
 be their length? 18. 
 
 5. A farmer wishes to put 182 bushels of wheat and 
 234 bushels of oats into the least number of bins possi- 
 ble, that shall contain the same number of bushels. How 
 many bushels must each bin hold? 26. 
 
 6. <A grocer had 126 apples and 189 pears which he 
 wished to divide into smaller heaps, each containing the 
 same number. What is the largest number that each 
 heap may contain? 63. 
 
 LEAST COMMON MULTIPLE. 
 
 92. The Least Common Multiple of two or 
 more numbers is the least number exactly divisible by 
 them. 
 
40 PROPERTIES, 
 
 93. 1. Find the least common multiple of 12, 30, 
 and 70. 420). 
 
 PROCESS. 
 
 2/12 - 30 - 70 
 
 S106 2 se eenanl 
 
 Be LR BS 
 
 Peay bee ie 
 2 Xb Oe ed ee AeO 
 
 Exprianation:— Write the numbers one beside another, 
 with a vertical line at the left, and divide by any prime 
 number which will exactly divide rwo or more of them. 
 Write the quotient and undivided numbers beneath. Di- 
 vide these in the same way, and thus continue until no 
 two numbers in the lowest line have a common divisor. 
 The product of these numbers and the divisors will be 
 the least common multiple. A composite number may 
 be used for a divisor, when it is contained in ALL the 
 given numbers. 
 
 2. How long must a box be that no room may be lost 
 in packing in it books 6 inches, 8 inches or 12 inches 
 long? 2 ft. 
 
 3. The least common multiple of 12, 16, 21, and some 
 unknown prime number to these three, is 3696. What is 
 the unknown number? af 
 
 4. How many bushels will the smallest bin contain 
 that can be emptied by taking out either 3 bushels, 5 
 bushels, or 14 bushels at a time? 210. 
 
 5. What is the least sum of money for which I can 
 purchase either sheep at 4 dollars; cows at 21 dollars, 
 oxen at 49 dollars, or horses at 72 dollars a head? 
 
 $3528. ° 
 
PROPERTIES. 41 
 
 6. Whatis the smallest number that is exactly di- 
 visible by 16, 24, and 30? 240. 
 
 7. Two men start at the same time to walk around a 
 field. One can go around in 3 hours, the other in 4 
 hours. In how many hours will they again meet at the 
 starting place? 12 
 
 ~_s 
 
 HANDY HINTS AND HELPS. 
 
 94. 'Twois an exact divisor of any even number. 
 
 95. Three is an exact divisor of any number, the 
 sum of whose digits is divisible by 3. 
 
 96. Four is an exact divisor of a number, if the 
 number expressed by its two right hand figures is divisible 
 by 4. 
 
 97. Five is an exact divisor of any number ending 
 with O or 5. 
 
 98. Six is an exact divisor of any EVEN number, 
 the sun of whose digits is divisible by 3. 
 99. Eight is an exact divisor of a number, if the 
 
 ‘number expressed by its three right hand figures is divi- 
 sible by 8. 
 
 109. Nine is an exact divisor of any number the 
 sum of whose digits is divisible by 9. 
 
 101. Ifan Kvea Number is divisible by an odd 
 number, it is also divisible by twice that number. 
 
 102. Any Number consisting of three equal digits, 
 as 111, 222, ete., is divisible by 37, and the sum of the 
 digits will be the quotient. 
 
 103. The Product of any three consecutive numbers 
 is divsible by 6. 
 
42 PROPERTIES. 
 
 104. The Product of any four consecutive numbers 
 is divisible by 24. 
 
 105. The Produet of any four consecutive numbers 
 plus 1 is a perfect square. 
 
 106. Odd Numbers wultiplied together always 
 give an odd number. 
 
 107. The Geometrical Mean of two numbers is 
 equal to the square root of their product. 
 
 108. The Product of the sum and difference of 
 of two numbers is equal tothe difference of their squares. 
 
 109. If the Difference of two numbers be added 
 to their sum, the amount will be double the larger num- 
 ber. If their difference be subtracted from their sum, 
 the remainder will be double the smaller number. 
 
= 
 
 COMMON FRACTIONS. 
 
 110. A Fraction is a part of a whole number. 
 
 Hit. Common Fractions are expressed by two 
 numbers, one written above the other with a horizontal 
 line between them. 
 
 112. The Numerator is the number which is writ- 
 ten above the line. 
 
 113. The Denominator is the number which is 
 written below the line. 
 
 114. A Proper Fraction is one in which the 
 numerator is less than the denominator. | | 
 
 1415. An Improper Fraction is one in which the 
 numerator equals or exceeds the denominator. 
 
 116. A Mixed Number is one consisting of an 
 integer and a fraction. 
 
 Li7. A Compound Fraction is a fraction of a 
 fraction. 
 
 118, A Complex Fraction is one having a_frac- 
 
 tion in one or both its terms. 
 
 119. To reduce fractions to Larger Terms, mul- 
 tiply both terms by the same number. 
 
44 COMMON FRACTIONS. 
 
 120. To reduce fractions to Smaller Termes. 
 divide both terms by the same number. 
 
 121. To reduce mixed numbers to Improper 
 Fractions, multiply the whole number by the given 
 denominator, to this product add the numerator, if any, 
 and write the result over the given denominator. 
 
 122. To reduce improper fractions to whole or 
 Mixed Numbers, divide the numerator by the denom- 
 inator. 
 
 £23. To reduce two fractions to a Common BDe- 
 
 nominator, multiply the denominators together for a 
 common denominator and cross multiply for the numer- 
 ators--that is multiply the numerator of the first fraction 
 by the denominator of the second, and the numerator 
 of the second by the denominator of the first. “ 
 
 124. If there are several fractions it is better to find 
 by inspection the Least Common Denominator, 
 divide this denominator by the denominator of each fraec- 
 tion and multiply both terms of the fraction by the 
 quotient. 
 
 125. To reduce Complex Fractions, write the 
 product of the means for the numerator and the product 
 of the extremes for the denominator of a simple fraction. 
 
 126. To Add or Subtract fractions, reduce them 
 to similar fractions, and add or subtract the numerators 
 as required and place the result over the common denom- 
 inator. When there are mixed numbers, add or sub- 
 tract the whole numbers and the fractions separately. 
 
 127. To Multiply fractions, multiply the numera- 
 tors together for the numerator, and the denominators 
 for the denominator of the product. Cancellation will be 
 found valuable in this case. Write the numerators on 
 the right of the line and the denominators on the left. 
 
COMMON FRACTIONS. 45 
 128. To Multiply two fractional numbers ending 
 with 4, take the product of the whole numbers, increase 
 this by $ their sum and annex 4 to to the product. 
 129. To Divide fractions. invert the divisor and 
 multiply. 
 130. Business Wen only care to have. the answer 
 correct to the nearest cent that is they disregard the 
 fraction. When it is a half cent or more they call it an- 
 
 other cent. 
 
 lL. Reduce ee ® to YAths. 8, 18 ae 
 2. Reduce +2, +2 to lowest terms. 2 3 
 3. Reduce 3, 3, to a common denominator. Br, te. 
 
 4. Reduce 3, \4, 2 to their least common denominator. 
 
 5 64 0 
 
 603 60> 0. 
 5. Five men share $11.56} equally. What is the 
 
 share of each? $2.314. 
 6. Change % to an equivalent fraction having 24 for 
 its denominator. oy. 
 
 7. Change #3 to an equivalent fraction expressed in 
 
 its lowest terms. - 
 
 8. Change 4% to an equivalent fraction, having 9 for 
 
 its denominator. 3 
 9. What will be the cost of 125 pounds of butter at 
 124 cents per pound? $1.564. 
 
 10. Ifa boy earn } of a dollar per day, how much 
 can he earn in 12 days? $9. 
 
 11. A man owned a boat worth 2700 dollars. How 
 much should he receive for ? of it? $1800. 
 
 12. A hunter shot 24 ducks one day, and } as many 
 the next. How many did he shoot in both days? = 44. 
 
46 COMMON FRACTIONS. 
 13. When wheat is selling at 3 of a dollar per bushel, 
 how many bushels can be bought for 24 dollars? 30. 
 l4. If from a bin containing 375% bushels of wheat, 
 2163 bushels are taken, how many bushels will remain? 
 159%. 
 15. James spent 4 of a dollar on Monday and 4 of a 
 dollar on Tuesday. What part of a dollar did he spend 
 both days? 3. 
 16. <A farmer received 21% dollars for hay, 343 dollars 
 for a cow, and 98? dollars for a horse. How much did 
 he get for all? $1555 
 17. What will be the cost of 74 yards of muslin at 
 at 12$ cents per yard, and 125 yards of gingham at 18} 
 cents per yard? $3,285. 
 18. What will be the cost of 8 loads of coal, each con- 
 
 taining 15? bushels, at 124 cents per bushel? — $15.75. 
 
 STATEMENT. 
 8 
 
 4 | 63 
 2| 25 
 
 19. A farmer sold ? of his cows, and ? of his sheep, 
 and then had 270 of each remaining. How many of 
 each had he at first? Cows, 450; Sheep, 945. 
 
 20. A man walked 254 miles on Monday, 28} on 
 Tuesday, 273 on Wednesday, 26% on Thursday. How 
 many miles did he walk? ° 1083. 
 
 21. A clerk earns $374 per month and pays $12} for 
 board, $23 for washing, and $43 for other expenses. 
 
 How much does he save? $173. 
 
COMMON FRACTIONS. 47 
 
 22. A lady having a 20 dollar bill paid $34 for dress- 
 
 voods, $9} for a bonnet, and #43 for sundries. How 
 much money had she left? $2. 
 
 23. A farmer received } of a dollar for eggs, 3 of a dol- 
 
 lar for butter and £ of a dollar for other produce. How 
 much did he receive for all? $234. 
 
 24. By selling 8 yards of cloth for 24 dollars, | made 
 
 * of the cost. TI paid for it with wheat at 3 of a dollar 
 per bushel. How many bushels did I give? 25. 
 
 25. James has ? of an orange. He gives Charles 3 
 
 of this and divides the remainder equally between three 
 
 boys. What part does each of the three boys receive? 
 1 
 
 Re 
 
DECIMALS. 
 
 131. A Decimal is one or more of the decimal di- 
 visions of a unit. 
 
 132. A Pure Decimal is one which consists of 
 decimal figures only. 
 
 133. A Mixed Decimal is one which consists of 
 « whole number and a decimal. 
 
 134. A Complex Decimal is one which contains 
 a common fraction at the right of the decimal. 
 
 135. A Cireulating Decimal is one in which a 
 figure or set of figures is continually repeated in the 
 same order. 
 
 136. Add and Subtract decimals as ordinary 
 numbers. 
 
 137. Multiply decimals as whole numbers, and in 
 the product point off as many decimals as are in both 
 factors. “4 
 
 138. Divide decimals as in whole numbers, and 
 point off in the quotient as many decimal places as the 
 dividend has more than the divisor. Decimal ciphers 
 may be annexed to the dividend if necessary. 
 
 139. Common Fractions may be reduced to 
 decimals by dividing the numerator by the denominator, 
 
DECIMALS. 49 
 
 140. A Circulating Decimal may be reduced 
 to a common fraction, by writing the repetend, or repeat- 
 ing part, for the numerator of a fraction, with as many 
 9s as there are places in the repetend for the denomi- 
 nator. 
 
 I41t. The various denominations of United States 
 Money have decimal relations, and for ordinary busi- 
 ness purposes two decimal places are sufficient. In writ- 
 ing results, count fractions amounting to five mills or 
 more as one cent and discard smaller fractions. 
 
 142. All Contractions used with integers, may 
 also be used with decimals. 
 
 1435. 1. Reduce 13 toa common fraction. sy. 
 
 2. What cost 33.21 yards of cloth at $4.41 per yard? 
 $146.46. 
 
 3. There are 7.92 inches in a link. How many inches 
 in 300 links? 2376. 
 
 4. When shingles are worth $2.50 per thousand, how 
 much will 3750 cost? $9.38. 
 
 5. A merchant sold 31.25 yards of muslin for $7.8125. 
 How much was that per yard? $0.25. 
 
 6. When land is worth $137.18 per acre, how much 
 must be paid for a farm of 38 acres? $5212.84. 
 
 7. Cowell & Co. sell galvanized wire at $3.75 per hun- 
 dred. How much must I pay fora bale containing 118 
 pounds? $4.24. 
 
 8. Bought 3 loads of hay, the first containing 1.06 
 tons, the second 1.05 tons, and the third 1.25tons. What 
 did it cost at $5.12 per ton? $17.22. 
 
 9. I bought 32) bushels of wheat at the rate of 16 
 bushels for $10.04, and sold it at the rate of 20 bushels 
 for $17.50. What was my profit? $79.20. 
 
 10. A dealer bought 3 loads of wood, the first contain- 
 
 ing 1.07 cords, the second 1 cord, and the third .945 
 cords. What did it cost at $3.60 per cord? $10.85. 
 
50 DECIMALS. 
 11. A grain dealer expended $70.15 in the purchase of 
 rye at $.95, wheat at $1.37, and corn at $.73 per bushel, 
 
 buying an equal quantity of each. How many bushels 
 did he purchase in all? 69. 
 
 CONTRACTED MU td) rishi Aas 
 
 144. When the Product of two decimal numbers 
 is not required to contain figures below a certain order, 
 the work may be shortened as illustrated below. 
 
 145. 1. Multiply 26.48352 by 12.452, retaining two 
 
 decimals in the result. 329.77 
 ORDINARY METHOD. CONTRACTED METHOD. 
 2648352 26.48352 
 12.452 254.2 1 
 51|296704 2648 4 
 132 \ 41760 a297 
 1059;'\)3408 1059 
 5296 :170A 132 
 26483 52 > 
 $329.7¢7\' 279104 29.04 
 
 Expianation:—In the contracted method the order of 
 the figures of the multiplier is ReEveRseD and the unit 
 figure written under the decimal of the multiplicand to 
 be retained. Begin with the right hand figure of the 
 multiplier to form partial products, dropping one figure 
 of the multiplicand, as we multiply by each successive 
 figure of the multiplier, making due allowance for the 
 units arising from the product of the neglected figures, 
 adding the nearest number of tens. Ciphers may be 
 annexed to the multiplicand if necessary. 
 
 2. Multiply 123.215263 by 15.16, reserving two deci- 
 mals in the product. 1867.94 
 
DECIMALS. D1 
 
 CONTRACTED DIVISION. 
 
 146. Contracted Division of decimals is the re- 
 verse of contracted multiplication. 
 147. 1 Divide 329.7727915 by 26.48352, retaining 
 two decimals. 12.45. 
 PROCESS. 
 
 2648 352)3829.77279 15 (12.45 
 264858 
 
 649 
 
 EXPLANATION: —In this case we see by inspection 
 that the first two figures of the quotient will be whole 
 numbers, and that four divisions must be made. Us- 
 ing as many figures of the divisor as there are divisions 
 to be made, we multiply by the first figure of the quo- 
 tient, making due allowance for units arising from the 
 product of it and the last omitted figure of the divisor. 
 At each successive division omit one figure of the divisor, 
 beginning at the right. 
 
 2. Divide 1867.94361727 by 123.215263, reserving two 
 decimal places in the quotient. 15.16. 
 
 iy. OF ILL LIB. 
 
PERCENTAGE. 
 
 f4%. The Base is the number on which percentage 
 is computed. 
 
 149. The Rate is the number which denotes how 
 many hundredths are to be taken. 
 
 150. The Percentage is the product obtained by 
 multiplying the base by the rate. 
 
 151. The Amount is the sum of the base and _per- 
 centage. 
 
 152. The Difference is the difference between the 
 base and percentage. 
 
 153. The Sign of Percentage is 4, and is read 
 PER CENT. 
 
 154. The following Equations are adapted to the 
 solution of all simple problems in percentage. 
 
 FORMULAS. 
 Percentage = Base x Rate. 
 Rate = Percentage + Base. 
 Base = Percentage + Rate. 
 Amount = Base x (1 + Rate.) 
 Difference = Base x (1 — Rate.) 
 
PERCENTAGE. 53 
 
 155. 1. A farmer who hada flock of 270 sheep, 
 sold 334% of them. How many did he sell? 9). 
 
 PROCESS. 
 3)270 
 90 
 
 Exputanation: Take sucha part of the given nuim- 
 ber as the rate per cent is of 100. 
 
 2, What per cent is lost by selling cloth at 75 cents 
 
 which cost 1 dollar? 25 
 3. What per cent is gained by selling cloth for 1 dol- 
 lar which cost 75 cents? 334 
 4. A farmer having a flock of 250 sheep sold 20¢ 
 of them. How many had he left? 200. 
 5. Bought corn at 50 cents a bushel. At what price 
 must it be sold to gain 20 per cent.? $.60. 
 
 6. A farmer sold a horse for 75 dollars, which was at 
 a loss of 25%. What was the cost of the horse? $100. 
 7. Bought cloth at 50 cents per yard and sold it at 
 60 cents per yard. What per cent did I gain? 20 
 8. A man sold a horse at a profit of 334%. If the 
 horse cost him 120 dollars, what did he get for him? 
 $160. 
 9, A farmer sold a horse for 100 dollars, which was at 
 a gain of 257. What was the cost of the horse? 
 SSO. 
 10. [send my agent 500 dollars to invest in goods. 
 After deducting 3% commission how much will be invest- 
 ed? $485. 
 ll. By selling cloth at $4.50 per yard I lose 104. 
 What will be my gain per cent, if I sell at 6 dollars per 
 yard? 20. 
 
INTEREST. 
 
 >< — 
 
 156. Interest is money charged for the use of 
 money. 
 
 157. The Principal is the sum on which interest 
 is counted. 
 
 158. The Amount is the sum of the principal 
 and interest. 
 
 159. Simple Interest is counted on the prin- 
 cipal only. 
 
 160. Compound Interest is interest on both 
 principal and unpaid interest. 
 
 161. The old Cancellation Method of comput- 
 ing interest is undoubtedly the best for general use. The 
 following diagram is intended to illustrate how the terms 
 should be arranged. 
 
 FORMULA. 
 
 ee 
 
 Principal. ' 
 
 ExpLaNnation:—We write the principal, rate per cent, 
 
—- 4 @ 
 
 INTEREST. Dy» 
 and number of days to run, on the right, and 860, orits 
 factors, on the left. If the time is in months, write #2 
 only on the left. Point off two places in the result when 
 the principal contains dollars only, and if it contain cents, 
 point off two more. The operation may be shortened in 
 various ways. Suppose the rate is 6 per cent, the result 
 would be the same if we omit the rate from the right 
 and write 60 only on the left. The divisor for any rate 
 may be found by dividing 360 by the rate. In count- 
 ing the time. regard 30 days as a month. 
 
 162. 1. Find the simple interest on 120 dollars for 
 1 month and 12 days, at &7. $1.12. 
 
 STATEMENT. 
 
 |1290 
 360 O08 
 | 42 
 2. Find the simple interest on 600 dollars for 24 days 
 at 7 per cent. $2.80. 
 3. Find the simple interest on $36.84 for 5 months, at 
 % per cent. #1.38. 
 4. What is the simple interest on 640 dollars for 21 
 days, at 6 per cent? $2.24. 
 >). Find the simple interest on 720 dollars for 1 year, 
 4 months, 15 days, at 6 per cent. $59.40. 
 6. Find the simple interest on 2500 dollars for 7 
 months and 20 days, at 5 per cent. $79.86. 
 7. Find the simple interest on 6 dollars for 6 years, 
 6 months and 6 days, at 6 per cent. $2.35. 
 8. What will be the amount of 720 dollars for 20 
 days at 6 per cent, simple interest? $722.40. 
 9. Find the simple interest on 125 dollars for 1 year, 
 2 months and 12 days, at 6 per cent. $9. 
 10. Find the amount of $256.20 for 1 year, 3 months 
 and 18 days, at 10 per cent, simple interest. $33.31. 
 
D6 INTEREST. 
 
 11. Find the amount of 200 dollars, for 2 years, at 6 
 
 per cent, compound interest, payable annually. 
 $224.72. 
 
 PARTIAL PAYMENTS: 
 
 163. Partial Payments made on notes, mort- 
 gages and other interest-bearing certificates of indebted- 
 ness are called INDORSEMENTS. 
 
 164. Nearly every Business Man has his own 
 method of computing interest when partial payments 
 have been made, and the subject has given rise to much 
 litigation. We give only the method which has been 
 adopted by the Supreme Court of the United States. 
 
 UNITED: STATES METHOD: 
 
 165. Find the Amount of the principal to the time 
 of the first payment, and subtract the payment for a new 
 principal. If the payment is less than the interest, 
 find the amount of the note to the time when the sum of 
 the payments shall exceed the interest due. Subtract 
 the sum of the payments and proceed as before. 
 
 166. This excellent Method is very simple. The 
 only difculty occurs when the interest is greater than 
 the payment. When this occurs, however, we can gener- 
 ally see by inspection that the payment is not sufficient 
 to discharge the interest,so that in such cases we 
 may compute the amount at once up to the time when 
 the sum paid is not less than the accrued interest, thus 
 saving unnecessary work, by mentally estimating the in- 
 terest in advance. 
 
INTEREST. 57 
 
 167. 1. A note for 150 dollars is dated June 12, 
 1888. Indorsed: October 12, 1889, 32 dollars; October 
 
 What was 
 $120.68. 
 
 12, 1790, $6.80; February 12, 1891, $21.60. 
 
 due May 27, 1892, interest at 6 per cent? 
 PROCESS. 
 
 Days. 
 i2 
 12 
 12 
 12 
 24 
 
 Years. Mon. 
 ss - 6 - 
 $9 10 
 90 10 
 91 - 2 - 
 
 92 _ | ee 
 ca ; Paid. 
 $32.00 
 
 It - 4 - 90 
 
 tt : 0 - Oo - eae 
 Oo - 4 - 0 21.90 
 
 pa eae ek et ies (1° Pw) 
 
 150.00 
 12.00 
 
 Principal. 
 Ent. lt yr. 4 mon. 
 
 162.00 
 32.00 
 
 Amount. 
 Payment. 
 
 130.00 New Principal. 
 
 10.40 
 
 140.40 
 28.40 
 
 112.00 
 §.68 
 
 Int, l yr. 4 mon. 
 
 Amount, 
 Payment. 
 
 New Principal. 
 Int. lL yr. 3 m. 15 d. 
 
 Balance due. 
 
 ExpLanation:—Write the dates under each other, 
 
 in 
 
5S INTEREST. 
 
 their regular order, subtract downward, and write the 
 intervals beneath with the payments opposite. 
 
 The first interval is 1 yr. 4 mon.; the interest on 150 
 dollars for this time is 12 dollars and the amount is 
 162 dollars; 162 dollars less 32 dollars is 130. dol- 
 lars, the second principal. 
 
 Consider mentally that the next interval of I year 
 would afford about 8 dollars interest, which is more 
 than the payment; hence we combine two intervals and 
 their payments, making the time I yr. 4 mon. and the 
 combined payment $28.40, which is enough to meet 
 accrued interest. The amount of 130 dollars for I yr. 
 4 mon. is $140.40; $140.40 less $28.40 is L12 dol- 
 lars, the third principal. 
 
 The amount of 112 dollars for the last interval, I yr. 
 $3 mon. 15 da. is $120.68, the balance due at date of 
 settlement. 
 
 2. A note for 120) dollars is dated October 15, 1889, 
 Indorsed: October 15, 1890, 1000 dollars; April 15, 1891, 
 200 dollars. How much remained due October 15, 1891. 
 interest at 6 per cent? $82.56. 
 
 3. A note for 600 dollars is dated July 14, 1890. In- 
 dorsed: May 26, 1891, $131,20; December 20, 1892, 40 
 dollars; September 14, 1893, 175 dollars. What was due 
 July 14, 1894, interest at 6 per cent? $371.70. 
 
 4. A note for 850 dollars is dated January 1, 1892. 
 Indorsed: July 1, 1892, $100.62; December 1, 1892, $15.28; 
 August 13, 1893, $175.75. What was due on taking up 
 the note January 1, 1894, interest at 6 per cent? 
 
 | $650.39. 
 
 5. A note for 400 dollars is dated March 4, 1888. In- 
 dorsed: September 4, 1888, 10 dollars; January 4, 1889, 
 30 dollars; July 4, 1889, 11 dollars; September 4, 1889, 
 80 dollars. What was due March 4, 1890, interest at 6 
 per cent? $313.33. 
 
INTEREST. 5Y 
 
 6. A note for 1750 dollars is dated November 23, 
 1892. Indorsed: November 26, 1894, 500 dollars; July 
 19, 1895, 50 dollars; September 2, 1895, 600 dollars, De- 
 cember 29, 1895, 75 dollars. What was due February 
 11, 1896, interest at 7 per cent? $879.71. 
 
 7. A note for 2150 dollars is dated September 20, 1893. 
 Indorsed: December 15, 1893, 75 dollars; February 4, 
 1894, 200 dollars; April 3, 1894, 150 dollars; July 1, 1894, 
 500 dollars: December 16, 1894, 1000 dollars. What was 
 due March 20, 1895, interest at 8 per cent? $439.28, 
 
DISCOUNT. 
 
 >< 
 
 168. Discount is a deduction made from a sum of 
 money to be paid. 
 
 169. Commercial Discount is a deduction from 
 the price of an article or from a _ bill without regard to 
 time. 
 
 170. The Net Price is the list price less the dis- 
 count. 
 
 17. 1. What is the net price of a parlor organ, 
 listed at 300 dollars, with 50, 20, and 10 off? $108. 
 PROCESS. 
 
 00 x .80 x .90 = .360600 x 300 = $108. 
 
 ExpLanation:—When several discounts are allowed 
 from list prices, subtract each from 100 and multiply 
 the remainders and the list price together to find the 
 selling price. . | 
 
 2. Find the value of a bill of goods amounting to 
 
 275 dollars at 5 per cent discount? $261.25. 
 3. What is the net price of a bill of hardware, listed 
 at 80 dollars, with 25, 20, and 10 off? $43.20 
 
 4, If shoes marked $2.50 per pair are sold at 10 per 
 cent discount, what is the net price? $2.50. 
 
DISCOUNT. 61 
 
 7. Sold county warrants amounting to 30 dollars at 
 
 5 per cent discount. What were the proceeds? 
 $28.50. 
 
 10. A merchant receives 2% off for cash. How much 
 does he save by discounting a bill of 240 dollars? 
 
 $4.80. 
 
 5. What is the cash value of a bill of goods amount- 
 ing to 500 dollars with 20 per cent discount, and 5 off for 
 cash? $380. 
 
 8. Ifamerchant sell goods at the catalogue price 
 from which he receives a discount of 20 per cent, what 
 per cent does he make? 25. 
 
 6. A merchant buys goods at a discount of 20 per 
 cent from list price and sells them at 20 per cent above the 
 list price. What per cent does he make? 50. 
 
 BANK DISCOUNT: 
 
 172. Bank Discount is simple interest, paid in 
 advance, for three days more than the specified time. 
 
 173. The Proceeds of a note are the face of the 
 note, less the discount. 
 
 i174. The Maturity of a note occurs on the last 
 day of grace, but if the last day of grace fall on Sunday 
 or a legal holiday, the note matures on the preceding 
 day. 
 
 1. What is the bank discount on $74.16 for 3 months, 
 
 18 days, at 5 per cent? $1.14. 
 2. What is the bank discount on 3860 dollars for 5 
 months, 8 days, at 6 per cent? $9.66. 
 
 8. Whatis the bank discount on 48 dollars for 4 
 months, 12 days, at 8 per cent? $1.44. 
 
62 DISCOUNT. 
 
 ERU EAD TSS@UNGE 
 
 175. True Discount is the difference between 
 the face of a debt and its present worth. 
 
 176. The Present Worth of a debt, due at some 
 future time without interest, is the sum which put at 
 interest at the specified rate, will amount to the debt 
 when it becomes due. 
 
 1L7¢@. To Find the present worth, divide the given 
 sum by the amount of one dollar for the given time and 
 rate. 
 
 1. When money is worth 6 per cent, what is the pres- 
 ent worth of 1300 dollars, payable in 5 years? $1000. 
 2. Bought 2895 dollars worth of goods, on two year’s 
 credit. What sum will pay the debt now, if money is 
 worth 5 per cent? $2631.82. 
 3. Bought a house for $975.50, payable in 18 months 
 without interest. How much will I gain by paying the 
 debt now, money being worth,6 per cent? $80.55. 
 4. How much will I gainif instead of paying 5400 
 dollars cash for a piece of property, I pay 6000 dollars 
 in 16 months, money being worth 9 per cent. 42.86. 
 
PROPORTION. 
 
 17s. Proportion is an equality of ratios. 
 
 179. Ratio is the relation of one number to another 
 of the same kind. 
 
 180. The Extremes of a proportion are the first 
 and fourth terms. 
 
 181i. The Means of a proportion are the second 
 and third terms. 
 
 182. The Sign of Proportion is the double colon, 
 ::, or the sign of equality, =. 
 
 183. Simple Proportion is employed for the 
 solution of problems in which three quantities are given, 
 so related that a fourth may be determined from them. 
 
 184. 1. If six menearn 75 dollars in a week, how 
 much will 10 men earn in the same time? $125. 
 
 PROCESS. 
 
 mh) 
 
 6 10 
 ExpbLanation:—-Write the third term, or that number 
 which is the same kind as the answer, on the right. 
 When the result is to be larger than the third term, 
 place the larger of the other two numbers on the right 
 and the smaller on the left, but reverse this order when 
 
64 PROPORTION. 
 
 the result is to be less than the third term. Apply can- 
 cellation and the result will be the required term. 
 
 2. If 12 yards of cloth cost 15 dollars, what will be 
 
 the cost of 16 yards? $20. 
 3. if an ocean steamer sail 1820 miles in 5 days, how 
 miles will she sail in 64 days? 2366. 
 
 4. If90 bushels of oats supply 40 horses 6 days, 
 how many days will 450 bushels supply them? 30. 
 
 5. If 28 men mow a field of grain in 12 days, how 
 many men will be required to mow it in 8 days? = 42. 
 
 6. Ifasum of money at interest produce 12 dollars 
 in 4 years, how much will it produce in 7 years? $21. 
 
 7. If 24 bushels of wheat can be bought for 27 dol- 
 
 lars, how many bushels can be bought for 45 dollars? 
 
 40, 
 
 8. If 20 bushels of wheat make 5 barrels of flour, 
 
 how many bushels will it require to make 16 barrels? 
 
 64, 
 
 9. If65 bushels of potatoes can be raised on 24 acres 
 
 of ground, how many bushels can be raised on 7 acres? 
 182. 
 
 10. If 35 head of cattle eat 36 acres of grass in a 
 
 - month, how many cattle would 468 acres keep the same 
 time. 455, 
 
 11. If 6 men can ean do a piece of work in 45 days, 
 
 how many days will it take 15 men to do the same work? 
 18. 
 
 12. If it require 12 men to lay a certain number of 
 
 bricks in 16 days, how many days will it take 8 men to 
 lay the same number? 24. 
 
 13. The shadow of a certain tree measures 100 feet, 
 
 while the shadow of a 4-foot stick measures 5 feet. 
 What is the height of the tree? 80 ft. 
 
 14. If 18 bushels of wheat is bought for 24 dollars, 
 
 and sold for 80 dollars, how much will be gained on 57 
 bushels, at the same rate of profit? $19. 
 
-PROPORTION. 65 
 
 15. If 20 men can perform a piece of work in 15 days, 
 how many men must be added that the work may be 
 
 performed in 4 of the time? 5, 
 16. If 4 of a bushel of peaches cost $0.52, what part of 
 a bushel can be bought for $0.3? ts. 
 
 17. If butteris worth 18 cents per pound, and 36 
 pounds of sugar is exchanged for 30 pounds of butter, 
 what is the price of the sugar per pound? $O.15. 
 
 COMPOUND PROPORTION. 
 
 185. Compound Proportion is employed in 
 the solution of problems in which the required term de- 
 pends on a compound ratio. 
 
 186. In compound proportion, all the terms are in 
 couplets or pairs of the same kind, except one. This is 
 called the Odd Term, and is always the same kind as 
 the answer. Each couplet should be considered sepa- 
 rately in making the statement. 
 
 87. 1. If8men earn 40 dollars in 3 days, how 
 
 much will 9 men earn in 4 days? $60. 
 PROCESS. 
 40 
 8/9 
 ; 3 | 4 
 
 ExpLanation:—Use the vertical form of cancellation, 
 writing the odd term on the right; then take the other 
 numbers in pairs, or couplets of the same kind, and _ar- 
 range them as in simple proportion. 
 
 2. If36 men earn 324 dollars in 18 days, how much 
 will 42 men earn in27 days? $567. 
 
 3. If 12 horses plow 11 acres in 5 days, how many 
 horses will plow 33 acres in 18 days? 10. 
 
66 PROPORTION. 
 
 4. If6 men, in 10 days build a wall 20 feet long, 3 
 feet high, and 2 feet thick,in how many days can 15 
 men build a wall 80 feet long, 2 feet high, and 3 feet 
 thick? 16. 
 
 5. Ifaman travel 130 miles in 3 days, when the days 
 are 15 hours long, how many days will it take him to 
 travel 390 miles, when the days are 9 hours long? 15. 
 
 6. If 12 men can mow 80 acres of grass in 6 days, 
 how many days will it take 15 men to mow 200 acres? 
 
 12. 
 
 7. If6men can dig a trench 20 rods long, 6 feet 
 deep, and 4 feet wide, in 16 days, working 9 hours per 
 day, how many days will it take 24 men to dig a trench 
 200 rods long, 8 feet deep, and 6 feet wide, working 8 
 hours per day? 90, 
 
COMPOUND NUMBERS. 
 
 a 
 
 188. Special rules are unnecessary for operations in 
 Compound Numbers as the method is the same as 
 the corresponding process in simple numbers, the only 
 difference, being in their scales of increase. The student 
 will readily understand the method of reduction upon 
 examining the tables. 
 
 AVOIRDUPOIS WEIGHT. 
 
 189. Avoirdupois Weight is used in weighing 
 all coarse and heavy articles, as hay, grain, groceries, etc., 
 and all metals except gold and silver. 
 
 190. The Avoirdupois Pound contains 7000 
 grains Troy, while the Troy pound contains 5760 grains. 
 
 TABLE. 
 T. LB. OZ. 
 1 2000 =. 32000 
 1 16 
 1, What cost 5 pounds of indigo at 10 cents per 
 ounce? $8. 
 
 2. What cost 25 lb. 8 oz. of butter at 16 cents per 
 pound? $4.08. 
 
68 COMPOUND NUMBERS. 
 
 3. What cost 4500 pounds of hay at 6 dollars per 
 
 ton? $13.50. 
 4, A horse weighs 1440 pounds Avoirdupois. How 
 much would he weigh by Troy weight? 1750. 
 TIME, 
 
 191. The Table given below is sufficiently accurate 
 for ordinary business purposes. 
 
 TABLE. 
 YR. MON. DA. HR. 
 1 12 360 8760 
 1 30 720 
 1 24 
 1. I was born April 25, 1869. How old was I, July 
 14, 1890? 21 yr. 2 mon. 19 da. 
 
 2. The Declaration of Independence was written 
 July 4, 1776. How many years had elapsed, March 1, 
 1860? 83 yr. 7 mon. 27 da. 
 
 LONG MEASURE. 
 
 192. Long Measure is used in measuring lenghts 
 and distances. 
 
 TABLE. 
 MI. RD. YD. FT. IN. 
 if 320 1776 5280 63360 
 1 at 163 198 
 1 3 36 
 1 12 
 1. Reduce 264 ft. to rods. 16. 
 
 2. Reduce 2 mi. 2 rd. 2 ft. to feet. 10595, 
 
COMPOUND NUMBERS. 69 
 3. How many steps of 2 ft. 8 in. each will a man take 
 
 in walking 2 miles? 3960. 
 
 SURVEYOR S MEASURE. 
 
 193. Surveyor’s Weasure is used in measuring 
 land, laying out roads, establishing boundaries, ete. 
 
 194. Surveyors use the Gunter’s Chain, which is 
 4 rods long and contains 100 links. 
 
 TABLE. 
 MI. CH. RD. LK. IN, 
 ] 80 320 8000 63360 
 1 4 100 792 
 1 25 198 
 1 7.92 
 1. Reduce 400 links to rods. 16. 
 2. Reduce 128 rods to chains. 32. 
 3. Reduce 640 chains to miles. 8. 
 4. Reduce 3 ch. 25 links to rods. 1% 
 
 SQUARE MEASURE 
 
 195. Square Measure is used in measuring sur- 
 faces. 
 
 TABLE. 
 A. SQ. RD. SQ. YD. * --->SQ. FUE SQ. IN. 
 1 160 4840 43560 6272640 
 ] 305° 2724 39204 
 1 9 1296 
 
 1 144 
 
70 COMPOUND NUMBERS. 
 
 1. Reduce 480 square rods to acres. 3. 
 2. Reduce 2560 square rods to acres. 16. 
 3. Express ~ of an acre in square rods. 60. 
 4. How many square feet in 27 square yards? 243. 
 
 ». How many square feet in 1728 square inches? 
 12. 
 
 6. How many square yards in 4545 square feet? 
 505. 
 
 7. A gentleman divided his farm of 328 A. 74 sq. rd. 
 equally among his 3 sons. What was the share of each? 
 109 A. 78 sq. rd. 
 
 CUBIC MEASURE. 
 
 196. Cubic Measure is used in measuring 
 
 solids. 
 TABLE. 
 CU. YD. CU. FT. CU. IN. 
 1 27 46656 
 1 1728 
 1. Reduce 20736 cubic inches to cubic feet. 12. 
 2. Reduce 482 cubic feet to cubic yards. 16. 
 
 3. Reduce 2 cu. yd. 2 cu. ft. to cubie inches. 
 . 96768. 
 
 DRY MEASURE. 
 
 197. Dry Measure is used for measuring grains, 
 vegetables, fruits, ete. 
 
 198. The Standard Unit of dry measure is the 
 bushel which contain 21502 cubic inches, or nearly 14 
 cubic feet. 
 
COMPOUND NUMBERS. 71 
 
 199. The Standard Gallon of the United States 
 
 contains 231 cubie inches, or about -"s of a ecubie foot. 
 The dry gallon contains 268% cubic inches. 
 
 TABLE. 
 BU. PK, GAL. QT. PT 
 1 4 8 3p 64 
 1 2 8 16 
 1 4 8 
 1 2 
 1. Reduce 448 pints to bushels. (ep 
 
 2. Reduce 1 bu. 1 pk. 1 gal. 1 qt. 1 pt. to pints. 
 91. 
 
 200. The weight of a Bushel of various articles 
 is given in the following 
 
 TABLE. 
 
 ARTICLES. ILB ARTICLES, LB. 
 
 | Apples 50 Hair, unwashed | 8 
 Barley 48) Hemp seed 44 
 
 | Beans 60 Hungarian seed 45 
 | Bluegrass seed 14) Lime 80 
 Bran 20, Millet 45 
 Buckwheat 52) Oats 132 
 Castor beans 46) Onions 157 
 Charcoal 22|| Onion sets ~~ =—«(14 
 Clover seed 60}, Potatoes 60 
 Coal 80), Potatoes, sweet 50 
 
 | Corn 56|| Rye 56 
 Corn, in ear 70), Salt 50 
 Corn meal 50) Timothy seed 45 
 
 | Flax seed 56 Turnips 56 
 | Hair, washed 4|| Wheat 60 
 
 1. How many bushels of corn in a load weighing 
 1344 pounds? 24. 
 
 2. How many bushelsin a load of wheat weighing 
 1290 pounds? 214. 
 
Lo COMPOUND NUMBERS. 
 
 3. How much’should I receive for 1536 pounds of 
 oats at 20 cents per bushel? ~ $9.60. 
 4. Bought 12 gallons of syrup at 75 cents a gallon, 
 and sold it at 24 cents a quart. How much did I gain? 
 $2.52. 
 5. At $2.40 per bushel what should I receive for a 
 load of timothy seed weighing 2271 pounds, deducting 
 the weight of the wagon, 1236 pounds? $55.20. 
 
 MISCELLANEOUS TABLE. 
 
 4 Inches, 
 
 1 Shingle. 
 
 4 Inches, 1 Hand. 
 6 Feet, Ll Fathom. 
 3 Miles, 1 Weague. 
 4 Gills, LEP ing 
 5 Bushels corn, 1 Barrel. 
 60 Seconds, 1 Minute. 
 60 Minutes, 1 Hour. 
 16 Drams, 1 Ounce. 
 24 Sheets, 1 Quire. 
 20 Quires, 1 Ream. 
 12 Things, 1 Dozen. 
 12 Dozen, 1 Gross. 
 20 Things, 1 Score. 
 15° Longitude, 1 Hour. 
 100 lb. Grain, 1 Cental. 
 100 lb. Fish, 1 Quintal. 
 196 lb, Flour, 1 Barrel. 
 200 Ib. Beef or pork, 1 Barrel. 
 280 lb. Salt, 1 Barrel. 
 39.37 Inches, 1 Meter. 
 
 4.8665 Dollars, 1 £ Sterling. 
 
INVOLUTION. 
 
 202. Involution is the process of raising a num- 
 ber to any given power. 
 
 203. A Power is the product arising from multi- 
 plying a number by itself in continued multiplication. 
 
 204. The First Power of a number is the number 
 itself. 
 
 205. The Second Power of a number is called its 
 SQUARE. 
 
 206. The Third Power of a number is called its 
 CUBE. 
 
 207. The Degree of a power is indicated by an 
 exponent, which is a small figure placed a little above 
 and at the right of the number. Thus, 3*, indicates 
 the fourth power of 3. 
 
 208. The Product of any two or more powers is 
 the power denoted by the sum of their exponents. Hence 
 if we multiply the third power of a number by the fourth 
 power the product will be the seventh. 
 
 209. 1. What is the fourth power of 3? 81. 
 PROCESS. 
 $3x3x3xS=SL. 
 
 ExpLanation:—Multiply the number successively by 
 
74 INVOLUTION. 
 
 itself till it has been taken as many times as a factor as 
 there are units in the exponent of the required power. 
 
 2, What is the square of 16? 256. 
 3. What is the cube of 9? 729. 
 4. What is the fourth power of 5? 625. 
 5. What is the fifth power of 6? RiA0; 
 
 NUMBERS ENDING WITH «3. 
 
 210. The following method of squaring a number 
 whose Unit Figure is 5 will be found valuable. 
 
 211° 1. What is the square of 25? 625. 
 PROCESS. 
 
 25 
 20 
 
 625 
 
 Expianation:—Multiply the part preceding 53 by 
 itself increased by I and prefix the result to 25. 
 
 2. What is the square of 75? 5625, 
 3. What is the square of 35? 1225. 
 4. What is the square of 45? 2025. 
 >. What is the square of 195? 38025. 
 6. What is the square of 115? 13225. 
 7. What is the square of 995? 990025. 
 
 ENTE GWE 2's: 
 
 212. The Square of numbers ending with 25 may 
 
—~l 
 A 
 
 INVOLUTION. 
 
 be readily written out by the method explained below. 
 
 213. 1. What is the square of 625? 390625, 
 PROCESS. 
 39 0625 
 
 ExpLaNnaTion:—Square the part preceding 25, add 
 
 half the same part to the result, discarding fractions, 
 and annex 0625, or if the part preceding 25 is odd, 
 annex 3625. 
 
 2. What is the square of-425? 180625. 
 3. What is the square of 925? 855625. 
 4. What is the square of 325? 105625. 
 5. What is the square of 1025? 1050625. 
 
 MIXED NUMBERS. 
 
 214. The method of squaring Mixed Numbers 
 ending with 4 is illustrated below. 
 
 215. 1. Find the square of 63. 42 
 
 vol 
 
 PROCESS. 
 6 
 62 
 
 A424 
 
 ExpLaNnaTion:—We say @ times 6 are 42 and annex 
 4 to the product. Always add 1 tothe multiplier. This 
 method is applicable in all problems like the above. The 
 student should study the method and apply the same 
 principle to other fractions. 
 
 2. Find the square of 73. d64. 
 
 3. Find the square of 93 905. 
 
 4. Find the square of 113. 1324, 
 
76 INVOLUTION. 
 
 5. Find the square of 493. 24504. 
 6. What cost 123 pounds of butter at 123 cents per 
 pound? $1564. 
 
 SOU ARE-OF PW @SDIGI Ts SE ie. 
 
 216. Small Numbers may be squared mentally, 
 by the following simple method. 
 
 217. 1 What is the square of 18? 324. 
 
 PROCESS. 
 16 x 20+4= 324 
 
 Expianation:—Take the product of two numbers, one 
 of which is as much less than the number to be 
 squared as the other is greater, and one of the numbers 
 a multiple of ten, and add the square of the difference 
 between the givén number and one of the asumed num- 
 bers. 
 
 2. What is the square of 27? 729, 
 3. What is the square of 21? | 441, 
 4, What is the square of 33? 1089. 
 ». What is the square of 79? 6241. 
 
 DOU ATs DOLE NT IN Bis 
 
 218. The Square of any number of nInES may be 
 written out without multiplying. 
 
 219. 1. What is the square of 999? 998001. 
 PROCESS. 
 998001 
 
 EXPLANATION :—Erase one 9 from the left of the num- 
 
> ae 
 bi Ree 
 i i 2 ae ed 8 
 a) 
 
 A ~ < 
 
 “ber to be Re a wtinted annex an $8, as many ciphers as there 
 e nines, anda I. ; 
 2. Find the square of 9. 81. 
 
 3. Find the square of 99. 9801. 
 4. Find the Square of 9999. 99980001. 
 
 " INVOLUTION. | 77 ie 
 y 
 : 
 
 poe 
 
 aa se 
 
EVOLUTION. 
 
 220. Evolution is the process of finding roots of 
 numbers. 
 
 221. A Root of a number is one of its equal fac- 
 tors. 
 
 222. The Sign of Evolution, \, is a modification 
 of the script letter r. Roots are also indicated by frac- 
 tional exponents. 
 
 223. The following table of Squares and Cubes 
 should be learned by the pupil. 
 
 A Maylide 
 
 Numbers 1, 2, 3, 4 5, 6, 4) othe 
 Squares I, 4, 9, 16, 25, 36, 49, 64, SI. 
 subes 1, 8, 27, 64, 125, 216, 343, 512,729. 
 
 224. It will be observed that Square Numbers 
 never end in 2, 3, 7, or 8, and that the cubes of no two 
 digits end with the same figure. Hence, in finding the 
 eube root of perfect cubes, we can easily determine the 
 unit figure of the root from the unit figure of the power. 
 
———— cle en SS aS ns eS OTS 
 
 Ee eee 
 
 EVOLUTION. 79 
 
 EVOLUTION BY FACTORING. 
 
 225. To find Any Root of a perfect power, resolve 
 the number into its prime factors, and for the square 
 roct take one of two equal factors, for the cube root * 
 take one of three equal factors, ete. 
 
 226. 1. What is the square root of 225? 15. 
 PROCESS. 
 225 =—-383x3xkK3x5 
 
 225 = $8 X53 = 15 
 
 2. Find the square root of 625. 25. 
 3. Find the square root of 1296. 36. 
 4. Find the cube root of 1728. 12. 
 5. Find the fourth root of 1296. 6. 
 6. Find the fifth root of 243. 3 
 
 SOUARE ROOT BY ANALYSIS. 
 
 227. Separate into periods of two figures each, 
 beginning at the right. The first figure is the root of 
 the greatest square in the left hand period. Subtract 
 its square from the first period, and bring down the 
 next period. Double the root found and divide, disre- 
 garding the right hand figure of the dividend, and place 
 the result in the root and at the right of the divisor. 
 Multiply the complete divisor by the last figure of the 
 root and proceed as before. If a cipher occur in the 
 
 . root, annex a cipher also to the trial divisor and bring 
 
 down the next period. 
 
80 EVOLUTION. 
 228. 1. What is the square root of 1296? 36. 
 PROCESS. 
 
 12.96(36 
 
 66 396 
 
 396 
 2. What is the square root of 2304. 48, 
 3. The area of asquare field is 6561 square rods. 
 How many rods in length or breadth? 81. 
 4, A man has a square field containing 4096 square 
 rods. How many rods in length or breadth? 64. 
 
 5. The length of a rectangular field containing 20° 
 acres is twice its width? What is the distance around 
 it? 240 rd. 
 
 6. A square field measures 6 rods on each side. What 
 is the length of the side of a square field which is 16 
 times as large? 24 rd. 
 
 7. If it costs $572 to enclose a field 72 rods long and 
 32 rods wide, how much less will it cost to enclose a 
 square farm of equal area with the same kind of fence? 
 
 SIMILAR SURFACES. 
 
 229. Similar Surfaces are to each other as the 
 squares of their like dimensions. 
 
 230. Like Dimensions of similar surfaces are 
 to each other as the square roots of their areas. 
 
 231. 1. A hole made by a 2 inch auger bit, is how 
 many times as large as one made by an inch auger bit? 
 4, 
 
EVOLUTION. 81 
 
 Cal 
 
 2. If the area of a circle, whose diameter is 7 feet is 
 38.5 sq. ft., what will be the area of a circle 21 feet in 
 diameter. 346.5 sq. ft. 
 
 3. A rectangular field is 12 rods wide and 20 rods 
 long. What must be the width of a road across one end 
 and one side to contain 4 the area of the entire field? 
 
 2 rd. 
 
 4. If one side of a triangle is 12 feet, and its area is 
 36 square feet, how many square feet in the area of a 
 similar triangle, the corresponding side of which is 8 
 feet? 16. 
 
 5. The area of a triangle, the length of whose base is 8 
 rods, is 92 square rods. How many square rods are there 
 in the area of a similar triangle, the corresponding side 
 of which is 4 rods? 13. 
 
 Cay Bs ROOTBY INSPECTION: 
 
 232. The Cube Root of perfect cubes of not more 
 than six figures can be easily found by inspection. 
 
 233. 1. Find the cube root of 15625. 25. 
 PROCESS. 
 ( 15,625 )* = 25 
 
 ExpLaNnation :—-For the first figure of the root, we write 
 the root of the greatest cube in the left hand _ period, 
 which is 2, and then by inspection, or reference to the 
 table, we see at once that the other figure of the root must 
 be 5, as the cube of no other digit ends with 5. 
 
 2. Find the cube root of 1728. 12. 
 3. Find the cube root of 12167. 23. 
 4. Find the cube root of 39304. 34. 
 5. Find the cube root of 91125. 45. 
 
82 EVOLUTION. 
 5. Find the cube root of 175616. 56. 
 6. Find the cube root of 300763. 67. 
 7. Find the cube root of 474552. 78. 
 8. Find the cube root of 704969. 89. 
 9, Find the cube root of 753571. 91. 
 
 CUBE; RG) Ochs bY aac eas less 
 
 234. Separate into periods of three figures each. 
 The first figure is the root of the greatest cube in the left 
 hand period. Subtract the cube and bring down the 
 next period. Square the root found, multiply by 300, 
 and divide to find the second figure of the root. To three 
 times the first figure of the root, annex the last. Multi- 
 ply this factor by the last root figure and add the result 
 to the trial divisor. Multiply the complete divisor by 
 the last figure of the root, subtract and proceed as_be- 
 fore. When the dividend will not contain the trial divi- 
 sor, write a cipher in the root and two at the right of the 
 trial divisor. 
 
 235. 1. What is the cube root of 13824? 24. 
 
 PROCESS. ) 
 
 13,824(24 
 ‘s 
 
 1200 3824 
 256 F 
 1456 5824 
 2. What is the cube root of 74088? 42. 
 
 8. What is the side of a cubical box which contains 
 873248 solid inches? 6 ft. 
 
EVOLUTION. 83 
 
 4. What is the depth of a cubical bin whose contents 
 
 are 79507 cubic feet? 43 ft. 
 5. What is the length of the side of a cubical box that 
 
 contains 15625 cubic feet? 25 ft. 
 6. What is the side of a cube equal to a pile of wood 
 
 81 feet long, 27 feet wide and 9 feet high? 27 it. 
 
 SIMILAR SOLIDS. 
 
 236. The Contents of similar solids are to each 
 other as the cubes of their like dimensions. 
 
 237. Like Dimensions of similar solids are to 
 each other as the cube roots of their contents. 
 
 238. The Side of a cube, whose solidity bears a giv- 
 en relation to that of a cube whose side is given, is found 
 by eubing the given side, multiplying the result by the 
 given proportion and extracting the cube root of the 
 product. 
 
 239. 1. What is the side of a cubical vat which 
 contains } as much as one whose side is 6 feet? 3 ft. 
 
 2. Ifa cubic inch of gold is worth 200 dollars, what 
 is the worth of a cube of gold whose side is 3 inches? 
 ) $5400. 
 
 3. Ifa cubical block of granite, whose side is 4 inches 
 weigh 12 pounds, what will a cubic foot of the same gran- 
 ite weigh? 324 Ib. 
 
 4, I have a cubical box whose side is 3 feet. I want 
 another which will contain 8 times as much. What will 
 be the length of its side? 6 ft. 
 
 5. Ifacannon ball 8 inches in diameter weigh 40 
 pounds, what is the weight of one of the same metal, 
 whose diameter is 4 inches? 5 Ib. 
 
MENSURATION. 
 
 240. For the practical convenience of those who have 
 occasion to refer to Mensuration, we give the follow- 
 ing principles, covering the whole ground of practical 
 geometry. 
 
 PRINCIPLES. 
 
 241. The Diagonal of a square is equal tothe side 
 of the square multiplied by 1.414. 
 
 242. The area of a Triangle is equal to half the 
 product of the base by the altitude. 
 
 2438. The side of an Inscribed Square is equal 
 to the diameter multiplied by .7071. 
 
 244. The area of any Parllelogram is San to 
 the product of the base by the altitude. 
 
 245. The areaof a Parabola is equal to the base 
 multiplied by two-thirds of the altitude. 
 
 246. The area of an Ellipse is equal to the product 
 of the two diameters multiplied by .7854. 
 
 247. The contents of a Sphere is equal to the 
 cube of the diameter multiplied by .5236. 
 
MENSURATION. 85 
 
 248. The contents of a Wedge is equal to the area 
 of the base multiplied by half the altitude. 
 
 249. The surface of a Sphere is equal to the 
 square of the diameter multiplied by 3.1416. 
 
 250. The area of a Sector of a circle is equal to the 
 length of the are multiplied by half the radius. 
 
 251. The area of a Trapezoid is equal to its alti- 
 tude multiplied by half the sum of its parallel sides. 
 
 252. The side of an Imnseribed Equilateral 
 Triangle is equal tothe diameter multiplied by .866025, 
 
 253. The contents of a Cylinder or Prism is 
 equal to the area of the base multiplied by the altitude. 
 
 254. The contents of' a Pyramid or Cone is 
 equal to the area of the base multiplied by one-third of 
 the altitude. 
 
 255. The convex surface of a Pyramid or Cone 
 is equal to the perimeter of the base multiplied by half 
 the slant height. 
 
 256. The entire surtace of a Cylinder or Prism 
 is equal to the area of both ends plus the product of the 
 length by the periphery. 
 
 257. The side of a Cube which may be cut from a 
 given sphere is equal to the square root of one-third of 
 the square of the diameter. 
 
 258. The diameter of a Cirele that shall contain 
 the area of a given square is equal to the side of the 
 square multiplied by 1.1284. 
 
 259. The area of any Regular Polygon is equal 
 to the perimeter multiplied by half the perpendicular 
 distance from the center to one of the sides, 
 
86 MENSURATION. 
 
 260. The convex surface of a Frustrum of a pyra- 
 mid or cone is equal to the sum of the perimeter of the 
 two bases multiplied by half the slant height. 
 
 261. The area of a Trapeziumi is equal to the di- 
 agonal multiplied by half the sum of the perpendiculars 
 drawn from the vertices of the opposite angles to the 
 diagonal. 
 
 262. The Ratio betewen the diameter and circum- 
 ference of a circle, expressed decimally and the approxi- 
 mation carried to thirty places. is 3.14159265358979323846 
 264338328, 
 
 263. The area of a Segment ofa circle is equal to 
 the area of a corresponding sector less the area of the tri- 
 angie, or plus the area of the triangle when the segment 
 is greater than a semicircle. 
 
 264. The side ofa Square that will contain the 
 area of a given circle is equal to the square root of the 
 area, or the diameter multiplied by .8862, or the circum- 
 ference multiplied by .2821. 
 
 265. The contents of a Frustrum of a pyramid 
 or cone is equal to the sum of the areas of the two ends 
 plus the square root of the product of these areas, multi- 
 plied by one-third of the altitude. 
 
 RECTANGISES. 
 
 266. A Rectangle is a figure that has four straight 
 sides and four right angles. 
 
 267. The Area of a rectangle is equal to the pro- 
 duct of its length by its breadth. 
 
MENSURATION. 87 
 
 268. 1 At 12 cents a square yard, what will it cost 
 to paint the walls of two rooms, each 16 feet square and 
 
 9 feet high? ~ $15.36. 
 STATEMENT. 
 | 128 
 9/9 
 12 
 
 ExpLanation:—-Multiply the entire distance around 
 the rooms by the height, divide by 9 to reduce to square 
 yards, and multiply by the price per yard. 
 
 2. How many square feet in a floor 16 feet long and 
 
 14 feet wide? 224. 
 3. How many acres in a field of land 96 rods long 
 and 80 rods wide. 48, 
 4. Ifa floor is 12 feet long, how wide must it be to 
 contain 132 square feet? 11 ft. 
 5. Find the difference between a floor 20 feet square, 
 and two others 10 feet square. 200 sq. ft. 
 6. One side of a rectangular field containing 63 acres 
 is 120 rods. What is the other? 84 rd. 
 7. How many yards of carpet 13 yards wide will cover 
 a floor 18 feet long, 15 feet wide? 20 yd. 
 
 8. At 10centsa square yard, what will it cost to 
 plaster the walls and ceilings of three rooms, each 15 ft. 
 6 in. long, 13 ft. 8 in. wide and 12 ft. high? $44.51. 
 
 PE tAN GPS, 
 
 269. A Triangle is a figure which has three an- 
 gles and three sides. 
 
 270. A Right Angle is the angle formed when 
 one line is drawn perpendicular to another. 
 
 271. The Hypotenuse of aright angled triangle is 
 the side opposite the right angle. 
 
88 MENSURATION.. 
 
 272. The Base of a triangle is the side on which it 
 is assumed to stand. 
 
 273. The Perpendicular is the side which forms 
 a right angle with the base. 
 
 274. The Area ofa triangle is equal to half the 
 product of the base by the altitude. 
 
 275. The Hypotenuse of aright angled triangle, 
 is equal the square root of the sum of the squares of the 
 other two sides. 
 
 276. The Base or Perpendicular is equal to 
 the square root of the difference of the squares of the 
 hypotenuse and the other side. It is also equal to the 
 square root of the product of the sum and difference 
 of the hypotenuse and the other side. 
 
 2@¢@. 1. What is the area of a triangle whose base 
 is 16 feet and whose altitude is 13 feet? 96 sq. ft. 
 2. ‘The base of a right angled triangle is 40 feet, and 
 the perpendicular is 30 feet. What is the eb aor 
 5 
 3. The hypotenuse of a right angled triangle is 73 
 feet and the perpendicular is 43 feet. What is the base? 
 6 ft. 
 4, The base of a right angled triangle is 34 feet and 
 the hypotenuse is 123 feet. What is the perpendicular? 
 ies 
 5. A rectangular field is 60 rods long and 45 rods 
 wide. What is the distance between two opposite corn- 
 ers? 75 rd. 
 6. What is the area of a triangular piece of ground 
 whose base is 40 rods, and whose perpendicular height 
 is 28 rods? 3d A. 
 7. A pole is 27 feet high. How many feet above the 
 ground must it be broken in order that the upper part, 
 clinging to the stump, may touch the ground 9 feet from 
 the base? 12. 
 
MENSURATION. 89 
 
 8. The main mast of a vessel .is 72 feet high. How 
 many feet above deck must it be broken in order that the 
 upper part, clinging to the stump, may touch the deck 
 16 feet from the base? 342, 
 
 CLERC lakes. 
 
 278. A Cirele is a plane figure bounded by a 
 curved line, every part of which is equally distant from 
 a point within, called the center. 
 
 279. The Circumferenee is the line which bounds 
 the circle. 
 
 280. The Radius of a circle is a straight line 
 drawn from the center to the circumference. 
 
 281. The Diameter ofa circle is a straight line 
 drawn through the center, and terminated by the cir- 
 cumference. 
 
 282. The Circumference of a circle is equal to 
 the diameter multiplied by 3+. Hence, the diameter is 
 equal to the cireumference divided by 3+. 
 
 283. The Area ofa circle is equal to the cireumfer- 
 ence multiplied by one-fourth the diameter; or, the 
 square of the diameter multiplied by +44. 
 
 284. 1. Whatisthe circumference of a cirele 28 
 
 inches in diameter? 88 in. 
 STATEMENT. 
 | 28 
 7\|22 
 
 EXPLANATION: Simply multiply by 3+, by reducing it 
 to an improper fraction and applying cancellation. 
 
 2. What is the circumference of a log 14 inches in 
 diameter? 3 ft. 8 in. 
 
 3. How far is it around a circular pond that is 45 feet 
 in diameter? 1413 ft. 
 
90 MFNSURATION. 
 
 4. What is the area of a circle 14 feet in diameter? 
 
 154 sq. ft. 
 5. What is the radius of a circle whose circumference 
 is 616 feet? 98 ft. 
 6. What is the area of a circular pond 70 rods in 
 circumference? 3850 sq. rd. 
 7. What is the diameter of a circle whose circumfer- 
 ence is 154 feet? 49 ft. 
 
 LUMBER MEASURE. 
 
 285. To measure Lumber, multiply the width in 
 inches, the thickness in inches, and the length in feet 
 together, and divide the product by 12. 
 
 286. To find the quantity of lumber in a Leg, mul- 
 tiply the square of the diameter in inches at the small 
 end by the length in feet, and divide the product by 24. 
 
 28¢. Whena board Tapers uniformly in width, 
 find the average by taking half the sum of the two ends. 
 If it taper also in thickness, the contents in board feet 
 may be found by multiplying the sum of the areas of the 
 two ends in square inches by the length in feet, and 
 dividing the product by 24. 
 
 FORMULA. 
 
 No. Pieces. 
 
 Te OO 
 atte % 
 Mc 
 
 Length. 
 | Width. 
 
 Thickness. 
 
MENSURATION. 9] 
 
 288. 1. How many feet in 3 pieces 2 x 4, 16 feet 
 
 long? 32. 
 STATEMENT. | 
 33 
 2 
 4 4 
 —«G 
 
 ExpLanation:—Write the given dimensions on the 
 right and the factors of 12 on the left. The price per 
 foot should also be written on the right when the cost 
 is required. 
 
 2. How many feet in 11 pieces 6 X 8, 14 feet long? 
 
 616. 
 3. How many feet in 16 boards 10 inches wide and 12 
 feet long? 160. 
 4. How many feet of lumber in a log 15 inches in 
 diameter and 16 feet long? 150. 
 5. How many board feet in a post 3 x 4 at one end, 
 4 x 6 at the other, and 10 feet long? 15. 
 6. How many feet in 56 fence posts 2 x 4 at one end, 
 and 4 x 4 at the other, and 8 feet long? ° 448, 
 7. What will it cost to floor a room 14 by 20 with 
 2 dollar flooring, allowing + for matching? $6.72. 
 
 8. <A farmer bought 75 boards of fencing 6 inches 
 wide, and 16 feet long. What did they cost at 2 dollars 
 per hundred feet? $12. 
 
 9. How many feet of boards will be required to en- 
 close two gable ends of a building 16 feet wide, allowing 
 that the roof is half pitch? 128. 
 
 LATH MEASURE. 
 
 289. To Compute the number of lath required to 
 cover a wall, multiply the number of square yards by 15. 
 
Q? MENSURATION. 
 
 290. In Computing the contents of walls, plas- 
 terers make no allowance for doors and windows. 
 
 * FORMULA. 
 
 { 
 9 Breadth. . 
 
 (15 
 
 291. 1. How many lath will be required for a wall 
 
 containing 250 yards? 3750. 
 2. How many lath will it take to cover the ceiling of 
 a room 24 feet long and 16 feet wide? 640. 
 
 3. How many lath will it take to cover the walls of a 
 room 22 feet wide, 32 feet long, and 10 feet high? 
 
 1800. 
 4, At $2.80 per thousand, what cost the lath for the 
 
 walls and ceiling of a room 12 by 15, and 8 feet 4 inches 
 high? $2.94. 
 
 SHINGLE MEASURE. 
 
 292. ‘To find the length of the Rafters, giving the 
 roof one-third pitch, take + of the width of the building: 
 for one-half pitch take } of the width of the building. 
 
 293. It Requires 9 shingles to the square foot if 
 exposed 4 inches; 8 if exposed 43 inches; 7 if exposed 5 
 inches, and 6 if exposed 53 inches to the weather. 
 
 i i 
 
MENSURATION. 93 
 
 294. 1. Find the number of shingles required to 
 make a roof 16 feet long and the rafters on each side 12 
 feet, shingles exposed 43 inches, 3072. 
 
 2. What length of rafters must | buy for a building 
 28 feet wide, giving the roof half pitch and allowing 1 
 foot for projections? 22 ft. 
 
 3. What length of rafters will be required for a 
 building 15 feet wide, giving the roof one-third pitch 
 aud allowing 1 foot for projections. 10 ft. 
 
 4. At $2.50 per thousand, what cost the shingles for 
 a third pitch roof on a building 20 x 382 allowing 6 in- 
 ches all around for projections, shingles exposed 44. in- 
 ches? 3 $16.50. 
 
 5. How many shingles will be required for a_ half 
 pitch roof on a building 16 x 20, allowing for projections 
 | foot on each side and six inches até each end, shingles 
 exposed 4 inches? 4914. 
 
 WOOD MEASURE. 
 
 295. A Cord of Wood contains 128 cubic feet, 
 To find how many cords in a pile of wood, multiply the 
 length, height, and width, together, and divide the 
 product by 128. 
 
 FORMULA. 
 
94 MENSURATION. 
 
 296. 1. How many cords in a pile of wood 32 feet 
 
 long, 8 feet high, and 4 feet wide? 8. 
 STATEMENT. 
 32 
 128 | 8 
 4 
 2. How many cords in a range of wood 32 feet long, — 
 6 feet high and 4 feet wide? 6. 
 3. How many cords ina pile of wood 28 feet long, 
 16 feet wide, and 12 feet high? 42. 
 4. What is the worth of a pile of wood 16 feet long, 
 4 feet wide and 5 feet high, at $22 per cord? $6. 
 5. Ifa pile of wood is 32 feet long, and 4 feet wide, how 
 many feet high must it be to contain 12 cords? 12; 
 
 6. A man sawed a pile of wood 40 feet long, 4 feet wide 
 and 53 feet high, for $1.50 per cord. How much did he. 
 earn? $10. 
 
 FLAY NER A Us 
 
 297. The only Accurate Method of measuring 
 hay is to weigh it; but the method explained below will 
 be found sufficiently accurate for ordinary purposes. 
 
 FORMULA. 
 
 Width. 
 
 700) Height. 
 
 Exprianation:—To find the number of tons in ordinary 
 
MENSURATION. 95 
 
 stacks or ricks, multiply the length, width and entire 
 height together and divide the product by 700. 
 
 298. 1. How many tons of hay in a rick 10 feet 
 
 wide, 14 feet high and 20 feet long? 4, 
 STATEMENT. 
 7| 10 
 1d) 14 
 10 | 20 
 2. How many tons in a stack of hay 12 feet wide, 15 
 feet high and 28 feet long? 73 
 3. How many tons in a stack of hay which measures 
 12 feet wide, 14 feet high and 15 feet long? 33 
 MASONRY. 
 
 299. Brickwork is commonly estimated by the 
 thousand bricks. 
 
 300. Stone Masonry is estimated by the perch, 
 which contains 25 cubic feet. 
 
 301. In Computing the contents of walls, masons 
 make no allowance for the corners, but estimate their 
 work by the entire distance around on the outside. They, 
 however, make a deduction of half the space occupied 
 by windows, doors, and other openings. 
 
 302. 1. How many bricks will it take to build a 
 wall 40 feet long, 84 feet high, and 1} feet thick, allow- 
 ing 20 bricks to the cubic foot, and deducting +) for 
 
 mortar? 9000. 
 STATEMENT. 
 AO 
 3 | 25 
 Bié< 
 20 
 
56 MENSURATION. 
 
 2. How many perches of masonry in a wall 35 feet long, 
 
 15 feet high, and 3 feet thick. 63. 
 STATEMENT, 
 3 
 25 | 15 
 3 
 3. How many perches in a wall 30 feet long, 165 feet 
 high, and 4 feet thick? 0, = 
 
 4. At $2.25 a perch, what will it cost to build the 
 walls of a cellar 160 feet in circumference, 12) feet high. 
 2 feet thick, allowing ;'5 for mortar and filling? $324. 
 
 5. How many bricks will be required to build an en- 
 closure 12 feet wide and 18 feet !ong, the walls of which 
 are 10 feet high, and 18 inches thick, allowing 20 bricks 
 to the cubic foot? 18000. 
 
 BOX MEASURE. 
 
 303. To find the contents of any regular vessel in 
 Bushels, multiply the product of its length, breadth, 
 and depth by .8; or, by .4 for bushels of corn in the ear; 
 or, by 74 for the contents in gallons. If the vessel is 
 cylindrical, proceed as if it were square and take 4 of 
 the result. 
 
 FORMULA. 
 5° 4 
 ; Length. 
 ; Breadth. 
 
 { 
 ) 
 ? | 
 ickmess. ; 
 Thickne pi 
 
 a 
 a 
 
 ExpLaNnaTion: —In the above formula, we have the com- 
 
MENSURATION. 97 
 
 mon fraction $ intead of .8, either being correct. For 
 corn in the ear, write 2 on the right instead of 4. 
 
 304. 1. How many bushels ina bin 10 feet long, 
 
 2 feet 6 inches wide, and 1 foot 3 inches deep? 25, 
 STATEMENT. 
 | 10 
 2,5 
 4| 5 
 a 
 
 ExpLanation:—Reduce inches to the fraction of a foot 
 and apply cancellation. Consider .8 as a whole number 
 and point off one figure from the right of the result. 
 
 2. How many gallons in a box 10 feet long, 5 feet 
 
 wide, and 4 feet deep? 1500. 
 3. How many bushels in a cylindrical tub 5 feet 
 in diameter and 6 feet deep? 96. 
 4. How many bushels in a wagon box 10 feet long, 
 3 feet wide, and 21 inches deep? 42. 
 5. How many bushels of corn, in the ear, ina crib 14 
 feet long, 10 feet wide, and 8 ft high? 448, 
 
 6. How many bushels in a box 7 feet 6 inches long, 
 4 feet 8 inches wide, and 2 feet 6 inches deep? 70. 
 
 7. What will be the cost of the wheat at 75 cents per 
 bushel, which would just fill a hogshead 3 feet 6 inches 
 in diameter, and 4 feet 8 inches deep? $27.44. 
 
 8. How much would the wheat be worth at $1.25 per 
 bushel which would just fill a bin, the base of which is 
 123 feet square, and the height 10 feet? $1562.50. 
 
Our CURIOSITY SHOP. 
 
 a a ee 
 
 305. The Following Paragraphs will be devoted 
 to a brief exposition of many interesting and valuable 
 properties of numbers, wonders, curiosities, ete. 
 
 LIGHTNING ADDITION. 
 
 306. This is the Method used by most lightning 
 calculators throughout the country. 
 
 PROCESS. 
 746327 
 
 253673 ¥ 
 170624 . 
 
 $29376 
 
 263405 
 
 724291 
 
 = SS. 
 
 2987696 
 
 ExpLaNnaTion:—The first and second lines are so ar- 
 ranged as to add to 9 all the way through, except in units 
 place, where they add to 10. The same is true of the 
 next two lines, and so on down till the desired number of 
 couplets are written. The last two lines are made up of 
 small figures, so arranged that in adding them there will 
 be nothing to carry. By adding the two bottom lines 
 
MISCELLANEOUS. 99 
 
 and prefixing the number of couplets above, the correct 
 sum is obtained. The method may be varied to add the 
 top and bottom lines, the two middle lines, ete. 
 
 NINE. 
 
 307. Every number consists of a certain number of 
 Nines, plus the sum of its digits. From the foregoing 
 axiomic principle, an instructive, as well as a very inter- 
 esting and amusing exercise may be deduced. If you 
 have a friend present, have him write a number with as 
 many digits as he may choose. Instruct him to subtract 
 the sum of the digits from the number, cancel one figure, 
 add the remaining digits, and tell you the result. By 
 deducting the result from the first succeeding multiple 
 of nine, the remainder will be the figure canceled. 
 
 PROCESS. 
 46532 
 20 
 
 Expn.anation:—If 16 be given as the sum of the 
 digits in the remainder, exclusive of the one canceled, 
 then, you would subtract it from 18, the first multiple of 
 nine above. Hence, 2 would be the digit canceled. 
 This property of nine affords short and accurate tests 
 for ordinary operations in Arithmetic. 
 
 Reo LN ON RIGU RE 
 
 308. To produce the result in One Figure, write 
 the nine digits, omitting the 8, in their regular order, as 
 the multiplicand; then multiply the figure desired in the 
 answer by 9, and use this product as the multiplier. 
 
100 MISCELLANEOUS. 
 
 309. 1. Multiply 12345679 by some number that 
 will produce the answer in fives. | 45, 
 
 PROCESS. 
 12345679 
 
 45 
 
 61728395 
 49382716 
 
 93953955555 
 
 2. -By what number must 12345679 be multiplied to 
 produce the answer in sevens? 63, 
 
 AN OU DOR WU AAI ie rob Wor lee 
 
 310. Leta person write any Unknown Number, 
 double it, add a given number, multiply by 5, cut off the 
 right hand figure, and subtract the first number. 
 
 31. 1. Take 12 for the starting number and 16 
 for the number to be added. 8. 
 
 PROCESS. 
 
 Expianation:—The result will be one-half the number 
 added, discarding fractions. After you have the result, 
 
MISCELLANEOUS. 101 
 
 other operations may be performed at pleasure. With 
 persons unacquainted with the principle upon which this 
 exercise is based, the successful operation of it will arouse 
 a vast amount of wonder and curiosity. 
 
 GREGORIAN CALENDAR. 
 
 312. To ascertain the Day of the Week for any 
 given date, divide the number of years elapsed since the 
 beginning of the century by 4, discarding fractions; to 
 this quotient add the dividend, the day of the month, 
 the complement of the month, and the complement of 
 the century; divide this amount by 7 and the remain- 
 der will indicate the day of the week. 
 
 313. The Complement of the present century is 
 0, and for 1900 it will be 5. The complements of January 
 and February, in leap years, are 2 and 5, respectively. 
 The other complements are exhibited in the following 
 
 AeA ie Br 
 Jan. 3. Apr. 2. Jul. 2. Oct. 3. 
 Feb. 6. May A. Aug. 5. Nov. 6. 
 Mar. 6. Jun. 0. Sep. 1. Dee. 1. 
 314. 1. On what day of the week did October 26, 
 1870, fall? Wednesday. 
 PROCESS. 
 70 Year 
 17 Leap Years 
 
 26 Day of Month 
 3 Comp. of October 
 0 Comp. of Century 
 7)116 Sum 
 16-4 Remainder 
 
 : 
 
102 MISCELLANEOUS. 
 
 EXPLANATION:—The remainder indicates the fourth 
 day of the week, or Wednesday. 
 
 2. On what day of the week did May 21, 1836 fall? 
 
 Saturday. 
 3. A gentleman was born April 25, 1869. On what 
 day of the week did his birthday fall? Sunday. 
 
 MARKING KEYS. 
 
 315. Merchants employ various combinations to 
 render cost and price marks unintelligible to all, except 
 their employes. Sometimes signs or symbols are used, 
 but most business men make use of words or phrases of 
 ten letters, each representing a figure. Thus take the 
 word 
 
 BRICK HOU SE 
 
 1238 45 6 7 8 9 O 
 
 EXPLANATION :—Suppose we wish to mark a piece of 
 goods at, say 15 cents, we would, using the above key, 
 write bk. The following keys are used in some of our 
 best business houses: 
 
 HANDY GIRLS 
 BLACK HORSE 
 THUMB SCR EW 
 ®ULcCK PAYER 
 CUMBE RLAND 
 SILVE RTOWN 
 VANDE RBILT 
 SPECU LATOR 
 IMPOR TANCE 
 CHARL ESTON 
 
MISCELLANEOUS. 103 
 
 TO FIND HER AGE. 
 
 316. Have Your Girl write the day of the month 
 on which she was born, annex two ciphers, add the num- 
 ber of the month in which she was born, multiply by 2, 
 add 5, annex two ciphers, divide by 2, add her age in 
 years, and subtract 250, and tell you the result. 
 
 317. 1. Suppose the person is 25 years old and was 
 born October 26. 
 PROCESS, 
 
 2600 
 10 
 
 2610 
 2 
 
 3220 
 5 
 
 2) 522500. 
 261250 
 25 
 261275 
 250 
 
 261025 
 
 ExpianaTion:—The two right hand figures will show 
 her age in years; the next two the month, and the re- 
 mainder, the day of the month on which she was born. 
 
 MAGIC WORDS. 
 
 318. The Magie Words are composed of ten 
 letters, each occuring twice. They are here appended 
 
104 MISCELLANEOUS. 
 
 as a matter of interest and discipline for the pupil. 
 
 ae) 
 
 DE TI D | 
 |) NOMEN | 
 | 
 
 ExpLaNnation:—Take twenty small articles of any kind 
 and divide them into couplets. Let each person present 
 choose a couplet. Then place the pieces in four rows, 
 like the words, being careful to place the pieces of each 
 couplet upon corresponding letters of the imaginary 
 words. It will now be an easy matter to pick out any 
 couplet, when you know the row or rows in which its 
 members are found. 
 
 MAGIC SQUARES. 
 
 319. A Magic Square is a square divided into a 
 number of smaller squares in which are inserted numbers, 
 generally in arithmetical progression, in such an order 
 that each line, whether added vertically, horizontally, or 
 diagonally shall amount to the same sum. 
 
 320. 1. Make a magic square, using the nine digits. 
 
 PROCESS. 
 2 | 716 
 Se ea os | 
 4|3|8| 
 
 Expianation:-—Write the numbers according to the 
 
MISCELLANEOUS. 105 
 
 accompanying diagrams, which explain themselves. 
 In constructing a square of nine figures, divide the num- 
 ber to which you wish the square to add by 3, and it 
 will give the average or middle number. By careful 
 study and a little practice the student will have but 
 little difficulty in making a square which will add to any 
 number he wishes. 
 
 ARRANGEMENT, KEY, 
 1234 em 
 abed se Sia alge 
 «5 A| x | a | 
 ABCD eis a 
 98S 76 djec |B 
 
 2. Make a magic square, using all the numbers 
 from | to 16 inclusive. 
 
 PROCESS, KEY. 
 | 2/15|14| 4 a|B/C/| da) 
 12, 6| 7| 9 E|ft|/¢|H) 
 8 10 1b 5 [h|@| Fle 
 / 7 : = 
 13 3) 2,16 Dic |b/|A 
 ARRANGEMENT. 
 12345678 
 a bedefgih 
 ABCDEFGH 
 1615 1413 121110 9 
 
 Nore:—Although this puzzle has occupied the attention 
 of many celebrated mathematicians from the earliest 
 
106004 MISCELLANEOUS. 
 
 times down to the present day, the above simple method 
 is now published for the first time. The magic square is 
 of but little use to the ordinary business man. Itis, how- 
 ever, a very interesting diversion and well worth investi- 
 gation by all who appreciate the wonderful. Any set 
 of numbers increasing or decreasing by a common 
 difference, may be arranged according to the above 
 formulas. 
 
 3. Make a magic square for the year 1888, using all 
 the odd numbers from 457 to 487 inclusive. 
 
 4, Using nine numbers, construct a magic square 
 which will add to my age, 21 years. 
 
 5. Construct a magic square for the year 1890, using 
 nine numbers. 
 
 6. Construct a magic square which will add to 100, 
 using all the even numbers from 10 to 40 inclusive. 
 
PROMISCUOUS PROBLEMS. 
 
 321. 1. Whatisathirdandhalfathirdof10? 5. 
 
 2. Express 100 by using the figure 9 four times. 
 
 998; 
 3. How many hands high is a horse which measures 
 5 feet? Le 
 4, How many steps of 2 ft. 8 in. each will a man take 
 in walking 2 miles? 3960. 
 5. What is the product of the complement of 80 
 and the reciprocal of 35? 500. 
 6. How many tons of hay in a stack 20 feet long, 
 15 feet high, and 14 feet wide? 3. 
 7. The sum of four equal numbers less 75 is 455. 
 What is one of the numbers? 132. 
 
 8. The sum of three equal numbers divided by 4 is 
 
 81. What is one of the numbers? 108. 
 9. If 2 cats can kill 2 rats in 2 minutes, how many 
 
 eats can kill 100 rats in 100 minutes? 2. 
 10. Multiply the sum of 2 and 2} by their difference 
 
 and extract the square root of the product. 1}. 
 
 11. Sold wheat at 63 cents and lost 123 per cent. At 
 what should I have sold to gain 84 per cent? 78¢. 
 
108 MISCELLANEOUS. 
 
 12. How many feet, board measure, in 6 pieces 4 x 4 
 at one end, 6 x 6 at the other, and 18 feet long? 234. 
 
 13. Ifa hen and a half lay an egg and a half in a 
 day and a half, how many eggs will one hen lay in 30 
 days? 20. 
 
 14. Bought bonds at 75 and sold them the same 
 day at 102 What was my per cent of gain on the in- 
 vestment? 36. 
 
 15. If you buy a saddle for 12 dollars, sell it for 15 
 dollars, and buy it back for 9 dollars, what per cent do 
 you make? 50. 
 
 16. How many bushels of wheat will a bin hold that 
 is 183 feet 6 inches long, 8 feet 4 inches wide, and 6 feet 
 8 inches deep? 600. 
 
 17. A man was married at the age of 25. If he lives 
 16 years longer he will have been married 47 years. 
 What is his age? 56. 
 
 18. When wood is selling at $4.50 per eord, how 
 much must I pay for a range 73 feet long, 43 feet wide, 
 and 13 feet high? $2.08. 
 
 19. If a cow and a calf eat a pumpkin and a half ina ~ 
 day and a half, how many pumpkins will they eatin a 
 month and a half? 45, 
 
 20. <A clerk spends 3 of his salary for board, and 3 
 of the remainder for clothes, am, saves 75 dollars a year. 
 What is his salary? 2 SO h Ds 
 
 21. Ifa sounding line is 100 Pr oehs long and all 
 but 24 fathoms and 3 feet is let out, what is the depth 
 of the water, in yards? | 151. 
 
 22. What must be the asking price of cloth, costing 
 $3.92 a yard, that I may deduct 123 per cent from it and 
 still gain 12} per cent? $5.04. 
 
MISCELLANEOUS. 109 
 
 23. How many shingles will be required to cover a 
 house 34 feet long, the rafters 12 feet long, showing 
 4 inches to the weather? 7344. 
 
 24. A contractor having | mile of road to grade, has 
 completed 319 rods, 5 yards, 1 foot, and 6 inches. How 
 much more has he to grade? 
 
 25. Ifa pole 10 feet long casts a shadow 16 feet long, 
 what is the height of a steeple which will cast a shadow 
 200 feet long at the same time? 125 ft. 
 
 26. A man being asked how many sheep he had, re- 
 plied: “If I had 3 times as many as I have and 5 more, 
 I would have: 260.” How many had he? 85. 
 
 27. A giant being asked his height replied: “If from 
 ~ 4 my height, in inches, 4 be subtracted, 4 of the remain- 
 der will be 4.” What was his height? 6 ft. 8 in. 
 
 28. A grocer claimed to sell sugar at cost, but used a 
 false pound in selling which contained but 14 ounces. 
 What per cent did he gain by the cheat? 144, 
 
 29. A snail, at the bottom of a 20 foot well, climbs 
 8 feet aday and slips back 4 feet at night. At that 
 rate, how many days will it be in reaching the top? 4. 
 
 30. A fence five boards high is built around a square 
 field containing 10 acres; each board is 6 inches wide. 
 What will the lumber cost at 3 dollars per hundred? 
 
 $198. 
 
 31. <A can doa piece of work in 30 days, B in 40 days. 
 After both werk 5 days, A leaves. How many more 
 days should A work, that the work may occupy but 20 
 days? 10. 
 
 32. The distance between St. Louis and Kansas City 
 is 283 miles. A postal clerk makes the round trip be- 
 tween these places, twice a week. How many miles will 
 he travel in a year? 58864. 
 
110 MISCELLANEOUS. 
 
 83. <A flag pole is 63 feet high. How many feet above 
 the ground must it be broken in order that the upper 
 part, clinging to the stump, may touch the ground 21 
 feet from the base? 28. 
 
 34. A jockey bought a horse for 90 dollars, sold him 
 for 100 dollars, bought him back for 95 dollars, and 
 sold him again for 100 dollars. How much did he make 
 by both transactions? $15. 
 
 39. A grocer claimed to sell molasses at cost, but used 
 a false gallon measure in selling, which contained 3 
 quarts and 1} pints. On a sale of 240 dollars, what did 
 he gain by the cheat? $16. 
 
 36. A log 20 feet long, 18 inches in diameter at one 
 end and 6 inches at the other, starts rolling on level 
 ground. How many revolutions will it make in rolling 
 around to the starting point? 40. 
 
 37. A and B together had 640 acres of land. A sold 
 B 100 acres and afterwards bought 150 acres of B, and 
 then sold C 160 acres, so that he had 60 acres less than 
 B. How many acres had each? 320. 
 
 38. Bought a horse, carriage and harness for 288 
 dollars. The horse cost 5 times as much as the harness, 
 and the carriage cost half as much as both horse and 
 harness. What was the price of the horse? $160. 
 
 39. A pile of wood is 114 feet 8 inches long, 4 feet 
 wide, and 6 feet and 3 inches high. What must be paid 
 for sawing and splitting, allowing 1 dollar per cord for 
 sawing and 50 cents per cord for splitting? $33.59.+ 
 
 40. A and B having started at the same time to travel 
 toward each other, met in 5 hours. B traveled 3 miles 
 an hour faster than A, and both together traveled 55 
 miles. How many miles per hour did each travel? 
 
 A 4, B7. 
 
MISCELLANEOUS. 111 
 
 41. ‘Two boats, leaving places 143 miles apart, sail 
 toward each other, one at the rate of 15 miles an hour 
 and the-other at the rate of 18 miles an hour. When 
 they meet, how many miles has each traveled? 
 
 j 65, 78. 
 
 42. Sold-a house for 720 dollars, taking a note draw- 
 ing simple interest at 7 per cent. Not needing the 
 money, I did not collect the note until the end of two 
 years 5 months and 27 days. What amount did I col- 
 _ leet? $845.58. 
 
 43. What must I pay for 1} gross of clothes pins at 
 4 cents a dozen, a quintal of fish at 2 cents a pound, 
 a half barrel of pork at 7 cents a pound, a half ream of 
 paper at 12 cents a quire, anda score of knives at 40 cents 
 apiece? $18.92. 
 
 44. A man was hired fora year for 100 dollars and a 
 suit of clothes; but at the end of 8months he left, and 
 received the suit and 60 dollars in money as full com- 
 pensation for the time he had worked, What was the 
 value of the suit? $20. 
 
 45. James and John agree to mow a field of grass in 
 11 days for 62 dollars. James claims that he can do 
 * of the work in the required time, and John claims that 
 he can do 4 of the work in the required time. How 
 much should John receive? $32. 
 
 46. A boy was engaged to work 30 days on these con- 
 ditions: For every day he worked he should have one 
 dollar and his board, and for every day he was idle he 
 should forfeit 25 cents for damage and board. At 
 the end of the month he received 20 dollars. How 
 - many days did he work? 22. 
 
 47. A bought 5 peaches for 5 cents, and B 9 peaches 
 for 9 cents. They were joined by C, and each one ate 
 an equal share. When C left he gave them 14 cents to 
 pay for his share, but they quarreled about the division of 
 it. How should this amount be divided? <A 1, B 13. 
 
FIZS MISCELLANEOUS. 
 
 48, A thief bought a coat for 5 dollars and tendered 
 a 50 dollar bill in payment. The merchant could not 
 make the change, but took the bill to a banker, who 
 changed it for him. After the thief had gone, the bank- 
 er discovered that the bill was counterfeit. He at once 
 demanded good money from the merchant, who was | 
 obliged to return him 50 dollars. How much did the 
 merchant lose? Coat and $5. 
 
 49, Three travellers met on the plains of Arabia; and 
 two of them brought their provisions with them; but the 
 . third, not having provided any, proposed to the others 
 that they should eat together, and he would pay the value 
 of his portion, This being agreed to, A produced 3 
 loaves, and B 5 loaves, all of which the travellers ate to- 
 gether, and C paid 8 pieces of money as the value of his ~ 
 share, with which the others were satisfied, but could not 
 agree about the division of it. Upon this, the matter was 
 referred to a judge, who decided impartially. How 
 
 many pieces did each receive? ASE aise