^^?^^^ UC-NRLF *B 3Db 33E ^'^^^W r^vl|?^M|^'#^»ti^i ^^ Jndujctiuje $^ri^s. KEY PRACTICAL ARITHMETIC CONTAINING THE SOLUTION MORE DIFFICULT EXAMPLES. BY WILLIAM J. MILNE, Ph. D., LL D., PRlNCIPAIi OF THE STATE NORMAL, SCHOOL, GENESEO, N. Y VAN ANTWERP, BRAGG & CO. CINCINNATI. NEW YORK. Eclectic Educational Series. McGvffey^s Revised Readers, Speller cmd Charts, Milne's Inductive Arithmetics and Algebra. Eclectic School Geometry. Harvey^s Revised Grammars. Holbrookes Normal Grammars. Ridpath's Inductive Grammar. New Eclectic Geographies. New Eclectic Penmanship. Eclectic System of Drawing. Forbriger's Drawing Tablets. Eclectic Histories of the United States. Thalh^imer's Historical Series. Ridpath's Histories of the United States. Eclectic Complete Book- Keeping. Murdoch's Analytic Elocution. Kidd's Neiv Elocufion. Etc., Etc., Etc. Descriptive Catalogue and Price List on application, IdUCAIIOS IIE3 ji Copyright, 1878, by John T. Jones. ECLECTIC PRESS, Van Antwerp, Bragg & Co., Cincinnati, 0. This work is designed to aid the teacher in the prepa- ration of his class exercises, and to assist him in detecting errors in the processes of the pupils. It contains concise analyses and brief solutions of all the difficult problems, thus relieving the instructor from the burdensome neces- sity of devising forms of analyses for his classes. Many of the problems can be solved in several ways, but the author has deemed it best to give that form of solution which is most natural to the student, although it may be occasionally longer than some other form. The student who is pursuing his arithmetical studies without the assistance or direction of an instructor, will be able to economize his time by a judicious use of this work. W. J. M. April, 1S78. M&T70:i9 (HI) Jhe Inductive ^erie^ OF MATHEMATICAL TEXT-BOOKS. I. FIRST I.ESSONS IN ARITHMETIC. II. PRACTICAL ARITHMETIC. III. KEY TO PRACTICAL ARITHMETIC. IV. ELEMENTS OF ALGEBRA. V. ELEMENTS OF GEOMETRY. {In preparation:} (iv) KEY MILNE^S PEAOTIOAL AEITHMETIC. NOTATION AND NUMERATION. Page 13. Example 2. 36,218,846 : Thirty-six million, two hundred eighteen thousand, eight hundred forty-six. 3. 84,540,600,040 : Eighty-four billion, five hundred forty million, six hundred thousand, forty. 4. 201,075,562,012,001: Two hundred one trillion, sev- enty-five billion, five hundred sixty-two million, twelve thousand, one. 5. 60,402,333,200,111 : Sixty trillion, four hundred two billion, three hundred thirty-three million, two hundred thousand, one hundred eleven. 6. 73214070. 7. 80040612788. 8. 225641351. 9. 354604892036. Page 14. 1. One hundred sixteen thousand, two hundred thirty- four. 2. Sixty-five thousand, two hundred thirty-one. (5) 6 NOTATION AND NUMERATION. 3. Twenty thousand, seven hundred three. 4. Seventy-one thousand, five. 5. Three thousand, one hundred four. 6. Forty-eight thousand. 7. Sixty thousand, twenty-nine. 8. One hundred forty-one thousand, one hundred twenty. 9. One hundred one thousand, two hundred seven. 10. Sixty-eight thousand, nine hundred seventy-eight. 11. Seventy-two thousand, twenty. 12. Eighty thousand, one. 13. Eight hundred fifty-seven thousand. 14. Ninety-one thousand, twenty-nine. 15. Seven thousand, six hundred forty. 16. Eight hundred thousand, nine hundred. 17. Two million, ^ye hundred sixty-eight thousand, two hundred forty-two. 18. One million, eight thousand, three. 19. Two hundred twelve million, three hundred seventy- five thousand, six hundred forty-seven. 20. Six hundred nine million, three thousand, five hun- dred eighty-eight. 21. Eight hundred ninety-seven million, eight hundred fifty-six thousand, eight hundred forty-six. 22. 200 : Two hundred. 23. 60,002 : Sixty thousand, two. NOTATION AND NUMERATION. / 24. 700,000,000 : Seven hundred million. 25. 230,000,060 : Two hundred thirty million, sixty. 26. 81,501,007,012 : Eighty-one billion, five hundred one million, seven thousand, twelve. 27. 30,000,000,000,603 : Thirty trillion, six hundred three. 28. 700,080,000,000,000: Seven hundred trillion, eighty billion. 29. 8,007,014,010 : Eight billion, seven million, fourteen thousand, ten. 30. Ie5,000,018,207,000,081 : Fifteen quadrillion, eighteen billion, two hundred seven million, eighty-one. Page 15, 1. Sixty million, seven hundred one thousand, eight hun- dred ninety-two. 2. Fifty million, six hundred seven thousand, eight hun- dred one. 3. Six hundred thousand. 4. Forty-nine million. 5. Five hundred ninety-three billion, six million, seventy thousand, five hundred. 6. Nineteen quadrillion, nineteen trillion, one hundred ninety million, nineteen thousand, nineteen. 7. One hundred sixty-three million, one hundred ninety- four thousand, five hundred sixty-eight. 8. Three billion, fifty million, fifty thousand, one hundred eighty-three. 8 NOTATION AND NUMERATION. 9. Five million, two hundred four. 10. Five hundred ninety-four thousand, nine hundred. 11. Twelve million, twelve. 12. Two hundred trillion, seven hundred ninety-eight bilh ion, thirteen million, four hundred thousand, nineteen. 13. Two hundred twelve quintillion, five hundred six quad- rillion, sixty-seven trillion, ninety-three billion, twelve million, sixty-three thousand, sixty-seven. 14. 2060153. 15. 60060060060. 16. 60200500. 17. 402348213020. 18. 78640009006016. 19. 6542000025. 20. 6542000025. 29. 941000000116022. 30. 23023023023. 21. 402348213020. 22. 5268949. 23. 200300800. 24. 29599000601. 25. 4000558244070. 26. 32061343404. 27. 555777669. 806070385206. Page 16. 31. 600075. 32. 12008988013. 33. 29757000480013565. 1. $2,235. 2. $202,025. 3. $112.25. 4. $602.09. Page 17. 5. $20000.32. 6. $12700000. 7. $6000000.88. 8. $12300.15. SUBTRACTION. Page 18. Fifteen ; twenty-four ; thirty-nine ; forty ; forty-nine ; ninety-nine ; seventy-seven ; three hundred eighty- nine ; seven hundred thirty-six ; five thousand, five hundred fifty-five ; five hundred fifty thousand, six hundred ; two hundred ten thousand, five hundred six ; seventy-three thousand, eight hundred ninety- nine ; one million, five hundred ninety-five thou- sand, eight hundred sixty-four. XV; XVIII; XXVII; LXXXI; XCV; LXXXVI; DXXXIV; DCLXXXIV; ML; VHllV; VTI; LXXVDCCCLXIX. SUBTRACTION. Page 41. 33. 297 + 308 == 605 mi. ; 861 — 605 = 256 mi., Am. 34. $584 + 8759 + $463 = $1806 ; $1806 — $1298 = $508, Am, Page 42. 35. 1235 + 1317 = 2552 pounds ; 3715 — 2552 == 1163 pounds, Am, 36. $637 + $317 = $954 ; $729 + $356 =. $1085 ; $1085 — $954 --$131 gain, Am, 10 MULTIPLICATION. 37. 1317 + 2357 = 3674 bricks in first two ; 3674 — 1719 = 1955 bricks in third, Ans, 38. 874,120,005 — 264,146,900 = 609,973,105 bu., Ans. 39. 351 + 27 = 378 acres, B's land ; 537 — 378 = 159 acres C had at first, Am. 40. $308.40 — $106.28 =- $202.12, cost, Ans. 41. 1870 — 378 = 1492, Ans. 42. $2191 + $3256 =: $5447, total deposited; $3412 + $2164 = $5576, total drawn out; $5576 — $5447 = $129 more drawn than deposited ; $1826 — $129 =r $1697 on hand Wednesday morning. 43. $2895 + $3864 = $6759 ; $15795 — $6759 = $9036 invested in land, Am. 44. 68754 — 56849 = 11905; 89346 — 68754 = 20592 ; 20592 — 11905 = 8687, Ans. 45. 38944 - 35442 = 3502 pupils, Am. 3IUL TIP Lie A TION. Page 55. 1. 896 X 58 = 51968 lb. ; $ .63 X 51968 = $32739.84. 2. $3.50 X 3923 = $13730.50, Ans. 3. 119 X 6 = 714 bu. ; 714 + 515 = 1229 bu.. Am. 4. 64 X 113 = 7232 ; 47 X 7232 = 339904 yd., Ans. MULTIPLICATION. H 5. $85 X 25 = $2125 ; $4.50x316 =-$1422;. $8 X 94 = $752 ; $2125 + $1422 + $752 = $4299, Ans. 6. $3750 + $4650 = $8400 ; $8400 = 84 hundred dollars ; $2 X 84 == $168, Ans. 7. 47 X 219 =-: 10293 bu., Ans, 8. 6 X 2 = 12 quills furnished by each goose; 12 X 398 = 4776 quills, Ans, 9. $.26 X 81 ==$21.06; $.28 X 53 = $14.84; $21.06 + $14.84 =: $35.90, Ans, Page 56. 10. 1716 X 4 = 6864 pickets for one lot; 6864 X 13 = 89232 pickets for all, Ans. 11. 56 X 13 = 728 ; $ .34 X 728 = $247.52, Ans. 12. 10 X 13 = 130 tons ; $6.85 X 130 = $890.50, cost; $7X48 = $336; $8.25 X 28 -= $231 ; $8.75X27 = $236.25; 48 + 28 + 27 = 103; 130 — 103 = 27 tons ; $9.50X27 = $256.50; - $231 + $236.25 + $256.50 = $1059.75, selling price ; $1059.75 — $890.50 = $169.25 profit, Am, 12 DIVISION. 13. 315 + 417 -= 732 ; 732 X 13 = 9516, enemy's loss; 9516 + 732 ::^ 10248, total loss. Am. 14. 18 X 36 = 648 miles sailed by first; 15 X 36 —- 540 miles sailed by second ; 648 — 540 = 108 miles apart, Am, 15. $.65X350 = $227.50; $1.35X215 = $290.25; $.43X273 = $117.39; $227.50 + $290.25 + $117.39 == $635.14, Am. 16. $65.50 X 325 = $21287.50 ; $3.25 X 345 = $1121.25; $2684.95 + $1121.25 = $3806.20 ; $21287.50 — $3806.20 = $17481.30, Am. 17. $1.75X28X17 = $833; $1.85 X 29 X 23 = $1233.95 ; $833 + $1233.95 = $2066.95, Am. 18. $3.25 X 37 X 13 = $1563.25, Am. 1 DIVISION. Page 68. 48. 6318 -^ 39 = 162 lots. Am. 49. $4386 -^ 17 = $258, Am. 50. 41600 -^ 320 = 130 mi., Am. 51. $6.84 -^- $ .38 = 18 doz., Am. f ANALYSIS AND REVIEW. 13 62. 60 X 24 == 1440, minutes in 24 hours ; 1575000 ~ 1440 = 1093|f|^ mi. per minute, Am. 53. 29100 -^ 5280 = 5|||^ mi., Ans. 54. 5X5 = 25, cumber of loads drawn per day ; 25 X 1250 = 31250 bricks drawn per day ; 4375480 -^ 31250 = 140 da. ; 480 bricks left. Am. 55. 91500000 -f- 185000 = 494||f|f^ sec, Ans. 56. $63X278 = $17514; $17514 — $1275 = $16239, amount unpaid; $16239 -^ 8 = $2029|, each annual payment. Am. 57. $3681452 -^ 365 -= $10086^2_^ av. da. income, Ans. 58. 221760 -^ 42 = 5280 ft., Am. 59. $50000 ~ $500 = 100, number of men, Am. 60. 1071400 -^ 50704 = 21-5^6^, average number, Ans. ANALYSIS AND JREVIEW. Page 72. 11. ( 12 + 7 — 9 ) = 10 ; 10X5 = 50, Ans. 12. ( 13 — 6 + 8 ) = 15 ; 15 X 6 = 90, Am. 13. (11— 2 + 5) = 14; 14X8 = 112,^718. 14. (3 + 4)x9 = 63; (3 + 6)-^3 = 3; 63 — 3 = 60, Am. 15. (54_7_3)X3 = 27; (3 + 5 — 4) -^4 = 1 ; 27 + 1=28, Am. 14 ANALYSIS AND REVIEW. 16. (36-7) X 5-: 145; (102 + 6) -^ 9 = 12; 145 + 12 :=: 157, Ans. 17. (99 — 3)-^8 = 12; (86 + 10) -M2 ^ 8; (3 + 6)-^3 = 3; 12 — 8 + 3 = 7, ^?is. 18. (45 + 3)-^6 = 8; ( 10 H- 15) -f- (7 — 2) = 25 -^ 5 =- 5 ; 8 + 5 + 6 = 19, Aiis. 9. 1118 A. =:. 1st tract; 3 times 1118—193 == 3161 A. == 2d tract; 2 times ( 1118 + 3161 ) — 105 = 8453 A. = 3d tract ; 1118 + 3161 + 8453 = 12732 A. ; 12732 -^ 5 = 2546| A., the share of each, Ans, Page 73. 20. $39 X 1516 = $59124, what he paid for cattle; $59124 + S1819 = $60943, what he sold them for; 1516 — 97 = 1419, number of cattle sold ; $60943 ^ 1419 = $42|ff I, the price per head, Ans. 21. $30 — ($7.25 + $4.25) = $18.50, his weekly savings; $1500 -^ $18.50 =: 81 J-g^^o weeks, Ans. 22. $115 + $30 = $145, entire expense of horse ; $155 + $50 =: $205, entire returns from horse; $205 — $145 = $60, gain, Ans. 23. $9215 — $8735 r=.r $480, the entire gain ; $480 -^ $ .25 = 1920, the number of bushels, Ans. 24. $200 + $50 = $250, the entire selling price ; $250 -^ 25 = $10, the selling price per barrel ; $10 — $8, first cost, = $2, gain per bbl., Ans. ANALYSIS AND REVIEW. 15 25. Since a yard of both kinds of cloth would cost him $13, he could buy a yard of both kinds as many times as $13 are contained times in $585, which is 45 times ; therefore he bought 45 yd. of each, Ans. 26. 48 -f 52 = 100 mi., the distance apart each day; 100 mi. X 5 = 500 mi., the whole distance, Ans. 27. 31 X 20 = 620, No. days required for 1 man to do it ; 620 -^ 31 =: 20, No. days it will take 31 men to do it. 28. $3.50 X 16 = $56, am't received for apples; $120 — $56 = $64, am't to be realized from barley ; $64 -^ $ .80 = 80, number of bushels of barley, Am. Page 74. 29. $9600 -^ $120 = 80, number of acres sold first; 140 — 80 = 60, number of acres sold at cost ; $17500 -^ 140 = $125, cost per acre ; $125 X 60 = $7500, amount of second sale ; $9600 + $7500 = $17100, am't for which it was sold ; S17500 — $17100 = $400 loss, Ans, 30. $628 + $350 + $262 + $700 + $175 == $2115, his an- nual expenses ; $2115 X 4 = $8460, expenses for 4 years; $53 X 130 = $6890, entire savings in 4 years; $15350, entire earnings in 4 years. $15350 — 4 = $3837.50, his annual income, Ans. 31. $196,000,000 -f- 2124 = $92278if||, the average loss per acre, Ans. 32. 140 -^- 4 = 35, number of rods he can run per min. ; 630 -^ 35 = 18, number of minutes required, Ans. 16 DIVISION BY FACTORS. 33. 315 + 175 + 300 = 790, number in the three fields ; 1000 — 790 = 210, number equally distributed in the other two fields; therefore, 210 -^- 2 = 105, the number in the fifth field, Am, 31 $45000 -^ 3 = $15000, amount given to grandchildren ; $15000-^11500 = 10, number of grandchildren. Am, 35. 22,300,000 ^ 6000 = 3716f^^^ pounds. Am, 36. 872,320 + 37,344 + 26,344 = 936,008, entire area ; 936,008 -^ 43,560 = 21|-i|ff, number of acres. Am. m VIS I ON BY FACTORS. Page 81. 14. The factors of 72 are 8 and 9. 8 )3528 9)441 49 Quotient, 49. 16. The factors of 32 are 4 and 8. 4)3824 4X4 = 16, true Rem. 8)956 119 ... 4 Eem. Quotient, 119i-|. 16. The factors of 49 are 7 and 7. 7)2184 4x7 = 28, true Rem. 7 )312 44 ... 4 Rem. Quotient, 44|^. DIVISION BY FACTORS. 17 17. The factors of 56 are 7 and 8. 7 )3275 3X7 = 21; 8 )467 . . . 6 Rem. 21 + 6 = 27, true Rem. 58 ... 3 Rem. Quotient, 58||-. 18. The factors of 27 are 3, 3, and 3. 3 )3276 3 )1092 1X3X3 = 9, true Rem. 3)364 121 ... 1 Rem. Quotient, 121^V 19. The factors of 45 are 3, 3, and 5. 3)4104 3 )1368 1X3X3 = 9, true Rem. 5)456 91 ... 1 Rem. Quotient, 91^V 20. The factors of 24 are 2, 2, 2, and 3. 2 )7304 2)3652 2)1826 1X2X2X2 = 8, true Rem. 3)913 304 ... 1 Rem. Quotient, 304-2^. Page S*>. 21. 1120 -f- 5 = 224, number of canisters ; 224 -f- 7 = 32, number of packages, Ans. 22. 912 -^ 6 =: 152, number of packages ; 152 ~ 4 = 38, number of quires, A^is, 18 GREATEST COMMON DIVISOR. CA N CELL A TION. Page 84. Ao. = v4off , the price per acre, Ans, 5 X 95 17. ^X 44x$.ll ^r,^ ^^^ number of bushels, Ans, $2.00 '' 18. 13X39X^$.32 ^^3^.^^^ ^^^ ^^^^ ^^^^ ^^^^^ ^^^^^ GREATEST COMMON DIVISOIl. Page 87. 20. The widest flagging that will suit all the walks Avill be equal to the greatest common divisor of the given widths, which is 14 inches, Ans, Page 88. 21. The largest packages will be equal to the greatest com- mon divisor of the given amounts of tea, which is 5 pounds, Ans, 22. The largest equal fields that can be formed from 324 acres and 78 acres will be equal to the greatest com- mon divisor of these numbers, which is 6 acres. There will therefore be 54 fields in one farm and 13 in the other, or 67 fields in both, Am, LEAST COMMON MULTIPLE. 19 LEAST C031M0N 3IULTIPLE. Page 91. 30. The box must be some number of times 6 inches, 8 inches and 12 inches in length, or a common multiple of those numbers, which is 24 inches, Aiis, 31. It must contain a number of yards which is some num- ber of times 4, 5 and 6 yards, or a common multiple of those numbers, which is 60 yards, Ans, 32. The number of pennies must be some number of times 4, 6, 8, 10 and 12, or the least common multiple of those numbers, which is 120 pennies, Ans. 33. The number of bushels in the bin will be equal to the least common multiple of 7, 10 and 30, which is 210 bushels, Ans, 34. The next time Avhen they wdll again start out together must be a number of times 8, 9, 15 and 20 weeks, or the least common multiple of those numbers, which is 360 weeks, Ans. Page 92. 35. A walks around the island in 24 days, B in 20 days, and C in 15 days, and therefore the time that will elapse before they are all together again will be some number of times 24, 20 and 15 days, or 120 days, Ans, 37. The length of the longest boards will be equal to the greatest common divisor of 48, 60, 96 and 108 feet, wdiich is 12 feet, Ans. 20 ADDITION OF FKACTION8. 39. The smallest number that will contain 250, 350 and 525 is their least common multiple, which is 5250, which must be increased by 25, since there is to be a remainder of 25. Therefore tlie answer is 5275. 41. The amount to be invested in each will be the least common multiple of the given numbers, which is $336. 42. If 9 is added to the least common multiple of 24, 28, 32 and 36, the result will be a multiple of 25. The least common multiple of those numbers is 2016, to which 9 is to be added, making 2025, Ans. 43. The size of the largest lots will be equal to the great- est common divisor of the numbers 152, 288 and 184 rods, which is 8 rods, Ans, ADDITION OF FRACTIONS. Page 107. 12. f + l+l + f + i + f- 2^/^, ori|; 4 + 5 + 8 + 2 + 7 + 4 = 30 Sum = 32H, Am. 13. -l + l + § + T% + ^ = 9 + 7 + 8+ 7 + 8 = Sum = 39 41U,Ans. ADDITION OF FRACTIONS. 21 14. -h + A + cV + A = m^ or If; Sum = 29||, Ans. 15. li + H + A + A + if^llMI 2+3+2=7 16. | + t\ + A + A= IftH 3+4 + 6 + 9 -=22 Sum =- 23|fi-|, ^?is. 17. He received the sum of $18f, $65f and U(Sl\. «f +»f + S| = $22%, or^; $18 + $65 + $161 = $244_ Sum -= «246yV, ^ns. 18. They all earn the sum of $67|, $23f and S23f. $1 +$| +$| = $2i $67 + $23 + $23 -= $113 Sum — $115|^, Aiis. 19. He walked the sum of 45| mi., 47f mi. and 50f mi. 2 I 3 4_ 5 110.3 ■§■ -r ^ ^ T — -^105 45 + 47 + 50 r= 142 Sum =- 143|^f , Am. 20. Since B has lOf acres more than A, he has the sum of 5^ and lOf acres, which is 15^^ acres. Smce C has as much as A and B, he has the sum of 5^ and 15^^ acres, which is 212% ^<^^^s. The sum of B's and C's will be the sum of 15i^ and 21^, which is 36^ acres, Am. 22 SUBTRACTION OF FRACTIONS. SUBTRACTION OF FRACTIONS. Page 109. 15. 10i| = 10if| 13 — 52 ¥9 — To^e- Rem. =r lOiff , or lOfJ, Jns. 16. 66| = m^ Eem. = 33^13^, ^ns. 17. 2101 = 210^-98 =r 209f| 109| = 109if ^ 109|f Rem. = lOOII, ^ns. 18. 112 ==111| 75i- 75i Rem. = 36|^, Ans. 19. 606f = 6061 70i=._70| Rem. = 536^, Am. 20. 589f = 589^ == 588ff 67f- 67^^:^ 67A R^m. = 521||, ^m. 21. There is left the difference between 506|T. and 418|-T. 506| -= 506f T. 418i = 418f T. D\l^ 88iT., Am. MULTIPLICATION OF FRACTIONS. 23 22. She paid for the articles purchased the sum of $2^ $15f and $3f , which is $22. Therefore, $25 — $22 --=^ $3, the amount she had left, Ans. 23. He sold the sum of 60| A., 45^ A. and 116|: A., which is 222i A. Therefore he had left the difference be- tween 412 and 222^ A. 412 :=.411f A. 222|^ = 222^ A. Di£ = 189|- A., Ans. 24. His entire expenses were equal to the sum of and $4|-, which is SSly^^. Therefore, $50^ — $31y^^ = the amount saved ; Dif. = JQJ Ul ^±0-^, MVLTIPLICATIOK OF FRACTIONS. Page 114. 24. The product is equal tofXfXfXi^XfX|-, which, cancelling, becomes -i^ = 3^, Ans. 25. The product is equal to f X yV X -f- X -f- X i^ X ^, which, cancelling, becomes ^ = 27, Ans. 26. The product is equal to-|XfXtXfXf, which, can- celling, becomes ^ = 31^, Ans. 24 DIVISION OF FRACTIONS. 27. The product is equal to V X -^- X ^ X t X f- X J- X f , which, ciiiicelHng, becomes ^-f-'^ = 147, Ans. 28. The cost will be j% of $f , which is $|, ^?is. 29. There are 5|- times 161 ft. xi X ¥ == "F" = ^^1 ^' 30. He sold f of f or f of the mill, Ans. 31. There are 121 times 42| yards. 2_5 s^ i^i ^ .4.2_z.5 =- 534| yd., Ans. 32. The amount realized will be 44 times $16|-. -2/ X ^-F := §81, ^ns. Page 115. 33. iXrVxif x^x¥=-lll-=i^, ^^i^- 31 iV x ¥ X 14 X ¥ =- ^Iff ^ - ^2H-|, ^n^. DIVTSIOJV OF FRACTIONS. Page 121. 15. The quotient is equal to f X | X V X | X | X A = 2 5 1 9 yJ^jo 16. The quotient is equal to f X H X V^ X f\ X f X A == 2V, Ans. 17. The quotient is equal to|XfXfXfX-| = | = 2|, ^?is. 18. The quotient is equal to|X|Xi\XiXfX^== DIVISION OF FRACTIONS. 25 19. The quotient is equal tofXfofi^X|XfXi = ■28" ^^ ^2 8' ■^'^^' 20. The quotient is equal to f X it X -^^ X -^# X | X tV 3_5 9_3_ J^.o 16 ^ 1 6 » -^'«"^' 21. The quotient is equal to y\ X V- X f X f X i X f = 22. The quotient is equal to ^^XiXlX|XfXi== 1-1 = 7^-^, Ans. 23. As many pieces as y\ of a yard is contained times in i of j% of a yard. | of /^-f- ^^^ = i X A X ^2^-= 2^1::^ 3 1 pieces, ^n8. 24. He will spend SI 7^ in as many days as $| is contained times in $171. 171 _^ 2 ^ 3_5 s^ | ^ 1 05 ^ 26^ da. 25. As many yards as $3f are contained times in $317f. 317| ^ 3f = 953 X 3-V =- 4|65 _ 9321 yd.^ ^,,3. Page 122. 26. As many bushels as $| is contained times in $17^. 17^ -^ I = 6^9. X I = 46 bu., ^918. 27. As many weeks as | of a barrel is contained times in 51 barrels. 5| -^ | = V X t = "^s^" "^ '^ J ^^^- ' ^*^^- 28. As many days as $f is contained times in $3f . 3| -^- f = ¥ X i==W = ^^ cla., ^ns. 29. Since the number when multiplied by f gives a product of li, li -f- 7 will give the number. 1 1 -^ f = f i- X |- = |- = 1| the number, ^?is. 30. As many days as 1^ cords are contained times in 17^ cords. 17i -^ li ^ -V- X t == ¥/ ^ l^il da. , Ans. 26 FRACrnONAL, FORMS. 31. Since there are 35 days in 5 weeks, the horse will eat 3V of 12| bushels daily. 3V of 12^ bu. = 3V of 2/ bu. = -fi bu., Ans, 32. As many bushels can be bought for $3168 as $1| are contained times in $3168. 3168 -^ 1|- = ^V^ X A = &^ = 1689-1 bu., Ans. FRA CTIONAL FORMS. Page 123. »• -^ — -7- — - y — y A -9 — o3> ^^'^• _ 3 •'X^^^e — 4X3- — rxr> ^^^^• G 5- ¥ = M-^A=-If XJ^ = M = 1A, Am. 13 6- ^ = 4-^61 = 4 X^T = A. ^'»«- ^ i 7. -^ 1_!L_42 Iv 3 1 J^c 4|" 8. ^ — 5i-l-lr=5-L 9. g = 6f^5i = i^X A = i|| = l,¥^, J«' FRACTIONAL RELATION OF NUMBERS. ai 16^/ 4 4 Jniq ^4 ^r X 2 5 7 5' ^^^^• 11. ft 13 • 5 13v19 247 9-7. J«o ^^' -5- 2"¥ "^ T9 2 4 X -3^- T2'0 ^ 1 2ir> ^'''^• 27 10. ^ = dl 5 19 13. ^ = 51- of I 12. 19 24 • 12 24v/ 7 14 J-,,0 11 1_6 V ^ — i-9-2 — 1 1-V Ans 14. |-^| = |offdividedby|of9=|X|XfXi = ^f. 15. M-? r^- 2 of 8 divided by6 = |XfXl = i, Am. 16. iof| = 3 divided by J of f -= f X | X f = 6f , ^?is. ^^* l|^==^ ^^* ^^'"^"^ ^'^ 4ix3-f Xf Xi^Xi-^. FRACTIONAL RELATION OF NUMBERS. Page 124. 29. J is I of |, and 1 is 5 times | of | or | of |; and since 1 is I of f, I of 1 is I of I of I or I of |. Therefore, I is f off. 30. 1^ is I of |, and 1 is 8 times | of f or f of |- : and since 1 is I of -|, f is f^ of I of I or ^ of |. Therefore, f is Vof|. 28 FRACTIONAL RELATION OF NUMBERS. 31. 4" is ^ of ^, and 1 is 7 times \ of -f, or } of 4; and since 1 is -I of ^, A of 1 is TT of i of f or |i of f There- fore, -^ is 1^ of f . 32. jlj- is |- of -j^, and 1 is 11 times ^ of ^\, or y- of -^j ; and since 1 is y- of y\, f of 1 is f of V- of ^, or f| of y\. Therefore, f is ffofyV 33. ^ is ^ of |, and 1 is 3 times ^ of |, or f of | ; and since 1 is I of I, I of 1 is I of f of f , or |f of |. Therefore, -g- is ig OI 3. 34. 4 is ^ of f, and 1 is 5 times ^ of f , or | of f ; and since 1 is I of I, I of 1 is f of I of f , or f of f . Therefore, f is I of f . 35. -^ is -|- of f , and 1 is 7 times ^ of -f-, or |- of f ; and since 1 is |- of -f-, f of 1 is f of ^ of f, or j'^^ of f . Therefore, I is Ve of y. 36. -I" is ^ of |, and 1 is 5 times ^ of f , or f of f ; and since 1 is I of f, f of 1 is f of I of I, or if of |. Therefore, f is II off. 37. "I" is ^ of f , and 1 is 9 times -I- of 4, or | of |; and since I is f of I, y% of 1 is -i§j of f of I, or fj of |. Therefore, y\is|^of|. Page 125. 13. Since |f is f of some number, I of the number is \ of i|^, or y^y; and since ^ of the number is ^, the number is 5 times ^, or i|-. Hence, j^ is f of ||. Since || is ■5^ of some number, ^ of the number is | of ||, or ^\, or \ ; and since ^ of the number is |^, the number is II times \ or -y-. Hence, |f is -^^ of -y-. REVIEW EXERCISES 29 14. Since -|-| is -f- of some number, ^ of the number is -J- of -|^, or -g^^- ; and since ^ of the number is -3^-5 , the number is 7 times ^%, or |-|, or f. Hence, -i-f is f of f. Since f^ is ^ of some number, ^ of the number is ^ of |-|, or ^; and since } of the number is -^-^^ tlie number is 7 times y^- , or -}-|- Hence, ^ is -^^^ of j^, 16. Since ff is ^^^ of some number, ^ of the number is 4- of |-|, or ■^; and since ^ of the number is -gV, the number is 13 times -^j, or f|, or |^. Hence, ff is -^^ of I". Since ff is |- of some number, ^ of the num- ber is I of If, or ^'^, or | ; and since -f of the number is I, the number is 7 times }, or ^, or 1. Hence, || is f of 1. 16. Since || is -j^ of some number, -j-V ^^ ^^^ number is •1- of II, or 2t; and since y^ ^f the number is 2^? the number is 15 times -225-, or f. Hence, || is ^ of f . Since f| is ^^^ of some number, yV of the number is | of 49. or -^Q ; and since y^ of the number is -^, the number is 10 times -^, or |^, or f. Hence, |f is y^^- off UEVIEW EXERCISES. Page 125. 1. $7f + $5|- = $13^, value of purchases; $20 — $13^ = $6f , the change he should receive. 2. $ .Hi- X 360 = $40.05, cost of sugar; $.621 X 50 =z 31.25, cost of tea; $71.30, cost of both. Arts, 30 REVIEW EXEUCISES. 3. Since he can cut ^ ^^ '^ acres, or o^ acres, in 1 day, in ■f of a clay he can cut j- of o^ acres, or 2^ acres, Am^. Page 126. 4. They will cost 3^^ times $.18f, which is $ Mj\, Am. 5. If a man can hoe a field in 7\ days, he could hoe one 2| times as large, in 2f times 7\ days, or ^ days; and 3 men can do it in ^ of the time required for 1 man to do it, or in ^ of -^J- days, or ^ = 6f days, Ans, 6. Since ]- of the quantity leaked out, there was left but -f of 41-J- gallons, or 35^ gallons; and since he wishes to obtain $6.15 for 354 gallons, the price of 1 gallon is found by dividing S6.15 by 35f , which gives 17f| cents. 7. Mr. Banker bought | of f of 155 A., which is 37|^ A. 8. The entire cost of the block was the sum of 83122^, $13684, $3258f and S1325y^2> ^^'J^ich is $9074^^; and $10000 — 9074y\V = S925yVa. the gain, Ans. 9. f of 272^ sq. ft., or 1632V n- ft., Ans, 10. Since $3215 is If, or y-, of the cost, -J- of the cost is -^^ of $3215, which is $^-f-|-'^ ; and since ^ of the cost is $^j\^-, the cost will be 9 times $4p, or $^1^:^^, or $2066fi^. 11. They will earn 8 X 6| X $2|, or $153^, Ajis. 12. He will walk 30^ miles in as many hours as 3-J- is con- tained times in 304-, which is 9y% hours, Aiis. 13. f of $19000 = $11400, am't divided among 4 sons; $11400 -^ 4 == $2850, the share of each son ; I of $19000 = $7600, am't divided among 3 daughters ; $7600 -f- 3 = $2533 1, the share of each daughter, Ans. EEVIEW EXERCISES. 31 14. S .20 X 3140 = 8628, the amount received this year. Since SB28 is } of what would have been received for it last year, J- of what would have been received for it last year is } of $628, which is $^|«; and the whole amount received last year would have been 8 times $^-\ or 8-5-V-4 = 8717f. S717f - $628 -- $89f more than this year, Ans, 15. Since | of the quantity is taken for grinding, he brings back but | of the amount taken to the mill. Therefore, 14 bu. = |- of amount taken to mill ; 2 bu. = |- of the amount taken to mill; 16 bu. = whole amount taken to mill, Alls, 16. Since $45 = -^ of my money, $9 = 5^ of my money ; $63 r=r whole of my money; $45 + $4| == $49.i ; and — ^ = = — , the part that $49^ is of my money. 63 126 14 ^ . J J Page 127. 17. Since he kept | of his sheep in one field, and the rest in another, the other field must have had | of his sheep. Therefore, | of his sheep = 148 sheep; ^ of his sheep = 74 sheep ; the entire number of his sheep =^ 222 sheep. 18. $7| X 21 =r $159|, the value of flour; $159| ~ 243 ==: $6if I, cost of wood per cord, Ans, 19. Since he gave the sum of |, ], and J^, or 3^2* ^^^ ^^*^^ j\ or I left, Ans. 20. Since A sells | of his share, he will have ^ left ; and since he owned ^ of the vessel, he will have \ of 7, or 2t of the vessel left; B will have | of f, or l^ of the ves- 32 REVIEW EXERCISES. sel. The value of A's share is -f-^ of $18826, or $4363^ ; the value of B's share is ^ of 118326, or $8726|-, Am, 21. Since I had |- of my money left after buying my clothes, my clothes must have cost ^ of my money. Therefore, ^ of my money =: $60 ; \ oi my money =: $15 ; entire amount = $105, Ans, 22. f of 8i^i of 3i = f X-*^X|X i^ = W = »3if 23. If 9 horses eat 16^ tons, 1 horse will eat -|^ of 16^ tons, which is Iff tons; and 7 horses will eat 7 times Iff tons, or 124^7- tons, Arts, 24. If f of a farm is worth $8516, ^ of a farm is worth $2838|; the whole farm is worth $11354|, An§, 25. Since he spent \ of his income traveling, and -J- of f , or -5^, of his income for books, he spent for both, -I- -f- t\ ^^ iV ^^ ^^^ income. Since he spent the rest of his money, which was y\ of his income, for paintings and other works of art, -^ of his income is $8526 ; -^^ of his income is $1065f ; his entire income is $15986^. 26. 14i -f- 3f = 4f , number of days the second worked ; - 7y\, number of days both worked ; the part of the money the first earned ; 141- ^^- H + 4i = 3i 7tV =--T*#7. 7A = T¥r, the part of the money the second earned; ^ of $53 = $24y%9^, the share of the first, Am, ■^ of $53 = $28yVV» t^e share of the second, Ans. 27. I of $13000 =^ $6500, the value of the brothers' shares. Since one owns f as much as the other, the share of the larger owner -f f of that amount, or f of the share REVIEW EXERCISES. 33 of the larger owner, is equal to the shares of both. Therefore, | of the share of the larger owner = $6500 ; 1 of the share of the hirger owner =:: S1300; entire share of the larger owner = 83900, Am. f of S3900 =i 82600, the other's share, Ans. 28. I value of stock = $3865 ; | value of stock == §1932 J ; entire value of stock = 85797|, Am, Page 128. 30. Since A can do it in 5 days, he can do i of the work per day. Since B can do it in 8 days, he can do |- of the work per day. Both working together can do I + 1^, or ^'J, of the work per day ; and since if they could do but -^Q per day, it would require 40 days, when they do 11 it will require but ^^ of 40 days, or o^o days, Ans, 31. Since A and B can do J^ of the work per day, and A can do yV ^^^ ^^^^ work per day, ^V ~ tV "^ sV? ^^^^ P^^^*^ B can do per day ; and since he can do -^^ of the work per day, it will require 30 days to do the work, Ans, 32. Since | of shorter piece rr=i 2 Qf longer, 4- of shorter piece =11 ^ of longer, and the shorter piece = ^ of longer. Therefore, the longer piece -f H ^^ ^^^^ longer piece, or |4- of the longer piece, is the whole length. Therefore, |^ of the longer piece = 124 ft. ; ^y ^^^' ^^^® longer piece =^ 4 ft. ; the longer piece = 84 ft. ; 124 ft. — 84 ft. -^^ 40 ft., the shorter piece, Ans, 33. Since after spending ^ his money and 8^, he had 821 left, before he si)ent the $h he had | of his money left. Therefore, $21 ^ = | of his money, and 843 = the whole of his money, Ans, 3 34 REVIEW EXERCISES. 34. Since A can do -^^ of the work in 1 day, and B can do 3V in 1 day, both can do 2V + t^» or yVo, in 1 day. Smce A, B, and C can do ^ per day, i — -^^^ will leave the part of the work C can do in 1 day. ^ — yU^ = ■^-Jqj the part C can do in 1 day. If he could do ^^ per day, it would take him 450 days; but since he can do ^^, it will take him -^j of 450 days, or 26y^7 days, Ans, 35. Since the shorter ladder is f of the longer one, the longer ladder -|- f of the longer ladder, or f of the longer lad- der is the length of both ladders. Therefore, ^ of the longer ladder = 75 ft. ; | of the longer ladder ==15 ft. ; the longer ladder = 45 ft. ; the shorter ladder = 75 ft. — 45 ft., or 30 ft., Ans. 36. Since the less number is f of the greater, the greater -f f of the greater, or ^ of the greater, is the sum of the numbers. Therefore, f of the gieater = 140 ; ^ of the greater = 20 ; the greater = 80 ; the less = 140 — 80 = 60, Ans, 37. His profits were decreased ^ of $2756^, or $551y%, which sum was ^ of his profits in 1875; therefore his profits in 1875 were 3 times $551y% or S1653yV His profits at that time Avere ^ of his receipts; therefore, $1653y^^ is I of his receipts, which are 5 times $1653y%, or §8269^. 38. Since f of A's money == f of B's, ^ of A's money = -^Q of B's, and A's money = y^^ of B's ; and since the sum of A's and B's money is equal to $5700, y^^ of B's money + B's money, or |f of B's money === $5700 ; J^ of B's money = $300 ; B's money = $3000 ; A's money ^ $5700 — $3000 ■= $2700, Ans, REDUCTION OF DECIMALS. 85 39. After working 9 months he would have been entitled to I of $240 and f of a suit of clothes, or $180 -f f suit of clothes. He received $168 +^a suit of clothes in equitable settlement; therefore, ^ of the value of the suit of clothes must have been $12, and the whole value of it, 4 times $12, or $48, Ans, 40. A and B can do -^^ ^^ ^^^ work per day ; and since A can do f as much as B, when both work together, A does 3 parts while B does 4 parts, or A does f and B ^ of the work, f of -^, or ^V? is the part A does per day, and therefore it will take him 28 days to do the work ; ■f of -^j or 2V is the part B does per day, and therefore it will take him 21 days, Aiis, REDUCTION OF DECIMALS. Page 136. 191 25 9^ ' 100 100 200 ^ 20. .334 = — 3___3___ — __i ^j^g^ ^ 100 100 300 ^ ^ 100 100 300 ' 22. .871 = ^ = ^ = ^ = 1, Ans. ' 100 100 200 ' 42. 2_2 90 23. .041 = -^ = ^^ — = 2Vo. ^^^«- ' 100 100 500 '^' 36 ADDITION OF DECIMALS. 24. .0371- = M= _^ _ _IL _ ^ Ans, 1000 1000 2000 '' 25. M2^=.^^=-^ = t^ = ^\, Am. 1000 1000 2000 '' 26. .0031 = 41- = ^- = — = 8! 0. ^^. ^ 1000 1000 4000 ''' 27. .078f% =. .0783 = ^-iff^, Am. 28. .0003i=r-^=:-J— =— 1-_, ^77^. 10000 10000 20000 29. .7561=-'^-^ = ^^ = ^^^ Therefore, 2.756i = 1000 1000 5000 30. .8U = ^^lil = — . Therefore, 13.814 = m^. ^ 100 100 500 o 50 ADDITION OF DECIMALS. Page I»9. 11. 6^ expressed as a decimal = 6.25; of expressed as a decimal =: 3.4; 5 1 expressed as a decimal =: 5.375 ; 64- expressed as a decimal = 6.2 ; 9f expressed as a decimal = 9.75 ; 30.975, Am. ADDITION OF DECIMALS. 37 12. He earned the sum of the amounts given : $7.25 $7.12i-= $7,125 $9.18f=: $9.1875 $8f =$8,625 $32.1875, or $32.18f, Am. 13. j-rlir expressed as a decimal = .018 ; TWoWo expressed as a decimal = .000015 ; jY^ expressed as a decimal = .81 ; yi^f Q- expressed as a decimal = .0146 ; ToWoo" expressed as a decimal = .00834; .850955, Ans. 14. 8 dollars 5 cents :r:r:$8.05; 13 dollars 19 cents = $13.19 ; 18 dollars 3 cents 8 mills == $18.038 ; 25 dollars 37 cents 5 mills =^ $25.375 ; $12f = $12,625; $3^^ = $ .4375 $77.7155, Ans, 15. He paid for repairs, the sum of the following: $381.45 215.385 323.94 181.57 $1102.345, Aiu, 38 SUBTEACTION OF DECIMALS. 16. He paid for all, the sum of the following: $13 1 =«13.875 8j\= 8.3125 $10}!.= 10.6875 15.49 $48,365, Am. SUBTRACTION OF BECIMALS. Page 140. 9. From .000084 10. From 80000. Take .0000084 Take .080 .0000756, Am. 79999.92, Am. 11. From $29.03 12. From $27.08 Take 17.09 Take 9.375 $11.94, Am. $17,705, Am. 13. $50. 45.895 $4. 105, the amount left. Am. 14. $12384.16 9864.18 $2519.98, part of income left. Am. 15. $1,374,837.64 1,298,369.58 $76,468.06, the surplus. Am, SHORT PROCESSES. 39 MULTIPLICATION OF DECIMALS. Page 142. 25. f of .55 X I of 6.5 == .75 of .55 X A of 6.5 = .75 X .55 X .4 X 6.5 -= 1.0725, Ans, 36. $ .131- X 37-=$ 4.995 $ .37-1- y^ 8 = $ 3. $2.35*^ X 27 ==^ $63.45 $71,445, cost of purchases, Am. SHORT PROCESSES. Page 149. 3. Since 12| is \ of 100, we first multiply by 100, and divide the result by 8. 8 )68800 8600, Am. 4. 16| == 1 of 100. Therefore, 402 X 161 = ^-^^^ = 6700, Am. 5. 25 -= 1 of 100. Therefore, 5056 X 25 -- ^^-^^-«- -- 126400, Ans. 6. 331 ^ ^ of 100. Therefore, 75630 X 33^ = x 5 (i|ooo ===2521000, Am. 7. 50 = i of 100. Therefore, 8404 X 50 = ^-«/^« = 420200^, Am. 40 SHOET PROCESSES. 8. 37i = f of 100. Therefore, 2160 x 37^ == | of 216000 = 81000, Am. 9. 66f --= I of 100. Therefore, 4236 X 66| -- 1 of 423600 = 282400, Ans. 10. 75 = f of 100. Therefore, 7288 X 75 = f of 728800 = 546600, Ans. 11. $1 X 27 = $-2^^ = $6.75, Ans. 12. $ .331 = i of $1. Therefore, $ .33i X 824 = $i X 824 = $^^ = $274Ml Ans. 13. $ .75 = f of $1. Therefore, $ .75 X 216 = «f X 216 14. $ .371 = f of $1. Therefore, $ .37^ X 287 = f X 287 ^$8|i^ $107,625, Am. 15. « .621- = I of $1. Therefore, $ .621 X 394 = $| X 394 ==$19_7^=^ $246.25, Am. 16 Since $ .37^ = | of $1, $1,371 = $1|. Therefore, f$l X 319 = $319. $^X319 = $ 79.75 $1X319 = $ 39.875 $1|X319 = $1|X319 = $438,625, Ans. Page 150. 2. 6075 pounds = 60.75 hundred-weight; and $ .35 X 60.75 = $21,261, Ans. 3. 8609 shingles = 8.609 thousand shingles; and $4.75 X 8.609 = $40.89275, An^. SHORT PROCESSES. 41 4. 1925 pounds = 1.925 thousand pounds; and since the hay costs $9.50 for 2000 pounds, it will cost $4.75 for 1000 pounds. Therefore, $4.75 X 1.925 = $9.14375, the cost, Ans, 5. 16795 pounds =^ 167.95 hundred-weight; and $4.50 X 167.95 =r $755,771 Ans. 6. 129765 laths = 129.765 thousand ; and $2.75 X 129.765 -=$356,853+, Am, 7. 6975 bricks ==6.975 thousand; and $3.25x6.975=. ^.668+, Ans, 8. 1825 pounds = 1.825 thousand pounds; and since the iron costs $45 per ton, it will cost ^ of $45, or $22.50, , per thousand pounds. Therefore, $22.50x1.825 = $41.0625, Am. 9. 6780 envelopes ==6.780 thousand; and $2.75 X 6.780 ==$18,645, Am, 10. 550 pine-apples =^ 5.50 hundred ; and $13.25 X 5.50 = $72,875, Am. 11. 1592 pounds = 15.92 hundred pounds; and $4.50 X 15.92 = $71.64, Am. 12. Since the coal costs $7.50 per ton, it will cost ^ of $7.50, or $3.75, per thousand pounds; and $3.75 X 15 = $56.25, Am. 13. Since the broom-corn costs $55 per ton, it will cost ^ of $55, or $27.50, per thousand pounds; and $27.50 X 2.294 = $63,085, Am. 14. 1964 pounds = 19.64 hundred-weight; and $13.45 X 19.64 = $264,158, Am. 42 ACCOUNTS AND BILLS. ACCOUNTS AND BILLS. 2. 75|- yd. carpeting, 37 " drugget, « rugs, 5 mats, 18 yd. oil-cloth, 9 '* carpet lining, 3 carpet-sweepers, 2 doz. stair-rods. Pago 15». @ «2.12| 1.20 4.16 2.371 1.08 121- 2.00 8.25 8160.437 44.40 33.28 11.875 19.44 1.125 6.00 16.50 S293.057, Ana. 3. 37 bbl. pork, 127 '' flour, 3 hhd. molasses, 169 gal., 29 firkins butter, 2120 lb., 3 boxes raisins, 5 bbl. kerosene, 207 gal. 25 doz. cans fruit, 3 pkgs. tobacco 318 lb., 13 doz. spices, •} @ $24.35 = 8.15 =: .43 $900.95 1035.05 72.67 31 --= 657.20 65 = 13.95 18i = 37.78 40 = 60.00 45 = 143.10 10 = 14.30 $2935.00, Am. 2.40 = 1.10 = ACCOUNTS AND BILLS. 43 Page 154. -Dr.- 3 gold watches = % 242.92 437 pwt. gold chains, @ » 1.15 = 502.55 35 sets tea-service, a 43.10 = 1508.50 7 sets tea-service. li 51. = 357.00 5 silver pie-knives. li 12. = 60.00 12 plated ice-pitchers, li 12.50 =: 150.00 -Or. $2820.97 427.37 Bal. due, $2393.60 Chicago, June 3, 1878. Mrs. M. T. Dana, Bought of G. C. Smith & Co. : 25 yd. calico @ $ .10 37 yd. sheeting " .18^ 2 pairs gloves " 1.50 1 sun-umbrella 5 yd. Hamburg edging . . . @ $ .25 7 pairs hose " .85 Received Payment^ $2 50 6 75 3 00 6 75 1 25 5 95 $26 20 G. C. Smith & Co., Per Martin. 44 6. ACCOUNTS AND BILLS. Olean, N. Y., Jan. 10, 1878. Mr. C. C. Lovell, Bought of R. P. Lawton : 7568 ft. hemlock . . @ $12.75 per M $96 49 8539 " pine flooring " 23.50 " 200 67 5608 " clear pine . . " 45.00 " 252 36 3815 " oak joists . " 32.00 " 122 08 7346 " ash flooring " 34.00 " 249 76 $921 36 Heceived Payment by note at 30 daySj R. P. Lawton. 7. KocHESTER, N. Y., May 5, 1878. Mr. Geo. M. Line, Bought of Steele & Avery : 15 reams comm'l note . @ $1.25 $18 75 7500 envelopes . . . . " 3.65 per M 27 38 18 gross steel pens. . . '* .75 13 50 24 Eidpath's Histories . " 1.25 30 00 9 Webster's Dictionaries " 10.25 92 25 $181 88 Heceived Payment, Steele & Avery. REVIEW EXERCISES. 45 REVIEW EXERCISES. Page 155. 1. $ 1 23. 59 -^ $.34 = 3631^, the number of pounds, Am. 2. 8.339 lb. X 15i == 127.16975 lb., Am. 3. 272-1- sq. ft. X 7| = 2075.90625 sq. ft, or 2075|f sq. ft. 4. 2000 lb. -^ 55.32 -^ 36.153 +, the number of cu. ft. 5. 2150.42 cu. in. X 1000 = 2150420 cu. in., Am. 6. 3-^.003=:::.003)3.000(=-1000, Am. 7. 300 -^ .00003000 ^- .00003 ) 300.00000 ( == 10000000. 8. 2l'82565 ft. X .20 = 436513 ft., the part sold, 2182565 ft. — 436513 ft. == 1746052 ft., amount left ; 1746052 ft. X .15 -= 261907.8 ft. burned. Am. 9. 385 lb. = 3.85 cwt. ; and $4.25 X 3.85 = $16.36i. 10. $31.25 ^$.11^- = 277|-, the number of pounds. Am. 11. 4.37^ X 26 =: $113.75, cost of broadcloth $113.75 -^ $7.25 = 1568.96 + lb.. Am. 12. If 15 tons cost $125.25, 1 ton will cost -^ of that sum, which is $8.35; and 35 tons will cost 35 times $8.35, which is :=-- $292.25, Am. 13. $1.25 X 350 = $437.50, Am. 14. $325 -^ $6.25 = 5200, the number of pounds. Am. 15. 8000 +-.004 = . 004) 8000.000 (=-2000000, Am. 46 REVIEW EXERCISES. Page 156. 16. .0008 -^ 40000 = 40000 ) .00080000 ( = .00000002. 17. $4.43| X 8 = $35.50, the cost of wood ; $35.50 -^ $1.37i = 25x\, or 25.818 -f days, Am. 18. $2.75 X 27 = $74.25, the cost of silk; 6.371 X 11 =- 70.125, the cost of lace; 2.15 X 9 ==: 19.35, the cost of the gloves; 1.10 X 10:= := 11.00, the cost of the hose; $174,725, the entire cost, Ans. 19. $12.50 — $7,625 = $4,875, amount saved per week; $500 -^ $4,875 = 102.56 +, the number of weeks. 20. 95150 bricks = 95.150 thousand; and $7.25 X 95.150 = $689.83f, Ans. 21. $6688.50 -f- 91 =$73.50, price per acre of 91 acres; $73.50 + $1.12i ==: $74,625, price per acre of the rest; 195 — 91 = 104, number of acres worth $74,625 per acre. Therefore, $74,625 X 104 = $7761, the value of 104 acres ; $6688.50 + $7761 = $14449.50, the value of farm, Ans. 22. $ .20 X 200 =: $40, the loss on sheep ; $4.50 X 175 = $787.50, the cost of 175 sheep; $787.50 + $40 = $827.50, what he sold them for; $827.50 ^ 175 = $4,728, the price per head, Ans. 23. Since .40 of profits were expenses, .40 of .15 of the value of the goods sold, or .06 of their value, was the expenses ; .15 of the value of the goods sold minus .06 of their value equals .09 of the value of the goods sold, which is equal to net profits. Therefore, .09 of value DENOMINATE NUMBERS. 47 of goods sold = $9000 ; .01 of value of goods sold = $1000 ; the value of the goods sold = $100000, Aiis. 24. (|--TV)x(3 + i) = MxV-; and (i| + f) + ¥ X m = mU = -2169 +, Ans. 25. $ .65 X 5000 = $3250, the cost of corn; .25 of 5000 bu. = 1250 bu. ; $ .70 X 1250 == $875, value of .25 sold ; $3250 + $447.50 = $3697.50, entire receipts; $3697.50 — $875 = $2822.50, am't rec'd for 2d sale; 5000 bu. — 1250 bu. == 3750 bu. , quant'y sold at 2d sale ; $2822.50 -^ 3750 = $ .75y4g., price per bu. of remainder. 26. He sold If^f =- 2V0 =- -3^' ^^• 27. 645iF= 645.2; and.37i-==f; 645.2 bu. X I -=241.95 bu., A's share; 645.2 bu. X t\ = 120.975 bu., B's share; 241.95 bu. + 120.975 bu. == 362.925 bu. ; 645.2 bu. — 362.925 bu. = 282.275 bu., C's share. DENOMINATE NUMBERS. Page 161. 2. £2 X 20 + 10s. = 50s ; 50s. X 12 + 6d. = 606d., Ans. 3. £13 X 20 + 5s. = 265s., Ans. 4. £4 X 20 + 6s. = 86s. ; 86s. X 12 + 5d. = 1037d. ; 1037d. X 4 = 4148 far., Ans. 48 d:enomi]sate numbers. 6. £f X 20 X 12 =- 90d., Am, 7. £^ X 20 X 12 = 210d., Am, 8. |s. X 12 X 4 =- 26| far., Am, 9. £^ X 20 X 12 = i|^^ = 109 iVd., Am, 10. £5 X 20 + 6s. =^ 106s. ; 106s. X 12 X 4 = 5088 far. 11. 12s. X 12 + 5d. = 149d. ; 149d. X 4 + 2 far. = 598 far. 12. £7 X 20 + 9s. = 149s. ; 149s. X 12 + 5d. = 1793d. 13. 17s.Xl2 + 6d.=::210d.; 210d. X 4 + 3 far. ^ 843 far. 14. £f X 20 ^ -6/s. , or 7|s. ; -^s. X 12 = -V'd- = ^d. There- fore, £f -=7s. 6d., Alls, 15. £5 X 20 + 13s. = 113s. ; 113s. X 12 + 3d. = 1359d. ; 1359d. X 4 -= 5436 far., Am, 16. £35X20 + 6s. =-. 706s.; 706s.Xl2 + 8d.=-8480d., Ajis. 17. £45 X 20 + 3s. = 903s. ; 903s. X 12 + 9d. == 10845d. ; 10845d. X 4 + 3 far. = 43383 far., Aiis, 18. £29 X 20 + 18s. ^ 598s. ; 598s. X 12 + 5d. = 7181d. ; 7181d. X 4 = 28724 far.. Am, Page 16S. 2. 345 far.-^ 4 =^ 86d. 1 far. ; 86d. -^ 12 -- 7s. 2d. There- fore, 345 far. = 7s. 2d. 1 far., Am, 3. 456s. -^ 20 = £22 16s., Am. 4. 1586d.-^12 = 132s. 2d.; 132s. ^20 = £6 12s. There- fore, 1586d. =£6 12s. 2d., Am, 5. 3864far.-^4 = 966d.; 966d. -^ 12 = 80s. 6d.; 80s.-^ 20 = £4. Therefore, 3864 far. = £4 Os. 6d., Am, DENOMINATE NUMBERS. 49 8. f flir. X i X tV = tttSm ^^' 9. 384d. -^ 12 = 32s. ; 32s. -^ 20 = £1 12s. Therefore, S84d. = £1 12s., Alls. 10. 3146s. -^ 20 -=£157 6s., Am, 11. 3596d.-M2==299s. 8d.; 299s.-f-20==£14 19s. There- fore, 3596d. =-- £14 19s. 8d., Ans. 12. 3846 far. -^ 4 = 961d. 2 flir. ; 961d. -M2 = 80s. Id. Therefore, 3846 far. ==. 80s. Id. 2 far., Ans, 13. 4856s. -f- 20 == £242 16s., Ans. 14. 5968 far. -^ 4 =- 1492d. ; 1492d. -M2 = 124s. 4d. ; 124s. -^ 20 == £6 4s. Therefore, 5968 far. = £6 4s. 4d. 15. 3984d. -^ 12 = 332s. ; 332s. -^ 20 = £16 12s. There- fore, 3984d. =£16 12s., Aiis. 16. 4685 far. -^ 4 = 1171d. 1 far. ; 1171 -M2 = 97s. 7d. Therefore, 4685 far. = 97s. 7d. 1 far., Ans. 17. 48567 far. -f- 4 = 12141d. 3 far. ; 12141d. -^12 = 1011s. 9d. ; 1011s. -f- 20 = £50 lis. Therefore, 48567 far. =£50 lis. 9d. 3 far., Ans. 18. £3 X 20 +14s. = 74s. ; 74s. X 12 + 5d. = 893d. ; 893d. X 4 = 3572 far., Ans. 19. 48596far.-^4 = 12149d.; 12149d.-M2=: 1012s. 5d. ; 1012s. -^ 20 = £50 12s. Therefore, 48596 far. = £50 12s. 5d., Ans. 20. £15 X 20 + 8s. = 308s. ; 308s. X 1 2 = 3696d. ; 3696d. X 4 = 14784 far., Ans. 21. $4.8665 X 15 = $72.9975, Ans. 4 50 LINEAR MEASURES. 22. $456-^$4.8665 = £93f|f||. Eeducing the fraction to integers of lower denominations, £|| ^ || = ^f 1 3 \ 6831 V 90 13 6 6 20 q 1 A 3 08 o. . 358 V 1 9 4296 9T3 3" A ^^ ~9TT3~^ J-^TST^- » 9733 A -l^ 9733 = |d. , nearly. Therefore, $456 = £93 14s. ^d. , nearly. 23. $394.45 -^ $4.8665 = £8l4W(fV; ^AWo = ^VVs X 20 = ¥rWs. = l9W3S-; W^3^Xl2 = |f|f-ld., nearly. Therefore, $394.45 = £81 Is. Id., nearly, Am. 24. $37.50 -- $4.8665 = £7fM|| ; £fH-|f = f f|| X 20 13738o 1.|1118q . 1118 V 19 13A±3A 13 6_8.3 9"T3 3'^* -^^9 7 3 3^'> ¥T3 3 /\ "^^ 9 7 3 3^* -^9733> or lid., nearly. Therefore, $37.50 = £7 14s. l^d., nearly, Aiu. 25. $4.8665X25 = $121,661 Ans. 26. £15 X 20 + 10s. = 310s. ; 310s. X 12 = 3720d. ; 3720d. X 4 = 14880 far.. Am. 27. $973.30 -^ $4. 8665 = £200, Am. 28. $1216.625 ^4.8665 = £250, Am. LINE An MEASURES. Page 166. 14. 5mi.x320 + 18rd. = 1618rd.; lG18rd.X 5i + 4yd. = 8903 yd., Am. 15. 7 rd. X 16^ + 5 ft. = 120^ ft. ; 120|^ ft. X 12 + 6 in. = 1452 in., Am. LINEAR MEASURES. 51 16. 63360 in. X 7 = 443520 in., Ans. 63360 in. X 9 = 570240 in., Am, 17. 327 ft. -^ 16i- = 327 X 3¥ = Vt" = l^x^x rd.. Am, 18. 36828 in. -M2 =- 3069 ft. ; 3069 ft. -^ 16i^ =: 186 rd. 19. 3960 rd. -^ 320 = 12f mi., Am. 20. 15 mi. X 320 + 8 rd. =- 4808 rd. ; 4808 rd. X 5^ + 5 yd. = 26449 yd. ; 26449 yd. X 3 -f 3 ft. = 79350 ft. ; 79853 ft. X 12 4- 4 in. =952204 in.. Am. 21. 8 mi. X 320 + 14 rd. = 2574 rd. ; 2574 rd. X 16^ + 5 ft. = 42476 ft. ; 42476 ft. X 12 + 4 in. = 509716 in. 22. 66454 in. -i- 12 = 5537 ft. 10 in. ; 5537 ft. -^S = 1845 yd. 2 ft. ; 1845 yd. -^^i = 335 rd. 2^ yd. ; 335 rd. -^ 320 = 1 mi. 15 rd. Therefore, 66454 in. = 1 mi. 15 rd. 21 yd. 2 ft. 10 in. ; or, since i yd. = 1 ft. 6 in., it may be expressed : 1 mi. 15 rd. 3 yd. 1 ft. 4 in., Am. 23. 158964 in. -f- 12 = 13247 ft. ; 13247 ft. -^ 3 = 4415 yd. 2 ft. ; 4415 yd. -^ 5^ = 802 rd. 4 yd. ; 802 rd. -^ 320 = 2 mi. 162 rd. Therefore, 158964 in. =:: 2 mi. 162 rd. 4 yd. 2 ft., Am. Page 167. 24. 5280 ft. X 7912 = 41775360 ft. Am. 25. 4 in. X 15 r= 60 in. ; 60 in. -^ 12 = 5 ft., Am. 26. 67 ch. 83 1. = 6783 1. Since 25 1. :^ 1 rd. , 6783 1.-^-25 = the number of rods, which is 271 rd. 8 1., Am. 27. 59 ch. X 100 + 75 1. = 5975 1. ; 5975 1. X 7.92 = 47322 in., A')is. 52 SURFACE MEASURES. SURFACE MEASURES. Page 169. 10. 9 sq. yd. X 9 + 3 sq. ft. -= 84 sq. ft. ; 84 sq. ft. X 144 + 15 sq. in. := 12111 sq. in., Aiis. 11. 3 sq. mi. X 102400 -f- 15 sq. rd. = 307215 sq. rd. ; 307215 sq. rd. X 301 ^ 9293253| sq. yd. ; 9293253| sq. yd. X 9 -= 836392831 sq. ft. ; 83639283| sq. ft. X 144 = 12044056860 sq. in., Ans, 12. 262685 sq. ft. ^ 9 = 29187 sq. yd. 2 sq. ft. ; 29187 sq. yd. -^ 30^ = 964 sq. rd. 26 sq. yd. ; 964 sq. rd. -^ IGO == 6 A. 4 sq. rd. Therefore, 262685 sq. ft. .== 6 A. 4 sq. rd. 26 sq. yd. 2 sq. ft., Ans. 13. 2 A. X 160 + 37 sq. rd. =-- 357 sq. rd. ; 357 sq. rd. X 30^ -f 5 sq. yd. = 108041 sq. yd.; 10804^ sq. yd. X 9 + 7 sq. ft. == 97245i sq. ft.; 97245^ sq. ft. X 144 = 14003316 sq. in., Ans. 14. 184265 sq. in. ~ 144 = 1279 sq. ft. 89 sq. in.; 1279 sq. ft. -f- 9 = 142 sq. yd. 1 sq. ft.; 142 sq. yd. -f- 30^- = 4 sq. rd. 21 sq. yd. Therefore, 184265 sq. in. = 4 sq. rd. 21 sq. yd. 1 sq. ft. 89 sq. in., Ans. Page 170. 16. I A. X 160 =: 88f sq. rd. ; f sq. rd. X 30^ = 26f sq. yd. ; I sq. yd. X 9 == 8 sq. ft. Therefore, -| A. = 88 sq. rd. 26 sq. yd. 8 sq. ft., Ans. nmi) A • _8_0_ i . 12X1. 3^ 3 J^g • T6 8^> T¥0 2 » 16 4- J ^'^^' SURFACE MEASURES. 53 18. f sq. rd. X 301 =. n^^ sq. yd ; H sq. yd. X 9 =- ff = 83^ sq. ft.; "3^ sq. ft. X 144 = 13|- sq. in. Therefore, f sq. rd. = 11 sq. yd. 3 sq. ft. 13^ sq. in., Ans, 23. 18X24 = 432 sq. ft.; 432 sq. ft.-^9 = 48 sq. yd.; $1.15 X 48 = $55.20, the cost of carpet, Aiis, 24. 18 X 17 = 306 sq. ft.; 306 sq. ft. -^ 9 = 34 sq. yd. 25. 18 X 15f == 2831- sq. ft.; 283i- sq. ft. -^ 9 = 31i sq. yd.; 31^ sq. yd. -f- f = 42, the number of yards of carpet ; $1.90 X 42 = $79.80, the cost of carpet, Arts, 26. 1 A. = 40 sq. rd. X 304- X 9 = 10890 sq. ft. Since the area is the product of the length by the breadth, if 10890, the area, is divided by 66, the breadth, the quotient will be the length. 10890 -i- 66 = 165 ft., the length; $3.25 X 10890 =- $35392.50, cost of lot, Ans, 27. 10 A. X 160 = 1600 sq. rd. Since the area, 1600 sq. rd., is the product of the length by the breadth, if it is divided by 20 rd., the breadth, the quotient will be the length. 1600-^20 = 80 rd., the length, Ans. Page 171. 28. 80 X 20 = 1600 sq. rd.; 1600 sq. rd. -^ 160 = 10 A.; $47.25 X 10 = $472.50, Ans, 29. A surface 10 ft. sq. contains 100 sq. ft.; 100 sq. ffc.— 10 sq. ft. = 90 sq. ft., Ans, 30. 48 X 22 = 1056 sq. ft.; 1056 sq. ft. -^ 9 = 1171 sq. yd.; $.30X1171 = $35. 20, Ans, 31. 45 X 32 = 1440 sq. ft.; 1440 sq. ft. -^ 9 = 160 sq. yd.; $.30X160 = $48., Am, 54 MEASURES OF VOLUME. 32. 18 X 11 = 198 sq. ft., the area of one side ; 198 sq. ft. X 2 = 39G sq. ft., the area of two equal sides ; 17 X 11 = 187 sq. ft., the area of another side ; 187 sq. ft. X 2 = 374 sq. ft , area of other two equal sides ; 396 sq. ft. + 374 sq. ft. = 770 sq. ft., in sides ; 770 sq. ft. -f- 9 — So^ sq. yd. , in sides ; 18 X 17 =- 306 sq. ft., in ceilnig ; 306 sq. ft. -f- 9 = 34 sq. yd., in eeiling ; 85f sq. yd. + 34 sq. yd. = 119f sq. yd., whole area; $.37 X 119| = $44.23, the cost, Am. 33. $.25X851 = $21.39, Ans, MEASURES OF VOLUME. Page 173. 4. 418 cu. ft.-- 24f = 16f perch; $1.75 X 16f = $29.56. 5. 38x4Xli- = 228cu.ft.; 228 cu. ft.-^24|==9373 perch. 6. 35 X 20 X 8 =- 5600 cu. ft.; 5600 cu. ft. -i- 27 -- 207-^ cu. yd., or loads, Ans, 7. 32 cu. ft. X 1728 + 114 cu. in. = 55410 cu. in., Ans, 8. 13 cu. yd. X 27 + 18 cu. ft. = 369 cu. ft., Ans, 9. 15 perch X 24f + 13|- cu. ft. = 384| cu. ft., Ans, 10. 8 X 8 X 8 r^ 512 cu. yd.; 512 cu. yd. X 27 == 13824 cu. ft., the contents of the cube; or it contains 13824 blocks, each a cubic foot, Ans, 11. 9 X 5 X 31- = 157^ cu. ft., Ans, BOARD MEASURE. 55 Page 174. 12. 40 X 4 X 5i =: 880 cu. ft.; 880 cu. ft. -f- 128 = 6|- C; $1.50X61^ = $10,311 A7U, 13. 8 X 7 X 5 == 280 cu. ft.; 280 cu. ft. X 1728 = 483840 cu. ill.; 483840 cu. in. -- 2150.4 = 225 bu., Ans. 14. 80 X 35 X 8 == 22400 cu. ft. ; 22400 cu. ft. -- 27 = 829|-^ cu. yd. ; $ .42 X 829^^ = S348.44 ; 80 + 35 = 115 ft., half the length of wall according to measure- ment of masons. Tlierefore the length of wall is 230 ft. 230x8Xli = 2760cu.ft.; 2760cu. ft.-^24f ==111^ perch ; $3.75 X IH^ =" $418.18, the cost of wall, Ans. 15. 35i X 19 X 3 = 2023-1- cu. ft.; 20231X22 = 44517 bricks, Ans. BOARD MEASURE. Page 174. 1. 16 in. = 11 ft.; 18 X 4== 24 ft, Ans. 2. 15 X 11 = 165; 165 -M2 = 13f ft., Ans. 3. 10 X 13 = 130 ; 130 -^ 12 = 10| ft., Ans. 4. 13 X 15 = 195 ; 195 -M2 = 16i ft., Ans. Page 175. 5. 40X9 = 360; 360-M2 = 30 ft; 30 ft. X 6 = 180 ft. 6. 16X8 = 128 ; 128 -M2 = lOf ; lOf X 318 = 3392 ft. , or 3.392 thousand ; $11 X 3.392 = $37,312, the cost. 56 LIQUID MEASURE. 7. 22 X 16 = 352; 352 -f-12 = 29^ ft.; 29^ X 3 ^ 88 ft. in each plank ; 88 X 35 = 3080 ft, or 3.08 thousand ; $17.50 X 3.08 = S53.90, the cost, Arts. a 35 X 18 = 630 ft.; -J- of 630 = 105, and 630 ft. + 105 ft. = 735 ft.; 735 XH = 918| ft., board measure, or .9181 thousand; $30 X .918f = $27.56i-, Ans, 9. 20 X 25 = 500 ft.; i of 500 = 62|, and 500 ft. + 62 1 ft. = 562A ft. ; 562i x li = 843f ft., board measure, or .843f thousand; $25 X.843f = $21.09f, Am. LIQUID MEASURE. Page 176. 5. 684pt.-f-8 = 85igal.; $.20 X 85^ = $17.10, Ans, 6. 3846 gi.^4 = 961 pt. 2 gi.; 961 pt.-^2 = 480 qt. 1 pt.; 480 qt.^4 = 120 gal. Therefore, 3846 gi.= 120 gal. 1 pt. 2 gi., Am. 4869 pt. -^ 2 = 2434 qt. 1 pt.; 2434 qt. -^ 4 =- 608 gal. 2 qt. Therefore, 4869 pt.=:608 gal. 2 qt. 1 pt., Am. 7. 3 gal. X 4 + 4 qt. = 16 qt; 16 qt X 2 + 1 pt = 33 pt; 33 pt X 4 4- 3 gi. = 135 gi., Am. 8. 4bbl.X31i + 6gal.=-:132gal.; 132 gal. X 4 = 528 qt; 528 qt X 2"^-= 1056 pt.; 1056 pt X 4 =: 4224 gi., Am. —484 pt -^ 2 =- 242 qt; 242 qt 4- 4 = 60 gal. 2 qt, Ans. DRY MEASURE. 57 9. 24 gal. X 8 -- 192 pt., Ans. 8459 gl -^ 4 = 2114 pt. 3 gi.; 2114 pt. -^ 2 =- 1057 qt.; 1057 qt. -f- 4 = 264 gal. 1 qt.; 264 gal. -^ 31i = 8 bbl. 12 gal. Therefore, 8459 gi. = 8 bbl. 12 gal. 1 qt. 3 gi., Ans, 10. 231 cu. in. X 7 = 1617 cu. in., Ans, 11. 3846 cu. in. -f- 231 cu. in. == 16f^, number of gallons. 12. 15 X 10 X 8 = 1200 cu. ft.; 1200 X 1728 = 2073600 cu. in.; 2073600-^231^:8976.623 gal.; 8976.623 -f- 31|- = 284 bbl. 30.623 gal., Am. nBY MEASUBJE. Page 177. 4. 3 bu. X 4 + 3 pk. = 15 pk.; 15 pk. X 8 + 5 qt. = 125 qt.; 125 qt. X 2 + 1 pt. = 251 pt., Ans, 5. 8 bu. X 4 -= 32 pk.; 32 pk. X 8 + 5 qt. = 261 qt.; 261 qt. X 2 + 3 pt. = 525 pt., Ans. 6. 16845 qt.^ 8 = 2105 pk. 5 qt.; 2105 pk.-^ 4 = 526 bu. 1 pk. Therefore, 16845 qt. = 526 bu. 1 pk. 5 qt., Ans, Page 178. 7. 13965 pt. -^ 2 == 6982 qt. 1 pt.; 6982 qt. -^ 8 :- 872 pk. 6 qt.; 872 pk. -^ 4 -- 218 bu. Therefore, 1365 qt. = 218 bu. 6 qt. 1 pt., Ans, 58 measurp:s of weight. 8. 57364 qt. ^ 8 =: 7170 pk. 4 qt.; 7170 pk. -f- 4 ^ 1792 bu. 2 pk. Therefore, 57364 qt. = 1792 bu. 2 pk. 4 qt. 9. 35 bu. X 4 + 3 pk. = 143 pk.; 143 pk. X 8 + 6 qt. = 1150 qt.; 1150 qt. X 2 + 1 pt. = 2301 pt., Ans, 10. 2150.4 cu. in. X 7 = 15052.8 cu. in., Ans, 11. 13846 cu. in. -^ 2150.4 = 6.4388 + bu., Ans, 12. 8 X 7 X 5 = 280 cu. ft.; 280 cu. ft. X 1728 == 483840 cu. in.; 483840 cu. in. -^ 2150.4 cu. in. = 225 bu., Ans, 13. 9 X 6 X 6 = 324 cu. ft. ; 324 cu. ft. X 1728 = 559872 cu. in.; 559872 cu. in. -f- 2150.4 cu. in. ^ 260.357 bu. MEASURES OF WEIGHT. Page 179. 4. 3 T. X 20 + 2 cwt. := 62 cwt. ; 62 cwt. X 100 + 5 lb. = 6205 lb., Ans. 5. 5 T. X 2000 + 216 lb. = 10216 lb., Am, 6. 5 lb. X 16 + 7 oz. == 87 oz. ; $ .121 x 87 = $10,875. 7. ^ lb. X 16 = 56 oz. ; $ .04|- X 56 :3= $2.52, Ans, 8. 5 cwt. 28 lb. = 528 lb. ; $ .08 X 528 ^ $42.24, Ans, Page 180. 10. \ bbl. = i of 196 lb. , or 49 lb. Flour at $8.50 per cwt. is $ .085 per lb. ; $ .085 X 49 -= $4,165, Ans, APOTHF.CAKIES' WEIGHT. 59 11. I quintal =- J of 100 lb., or 50 lb. ; and $ .06^ X 50 =. $3.25, A)is. 12. 13 cwt. 18 lb. = 1318 lb., or 1.318 thousand pounds; and since hay is $15 for 2000 lb., it is i of $15, or $7.50, for 1000 lb. Therefore, $7.50 X 1.318 = $9,885. 13. When flour is $10 per bbl., 1 lb. ^ j^ of $10, or $.05^%. Therefore as many pounds can be bought for $2.80 as $.05^ is contained times in that sum, which is 54.8 + lb., Ans. 14. 3cwt. 191b. 9 oz.=- 319^2^ lb.; $.17x3193^=-$54.32yV 15. $9 X 15 -= $135, the cost of flour ; 196 X 15 = 2940 lb., or 29.40 centals, the quantity of flour; $5 X 29.40 = $147; and $147— $135 = $12, the gain, Ans. 18. 275000 lb. -+ 280 lb. = 9821 bbl., Ans. 17. 3T. 4 cwt. 20 lb. = 6420 lb. ; 2bu. = 1201b.; 64201b. -f- 120 lb. ^ 53|, or 54 bags, Aris. APOTHECARIES^ WEIGHT. Page 181. 3. 7 oz. X 20 + 5 pwt. = 145 pwt. ; 145 pwt. X 24 + 18 gr. = 3498 gr. , Ans. 4. 3456 gr. -+ 24 = 144 pwt. ; 144 pwt. -h 20 == 7 oz. 4 pwt., Ans. 6. 2 oz. 15 pwt. -^ 55 pwt. ; $1.35 X 55 == $74.25, Ans. 60 MEASURES OF TIME. 6. 3 lb. 5 oz. = 41 oz. ; 41 oz. -^5 oz. = 8 spoons, and 1 oz. remaining, Aiis. 7. 5 oz. 7 dr. = 47 dr. ; 47 dr. X 3 = 141 sc. ; 141 sc. X 20 = 2820 gr. ; 2820 gr. ~ 5 = 564, number of ijowders. MEASURES OF TIME. Page 183. 8. 5 hr. X 60 + 15 min. = 315 min. ; 315 min. X 60 + 12 sec. = 18912 sec, Ans. 9. 6 hr. X 60 -h 27 min. =:- 387 min. ; 387 min. X 60 + 38 sec. = 23258 sec, Ans, 10. 48695 sec -^ 60 = 811 min. 35 sec. ; 811 min. -^ 60 = 13 hr. 31 niin. Therefore, 48695 sec. --- 13 hr. 31 min. 35 sec, Ans. 11. 38497 sec ^ 60 = 641 min. 37 sec ; 641 min. -!- 60 = 10 hr. 41 min. Therefore, 38497 sec =--- 10 hr. 41 min. 37 sec, Ans, 12. 365 da. X 5 = 1825 da. ; 1825 da. X 24 = 43800 hr. ; 43800 hr. X 60 = 2628000 min., Ans, 13. Jan., 31 da. + Feb., 28 da. + Mar., 31 da. + April, 30 da. = 120 da.. Am, 14. April, 30 da. + May, 31 da. + June, 30 da. + July, 31 da. + Aug., 31 da. + Sept., 30 da. + Oct., 14 da. = 197 da.. Am. CIECULAR MEASURE. 61 15. 2 wk. X 7 + 5 da. = 19 da. ; 19 da. X 24 + 13 lir. rrr 469 hr., Am. 16. 5 da. X 24 + 10 hr. = 130 hr. ; 130 hr. X 60 + 15 min. = 7815 mm., Am, 17. 384600 sec. -f- 60 = 6410 mm. ; 6410 min. -^ 60 == 106 hr. 50 min. ; 106 hr. -^ 24 = 4 da. 10 hr. Therefore, 384600 sec. = 4 da. 10 hr. 50 min., Am. 18. 15 hr. X 60 + 12 min. = 912 min. ; 912 min. X 60 + 18 sec. = 54738 sec, Am. 19. 32965 min. -- 60 = 549 hr. 25 min. ; 549 hr. -- 24 = 22 da. 21 hr. ; 22 da. -f- 7 = 3 wk. 1 da. Therefore, 32965 min. = 3 wk. 1 da. 21 hr. 25 min., Am. CIBCULAB MEASURE. Page 185. 2. 21° X 60 + 12'=- 1272'; 1272' X 60 + 18"== 76338". 3. 34° X 60 + 12'= 2052'; 2052' X 60 + 43"= 123163". 4. 468560" -f- 60 = 7809' 20"; 7809'-- 60 = 130° 9'. Therefore, 468560"= 130° 9' 20", Am. 5. 384500" -^ 60 = 6408' 20"; 6408' -^ 60 = 106° 48'. Therefore, 384500"= 106° 48' 20", Am. Page 186. 6. 20 X 3 = 60 quires ; $ .15 X 60 = $9.00, Am. 62 DENOMINATE FRACTIONS. DENOMINATE FRACTIONS. Page 187. 8. f of a pound Troy = f of 12 oz. = -^/ oz. = 7| oz. ; \ oz. = i of 20 pwt. , or 4 pvvt. Therefore, f of a pound Troy ^^ 7 oz. 4 pwt. , Aiis, .9. f of a ton = f of 2000 lb. = 857| lb. ; i lb. -= i of 16 oz. = 2f oz. Therefore, f of a ton =- 857 lb. 2f oz. 10. I of a furlong = | of 40 rd. =- 26| rd. ; | rd. = f of 51 yd. =: 3| yd. ; | yd. = f of 3 ft. == 2 ft. Therefore, 1 of a furlong = 26 rd. 3 yd. 2 ft., Am. 11. y\ of an acre = ^ of 160 sq. rd. = 43^ sq. rd. ; -^ sq. rd. = 1^ of 30^ sq. yd. = 19^ sq. yd. ; \ sq. yd. = ^ of 9 sq. ft. = 2\ sq. ft. ; \ sq. ft. r= i of 144 sq. in. ^ 36 sq. in. Therefore, -^ of an acre = 43 sq. rd. 19 sq. yd. 2 sq. ft. 36 sq. in., Ans. 12. f of a peck --= |. of 8 qt. -- ^ qt. ; | qt. =: | of 2 pt. = H P^- 5 ^ pt. = 1^ of 4 gi. -- \\ gi. Therefore, f of a peck = 6 qt. 1 pt. 1 1^ gi. , An^. 13. y\ of a day = ^ of 24 hr. ^ 1\ hr. ; \ hr. = i of 60 min. = 12 min. Therefore, y% of a day =-- 7 hr. 12 min. 14. I of a sq. rd. = f of 301 gq. yd. = ISfl sq. yd. ; f| sq. yd. =- If of 9 sq. ft. = 8^V ^q. ft. ; ^ sq. ft. - ^ of 144 sq. in. = 22^ sq. in. Therefore, | of a sq. rd. = 18 sq. yd. 8 sq. ft. 22^ sq. in., Ans. 15. -^Q of a cu. yd. =: ^^^ of 27 cu. ft. ~ 9/^ cu. ft. ; -^-^ cu. ft. = -^\ of 1728 cu. in. = 777f cu. in. Therefore, ^^ of a cu. yd. = 9 cu. ft. 777f cu. in., Am. DENOMINATE FRACTIONS. 63 17. Since there are 64 pints in a bushel, ^^ of a bu. = ^ of 64 pt. = ^ pt. -= II pt., Am, 18. Since there are 5280 feet in a mile, 2tIt¥ ^^ ^ ^*^- "= ^yf^^ of 5280 ft. = Mff^ = f ft., Ans. 19. Since there are 288 scruples in a pound, yg^^ of a lb. = -i-y\y of 288 sc. = im sc. = 1^ sc, Ans, 20. Since there are 64 pints in a bushel, .006 of a bu. = .006 of 64 pt. = .384 pt., Ans, Page 188. 22. £.575X20 = 11.5s.; .5s.Xl2 = 6d. Therefore, £.575 ~ lis. 6d., Ans. 23. .1935 lb. Troy X 12 == 2.322 oz. ; .322 oz. X 20 rr. 6.44 pwt. ; .44 pwt. X 24 == 10.56 gr. Therefore, .1935 lb. Troy^ 2 oz. 6 pwt. 10.56 gr., Ans. 21 .436 of a ream X 20 = 8.72 quires ; .72 quires X 24 = 17.28 sheets. Therefore, .436 of a ream = 8 quires 17.28 sheets, Ans. 25. .1845 of a gallon X 4 --- .738 qt. ; .738 qt. X 2 = 1.476 pt; .476 pt. X4 = 1.904 gi. Therefore, .1845 of a gallon — 1 pt. 1.904 gi., Ans. 26. .135 of a rod == .135 of 16i ft. = 2.2275 ft. ; .2275 ft. = .2275 of 12 in. =- 2.73 in. Therefore, .135 of a rod = 2 ft. 2.73 in., Ans. 27. .455 of a mile = .455 of 320 rd. = 145.6 rd. ; .6 rd. = .6 of 161 ft. == 9.9 ft. ; .9 ft. = .9 of 12 in. = 10.8 in. Therefore, .455 of a mile = 145 rd. 9 ft. 10.8 in., Ans. 64 DENOMINATE FEACTIONS. 28. .4832 of a bushel = .4832 of 4 pk. = 1.9328 pk. ; .9328 pk. = .9328 of 8 qt. = 7.4624 qt. Therefore, .4832 of a bushel = 1 pk. 7.4624 qt, Am. 29. .684 of a league = .684 of 3 mi. = 2.052 mi. ; .052 mi. = .052 of 320 rd. =. 16.64 rd. ; .64 rd. = .64 of 16-i- ft. = 10.56 ft. ; .56 ft. =:::. .56 of 12 in. = 6.72 in. There- fore, .684 of a league = 2 mi. 16 rd. 10 ft. 6.72 in.. Am. 8. 1 inch = 3^g of a yd. ; and j\ of an in. = -j^ of -^ of a yd., or 3!^ yd.. Am. 9. 1 second = -g-gVo^ of an hr. ; and f of a sec. = f of -g-eVo" of an hr. , or -gyV o" ^^ ^^^ ^^- ' ^^^' 10. 1 day = -3-I-5 of a yr. ; therefore, 1 wk. =rr ^^ of a yr. ; and . 375 of a wk. :::^ yVo^, or I of 3f5 of a yr. , or 2|lo- of a y^- 11. 1 pound = 2 0^0 of ^ ton; and .35 or -^ lb. =^ of 2 0'^oo of a ton, or 4^qIqq of a ton, Am. 12. 1 cubic inch = yyVs" of a cu. ft. ; and |- cu. in. = f of TT28^ or yJ3 2 CU. ft.. Am. Page 180. 13. 1 square yard = y-gVo of an acie ; and f sq. yd. = | of Winr == 242¥o aore, Am. 14. 1 pint = 2"^ of a bbl. ; and | pt. =^ f of -^^ or -^q-^ bbl. 6. Since 3 yd. 2 ft. ^- 11 ft., and 2 yd. 2 ft. = 8 ft., 8 ft. = ^of 11 ft.. Am. 7. Since 5 gal. 3 qt. 1 pt. — 47 pt. , and 2 gal. 1 qt. 1 pt. = 19 pt., 19 pt. =. ^ of 47 pt., Am. DENOMINATE FRACTIONS. 65 8. Since 2 lb. =^ 480 pwt., and 3 oz. 10 pwt. = 70 pwt., 70 pwt. = -^\% or 4^ of 480 pwt., Am, 9. Since 3 pk. = 48 pt. , and 2 qt. 1 pt. = 5 pt. , 5 pt. = ^ of 48 pt. , Ans, 10. Since 3 bbl. = 3024 gi., and 13 gal. 3 qt. 2 pt. 2 gi. = 450 gi., 450 gi. = 3^^, or ^^ of 3024 gi., Ans. Page 190. 12. 60 24 14. 12 161 16. 4 12 20 18. 12 15.00 min. 13. 4.25000 hr. .17708 + da., ^?25. 6.0 in. 3.5000 ft. .2121 + rd., Ans. 3.00 far. 5.7500d. 18.47916s. £.923958+, Am, 3.00 in. 2.00 qt. 3.2500 pk. .8125 bu.. Am. 320 14.2500 ft. 37.8636 + rd. .1183 + mi., Am, 15. 18s. 5fd. ::= 887 far., and £1 izz: 960 far. ; therefore, 18s. 5f d. = III of a £, or £.9239+, Am, 17. 16 1b. 11 oz. ==267 oz., and 1 cwt. = 1600 oz. ; therefore, 16 lb. 11 oz. = T¥(nrcwt., Am, 19. 60 24 7 14.0000 min. 5.2333 + hr. 3.21805 + da. .45972 + wk. ^725. 66 20. 24 20 22. 36 1760 REVIEW EXERCISES. 15.000 sheets. 21. 16 8.62500 quires. .43125 ream, Am. 9.00 in. 7.0000 cu. ft. 3.4375 cd. ft. .4296+ C, Ans. 23. Since 4 oz. 7 pwt. 13 gr. = 2101 gr., and 1 lb. = 5760 gr., 4 oz. 7 pwt. 13 3717+ ml, Ans. gr. = |i|^ lb., Am. 654.2500 yd. ME VIEW EXERCISES. Page 190. 1. 15 lb. 8 oz. = 151 lb. ; $ .31 X 151 == $4,805, Am. 2. 3 pk. 2 qt. = 26 qt. ; $ .09 X 26 = $2.34, Am. 3. 18 bu. 3 pk. =r 18f bu. ; $1.05 X 18f = $19.68f , Am. 4. 42 rd. 7 ft. 8 in. = 700| ft. ; $ .75 X 700| == $525.50. 5. 12 gal. 3 qt. = 12f gal. ; $ .50 X 12f = $6,375 ; and $6,375 -^ $ .30 ::^ 21.25, or 21^, No. lb. of butter, Am. 6. $ .371 X 15 = $5,625, the cost of oats. In 15 bushels there are 120 half-pecks ; therefore, $ .15 X 120 = $18, the selling price ; $18 — $5,625 == $12,375, the gain. 7. 4 X 6 X 60 == 1440 cu. ft. ; 1440 cu. ft. -^ 128 = n\ cords ; $4.25 X Hi = $47.81^, the cost of wood. Am. 8. Since there are 231 cu. in. in a gallon, a barrel will con- tain Zl\ times 231 cu. in., or 7276|^ cu. in.; and 100 barrels will contain 727650 cu. in., the volume of the cis- REVIEW EXERCISES. 67 tern. Since the volume is the product of three dimensions, when the product and two are given, the third dimension may be found by dividing the product by the product of the two given dimensions. 6 ft. =: 72 in. ; 10 ft. = 120 in. ; 72 X 120 = 8640 sq. in. ; and 727650 -^ 8640 ^ 84.2187 + in. ; 84.2187 + in. -^ 12 == 7.0182 + ft. Page 101. 9. Since 1 oz. Troy =^480 gr., and 1 oz. Avoirdupois ^= 437|- gr., it will require 1.0971 oz. Avoirdupois to equal 1 oz. Troy, which, at $ .75 per oz., would cost S .8228 -f ; and $1 — $.8228 + ^ .1772 +, gain per oz. Troy, Am. 10. 15 rd. 8 ft. = 2551 ft.; 27 rd. 9 ft. = 454^ ft.; 255^ X 454i = 1161241 square feet ; 116124f -^ 43560 =. 2.6658+ A.; $150 X 2.6658+ ^ $399.87-f, Am. 11. 10 hr. = 600 min. = 36000 sec. ; and since 2 marks are made per sec, 2 times 36000, or 72000 marks, are made per day ; and to make a million marks it will require as many days as 72000 is contained times in 1000000, which is 13| days. Am. 12. 18 X H X 4 ^= 96, the number of feet, board measure ; 96 ft. = .096 of 1000 ft. ; and $18 X .096 .:= $1,728. 13. $1689600.000 --$.10 ^16896000, No. of ft. in length; 16896000 -^ 5280 ft. ■= 3200, No. of miles. Am. 14. 7 oz. 3 dr. 4 sc. = 3620 gr. ; and 3620 gr. +- 2 gr. ~ 1810, the number of pills, Am. 15. $ .20 X 20 = $4.00, the selling price ; 2.55, the cost; $1.45, the gain per ream, An8, 68 ADDITION OF DENOMINATE NUMBERS. 16. 13 lb. 7 oz. = 13^^ lb. ; $ ,21\ X 13^V = ^^-^"^^ ; and $3,695 -^ $ .12 = 30.794 lb., or 30 lb. 12.7+ oz., Am, 17. 73f X 6 X 4 -M28 = 13f| C, amount of first pile ; 30 X 71 X 4 +128 = 6ff C, amount of second pile ; 37 X 3|- X 4 + 128 = 4g\ C, amount of third pile ; 24^ C, entire amount. $4.60 X 24if = $113.13 +, entire value, Am. 18. 4 reams 8 quires 12 sheets = 2124 sheets ^ 4248 half- sheets =- 4.248 thousand ; $6.50 X 4.248 =: $27,612. 19. $18 + $6.85 ==2.627+, Am, 20. \ of 327 bu. 3 pk. 5 qt. =::. 5246^ qt. ; 167 bu. 3 pk. =. 5368 qt. ; 5368 qt. — 5246 1- qt. = 1211 qt. === 3 bu. 3 pk. 1 qt. 1 pt. too much, Ans. 21. 20 fath. = 120 ft. ; 62^ lb. X 120 = 7500 lb., the press- ure on 1 ft. of surface ; 7500 X 9 = 67500 lb., the press- ure on a square yard, Aiis. A 1} DITTO J^ OF T>ENOMTNATE NUMBERS. 10, Page 193. 12f yd. = 12 yd. , 2 ft. 3 in. 8|yd. = 8 2 n m yd. = 37 1 6 39| yd. = 39 2 98 2 4J-, Am, ADDITION OF DENOMINATE NUMBERS. 69 11. I A. = 32 sq. rd. sq.yd. sq. ft. sq. in. fA. = 60 129i sq. rd. r== 129 7 5 9 1181 sq. r d. = 118 7 5 9 2 A. 19 15 1 18, Am, 12. 24 gal. 2 qt. 3 pt. 45i gal. = 45 1 39| gal. =: Sd 1 1 109 1 1, Ans. Page 104. 13. 3 yr. 4 mo. 18 da., age of James; _2 8 6_ 6 24, age of Henry ; 7 10 14 13 11 18, age of William ; 1 8 15 7 18, age of Herbert, Ans. 14. 20| cwt. = 20 cwt. 33 1b. 5ioz. 16|T. = 16T. 2 85 llf 17ilb. = 17 2 19 18 15 7 8 2 7 5 fib. 6f |T. = 11 11 1^ 2 3 57 4 22 17 46 Hh Ans. 70 SUBTRACTION OF DENOMINATE NUMBERS. SUBTRACTION OF DENOMINATE NVMBEIiS. Page 195. 9. f A. = 120 sq. rd. sq. ft. sq. in. 72 160 39 47 lll(i) 105 Or since \ sq. ft. =^ 36 sq. in., the answer may be writ- ten : 47 sq. rd. Ill sq. ft. 141 sq. in. ; or since 111 sq. ft. = 12 sq. yd. 3 sq. ft., the difference may be expressed as 47 sq. rd. 12 sq. yd. 3 sq. ft. 141 sq. in., Ans, 10. £384 6s. 5d. 2 far. 297 9 8 3 86 16 8 3, Am. 11. 97|A. =97 A. 133sq.rd. lOsq.yd. Osq.ft. 108sq.in. 38 A. 89isq. ^^ ^- ,1 = 38 39 a.rd.J 136 12 17 5 117 285A. Osq.rd. Osq. yd. Osq.ft. Osq.in. 136 12 17 5 117 148 147 12(1) 3 27 ^sq.yd.= 2 3^ 148 147 12 5 63,^718. 12. 9R. 18qu. 15sh. 14, 1851 yr, 3mo. 15da. 8y% R. = 3__^___8 1843 1 3 6 10 7 8 2 12, ^ns. MULTirLICATION OF DENOMINATE NUMBERS. 71 Page 196. 15. 1869 yr. 12 mo. 15 da. 16. 1871 yr. 1 mo. 5 da. 1803 4 2 1860 5 15 66 8 13, Ans, 10 7 20, Ans, 17. 1837 yr. 12 mo. 8 da. 3 2 5 1841 2 13 Feb. 13, 1841, Am. 19. 1865 yr. 4 mo. 9 da. 20. 1876 yr. 10 mo. 10 da. 1861 4 11 1871 7 9^_ 3 11 28, Ans, 5 3 l^~A7is, MULTirLICATION OF DENOMINATE NUMBERS. Page 197. 5. The farm consists of 7 times 18 A. 25 sq. rd., which is 127 A. 15 sq. rd., Ans. 6. Since the field is square the sides are of equal length, and the entire length of the fence will be 4 times 28 rd. 5 yd. 2i ft., which is 116 rd. 1 yd. 1 ft, Ans. 7. The entire quantity of wood will be 7 times 13 C. 7 cd. ft. 24 cu. ft., which is 98 C. 3 cd. ft. 8 cu. ft., Ans. 8. The cost will be 14^ times £2 5s. 6d. \ of £2 5s. 6d. is £1 2s. 9d., and 14 times £2 5s. 6d. is £31 17s. Th^ sum of £1 2s. 9d. and £31 17s. is £32 19s. 9d., Am, 72 DIVISION OF DENOMINATE NUMBERS. 9.* The whole quantity of potatoes was 4 times 27 bu. 3 pk., which was 111 bu. $ .45 X HI = $49.95, Am, nivisiojsr of jdekominate nvmbebs. Page 198. 3. 5 )427 A. 131 sg. rd. sq. yd. 85 90 62V To ^^- J^' "^ ^^f ^^* ^"- Therefore the result is 85 A. 90 sq. rd. 6 sq. yd. 64i sq. in., Ans. 4. 4 )315 gal. 3 q t. 78 3f f qt. = 1 pt. 2 gi. Therefore the result is 78 gal. 3 qt. 1 pt. 2 gi., Ans, 5. 9 )16 T. 1300 lb. 6. 3 )8 C. 100 cu. ft. 1 1700, Ans, 2 118|, Ans. 7. 10 ) £31 5s. 8d. 3 2 6|, Ans. 9. 2 qt. 1 pt. = 5 pt. ; 31 gal. =::248 pt. ; 248 pt. -f- 5 pt. = 49f. Therefore he must dip 50 times., Ans. 10. 23 mi. 160 rd. 4 yd. 2 ft. = 124094 ft. ; 100 mi. = 528000 ft. Therefore, 528000 -^ 124094 ^ 4^i^\\ da. ^= 4 da. 6 hr. 6 min. 58 + sec. , Ans, LONGITUDE AND TIME. 73 11. 13 )300 mi. 23yV mi. = 23 mi. 24 rd. 10 ft. 1|| in., Am. 12. 2 cwt. 35 lb. = 235 lb. ; 3 T. 4 cwt. 18 lb. =. 6418 lb. Therefore, 6418 -f- 235 = 272%-, ^^^ number of barrels. 13. 2 oz. 10 pwt. =- 50 pwt. ; 13 lb. 7 oz. 15 pwt. ^- 3265 pwt. Therefore, 3265 -^ 50 = 65, the number of spoons, and 25 pwt. or 1 oz. 5 pwt. remaining. Ans. 14. Since the boards were 8 in. wide, they were 4 pickets wide, and they were as many pickets in length as 2 ft. 4 in. is contained in 11 ft. 8 in., which is 5 times. Therefore, 4X^X5 = 100, number of pickets. Arts. LONGITUDE AND TIME. Page 201. 3. 15 )32° 18^ 24^^ 2 hr. 9 min. 13y^^ sec, Aiis, 4. Since the difference in time is 3 hr. 9 min. 7 sec, the difference in longitude is 15 times as many degrees, etc., which is 47° 16' 45''. Therefore the longitude of San Francisco is 75° 10'+ 47° 16' 45", which is 122° 26' 45" west, Ans. 5. Since one is east longitude and the other is west longi- tude, their difference in longitude is the sum of 74° 3' and 2° 20', which is 76° 23'; and the difference in time is ^ as many hours, etc : 5 hr. 5 min. 32 sec 74 METHIC MEASUHES. 6. The difference in time is 3^- us many hours, etc., as there are degrees, etc. : j^- of 77 is 5 hr. 8 min., Aiis. 7. The difference in degrees is 2° 57', and the difference in time is -^ as many hours, etc., as there are degrees, etc., which is 11 min. 48 sec. And since New York is east of Washington, the time at New York is 11 min. 48 sec. past 12, Ans. 8. Since the difference in time is 1 hr. 5 min. 8 sec, the difference in longitude will be 15 times as many de- grees, etc., which is 16° 17', Ans. 9. The difference in longitude between the places, counting from Pekin west to Washington, is 193° 27' 30", and therefore the difference in time is 12 hr. 53 min. 50 sec. Therefore, when it was noon at Washington, it was 53 min. 50 sec. past midnight, on January 2d, at Pekin. 10. 15 times 1 hr. 11 min. = 17° 45'. And since his time was too slow, he must have been traveling toward the sun ; that is, toward the East, Ans, METBIC MEASURES. Page S06. 8. Since 2.47114 acres = 100 ares, 1 acre :== 100 ares -f- 2.47114 = 40.4671 + ares, Ans. 9. 1 hectolitre = 2.8372 bu. ; ^ hectolitre =:=: .9457, or nearly a bushel, Ans. METRIC MEASURES. 75 10. Since there are 10 decimetres in a metre, in 586.431 metres there are 10 times 586.431, or 5864.31 deci- metres, Ans. 11. §.22X324.16 = $71,315, Ans. 12. »2.15 X 38 = $81.70, the cost. Since 1 dekalitre = 2.6417 gal., 38 gal. -f- 2.6417 = 14.38 dekalitres; $5 X 14.38 = $71.90, the selling price ; $81.70 — $71.90 = $9.80, the entire loss ; $9.80 -- 38 = $ .257 +, per gal., Ans. 13. Since a metre is 39.37079 in., at the price per metre 1 in. = $3 ~ 39.37079, and 36 in. or 1 yd. = 36 times that result, which is $2,743. And $2.90 — $2,743 = $.157, tlie difference per yard in favor of buying by the metre, Ans. 14. M. 4.2 X M. 3.8 = M. 15.96, Ans. 15. M. 3.5 X M. 3 rr. M. 10.5 ; S. 12 -^ 10.5 = M. 1|, An^. 16. $2 X 100 = $200, the cost ; 39.37079 X 100 -^ 36 = 109.363, No. of yd. in 100 M. ; $2.25 X 109.363 = $246,067, the selling price ; $246,067 — $200 = $46,067, the gain, Ans. 17. $2.50x1400 = $3500, Am. 18. Since a kilogramme, or kilo, is 2.20462 lb., there will be as many kilos in 196 lb. as 2.20462 is contained times in 196, which is 88.904 -|- kilos, Ans. 19. 1000 X 180 = 180000 sq. metres ; 180000 sq. metres -^ 100 = 1800 ares ; 1800 ares -i- 100 = 18 hectares, Ans, $250x18 = $4500, Ans. 76 PERCENTAGE. 20. 5 X5 X 2.5 = 62.5 cu. metres. Since a litre is a cube whose edge is -^^ of a metre, a cu. metre will contain 1000 litres, and 62.5 cu. metres 62500 litres, or 625 hectolitres, Ans. 21. 5.2 X 3.2 = 16.64 sq. metres, the area of one side ; 16.64 X 2 = 33.28 sq. metres, the area of two sides ; 4.5 X 3.2 = 14.40 sq. metres, the area of another side; 14.40 X 2 = 28.80 sq. metres, area of other two sides; 5.2 X 4.5 := 23.40 8:^. metres, the area of ceiling ; 33.28 + 28.80 -f- 23.40 -= 85.48 sq. metres, entire area; $ .35 X 85.48 = $29,918 +, Am. 22. Since a kilo is 2.20462 lb., $.23-^2.20462 will give the price per pound, which is $.104326, or $208,652 per ton. At $ .11 per pound, a ton would sell for $220. Therefore, $220 — $208,652 ==$11,348, the difference in favor of 11 cents per pound, Ans. 23. $ .18f X 200 = $37.50, am't received for the molasses; 26.417 gal. X 2 :== 52.834 gal., No. gal. in 2 hectolitres ; $.90X52.834 = $47.55; $47.55 — $37.50 = $10.05, the difference in favor of selling at 90 cents per gallon. Am. PERCENTAGE. Page 210. 17. 100% - 33i% = 66f % ; 66f %, or .66|, or |, of 450 sheep = 300 sheep, Am. 18. 100^ — 85^=15%; 15^,or.l5,or^,of$2000is$300. PERCENTAGE. 77 19. 37|-%, or .37i or f, of 816 bu. = 306 bu. ; $1.56 X 306=:$477.86; 816 bu. — 306 bu. = 510bu. ; $1.60 X 510 =:: $816. Therefore, $816 + $477.36 = $1293.36, amount of his sales of wheat, Ans. 20. 100^—5^ = 95%; 95^, or .95, of $318.57 is $302.64. 21. 100^ +25% = 125%; 125%, or 1.25, or |, of $30000 = $37500, Ans, 22. 12i%, or .124^, or i of $3000 = $375, the loss; $3000 — $375 = $2625, Aiis. 23. 15% + 27% = 42%, invested in bank-stock and bonds ; 100% — 42^ = 58% , the part invested in a mill ; 58% , or .58 of $40000 is $23200, Ans. 24. 24% of $18500 = $4440; $18500 + $4440 = $22940, the property of the elder ; 331^, or i, of $18500 = $6166. 66|, loss of the younger ; $18500 — $6166.661 = $12333.33^, younger's prop'ty Page SIS. 22. He sold m, or -^\, or .38f, or 38|^, Ans, 23. He spends |f^, or |, or .88f , or 88f % ; therefore he has left 100% — 88f%, or lli%, Arts, 24. Since he gained $.25 per pound, he gained -ff, or -|-, or 331%, of the cost, Ans. 25. 50 bu. are 3^Vo' ^^ eror o^ i%» of 30000 bu., Ans. 26. Since he gains $1.50 on each hat, he gains |-|^, or ^, or 27y\^, of the cost, Ans. 27. He receives -^l^-(i, or -^^, or 1^^ , of the sale, Ans. 78 PEBCENTAGE. Page S13. 28. The sum paid for insurance is y^f ^-q, or -j^-q, or 1\%, of the value of the cargo, Ans. 29. The eldest received ^, or y^Q^^, or 25%, of the whole; the next, ^, or ■^-^, or 20%, of the whole; and since the sons had 25% + 20%, or 45%, the daughters had each ^ of the remaining 55^, which is 18^%, Ans, Page 214. 17. Since 75%, or f , of whole number is 275 bbl., the whole number is 4 times -1^ of 275, or 366| bbl., Ans. 18. Since 25^, or |, of the value of the mill is $3750, the whole value is 4 times $3750, or $15000, Ans. 19. Since he sold 25^ , or \, of his share, he sold \ of 40% , which is 10^ , or y'^-, of the whole ; and since y^^- of the value of the foundry is $10000, the whole value is 10 times that sum, or $100000, Ans. 20. Since he had 90% left, the part sold must be 10^, or y^o", of the whole. Therefore, 110 A. 43 sq. rd. = -^^ of the whole, and the whole is 10 times that sum, or 1102 A. 110 sq. rd., Aiis. 21. i of $3000 =: $1000, which is 62^%, or |, of the cost of house and barn. Therefore the house and barn will cost 8 times ^ of $1000, which is $1600, Ans. 22. $80 is 81% of one-half the amount due, or 4^% of the amount due. Therefore the amount due is equal to - 4\ X 100, which is $1882^^, Ans. PERCENTAGE. 79 23. $1.60 X 4500 = $7200, amount received for the wheat ; $7200 was 90%, or y^^, of the cost, which was 10 times i of $7200, which is $8000, Ans. 24. Since $8000 is 40^ , or -|, of the amount paid for the lot, the cost of the lot was 5 times ^ of $8000, which is $20000, Am. 25. 75%, or |, of $600 is $450; and $450 is 33^%, or |, of one-half his income, or ^ of his income. Therefore his income is 6 times $450, which is $2700, Ans. 26. Since he owned -|^ of the vessel, and sold 25%, or J, of his share, he sold -^-^ of the vessel. Since he received $3350.50 for ^^ of the vessel, at that rate the vessel would be worth 12 times $3350.50, or $40206, Ans. 27. Since $7500000 is 37^%, or f, of the estimated loss, the estimated loss is 8 times ^ of $7500000, which is $20000000, Ans. 28. Since 25% of i of 60 = i of | of 60, which is 5, and 75% of ^ of the number is f of J of the number, which is \ of the number, 5 is ^ of the number. Therefore the number is 4 times 5, or 20, Ans. 29. I of 40% of 100 = 26|; 5% of 10 times ^ of the number = |- of the number. Therefore, 26|^ is ^ of the number, which is 213|^, Ans. Page 2ie. 13. $15400 -M.10 = $14000, value of his property, Ans. 14. $345 -^ 1.15 = $300, the cost of the horse, Ans, 80 PERCENTAGE, 15. $1950 ~ 1.30 = $1500, the previous salary, Ans. 16. $3750 -^ 1.25 = $3000, one-half the cost of the house. Therefore the house cost 2 times $3000, or $6000, Ans. 17. 872 4- 1.09 = 800, the attendance during 1875, Ans. Page ;317. 14. Since he paid out 75 % of his salary, he had but 25 % , or ^, of it left. Therefore, $450 is \ of his salary, and his salary is 4 times $450, which is $1800, Ans. 15. Since he sold 30% of his wheat, he had 70%, or -^q, of it left. Therefore, 350 bu. is -^-^ of the entire quan- tity, which is 500 bu., Ans. 16. Since he sold it at 30% less than his asking price, he sold it for 70% of his asking price. Therefore, 70% of his asking price is $29.24; and $29.24-^.70 = $41. 77^-, the asking price, Atis. 17. If It lost 15^ of its men, it had 85^ left. Therefore, 85^ of the number is 527 ; and 527 -^ .85 = 620, the entire number, Aiis. 18. Since he lost, during 1876, 10^ of what he had left after his losses in 1875, $40500 is 90% of the amount at the end of 1875 ; and $40500 -^ .90 = $45000, the amount he had at the end of 1875. Since he lost 10% of his money during 1875, $45000 must be 90% of the amount of his money; and $45000 -^ .90 ^^^ $50000, the amount of his money, Ans, COMPOUND INTEREST. 81 19. Since his profits were 23^ less than in 1875, $10318 is 11% of the profits in 1875. Therefore, $10318^ .77 =-$13400, the profit in 1875, Ans, COMPOUND INTEREST. Page :336. $315.00, Principal for 1st year; 18.90, Interest for 1st year; §333.90, Principal for 2d year; 20.03, Interest for 2(i year; $353.93, Principal for 3d year ; 10.62, Interest for 6 mo. ; $364.55, Amount for 2 yr. 6 mo.; 315.00, Given principal ; $ 49.55, Compound interest. Arts, 3. $324.18, Principal for 1st year; 22.69, Interest for 1st year; $346.87, Principal for 2d year ; 24.28, Interest for 2d year; $371.15, Principal for 3d year; 25.98, Interest for 3d year; $397.13, Principal for 4th year ; 11.58, Interest for 5 mo.; $408.71, Amount for 3 yr. 5 mo., Am. 6 82 COMPOUND INTEREST. 4. $525.75, Principal for 1st year; 31.55, Interest for 1st year; $557.30, Principal for 2d year; 33.44, Interest for 2d year; $590.74, Principal for 3d year; 35.44, Interest for 3d year; $626.18, Principal for 4th year; 12.52, Interest for 4 mo. ; $638.70, Amount for 3 yr. 4 mo.; 525.75, Given principal ; $112.95, Compound interest, Ans. 6. $1.191016, Amount of $1 for 3 yr.; .041685, Interest for 7 mo.; $1.232701, Amount of $1 for 3 yr. 7 mo.; 600.50, $740.23, Amount of principal for 3 yr. 7 mo. ; 600.50, Given principal; $139.73, Compound interest, Ans, $1.1449, Amount of $1 for 2 yr. ; .026714, Interest for 4 mo.; $1.171614, Amount of $1 for 2 yr. 4 mo.; 318.25 $372.86, Amount of principal for 2 yr. 4 mo.; 318.25, Given principal ; 54.61, Compound interest, Ans. COMPOUND INTEREST. 83 8. $1.191016, Amount of $1 for 3 jr.; .013895, Interest for 2 mo. 10 da.; $1.204911, Amount of $1 for 3 yr. 2 mo. 10 da.; 412.08 .51, Amount of principal for 3 yr. 2 mo, 10 da. ; 412.08, Given principal; $ 84.43, Compound interest, Am, 9. SI. 1664, Amount of $1 for 2 yr. ; .042768, Interest for 5 mo, 15 da.; $1.209168, Amount of $1 for 2 yr. 5 mo. 15 da.; 310.24 $375.13, Amount of principal for 2 yr, 5 mo. 15 da. ; 310.24, Given principal; $ 64.89, Compound interest, Ans^ 10. $328.00, Principal; $ 59.31, Simple interest for given time at 7% ; $1.1236, Am^t of $1 for 2 yr., comp. int.; .039326, Int. on that amount for 7 mo. ; $1.162926, Am'tof $1, 2yr. 7 mo,, comp. int.; 328 $381.44, Am't for 2 yr. 7 mo,, comp. int.; 328 $ 53.44, Compound interest ; $59.31 — $53.44 = $5.87, Ans, 84 ANNUAL INTEREST. 11. Since the interest is compounded semi-annually, there are 7 semi-annual periods for reckoning interest, which increase the principal by 3^ of itself each time. There-i fore the compound interest for 3| yr. semi-annually at 6^ is the same as compound interest for 7 yr. at S%. $1.229874, Amount of $1 for 7 yr. at 3^, or 3iyr. 300 at 6% semi-annually. $368.96, Amount due, Ans. ANNUAL IN TEH EST. Page 228. 2. Int. of $350 for 4 yr. at 8% = $112 ; Int. of $350 for 1 yr. at 8^ =^ $28 ; Int. of $28 for 3 yr. + 2 yr. -f 1 yr., or 6 yr., at Sfo = ^13.44; $350 + $112 + $13.44= $475.44, Amount due. Am, 3. 1877 yr. 4 mo. 15 da. 1873 1 1_ 4 3 14 Int. of $750 for 4 yr. 3 mo. 14 da. at 6% ==$193; Int. of $750 for 1 yr. at 6^ =$45; Int. of $45 for 3 yr. 3 mo. 14 da. + 2 yr. 3 mo. 14 da. -f 1 yr. 3 mo. 14 da. -f 3 mo. 14 da., or 7 yr. 1 mo. 26 da., at 6^ =$19.32; $750 + $193 + $19.32 = $962.32, Amount due, Ans. PARTIAL PAYMENTS. 85 rABTIAL PAYMENTS. Page 229. 1, Principal $850. Int. to Nov. 15, 1876 51.887 Am't of principal Nov. 15, 1876 . $901,887 First payment $200. Int. on payment to Nov. 15, 1876 . 8.361 Second payment 255.000 Int. on payment to Nov. 15, 1876 . 2.975 Am't of payments Nov. 15, 1876 . $466,336 Am't dne Nov. 15, 1876 .... $435,551, Ans. 2. Principal $1800. Int. to May 15, 1876 126. Am't of principal May 15, 1876 . $1926. First payment $300. Int. on payment to May 15, 1876 . 13.708 Second payment 200. Int. on payment to May 15, 1876 . 4.549 Third payment 1000. Int. on payment to May 15, 1876 . 4.861 Am't of payments May 15, 1876 . $1523.118 Am't due May 15, 1876 .... $402,882, Ans. 86 PARTIAL PAYMENTS. 3. Principal $585.25 Int. to Nov. 3, 1876 27.31 Amount at maturity 612.56 First payment $325. Int. to Nov. 3, 1876 9.163 Second payment 84.30 Int. to Nov. 3, 1876 1.458 Third payment 100. Int. to Nov. 3, 1876 1.186 Am't of payment at maturity . . $521,107 Am't due Nov. 3, 1876 .... $91,453, Ans. Page 331. Principal $2500.00 Int. to Sept. 15, 1871 27.083 Amount 2527.083 First payment 150.000 New principal 2377.083 Int. to Nov. 12, 1871 22.582 Amount 2399.665 Second payment 300.000 New principal 2099.665 Int. to Dec. 1, 1871 ..... 6.648 Amount 2106.313 PARTIAL PAYMENTS. 87 (Brought forward) .... $2106.313 Third payment 100.000 New principal 2006.313 Int. to April 3, 1872 40.795 Amount 2047.108 Fourth payment 325.000 New principal 1722.108 Int. to May 15, 1872 ..... 12.054 Amount 1734.162 Fifth payment 275.000 New principal 1459.162 Int. to Sept. 20, 1872 30.399 Amount 1489.561 Sixth payment 1000.000 New principal 489.561 Int. to Jan. 1, 1873 8.240 Amount due Jan. 1, 1873 . . . $497,801, Ans. 3. Principal $2150.00 Int. to Dec. 15, 1873 40.61 Amount 2190.61 First payment 75.00 New principal 2115.61 88 PARTIAL PAYMENTS. (Brought forward) .... $2115.61 Int. to Feb. 4, 1874 23.036 Amount 2138.646 Second payment 200.000 New principal 1938.646 Int. to April 3, 1874 25.417 Amount 1964.063 Third payment 150.000 New principal 1814.063 Int. to July 1, 1874 35.475 Amount 1849.538 Fourth payment 500.000 New principal 1349.538 Int. to Dec. 16, 1874 49.483 Amount 1399.021 Fifth payment . 1000.000 New principal 399.021 Int. to maturity 40.256 Amount due at maturity .... $439,277, Arts, 4. Principal $6725.00 Int. to May 5, 1875 * 95.27 Amount 6820.27 PARTIAL PAYMENTS. 89 (Brought forward) .... $6820.27 First payment 275.00 New principal 6545.27 Interest to Aug. 15, 1875 .... 109.09 Second pay't (less than int. due) . $50 Int. on previous prin. from Aug. 15. 1875, to Nov. 12, 1875 . . . 94.904 Amount 6749.264 Sum of second and third payments . 1050.000 New principal 5699.264 Int. to Jan. 3, 1876 ...... 48.443 Amount 5747.707 Fourth payment 184.25 New principal 5563.457 Fifth pay't (less than int. due) $84.10 Int. on same prin. fiom Sept. 13. 1876, to Dec. 23, 1876 . . . 92.724 Amount 5887.990 Int. to Sept. 13, 1876 ..... 231.809 Sum of fifth and sixth payments . . 1084.10 New principal 4803.89 Int. to Feb. 10, 1877 37.63 Amount due Feb. 10, 1877 . , . $4841.62, Am. 90 PARTIAL. PAYMENTS. 5. Principal $5825.00 Int. to May 15, 1871 519.07 Amount 6344.07 First payment 728.50 New principal 5615.57 Int. to April 8, 1872 403.073 Amount 6018.643 Second payment 1000.000 New principal 5018.643 Int. to Dec. 12, 1872 272.122 Third pay't (less than int. due) . $125 Int. on same prin. from Dec. 12, 1872, to July 9, 1873 .... 230.857 Amount - . . . 5521.622 Sum of third and fourth payments . 1105.000 New principal 4416.622 Int. to June 12, 1874 326.83 Amount 4743.452 Fifth payment 1000.000 New principal 3743.452 ^ Int. to April 4, 1875 242.90 Amount due April 4, 1875 . . . $3986.352, Ans. PARTIAL PAYMENTS. 91 6. Principal $895.75 Int. to Jan. 10, 1873 26.126 First pay't (less than int. due) . $25 Int. on same prin. from Jan. 10, 1873, to Oct. 12, 1873 .... 67.678 Amount 989.554 Sum of first and second payments . 225.000 New principal 764.554 Int. to Jan. 18, 1874 20.388 Amount 784.942 Third payment 75.000 New principal $709,942 Int. to March 25, 1874 13.212 Amount 723.154 Fourth payment 187.500 New principal 535.654 Int. to Jan. 1, 1875 41.066 Amount 576.720 Fifth payment 375.000 New principal 201.72 Int. to Nov. 15, 1875 ..... 17.59 Amount due Nov. 15, 1875 . . . $219.31, Ans. 92 PARTIAL PAYMENTS. 7. Principal $580.00 Int. to April 1, 1875 7.25 Amount 587.25 First payment . 85.00 New principal 502.25 Int. to July 1, 1875 6.278 Amount 508.528 Second payment 85.00 New principal 423.528 Int. to Oct. 1, 1875 5.294 Amount 428.822 Third payment 85.000 New principal 343.822 Int. to Jan. 1, 1876 4.297 Amount 348.119 Fourth payment 85.000 New principal 263.119 Int. to April 1, 1876 3.288 Amount 266.407 Fifth payment 85.000 New principal 181.407 Int. to July 1,1876 2.267 Amount 183.674 PARTIAL PAYMENTS. 93 (Brought forward) .... $183,674 Sixth payment 85.000 New principal 98.674 Int. to Oct. 1, 1876 1.232 Amount 99.906 Seventh payment 85.000 New principal 14.906 Int. to Jan. 1, 1877 .18 Amount due at maturity .... 815.086, Aiis. Page 2*^2. 8. Principal $10000.000 Int. to Feb. 23, 1876 157.777 Amount 10157.777 First payment 750.000 New principal 9407.777 Int. to July 17, 1876 301.048 2d pay't (less than int. due) $108.25 Int. on same prin. from July 17,1876, to Nov. 23, 1876 . . . 263.418 Amount 9972.243 Sum of second and third payments . 3108.25 New principal 6863.993 94 PARTIAL PAYMENTS. (Brought forward) .... $6863.993 Int. to Jan. 18. 1877 83.893 Amount 6947.886 Third payment 4000.000 New principal 2947.886 Int. to May 12, 1877 74.679 Amount due May 12, 1877 . . . $3022.565, Am. 9. Principal $3124.75 Int. to Dec. 23, 1874 213.87 Amount 3338.62 First payment 985.00 New principal 2353.62 Int. to Feb. 15, 1875 27.197 Amount 2380.817 Second payment 875.35 New principal ....... 1505.467 Int. to Feb. 20, 1876 122.11 Amount 1627.577 Third payment 1025.000 New principal 602.577 Int. to July 8, 1876 18.479 Amount due July 8, 1876 .... $621,056, Ans. PARTIAL PAYMENTS. 95 10. Principal $1885.75 Int. to June 30, 1874 51.336 First pay't (less than int. due) . $50 Int. on same prin. from June 30, 1874, to Nov. 8, 1874 ... 46.935 Amount 1984.021 Sum of first and second payments . 150.000 New principal 1834.021 Int. to Feb. 5, 1875 31.027 Amount 1865.048 Third payment ....... 125.000 New principal 1740.048 Int. to April 17, 1875 24.36 Amount 1764.408 Fourth payment 500.000 New principal 1264.408 Int. to Dec. 1, 1875 55.07 Amount 1319.478 Fifth payment 500. New principal 819.478 Int. to March 1, 1876 14.34 Amount due March 1, 1876 . . . S833.818, Am, 96 PROBLEMS IN INTEREST. PROBLEMS IN INTEREST. Page 23». 4. Since $250 loaned at 6^ produces $15 annually, it will require as many years to produce $30 as $15 are contained times in $30, which is 2 yr., Ans, 5. $600 at 8% will produce $48 annually; $24-^$48 = I". Therefore it will require ^ yr., or 6 mo., Aiis. 6. $115 at 6% will produce $6.90 annually; $13.80 -^ $6.90 = 2. Therefore it will require 2 yr., Ans. 7. $12.60 at 7% will produce $.882 annually; $4.15 -^ $ .882 = 4.705. Therefore it will require 4.705 yr., or 4 yr. 8 mo. 14 da., Aiis. 8. $35.25 at 1% will produce $2.4675 annually; $13.25-^ $2.4675 = 5.369. Therefore it will require 5.369 yr., or 5 yr. 4 mo. 13 da., Ans. 9. Since the annual interest is ^% , or i^-^, of the principal, it will require as many years to produce an amount of in- terest equal to the principal as yf^ is contained times in TW' ^^* ^% "^ 100%, which is 16| times. Therefore it will require 16|^ yr,, Aiis. Or, Since the annual interest on $100 at 6^ is $6, it will require as many years to produce $100 interest as $6 is contained times in $100, which is 16 1 times. Therefore it will require 16| yr. 10. Since 8% of the principal is added annually, to double itself or add 100^ of the principal would require as many years as 8% is contained times in 100%, which is 12^ times. Therefore it will require 12^ yr., Aiis. PROBLEMS IN INTEREST. 97 11. For any sum to double itself by adding 5% annually, will require as many years as 5% is contained times in 100%, which is 20 times. Therefore it will require 20 yr., Ans, 12. Since 5^, or yf q-, of the sum is added annually, to add a sum equal to twice the original sum will require as many years as yfo" ^^ contained times in f^^, or 5^ in 200%, which is 40 times. Therefore it will require 40 years, Am, Page S34. 5. Since the interest of $125 for 2 yr. at 1^ is $2.50, to produce $15 in the same time it will have to be loaned at as many times 1^ as $2.50 is contained times in $15, which is 6 times. Therefore the rate is 6%, Ans, 6. The interest of $250 for 6 mo. at 1^ is $1.25; $8.75 -^$1.25 = 7. Therefore the rate is 7%, Am. 7. The interest of $415 for 2 yr. 6 mo. at 1^ is $10,375; $56,025 -^ $10,375 -= 5|. Therefore the rate is 5|^. 8. The interest of $317 for 1 yr. 5 mo. at 1% is $4.4 of the investment; 6|^ — 64%^ = y^y % , in favor of New York 7's at 105, Am, 31. The dividend is jj^, or S^-^%, of investment, Am, 33. .15~.07 = 2.14f, or 214f%, Am, 34. .10 ^.08:= 1.25, or 125%, which is 25% premium. 35. .07-^.10^.70, or, 70%, Am. 36. .06-f-.07 = .85f, or, 85f%, Am. 37. 6% stock yields an income of -9%, or 6|% ; .15 -^ .06| =-2.25, or 225^, A71S, 38. $9280X1.071 = $9941.20, Am, 39. $7225 X 1.08i =- $7812.03 +, Am. Page 277. 40. $5000 -M.06i = $4694.84+, Am. 41. $7250 -T- 1.05i = $6888.36 +, Am. 128 INSUHANCEc 42. $135000 — S5000 3r:r $130000, the entire dividend ; $130000^ $2000000 = .06i or 6^^, Am. 43. $225000 X .40 = $90000, the expenses ; $90000 + $10000 ^ $5100000, entire deductions; $225000 — $100000 == $125000, entire dividend; $125000 -^ $1500000 ^ .08^, or 8^^, the per cent, of dividend. Therefore, A. makes 8^^, or an investment of 75%, or f|-=ll^^, Am. 44. $150 X 200 = $30000, par value of stock ; $30000 -^ .98 = $30612.25 +, par value of bonds ; $30000 X .06 = $1800, income from stock; 30612.25 X .06 =- $1836.73 +, income from bonds ; $1836.73 — $1800 =^ $36.73 +, gain. Am. 45. $5000 X 1.15 = $5750, market value of U. S. 6's; $5750 -M.05 =- $5476.19 +, par value of 10-40's; $5000 X. 06 = $300, income from U. S. 6's; $5476.19X.05 = $273.809, income from U. S. 10-40's; - $273,809 = $26.19, loss. Am. INSUIlAXCEo Page 280. 2. $15000 X.01i = $187.50, premium, Am. 3. $5000 X.00| = $37.50, prem. for insuring house; $3000 X .01 = $30.00, prem. for insuring furniture; §67.50, the entire premium, Ajis. INSURANCE. 129 4. $18000 X I == $12000, the amount insured; $1200 X .021 = $270, premium, Am. 5. $80000 + $65000 = $145000, value of ship and cargo; $145000 X I = $108750, the sum insured ; $108750 X.01i = $1359.375, premium, Am. $108750 — $1359.375 = $107390. 625, loss. Am. 6. $15850 X.03i=: $515. 125, premium, Am. 7. $275-^.001 = $55000, sum insured, Am, 8. $325 -^ $16250 = .02, or 2%, rate of premium. Am. 9. $47.50 -^ $9500 = .OOi or |^, rate of premium, Am. 10. $175-^.01^ =-$14000, sum insured, A^is. 11. $652.50 ^$43500 = .015, or 1|%, rate of prem., Am. 12. $180-^.03 = $6000, the amount insured; I of value of mill = $6000 ^ of value of mill = $3000 Entire value of mill = $9000, Am. 13. -$30000 X I = $22500, sum insured ; $22500 X .OOf = 168.75, the premium paid ; $22331.25, loss of insurance co's; $22500 + $5000 = $27500, secured by merch't; $30000 — $27500 = $2500, loss goods to merch't ; 1 68, 75, loss of premium paid ; $2668.75, entire loss to merch't. 130 INSURANCE. 14. $400 -^ .04^ == $8888. 88f, the sum insured ; I of value of silks = $ 8888. 88f I of value of silks =$ 4444. 44f Entire value of silks = $13333.33^, Ans. Page 281. 15. $1657. 50 -f-.02i =$66300, sum insured; f of value of works = $66300 i of value of works = $22100 Entire value of works = $88400, Am. 16. $2475 -f-.Oli-- $198000, sum insured; I of value of ship and cargo = $198000 ^ of value of ship and cargo =^ $ 99000 Entire value of ship and cargo = $297000, Ans. 17. $225 -^ .Oli = $15000, sum insured ; f of value of store and contents = $15000 ^ of value of store and contents = $ 5000 Entire value of store and contents = $20000 Since the stock was worth half as much as the store, both together were 1| times the value of the store. There- fore, $20000 -Mi r=:$l 3333.331, the value of store; ^ of $13333.33^ = $6666.66|, the value of stock, Ans. 18. $20000 X I === $15000, the property insured. Since the sum insured is such, that the remainder, after pay- ing li^ premium, is $15000, then $15000 is 98|% of the sum covered by insurance ; $15000 -^ .98^ = $15228.42, Ans. INSURANCE. 131 Page 2S2. 2. $31.30X3 ==$93.90, annual premium, Am. 3. $5500 = 5.5 thousands. Therefore, $26.38 X 5.5 = $145.09, annual premium, Am, 4. $ 28.90 X 5 = $ 144.50, annual premium; $144.50 X 9 = $1300.50, entire premiums ; $5000 — $1300.50 = $3699.50, more. Am. 6. $22.90X5 = $114.50, annual premium ; $114.50x30 = $3435, entire premium paid. The first payment was on interest for 29 years, the second for 28 years, the third for 27 years, etc., or the entire interest will be the same as the interest of the annual payment for 435 years. $114.50 X .06 = $6.87, interest for 1 year ; $6.87 X 435 = $2988.45, entire interest; $3435.00, entire premium ; $6423.45, payments and interest; $6423.45 — $5000 --$1423.45, loss. Am. 6. $54.90 X 5 =:: $274.50, annual payment; $274.50 X 15 = $4117.50, entire payments. Since the first pay- ment was on interest 15 years, the second 14 years, etc., the entire interest will be the same as the interest of the annual payment for 120 years. $274.50 X -07 = .215, interest for 1 year; $19,215 X 120 = $2305.80, entire interest; $4117.50, entire payments; $6423.30, payments and int. ; $6423.30 — $5000 = $1423.30, the loss. Am. 132 EXCHANGE. 7. $50 X 15 = $750, amount of premium paid. Since the first payment was made 14 years previous to the accident^ the second 13, the third 12, etc., the entire interest will be the same as the interest of the annual payment for 14 + 13 + 12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 ==r 105 years. $50 X .06 := $3, the interest for 1 year. $3 X 105 = $315, entire interest ; $750, entire premium; $1065, premium and interest; $30 X 20 = $600, amount of benefit; $1065 — $600 = $465, the loss, Am. EXCHANGE. Page 286. 3. 100^ + \% = 1001%, the rate of exchange ; $1000 X 1.001- ^ $1005, cost of draft. Am, 4. 100^ + 1\% = 101\fo, the rate of exchange; $3000 X 1.01^ = $3037.50, cost of draft, Am. 5. 100% + i^ rrr 100|%, the rate of exchange. There- fore, the rate of exchange for a dollar is $1.00125. The discount on $1 in Chicago for 33 days is $.0055. There- fore, $1.00125 — $.0055 =- $.99575, the cost of $1 of draft; $.99575 X 5000 = $4978.75, cost of draft. Am, EXCHANGE. 133 6. $1 -— $.00125 = » .99875, the rate of exchange for $1 of draft ; The discount on $1 in Buffalo for 93 days is $.018081; $.99875 — $.018081 = $.98066|, the cost of $1 draft; $.980661 X 1500 = $1471, the cost of draft, Am. 7. $1 _ $.0025 == $.9975, the cost of $1 of draft; $. 9975 X 5000 ^ $4987.50, the cost of draft, Am, 8. $1 + $.00125 = $1.00125, the cost of $1 of draft; $1.00125 X 3000 = $3003.75, the cost of draft, Am, 9. $1 + $ .00125 =-: $1.00125, the rate of exchange for $1 of draft ; The discount on $1 in Chicago for 93 days is $.0155; $1.00125 — $.0155 = $.98575, the cost of $1 of draft; $.98575 X 5000 -= $4928.75, the cost of draft. Am, Page 287. 10. $1 4- $.0025 = $1.0025, the rate of exchange for $1 of the draft; The discount on $1 in St. Paul for 60 days is $.0116|; $1.0025 — $.01161 = $.99081 the cost of $1 of draft; $.99081 X 1500 = $1486.25, the cost of draft, Am, 11. $1 + $.001 = $1,001, the rate of exchange for $1 of draft ; The discount on $1 for 63 days in Cincinnati is $.0105 ; $1,001 — $.0105 = $.9905, the cost of $1 of draft; $.9905 X 5000 = $4952.50, the cost of draft, Am, 14. $1 + $.0025 = $1.0025, the cost of $1 of the draft; $5725 -^ $1.0025 = $5710.72 +, the face of draft, Am. 134 EXCHANGE. 15. $1 + $.00125 =: $1.00125, the rate of exchange for $1 of draft ; The discount on U for 33 days at 6^ is $.0055 ; $1.00125 — $ .0055 = $ .99575, the cost of $1 of draft; $1500 -^ $.99575 = $1506.40 +, face of draft. Am. 16. $1 + $.005 = $1,005, the rate of exchange for $1 of draft ; The discount on $1 for 63 days at 9fo is $.01575; $1.005 — $.01575 = $.98925, the cost of $1 of draft; $1200 -^ $.98925 = $1213.04+, the face of draft, Arts, 17. $1 — $ .00125 = $ .99875, the cost of $1 of draft ; $10000 -^ $.99875 = $10012.51 +, face of draft. Am. 18. $1 + $ .00125 = $1.00125, the rate of exchange for $1 of draft ; The discount in St. Louis on $1 for 33 days is $.0055; $1.00125 — $.0055 = $.99575, the cost of $1 of draft; $3500 -^ $ .99575 = $3514.93 +, face of draft. Am, 19. $1 + $.00125 = $1.00125, the rate of exchange for $1 of draft ; $1750 -^ $1.00125 = $1747.81 +, face of draft, Am, Page 289. 3. $3762.50-M.05i=$3566.35, value of currency in gold; $3566.35-^$4.87 = 732.31 + , value in pounds sterling; £732.31 =£732 6s. 21d., the face of draft. Am. 4. $5928.75 -^ 1.06^ = $5580, value of currency in gold; $5580 ^ $4,851- = 1149.33 +, value in pounds sterling ; £1149.33 = £1149 6s. 7id., the face of draft. Am, EXCHANGE. 135 5. $5575.20-^1.071 3= $5174.1995+, value of currency in gold; $5174.1995 -^ $4,875 =:: 1061.374 +, the value in pounds sterling; £1061.374 ==£1061 7s. 5fd., face of draft, Ans, 6. 3269 fr. ^5.15 fr. == 634.757, the value in dollars in gold ; $634,757 X 1.05f== $668,875 +, cost of draft in cur- rency, Aris. 7. 8950 fr. -^ 5.19 fr. = 1724.47 +, the value in dollars in gold; $1724.47 X 1.061 = $1832.249 +, cost of draft in cur- rency, Ans. a $1575-+ 1.07i = $1468.531, value of currency in gold; 5.19 fr. X 1468.531 = 7621.67 fr., face of draft, Ans, 9. 5725 fr. -^ 5.20 fr. = 1100.961 +, value in dollars in gold; $1100.961 X 1.06i==: $1169.77 +, cost of draft, Ans. 10. £895 10s. = £895.5; $4.87 X 895.5 ==$4361.085 +, the gold value of draft; $4361.085 X 1.06| = $4650.00, the value of draft. Am. IL £585 10s. 5d. = £585.5208 + ; $4.8665 X 585.5208+ =r $2849.437 +, cost in gold; $2849.437 X 1.07^ = $3063.14 +, cost of draft in cur- rency, Ans. 12. £875 5s. 4d. = £875.2|; $4,885 X 875. 2| =: $4275.678, cost in gold ; $4275.678 X 1.04^ = $4484.11 +, Am. 136 AVERAGE OF PAYMENTS. AVERAGE OF PAYMENTS. Page 292. 2. $ 300 X = $1200 X 3 = 3600 $ 800 X 4 = 3200 6800 mo. -^ 2300 = 2|| mo., $2300 6800 mo. or 2 mo. 29 da., Am. a « 500 X 1 = 500 « 500 X 2 = 1000 1!^X 4 =!^ 4700 mo -1800 = 2H mo.. $1800 4700 mo. or 2 mo. 18 da., Am, 4. $1500 X -= $3000 X 1 = 3000 $2000 X 3 = 6000 9000 mo. -^ 6500 =: 1^ mo., $6500 9000 mo. or 1 mo. 12 da., Am. 5. $ 800 X 1 = 800 $ 800 X 2 = 1600 $ 800 X 4 = 3200 _„^ . ^ .„^ ^^ 5600 mo. -f- 2400 = 2^ mo. , $2400 5600 mo. or 2 mo. 10 da., Am. 6. $ 750 X 2 = 1500 $ 750 X 3 = 2250 $1500X4 = 6000 ^^_^ . ^.^. ^^ 9750 mo. -^ 3000 := 3^ mo., $3000 9750 mo. or 3 mo. 8 da., Am. AVERAGE OF PAYMENTS. 137 Page 294. 2. Due Mar. 1, 1877 . . . $200 X = ^^ June 10, 1877 . . . $500 X 101 = 60500 '' June 12, 1877 . . . $275 X 103 =. 28325 ** Sept. 1, 1877 . . . $400 X 184 ==. 73600 $1375 152425 da. 152425 da. --- 1375 = 110 + da. Therefore the average time of payment is 1 11 days after the standard date, March 1, which is June 20, 1877, Am. 3. Due Mar. 1, 1877 . . . $500 X = Apr. 15, 1877 . . . $850 X 45 -= 38250 May 20, 1877 . . . $375 X 80 =. 30000 June 28, 1877 . . . $650 X 119 = 77350 $2375 145600 da. 145600 da. -^ 2375 =- 61 + da. Therefore the average time of payment will be 62 days after the standard date, March 1, which is May 2, 1877, Aiis, 4. Due Jan. 10, 1877 . . . $300 X 26 = 7800 ^' Apr. 15, 1877 . . . $400 X 121 = 48400 '' Jan. 1, 1877 . . . $750 X 17 = 12750 " Dec. 15, 1876 . . . $300 X == $1750 68950 da. 68950 da. -^ 1750 == 39 -f- da. Therefore the average time of payment was 40 days after December 15, 1876, which was January 24, 1877, Am. 138 AVERAGE OF PAYMENTS. 5. Due Dec. 1, 1876 . . . $600 X = ^^ Feb. 3, 1877 . . . $400 X 64 == 25600 *' Dec. 20, 1876 . . . $250 X 19 = 4750 *^ Dec. 10, 1876 . . . $375 X 9 ^ 3375 $1625 33725 da. 33725 da. -M625 = 20 + da. Therefore the average time of payment was 21 days after December 1, 1876, which was December 22, 1876, Am. 6. Due Apr. 10, 1877 . . . $460 X 5 = 2300 *' Apr. 5, 1877 . . . $200 X := " May 30, 1877 . . . $200 X 55 = 11000 '' Apr. 25, 1877 . . . _$900 X 20 = 18000 $1760 31300 da. 31300 da. -^ 1 760 = 18 da. , nearly. Therefore the av- erage time of payment was 18 days after the stand- ard date, April 5, which was April 23, 1877, Arts, 7. Due May 25, 1877 . . . $850x10-- 8500 '^ May 15, 1877 . . . $600 X =- " July 20, 1877 . . $500 X ^^ = 33000 *' June 10, 1877 . . . $960 X 26 == 24960 $2910 66460 da. 66460 da. -^ 2910 = 22 + da. Therefore the average time of payment was 23 days after the standard date. May 15, which was June 7, 1877, Ans. 8. $ 600 X = $ 600 X 2 = 1200 8400 mo. ~ 2400 = 3^ mo. ; $1200 X 6 ^ 7200 May l + 3i mo. = Aug. $2400 8400 mo. 16, 1877, Ans. AVERAGE OF ACCOUNTS. 139 AVERAGE OF ACCOUNTS. Page 5296. 2. Dae. Amount. Days. Product. Paid. Amount. Days. Product. 1877. Mar. Mar. June 5 15 25 30 $375 200 800 450 117 107 5 43875 21400 4000 1877. Jan. Mar. Apr. 30 15 1 $200 600 200 151 107 90 30200 64200 18000 June 119400 1825 1000 69275 $1000 69275 43125 825 43125 -- 825 r=: 52 + da.; June 30, 1877 + 53 da. = August 22, 1877, Am>. 3. Due. Amount. Days. Amount. Paid. Amount. Days. Product. 1877. Mar. Mar. 1 3 15 1 $1600 3800 5500 1500 92 90 17 147200 342000 93500 1877. Mar. May 18 23 $2000 5000 $7000 75 9 150000 45000 May- June 195000 $12400 7000 $5400 582700 195000 387700 387700-^5400=: 71 + da.; June 1, 1877 — 72 da. = March 21, 1877, Ans. 140 AVERAGE OF ACCOUNTS. 4. Due. Amount. Days. Product. Paid. Amount. Days. Product. 18T7. 1877. Apr. 10 $150 75 11250 Apr. 12 $250 73 18250 Apr. 30 400 55 22000 May 1 200 54 10800 May 16 100 39 3900 June 7 400 17 6800 June 24 500 $850 35850 $1150 37150 850 35850 $300 1300 1300-^ 300 = 4 + da.; June 24, 1877 — 5 da. = June 19, 1877, Ans. 5. Due. Amount. Days. Product. Paid. Amount. Days. Product. 1877. Feb. Apr. Apr. 1 20 15 3 $500 850 1500 2500 183 105 110 91500 89250 165000 1877. Feb. Feb. June 3 28 18 $500 200 1200 181 156 46 90500 31200 55200 Aug. $1900 176900 $5350 1900 345750 176900 $3450 168850 168850 — 3450 = 48 + da.; August 3, 1877 — 49 da. = June 15, 1877, Ans. AVEEAGE OF ACCOUNTS. 141 6. Due. Amount. Days. Product. Paid. Amount. Days. Product. 1877. Feb. Apr. 1 15 20 3 $1800 3000 4800 6000 183 110 14 329400 330000 67200 1877. Feb. July 20 21 $3000 8000 164 13 492000 104000 July Aug. $11000 596000 $15600 11000 726600 596000 $4600 130600 130600 -f- 4600 = 28+ da. ; August 3, 1877 — 29 da. = July 5, 1877, Ans. 7. Due. Amount Days. Product. Paid. Amount. Days. Product. 1877. Sept. Nov. 10 1 8 20 $500 700 800 600 71 19 43 35500 13300 34400 1877. July Aug. 20 20 $400 1000 123 92 49200 92000 Oct. Nov. 141200 $1400 83200 $2600 1400 83200 58000 $1200 58000 -- 1200 = 48-1- da. Therefore the debt was due on Nov. 20 + 49 da., or Jan. 8, 1878. The cash balance will be the present worth of $1200 due in 7 da. @ 6%. $1200 -^- l.OOH = $1198.604-, the cash balance, Ans. 142 PARTNERSHIP. rA R TNE JB S HIP. Page 300. 2. $7000 +$7000 + «6000 = $20000, , entire capital ; "2'oiro'Tr> or 2V» of $6000 = $2100, A's share of gain ; yWA» or /o. of $^^^^ = $2100, B's share of gain ; 6000 2 ' or 2%. of $6000 = $1800, C's share of gain. 3. $3000 + $6000 + $4000 = $13000, , entire capital; 3000 13000J or ^, of $2600 = $600, A's share of gain ; 6000 13000> or y%, of $2600 = $1200, B's share of gain ; TTTOOJ or ^\, of $2600 = $800, C's share of gain. 4. $10000+ $8000 + $12000 = $30000, , entire capital; 10000 30000> or I-, of $6000 = $2000, A's share of loss ; Too'oo'y or 3«o, of $6000 = $1600, B's share of loss; ■3o"o"oTr> or i|, of $6000 = $2400, C's share of loss. 5. $2500 + $2000 + $3500 = $8000, entire capital ; 2 5 00 8000> or -j^, of $640 = $200, A's share of loss ; mi or i, of $640 = $160, B's share of loss ; "TO^^J or y7_» of $640 = $280, C's share of loss. 6. $8000 + $10000 + $9000- capital ; or \, of $3000 : or 1, of $3000 : or 4^0. of $3000 : or M, of $3000 : 8000 4 OlTllJ 10000 40 0ir' 9000 4T(T-0 0'» 130 00 T0T7Tro> $13000 = $40000, entire : $600, A's share of gain ; : $750, B's share of gain ; : $675, C's share of gain ; :$975, D's share of gain. PAETNERSHIP. 7. $8000 + $10000 : i of $5400 $5400 — $1800 T\moriof$3600: itmor|of$3600: 8. I of $46000 $17250 + $11500 : $46000 — $28750 : $48300 — $46000 : iet^orfof$2300: iil^^oriof$2300: iHf^orfof$2300: : $1800, : $3600, : $1600, : $2000, $17250, : $28750, : $17250, : $2300, $862.50, : $575, : $862.50, 143 capital of D & G ; L's gain; gain .of D & G; D's gain ; G's gain. furnished by E ; furnished by E & F ; furnished by G ; entire gain ; E's share of gain ; F's share of gain ; G's share of gain. 9. $6470 + $5420 + $3410 = $15300, entire capital; $2744.78 +, A's share of gain; $2299.33 +, B's share of gain; iWA of ^6490.75 3^VA of $6490.75 3410 1530Ty of $6490.75 = $1446.63 +, C's share of gain. 10. 125 A. 60 sq. rd. = 125.375 A. ; $3.75 X 125.875 =^ $470,156 +, expense of pasture; 125 + 145 + 175 + 340 :-= 785, number of sheep; III of $470,156= $74.86+, am't A should pay ; m of $470,156:=: $86.84+, am't B should pay; ii| of $470,156 = $104.81 — , am't C should pay; m of $470,156 = $203.63 — , am't D should pay. Page 301. 11. Since the entire gain was $4200, and C's gain was $1400, or ^ of the entire gain, C's stock was ^ of the entire stock, or ^ as much as A's and B's together ; 144 PARTNERSHIP. $6000 + $8000 = $14000, A\s and B's stock ; ^ of $14000 =$7000, C's stock; $4200 — Ji400 = $2800, A's and B's gain ; ttA% or^» «f ^2800 = $1200, A's gain; ■^%% or f , of $2800 == $1600, B's gain, Ans. 12. $500 + $600 + $800 + $1000 + $1200 = $4100, entire capital ; ■^% or ^, of $2750 = $335,365 +, A's gain; ■iiA^ or /j-, of $2750 = $402,439+, B's gain; ■i^, or ^, of $2750 = $536,585 +, C's gain ; ifw. or i^, of $2750 = $670,731 +, D's gain ; ^00^ or -^, of $2750 = $804,878+, E's gain, Ans, 13. Since A paid \ of the purchase money, he should re- ceive ^ of the gain ; since B paid ^ of the purchase money, he should receive ^ of the gain ; since C paid the rest, or j%, of the purchase money, he should receive ^ of the gain. J of $3000 = $750, A's gain ; I of $3000 = $1000, B^s gain ; ■^ of $3000 = $1250, C's gain, Ans. Page 302. 2. $4000 X 18 = $72000 = A's capital for 1 mo. ; $6000 X 15 = $90000 = B's capital for 1 mo. ; $8000 X 9 = $72000 = C's capital for 1 mo. ; $234000 = entire capital for 1 mo. AtoVo or A" of $9360 = $2880, A's gain ; A?oVoOri^of$9360 = $3600, B's gain ; ^¥wV or -3^ of $9360 =$2880, C's gain, Ans. PARTNERSHIP, 145 3. $4000 X 8 = $32000 = A's capital for 1 mo. ; $6000 X 7 = $42000 =^ B's capital for 1 mo. ; $3500 X 12 = $42000 = Cs capital for 1 mo. ; $116000 = entire capital for 1 mo. iVeVA <^^ TO of $2320 =: $640, A's gain ; _4_^o_o_o_. or 2 1. of $2320 = $840, B's gain ; iVe wo or fir of $2320 = $840, C's gain, Ans. 4 20% of $5875 =- $1175, B's capital ; 35^ of $5875 = $2056.25, C's capital ; 45 % of $5875 = $2643. 75, D's capital. $1175 X 15 = $17625.00, B's capital for 1 mo. ; $2056.25 X 9 = $18506.25, C's capital for 1 mo.; $2643.75 X 10 = $26437.50, D's capital for 1 mo. ; $62568.75, entire capital for 1 mo. $2502.75 -^ $62568.75 -= .04 ; or the loss is 4^ of the capital for 1 mo. ; 4% of $17625.00 = $ 705, B's loss; 4% of $18506.25 = $ 740.25, C's loss; 4% of $26437.50 = $1057.50, D's loss, Ans. 5. 20 men for 3 mo. = 60 men for 1 mo. , A's men ; 25 men for 3^ mo. = 87|^ men for 1 mo., B's men; 15 men for 4 mo. = 60 men for 1 mo,, C's men; Entire No. of men = 207-|- men for 1 mo. $1475 ^ 2071- = $7.1084 +, profit for 1 man 1 mo. ; $7.1084 X 60 =$426.50+, As profit; $7.1084 X 87-1 ::::. $621.98 +, B's profit; $7.1084 X 60^^ =$426.50+, C's profit. Am. 10 146 PABTNERSHIP. 6. $5000 X 12 = $60000 = A's capital for 1 mo. ; $8000 X 10 -= $80000 = Ws capital for 1 mo. ; $10000 X 6 -- $60000 = C's capital for 1 mo. ; $200000 = entire capital for 1 mo. 2%%%% ^r TU^ of $^^^^ ^ *2550 = A's loss ; ^8_o^o_oo_^ or I, of $8500 = $3400 = B's loss ; j%%%%%, or y%, of $8500 = $2550 = C's loss. 7. $4500 X 6 = $27000 =^ A's capital for 1 mo. ; $5000 XS = $40000 = B's capital for 1 mo. ; $6500 X 7 = $45500 = C's capital for 1 mo. ; $112500 = entire capital for 1 mo. $4500 -^ $112500 == .04; or gain is 4% of capital for 1 mo. ; 4fo of $27000 = $1080, A's gain ; 4% of $40000 = $1600, B's gain; 4% of $45500 = $1820, C's gain. 8. $1200 X 12 = $14400 = G's capital for 1 mo. ; $1500 X 12 =: $18000 = L's capital for 1 mo. ; $3000 X 6 = $18000] ^, . ^ ^ ^^^^^ ^ ^ ^^^^r=Fs capital for 1 mo.; $1000 X 61=^$ 6000] ^ $56400 = entire capital for 1 mo. UiU, or if of $2200 = $561,702+, G's profits; iffw' 01- if> of $2200 = $702,127+, L's profits; UUh or It> of $2200 -: $936,170+, Fs profits. SIMPLE PKOPORTION. 147 SIMPLE morOBTION. Page 310. 3. 6 men : 10 men : : $75 : ( ) ; !Ii-><12^X70>^ _ 153 da., Am. 25 X 8 X 80 X 60 X 10 17. 52 men 45 ft. 10 ft. 15 da. 355 ft. : ( ) ; 154 INVOLUTION. 355 X 45 X 60 X 8 X 25 52 X 45 X 10 X 15 ::::- 546y\ ft., AuS. INVOL UTION. Page 317. 2. 54^ = 50^ + 2 (50X4) + 4^ =-2916, Am. 3. 71^ = 70^ + 2 (70X 1) + l' = 5041, ^ns. 4. 68' = 60^ -t- 2 (60X8) +8^ = 4624, ^71^. 5. 47' = 40' + 2 (40 X 7) + 7' =- 2209, Ans. 6. 89' = 80' + 2(80x9) + 9' = 7921, ^ns. 7. 26' = 20' + 2(20x6) + 6'= 676, ^ns. 8. 74^ = 70' -f 2 ( 70 X 4) + 4^ -= 5476, Ans, 9. 95' -= 90' + 2 (90 X 5 ) + 5' =- 9025, Ans, 10. 82' = 802 + 2(80x2) + 2'=-6724, Am. 11. 39' == 30' + 2 (30 X 9 ) + 9' = 1521, Am. 12. 44' == 40^ -f 2 (40 X 4) + 4' = 1936, Am. 13. 67' = 60' 4- 2 (60 X 7) + 7' -= 4489, Am. 14. 16'=: 9' + 2( 9X7) + 7'= 256, ^ris. INVOLUTION. 155 15. 202 = 122 + 2(12x8)+8^= 400, ^ns. 16. 32^ = 30^ + 2 (30 X 2) + 2^ = 1024, Am. 17. 13^== 7^+2( 7X6) + 6^= 169, ^ns. 18.26^= 9^ + 2 (9 X 17) + 17^ = 676, ^ns. 19. 17^= 8^ + 2 ( 8x9)4-9^=-: 289, ^ris. Page 318. 2. 26=^ = 20^ + 3 (20^ X 6) + 3 (20 X 6') + 6^ == 17576. 3. 31^ = 30^ + 3 (30^X1) + 3 (30X1') + 1^ = 29791. 4. 28^ = 20^ + 3 (20^ x 8) + 3 (20 X 8^) + 8^ = 21952. 5. 42=^ = 40' + 3 (40^ X 2) + 3 (40 X 2^) + 2^ = 74088. 6. 27' = 20' + 3 (20'- X 7) + 3 (20 X 7^) + 7^-= 19683. 7. 36^ = 30=^ + 3 (30'X6) +3 (30X6') + 6^ = 46656. 8. 38' = 30^ 4_ 3 (302 X 8) + 3 (30 X g2) _|. §3 ^ 54372. 9. 39^ = 30^ + 3 (302x 9) + 3 (30 X 9') + 9^ = 59319. 10. 54^ = 50^ + 3 ( 502 X 4) + 3 ( 50 X 4' ) + 43 ^ 157464. 11. 52=* = 50=* + 3 (50^ X 2 ) + 3 (50 X 2') + 2=* = 140608. 12. 64^=60^+3 (eo^X 4)+3(60x 4')+ 4^ or ^ 48=*+3(48^Xl6)+3(48Xl62)+16^ or [ 262144. 35^+3(352X29)+3(35x29')+29^ etc. 156 EVOLUTION. 13. 66^=60^+3(602X 6)+3(60x 6^)+ 6^ or ] 52^+3(522Xl4)+3(52xl4')+14\ or 1- 287496. 40^+3(40^X26)+3(40x262)+26^ etc. J EVOLUTION. Page 320. The prime factors of 144 are 2, 2, 2, 2, 3, 3. Since the square root is sought, they are to be separated into two equal groups. Therefore the square root of 144 is 2 X 2 X 3, or 12, Ans. The prime fictors of 256 are 2, 2, 2, 2, 2, 2, 2, 2. Therefore the square root of 256 is 2x2x2x2 = 16, Am. The prime factors of 324 are 2, 2, 3, 3, 3, 3. There- fore the square root of 32 is 2 X 3 X 3 = 18, Am, The prime factors of 576 are 2, 2, 2, 2, 2, 2, 3, 3. Therefore the square root of 576 is 2 X 2 X 2 X 3 = 24, Am. The prime factors of 64 are 2, 2, 2, 2, 2, 2. Since the cube root is sought, they are to be separated into three equal groups. Therefore the cube root of 64 is 2 X 2 ==: 4, Am. The prime factors of 512 are 2, 2, 2, 2, 2, 2, 2, 2, 2. Therefore the cube root of 512 is 2x2x2== 8, Am. SQUARE ROOT. 157 The prime factors of 4096 are 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2. Therefore the cube root of 4096 is 2X2X2X2 = 16, ^?is. The prime factors of 13824 are 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3. Therefore the cube root of 13824 is 2X2X2X3 = 24, Ans. 4. The prime factors of 1296 are 2, 2, 2, 2, 3, 3, 3, 3. Therefore the fourth root is 2x3 = 6, Ans. The prime factors of 248832 are 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3. Therefore the fifth root of 248832 is 2 X 2 X 3 = 12, Ans. SQUARE BOOT. Page 3S5. i = .5. To extract the square root of .5, ciphers must be annexed and the square root extracted. .50-00-00 49 •(.707+, Am. 1407) 10000 9849 .9= . ,90-00 -00 81 (.948+, Ans. 184) 900 736 1888) 16400 15104 158 EVOLUTION. APPLICATIONS OF SQUABE MOOT. Page 325. 1. i/625 sq. ft. = 25 ft., Am. 2. 1/2025 sq. rd. = 45 rd., Ans, 3. I" of 5408 sq. ft. is 2704 sq. ft., area of each square; 1/27 04 = 52, the side of square or width of rectangle ; 2 times 52, or 104, is the length of rect- angle. Alls, 4. Since one field contains 4 times as much as the other, one will contain 1- of 50 A. and the other |- of 50 A. ; I of 50 A. = 10 A. ; | of 50 A. == 40 A. ; 10 A. X 160 :=: 1600 sq. rd., the area of smaller field ; 1/I6OO sq. rd. = 40 rd., the side of smaller field ; 40 X 4 = 160, rods of fence needed for smaller field ; 40 A. X 160 = 6400 sq. rd., the area of larger field; >/'6400 sq. rd. =: 80 rd., the side of larger field; 4 X 80 = 320, rods of fence needed for larger field ; 160 + 320 = 480, rods of fence for both fields, Ans, 6. Since the length is twice the width, if it is divided into two equal fields, each containing 10 acres, each will be a square whose side is half the length of the rectangle ; 10 A. X 160 = 1600 sq. rd., the area of each square; APPLICATIONS OF SQUAKE ROOT. 159 l/ISOO sq. rd. = 40 rd. , the side of square and width of the rectangle. Since the length was twice the width, the length will be 40 X 2 = 80 rd. The distance around it, there- fore, is 240 rd., Ans. 6. 72 X 32 = 2304 sq. rd., the area of the field ; l/2304 = 48 rd., the side of an equal square field; 48 rd. X 4 = 192 rd., the distance around squaie field; 2 (72 + 32) = 208 rd., the distance around rectangular field; 208 rd. — 192 rd. = 16 rd., the distance shorter around square field; $572 -^ 208 = $2.75, the price of fence per rd. ; $2.75 X 16 =$44, am't less in form of a square, Ans, Page S27. 2. \/'W+'W= 25 ft., the hypotenuse, Ans, 3. i/lW^=T(P = 113.137 + ft, the perpendicular, Ans. 4. |/50^— 30^ = 40 ft., the base, Ans, 6. Since the part broken off*, or hypotenuse, was 125 ft., the part standing, or perpendicular, was 25 ft. Therefore, 1/125^—25' = 122.474 + ft., width of stream, Ans. 6. In 6 hours, the first had traveled 90 miles, the second 108 miles. The distance they are apart is the hypotenuse of a right-angled tri- angle, whose sides are respectively 108 and 90 miles ; ' losHT l/lW+W = 140.584+ mi., the distance apart, Ans, 160 EVOLUTION. Page S2H. 7. The walk is the hypotenuse of a right-angled triangle whose sides are respectively 45 and 60 rd. ; \/~W+W=7d rd., the length of walk, Ans. 8. The angle A D B is a right an- ffle. Therefore, l/60^+ 100^= 116.619+ ft, the distance AB. The angle A B C is a right an- gle. Therefore, 1/116.619^ + 26^-= 119.482-f- ft., the distance AC, Aiis, 9. t/125^— 60^ = 109.658+ ft., the distance C E ; 1/125^- 80^=96.046+ ft., the distance E D ; 109.658 ft. + 96.046+ ft. = 205.704+ ft., the distance C D, width of the street. 10. 1/225^—145^=172.046+ ft., the height of the pole, Aiis. 11. 488 ft. 10^ in. = 488.875 ft. ; 1/488.875^—300^=386.003+ ft. , the perpendicular or the height of cathedral, Am, / / / ^--# / A A 60 ft. D SIMILAR FIGURES. Page 329. 2. 2^ : 4^ :: 6.2832 sq. ft. : ( ); 6.2832 X 4^ ^ 25.1328 + sq. ft., Am. 4 i/W : l/25 : : 80 rd. : ( ) ; ^^^;^^^ = 63.245 + rd., Am, l/40 5. i/3T4l6 : 1/113.0976 : : 20 ft. : ( ); 161 gg-Xl^ll^:09^^12ft.,Jln.. 1/314.16 6. 120 rd. : ( ) : : l/9 : /Si ; 120Xl/6i ^ ^QQ j,^^ ^j^^ length, Am, 1/9 12 : ( ) : : i/9 : l/S^; ^^ ^ ^"^^^ ^ 10 rd., the breadth. Am. 9 7. 7.13 rd. : ( ) : : i/T : i/5; V^5 X 7.13 == 15.94268 + rd. Am. 11 162 EVOLUTION. ArrLICATIONS OF CUBE HOOT. Page 33S. 1. f^91125 cu. ft. = 45 ft., the length of edge, Arts. 2. The cubical contents of a box 2 ft. 8 in. long, 2 ft. 3 in. wide, and 1 ft. 4 in. deep are 13824 cubic inches ; 1?/'^13824 cu. in. = 24 in., the edge of cubical box, Ans. Page 334. 3. #^2197 cu. ft. = 13 ft., the depth of cistern, Am, 4. Since there are 2150.4 cu. in. in a bushel, 1000 bu. will contain 2150400 cu. in. ; 2150400 cu. in. -^ 1728 = 1244.444 + cu. ft. ; if 1244.444 cu. ft. = 10.75+ ft., the depth of bin, Am. 5. Since there are 31^ gallons in a barrel and 231 cu. in. in a gallon, 300 barrels will contain 300 X 31|- X 231 cu. in., or 2182950 cu. in.; 2182950 ~ 1728 = 1263.28125+ cu. ft.; #"1263.28125= 10.81 + ft., the depth of cistern, Am, 6. Since the bin is twice as long as it is wide or high, it is equal to two cubical bins each containing 1000 bu. ; 2150.4 cu. in. X 1000 = 2150400 cu. in. in 1000 bu. ; 2150400 cu. in. -^ 1728 = 1244.444 -[- cu. ft. ; 1^1244444 cu. ft. =10.75+ ft., the side of cubical bins and width and depth of bin ; 10.75 X 2 = 21.50 + , the length of bin, Am. APPLICATIONS OF CUBE ROOT. 163 7. A bushel contains 2150.4 cu. in. Therefore, f^21dOA = 12.89 + in., the depth of box. Am, 8. 231 cu. in. X 31^ = 7276.5 cu. in., the cubical con- tents of a barrel ; lf7276.5 cu. in. = 19.37+ in., the depth, Am, 9. By referring to Ex. 5, the dimensions of such a cistern are 10.82+ ft.; 10.82 X 10.82 = 117.0724 sq. ft., area of each side ; 117.0724 sq. ft. X 5 = 585.362 sq. ft., surface plastered ; » 585.362 -f- 9 = 65.0402 + sq. yd. ; $.30 X 65.04 ■= $19.51 +, Am, 10. 2150.4 cu. in. X 100 == 215040 cu. in., the cubical contents. Since the length is to be 3 times the width and height, the box will be equal to 3 equal cubes, each contain- ing I of 215040 cu. in., or 71680 cu. in. ; 71680 cu. in. -f- 1728 -= 41.4814+ cu. ft. ; 1^41.4814+ cu. ft. = 3.46+ ft., the width; 3.46 ft. X 3 = 10.38+ ft., the length, Am, 11. 1^13824 cu. ft. = 24 ft., the length of side of cube; 24 X 24 = 576 sq. ft. , the area of the side of the cube ; 576 sq. ft. X 6 = 3456 sq. ft., the entire surface of the cube. Since the height of the rectangular solid is one-half its length, and the wadth f its length, if the solid be divided by the plane ABC cutting off | of the width, and by BEF cutting into equal parts lengthwise, there will be / ___ / / /A / / E/ / / F C / / 164 EVC^LUTIONo formed two equal cubes and two other parts, which together will be equal to another cube of the same size. Therefore the solid may be regarded as com- posed of three cubes, each of whose edges is equal to the height of the rectangular solid. Therefore, ^ of 13824 cu. ft., or 4608 cu. ft., is the volume of each cube; #"4608 cu. ft. = 16.64+ ft., the edge of cube and height of rectangular solid ; 16.64 + ft. X 2 = 33.28 + ft., the length of the solid; f of 33.28 + ft. = 24.96 + ft., the width of the solid; 16.64 X 24.96 = 415.33 + sq. ft., the area of one end; 415.33 + sq. ft. X 2 = 830.66 + sq. ft., the area of both ends ; 33.28X16.64 = 553.779+ sq.ft., tlie area of one side; 553.779 X 2 = 1107.558 + sq. ft., the area of two sides; 33.28 X 24.96 = 830.668+ sq. ft., the area of top ; 830.668 sq. ft. X 2 = 1661.336+ sq. ft., the area of top and bottom ; 830.66+ sq. ft. + 1107.558+ sq. ft. + 1661.336+ sq. ft. = 3599.55+ sq. ft., the entire surface of rect- angular solid; 3599.55 + sq. ft. — 3456 sq. ft. = 143.55 + sq. ft., dif- ference. Arts, 12. 1^61026.048 cu. in. = 39.37 in.. Am. Page 335. 2. According to Prin. 2, expressed in a different way, the cube roots of the volumes of the solids are to each other as the dimensions of the solids. Therefore, SIMILAR FIGURES. 5 ft. : ( ) ; ==10 ft, Ans. 3. P : 3^ : : $125 : ( ) ; U25 X 27 = $3375, the value, Ans. 165 1^800 lb. : 1^6400 5 X #"6400 4. 1^411.42 : if 1000 8X1^1000 #"411.42 5. f/TOOO : 1^2000 8 ft. : ( ); 10.75 + ft., Ans. () : 3ft.; 3 X if 1000 1^2000 = 2.38 + ft. diameter, Ans. 6. #^1000 10 1 #^8000 20 2 1st bin 1st bin 1st bin 2d bin ; or 2d bin ; or, 2d bin ; that is, The second bin is to be tmice the first, Aris. 7. 1st sphere 1st sphere 1st sphere 2d sphere 2d sphere 2d sphere 43 64 1 12^; or, 1728; or, 27 ; that is. The second contains 27 times as much as the first, Ans. 8. Since each woman is to wind oflT ^ of the yarn, the ball that is left after the first has wound ofi* her share con- tains f of the amount in the orignal ball ; and since the corresponding dimensions of similar solids are to each other as the cube roots of their volumes (Prin. 2), ifl : if I" : : 4 : ( ). Therefore, 166 PROGRESSIONS. 4 X 1^1 == 3.494 in., the diameter of the ball when the second begins to wind off her share. Hence, 4 in. — 3.494 f = .506, diameter wound off by fii*st, Am. VT : if r : : 4 : ( ). Therefore, 4 X #T= 2.773+ in., the diameter of ball when the third begins to wind, Ans. 8.494 in. — 2.773 in. = .721 in., the diameter wound off by second, Ans. 9. 1^8 : 1^60 : : 12 ft. : ( ) ; 6 X 1^60 = 23.48+ ft., height. Am, mOGRESSIOJ^S. Page 3S8. 2. 10 + (5 X 9) = 55, the 10th term, Am. 3. 6 + (8 X 24) = 198, the 25th term. Am. 4. His daily wages would form an arithmetical series of which the first term is $.25, the common difference $.03, and the number of terms 50. Therefore, $ .25 + ($.03 X 49) = $1.72, the 50th or last day's wages, Am. 6. The first term is therefore IGy^g ^^-j *^^ common differ- ence 2 times 1^y2 ^'^ » ^^ ^^i ^^-j ^^^ *^^ number of terms 7. Therefore, 16^ ft. + (32^ ft. X 6) = 209-j2^ ft., the distance it falls the seventh second, Ans. ARITHMETICAL PROGRESSION. 167 6. 75 -1 (5 X 999) = 5070, the 1000th term, Ans, 8. 2 4 (3 X 49) = 149, the last term ; 1±1^ X 50 = 3775, the sum. Am, 2 9. jLj. + (_i^ X 99) ::= 10, the last term ; Tg"+-^^ X 100 =3 505, the sum, Am. 2 Page 339. 10. 15 + (3 X 10) = 45, the distance traveled the 11th day; ^^ "^ ^^ X 11 = 330, the whole distance, Am. Li 11. Since it strikes one first and twelve last, 1-1-12 t-L — X 12 = 78, the whole number of strokes, Am. 12. Since the annual increase was $6 per year, the first term $100, and the number of years 21, an arithmetical series is formed: $100 + ($6 X 20) = $220, the last term ; 100 + 220 ^ 21 = $3360, the amount due, Ans. 2. 10 X 3' = 2430, the 6th term, Am. Page 340. 3. 10 X 4' = 10240, the 6th term, Am. 168 MENSUEATION. 4. His wages would form a geometrical progession in which the first term is 5, the ratio 3, and the number of terms 10. Therefore, $.05 X 3^ = $984.15, his wages for the last day, Ans. 5. $100 Xl.065 = $133.82, the sixth term, Ans. 6. Since the amount for the first year is 1.05 of $520, the first term will be $546. Therefore, $546 X 1.05' = $696,849+, the amount, Ans. 8. (1024X4) -4 ^.^ ^^^ ^^^ 4 — 1 9. V-TTn A^3; ^=z4||^, the sum, Ans. -5^-, MENS Un A TION. Page 343. 1. 10 X 3.1416 =^ 31.416, the circumference, Ans. 2. 45 X 3.1416 = 141.3720, the circumference, Ans. 3. 300X3.1416 = 942.48 rd. in circumference; 942.48 -^ 320 = 2 mi. 302.48 rd., Ans. MEASUKEMENT OF SURFACES. 169 4. Since the radius is 20 rd. , the diameter is 40 rd. ; 40 X 3.1416 = 125.6640 rd., Am. Page 344. 5. 5 ft. 6 in. X 2 = 11 ft., the diameter; 11 X 3.1416 = 34.5576, the circumference, Am. 6. 318.5-^3.1416 = 101.38+ rd., Am. 7. 1284-^3.1416 = 408.708+, the diameter; 408.708 -^ 2 = 204.354+ rd., the radius, Am. . 13x40 = 520sq. ft.. Am. 2. 3 ft. 8 in. = 3f ft. ; 7 X 3| = 25| sq. ft., Am. 3. 24 X 30 = 720 sq. rd., the area. Am. 4. 35 X 15 = 525 sq. ft., the area, Am. Page 345. 1. ^4X18 :::^ 216 sq. ft., ^HS. 2. 2ixi2 ^ 126 sq. ft.. Am. 3. Since the shortest side is 120 ft., and the sides are in a ratio of 5, 6 and 8, 120 ft. must be f of the length of the next side ; 120 ft. are f of 144 ft, the second side ; for similar reasons 144 ft. must be f of length of next side ; 144 ft. are f of 192 ft. , the length of third side ; 1 70 MENSUllATION. JL20+1 4 4±i.9 2 ^ 228 ; 228-1 20 .-. 108 ; 228 — 144^84; 228 — 192 = 36; l/228 X 108 X 84 X 36 -= 8629.19 sq. ft., the area; 8629.19 sq. ft. -^ 43560 = .19809 acres; $850 X .19809 =: $168,376+, the cost, Aiis. 4. 1^.0 + y 0+8 ^ 205 ; 205 — 180 -- 25 ; 205 — 150^=55; 205 — 80 = 125; l/205 X 25 X 55 X 125 = 5935.85 sq. ft., area, Am, 5. The perpendicular line A B divides the base into two equal parts. Therefore, one-half the ^ base is 16 ft. Therefore, ^^:v^^]\^ l/20' — 16^ = 12, the altitude A B ; XiL/\ ^ ^ 2 ^ ^ = 192 sq. ft., area of one end; ^ 192 sq. ft. X 2 = 384 sq. ft, area of both ends, and amount of lumber needed ; 384 sq. ft., @ $22| per M, cost $8.64, Am. 6. ^OQxioo ^ 15000 sq. ft., the area. Am. Page 346. 1. 110 X ^^^ = 5500 sq. ft., the area. Am. 2. (10 + 8) X f ^ 54 sq. rd., the area. Am. 3. 96 X -^ = 576 sq. ft., the area, Am. 4. (40 + 30)X^2-='^00sq.rd.; 700sq.rd.+-160=4f A.; $125 X4f =$546,875, the cost, Am. 5. 120X^^$^==3000sq.rd.; 3000sq.rd.-M60 = 18f A.; $110 X 18| = $2062.50, the cost, Am. MEASUREMENT OF SURFACES. 171 Page 347. 1. 52 X .7854 =- 19.635 sq. ft., the area, Ans. 2. S' X .7854 = 50.265+ sq. ft., the area, Ans. 3. 120 -^ 3.1416 = 38.197+ rd., the diameter; i of ( 120 X ^^ ) = 1145.7 sq. rd., the area, Ans. 4. 100-^-3.1416 = 31.8309+ ft., the diameter; I of ( 100 X ^J-|^-9 ) = 795.77 sq. ft., the area, Ans. 0. 320 — 3.1416 = 101.8589+ rd., the diameter, I of ( 320 X i^^8_58_9 ) ^ 8148.712 + sq. rd., the area; 8148.712 sq. rd. -^ 160 = 50.929 + A., Ans. 6. 15 rd. X 2 = 30 rd., the diameter; 30^ X .7854 = 706.86 sq. rd., the area, Ans. 7. Since the area is the square of the diameter multiplied by .7854, if the area is divided by .7854 the quotient will be the square of the diameter. The square root of this quotient will be the diameter, and half of the diameter will be the radius or the length of the rope. 160 sq. rd.-^. 7854 = 203.7178+ sq. rd., the square of the diameter; 1/203.7178 sq. rd. = 14.27-^- rd., the diameter; 14.27+ rd.-^2=r7.13+ rd., the radius, the length of the rope, Ans. 8. 113.0976 sq. rd. -^ .7854 = 144 sq. rd., the square of the diameter; l/l44 sq. rd. = 12 rd„, the diameter, Ans. 172 MENSURATION. 9. 350^ X .7854 = 96211.5 sq. ft. ; 96211.5 sq. ft. -^ 272^ = 353.3939+ sq. rd. ; 353.3939+ sq. rd. -i- 160 = 2 A. 33.3939+ sq. rd., tlie area, Am. 10. 352 X .7854 = 962.115 sq. ft., the area. Am. Page 350. L 2 ft. X 3.1416 == 6.2832 ft., the perimeter; 6.2832 X 5 = 31.416 sq. ft., the convex surface, Am, 2. 2-|- ft. X 4 = 10 ft., the perimeter of prism ; 10 X 4 = 40 sq. ft., the convex surface. Am. 3. 6 ft. X 3 = 18 ft. , the perimeter of prism ; 18 X 8 = 144 sq. ft., the convex surface, A7is. 4. 2 ft. X 3.1416 = 6.2832 ft., the perimeter; 6.2832 X 5 = 31.416 sq. ft., convex surface ; ( 2^ X .7854 ) X 2 = 6.2832 sq. ft., area of ends ; 37.6992, the entire surface. Am. 6. 10 in. + 14 in. + 18 in. = 42 in., or 31- ft., perimeter; 31. X 18 = 63 sq. ft., the convex surface. Am. Page 351. 1. 15 ft. X 4 = 60 ft., the perimeter of base ; 60 X ^ = 540 sq. ft., the convex surface, Am. 2. 12 ft. X 3.1416 = 37.6992 ft., the perimeter of base; 37.6992 X ^^ = 376.992, sq. ft., convex surface. Am. MEASUREMENT OF SOLIDS. 173 3. 20 ft. X 3.1416 = 62.832 ft., the perimeter of base ; 62.832 X -T = 628.32 sq. ft., the convex surface, Ans. 4. 6 ft. X 8 = 48 ft. , the perimeter of base ; 48 X ^ = 1920 sq. ft., the convex surface ; 1920 -f- 9 = 213i sq. yd. ; $ .30 X 213^ =- ?64, Am. 5. 6 ft. X 3.1416 = 18.8496 ft., the perimeter of base; 18.8496 X ^ = 89.5356 sq. ft., convex surface. Am. 6. 10 ft. X 4 =: 40 ft., the perimeter of base 40 X -^ = 400 sq. ft. , the convex surface, Ans. 7. 8 ft. X 3.1416 = 25.1328 ft, the perimeter ; 25.1328 X f = 75.3984 sq. ft., convex surface, A71S. 8. 10 ft. X 3.1416 = 31.416 ft., the perimeter of base ; 31.416 X ^ = 157.08 sq. ft., the convex surface. Am. Page 35S. 1. 12 ft. X 3.1416 = 37.6992 ft., perimeter of lower base ; 8 ft. X 3.1416 = 25.1328 ft., perimeter of upper base; I X (37.6992 + 25.1328) X 8 = 251.328 sq. ft., con- vex surface. Arts. 2. 40 ft. X 4 =: 160 ft., perimeter of lower base; 20 ft. X 4 = 80 ft. , perimeter of upper base ; ^ X (160 + 80) X 25 = 3000 sq. ft, convex surface. 3. 10 ft. X 3.1416 = 31.416 ft, perimeter of lower base ; 8 ft X 3.1416 rrrz 25.1328 ft, perimeter of upper base; i X (31.416 + 25.1328) X 12 =- 339.2928 sq. ft, con- vex surface; 174 MENSURATION. 339.2928 sq. ft. -4- 9 = 37.6992 sq. yd. ; $.15 X 37.6992 = §5.654+, the cost, Ans. 4. 9 ft. X 4 = 36 ft. , perimeter of lower base ; 8 ft. X 4 = 32 ft., perimeter of upper base; i X (36 + 32) X 10 -= 340 sq. ft., convex surface. 5. The area of sides is 340 sq. ft. Since the plank for the sides is 1^ in. thick, the lumber required will be 1^ times 340, which is 510 ft., board measure. 9 X 9 = 81 sq. ft., the area of the base. Since the plank is 2 in. thick, the lumber required for the bottom will be 2 times 81, which is 162 ft., board measure. 510 + 162=^672 ft. lumber; 672 ft. @ $40 per M = $26.88, Ans. 1. 152 X 3.1416 = 706.86 sq. in., the surface; 706.86 sq. in. -f- 144 =^ 4.908+ sq. ft., Ans. 2. 8^ X 3.1416 -= 201.0624 sq. in., the surface; 201.0624 sq. in. -^ 144 = 1.396+ sq. ft., Ans. a 9J in. -^ 3.1416 = 2.9045+ in., the diameter; dl X 2.9045+ = 26.50+ sq. in., the surflice, Ans. Page 353. 4. 12-^-3.1416 = 3.8197+ ft, the diameter; 12 X 3.8197 = 45.836+ sq. ft., the surface, Ans. 1. 1 X 1 == 1 sq. ft. , the area of base ; 1 X 2 = 2 cu. ft. , the solid contents, Ans. MEASUREMENT OF SOLIDS. 175 2. myx .7854 = 1.76715 sq. ft., the area of base ; 1.76715 X 4 = 7.0686 cu. ft., the solid contents, Am. 3. 20 X li X 1 = 30 cu. ft., the solid contents of stick; $.30 X 30 = $9.00, the cost, Ans. 4. 8 X 8 = 64 sq. ft., the area of base ; 64 X 9 = 576 cu. ft., the volume of the bin ; 1728 cu. in. X 576 = 995328 cu. in. ; 995328 cu. in. 4- 2150.4 = 462.85+ bu.. Am. 5. 8^ X .7854 = 50.2656 sq. ft., the area of base ; 50.2656 X 7 ==: 351.8592 cu. ft,, the volume of the vat; 1728 cu. in. X 351.8592 = 608012.6976 cu. in. ; 608012.6976 cu. in. -^ 231 = 2632.089+ gal.. Am. 6. 15 X 15 = 225 sq. ft., the area of base ; 225 X 12 = 2700 cu. ft., the volume of the bin ; 1728 cu. in. X 2700 = 4665600 cu. in.; 4665600 --■ 2150.4 ==2169.642+, number of bushels; $1.85 X 2169.642 = $4013.837+, value of wheat, Am. Page 354. 1. 6^ X .7854 = 28.2744 sq. ft., the area of base ; 28.2744 X I = 84.8232 cu. ft, the solid contents, Am. 2. 30 X 30 =. 900 sq. ft., the area of base ; 900 X -^ =^ 18000 cu. ft., the solid contents, Am. 3. 6' X .7854 =:= 28.2744 sq. ft., the area of base ; 28.2744 X I = 75.3984 cu. ft., the solid contents; 165 lb. X 75.3984 -- 12440.736 lb., Am. 176 MENSURATION. 4. 4 X 4 =:== 16 sq. ft., the area of 16 X f = 42| cu. ft. , the solid contents ; 171 lb. X 42| = 7296 lb., Am. 20 X 20 = 400 sq. ft., the area of lower base ; 10 X 10 = 100 sq. ft., the area of upper base ; l/400 X 100 = 200 sq. ft., the mean proportional; (400 + 100 + 200) X -^/ = 4666| cu. ft., the volume. Page 355. 2. 5' X .7854 = 19.635 sq. ft., the area of upper base; 8 2 X .7854 = 50.2656 sq. ft., the area of lower base ; 1/19.635 X 50.2656 = 31.416, the mean proportional; ( 19.635 + 50.2656 + 31.416 ) X -J = 236.405+ cu. ft., the volume, Ans. 3. 3' X .7854 = 7.0686 sq. ft., the area of lower base; 1^ X .7854 = .7854 sq. ft, the area of upper base ; 1/7.0686 X .7854 = 2.3562+ sq. ft., the mean propor- tional ; (7.0686 + .7854 + 2.3562) X -^^ = 136.136+ cu. ft., the volume, Atis. 4. 12' X .7854 = 113.0976 sq. ft., the area of lower base; 10' X .7854 = 78.54 sq. ft., the area of upper base; 1/113.0976 X 78.54 == 94.248+ sq. ft., the mean pro- portional ; (11.3.0976 + 78.54 + 94.248) X 1 = 857.6568 cu. ft., the volume ; MISCELLANEOUS EXAMPLES. 177 1728 cu. ill. X 857.6568 -^ 231 :==6415.7l8+gal., Ans, 1. 5^ X .5236 = 65.45 cu. ft., the volume, Ans, 2. 8^ X .5236 =: 268.0832 cu. ft., the volume, Aris. 3. 9.4248 -^ 3.1416 ^ 3, the diameter ; 3=^ X .5236 = 14.1372 cu. ft., the volume. Am, 4. 18 in. = li ft. ; (l^Y X .5236 = 1.76715 cu. ft. ; 450 lb. X 1.76715 = 795.217+ lb., weight of ball, Ans. 5. (21)^ X .5236 = 8.181 cu. ft, the volume, Ans. 6. (25)' X .5236 = 8181.25 cu. ft., the volume, Ans. MISCELLANEOUS EXAMPLES. Pa^e 356. 1. It will take 6 men f of 12 days, or 10 days, Ans. 2. They will cost | of $7.50, or $12, Ans. 3. Since 20 men are required to load the vessel in 6 days, it would require 120 men to load it in 1 day. There- fore as many men would be needed to load it in 1-|- days as 1^ is contained times in 120, which is 80 men, Ans. 12 178 MISCELLANEOUS EXAMPLES. 4. Since she sailed 42| miles in 2^ hours, 42i-f-2|^=: 17 miles, the rate per hour. Therefore, in 20 minutes, or ^ of an hour, she sailed -^^ of 17 miles, which is 5f miles, Ans. 5. 28 rd. Ida. : 56 rd. 1 ^ . . : Ida. } ^^^"^" ' ^^' 56 X 1 X 6 men „ . = 16 men, Ans. 28 XI 6. f yd. : I yd. : : $3| : ( ) ; «3|X| 3 = $5i. Am, 7. The board of 8 persons for 2| weeks is the same as the board of 1 person for 8 times 2i- weeks, or 20 weeks; and the board of 10 persons for 3 weeks is the same as the board of 1 person for 30 weeks. Therefore, since the board of 1 person for 20 weeks is $50, for 1 week it will be 2V ^f $^^> or $2| ; and the board of 1 person for 30 weeks will be 30 times $2i, or $75, Am. 8. Since 3 lb. tea are worth 14 lb. coffee, 1 lb. tea is worth 4f lb. coffee, and 7 lb. tea are worth 32| lb. coffee. Since 5 lb. coffee are worth 18 lb. sugar, 1 lb. coffee is worth 3| lb. sugar, and 32f lb. coffee are worth 117| lb. sugar. Since 21 lb. sugar are worth 60 lb. flour, 1 lb. sugar is worth 2^ lb. flour, and 117| lb. sugar are worth 336 lb. flour. Therefore, 7 lb. tea are worth 336 lb. flour, A)is. MISCELLANEOUS EXAMPLES. 179 9. $ .23 X 56 X 12 := $154.56, value of butter; $ .85 X 5 = $4.25, cost of tea; $.13 X60 = 7.80, cost of sugar; $1.12^X15 =■ 16.875, cost of cloth; B.925, cost of purchase. Therefore, $154.56 — $28,925 r^ $125,635, money received, Ans, 10. Since ^ of the provision was useless, they received f of the daily allowance, which was f of 15 ounces, or 13^ ounces, Aiis, 11. Since it required 5 more than 3 times the number of sheep to make 185, if 5 be subtracted from 185 the remainder will be 3 times the number of sheep. There- fore, 3 times the number of sheep = 180, and the num- ber of sheep = 60, Ans, Page 357. 12. Since ^ of last remainder = $5, last remainder .-^ $10. Since ^ of previous rem'r = $10, previous rem'r = $20. Since J of previous rem'r = $20, previous remV =- $40. Since |- of his money = $40, his money = $80. 13. A can do -^^ of the work in 1 day, B can do -^^ of the work in 1 day, C can do 2V ^^ ^^^ work in 1 day ; All working together can do -^^ --)- y^g -f 2V' ^^ rVo ^^ the work in 1 day. It will therefore require as many days for all to do the work as f\\j- is contained times in i|§, or 5|f days. 180 MISCELLANEOUS EXAMPLES. 14. 4ff bu. X 100 = 466| bu., the wheat needed for 100 barrels of flour. Smce the miller took -|- for grinding, the amount used for flour was only ^ of the quantity taken to mill. Therefore, 466| bu. -^ f = 533|- bu., the quantity taken to mill. $.021 X 533^- = $12, expense of taking it to mill; $.45 XlOO =r $45, cost of barrels; $.25 XlOO =$25, commission; $82, expenses. $550 + $165 + $100 = $815, the receipts ; $815 — $82 = $733, the net receipts ; $1.45 X 5331 = $773.33, amount offered ; $773.33 — $733 -= $ 40.33, loss, Ans. 15. Since 3 times the number of trees -f 5 = 1358 trees, 3 times the number of trees = 1353 trees, and the number of trees = 451 trees, Ans, 16. Since ^ of A's money = f of B's, The whole of A's money = f of B's. Therefore A's money exceeded B's hj ^, Therefore i of B ' s money = $8 ; The whole of B's money = $40, Ans, A's money =:= f of B's = $48, Am, 17. Since 2 cattle eat as much as 7 sheep, B's 24 cattle eat as much as 84 sheep, and C's 10 cattle and 35 sheep eat as much as 70 sheep. Therefore, MISCELLANEOUS EXAMPLES. 181 70 sheep for 6^ mo. = 455 sheep for 1 mo., A's; 84 sheep for 4^ mo. = 350 sheep for 1 mo. , B's ; 70 sheep for 5|- mo. = 385 sheep for 1 mo., C's; Whole No. for the time =.1190 for 1 mo. Therefore, ^\ of $170, or $65== A's share; _3_^o^ of $170, or $50:=:=B's share; -5^ of $170, or $55 = C's share, Ans, 18. Since a pole 10 feet long casts a shadow 13 feet long, a pole -^ of a foot in length would cast a shadow 1 foot long ; and therefore to cast a shadow 621 feet long, the pole would have to be 62|- times j-| ft. , or 4^8-^-^ ft. , Ans, 19. Since C's weight is equal to the sum of A's and B's, It must be ^ of the sum of their weights. ^ of 490 lb. = 245 lb., C's weight. Since A's weight is only f of B's, the Aveight of both would be If times B's weight. Therefore, If, or I, of B's weight = 245 lb. ; i of B's weight = 35 lb. ; B's weight = 140 lb. ; A's weight = 1 of B's = I of 140 = 105 lb., Ans. 20. Since f of A's money == | of B's, \ of A's money = ^ of f , or -g^, of B's, and The whole of A's money = 8 times /y, or f |^, of B's. Therefore, since A's money is |-f- of B's, his money exceeds B's by -^ ; and ■^ of B's money = $5 ,• 1 27 of B's money The whole of B's money = $27 ; A's money = $5 + $27 = $32, Ans. 182 MISCELLANEOUS EXAMPLES. 21. Since their ages are in the ratio of 8, 4 and 5, 3 times a certain number --:= A's age ; 4 times a certain number = B's age; 5 times a certain number = C's age. Therefore, 12 times a certain number = the sum of their ages, 136 yr., and The number = -^l of 136 yr., or 11^ yr. Hence, 3 times 11^ yr., or 34 yr. = A's age ; 4 times 11^ yr., or 45^ yr. = B's age; 5 times 11^ yr., or 56|yr. = C's age. Am. Page 358. 22. He paid 1^ cent'' each, and sold them for 1-|- cents each. He gained -^^ of a cent on each apple, and must sell 12 apples to gain 1 cent. Therefore, to gain 60 cents, he must have sold 60 times 12, or 720 apples. 23. A and B can do 3^2 of the work in 1 day; B and C can do 2V of the work in 1 day; A and C can do -^ of the work in 1 day ; ■^2" + A" + ^ ^^ 2 V/2 > twice the part that all can do in 1 day; Ifff "^ 5W4 = ISfl-f- days, time in which all can do it ; ■iN-? — Ts = tUt > Pa^t A can do in 1 day ; ■fill -r- -^8 2 -4- = ^^w days, time in which A can do it ; ■^Wt -- 2V = tUt^ P^^t S can do in 1 day ; ffrf ~^" "S^ff ¥ "^ "^^ii days, time in which B can do it ; tW4 — T2 = tVA' part C can do in 1 day ; ffff -;- -^^^ = 46j^ days, time in which C can do it. MISCELLANEOUS EXAMPLES. 183 24. A can do y^ of the work in 1 hour ; B can do -^-^ of the work in 1 hour ; TTO+ oV =¥6"o» P^^^ ^^^^ ^^^ ^0 i" ^ hour; fff -^-3|-0-==51|-, number of hours it takes both to do it; 51f-f- 8 =6f, number of days of 8 hours each, J.?is. 25. Since one is worth | as much as the otlier, the value of both would be If times the value of the better horse. Therefore, 1-|, or ^^, of value better horse =^ $390 ^ of value of better horse = $30 The value of better horse =$240, Am. The value of poorer horse = f of $240 ^$150, Am. 26. 41 X 3.1416 X 720 =~- 10178.784 ft., distance traveled; 4 X 3.1416 = 12.5664 ft., circumference of fore-wheel; 10178.78 -^ 12.5664 = 810, revolutions, Am. 27. Since a shadow 9 inches long was produced by a 6-foot pole, a shadow 1 inch long would be produced by a pole f , or |, of a foot in length. Therefore a shadow 9 feet, or 108 inches, must have been produced by a pole 108 times f foot, which is 72 feet, Ans. 28. $245.30 X .0155 = $3.80, Am. 29. He steps 60 inches in 3 seconds, or 20 inches per sec- ond. It will require as long for him to walk 10 miles as 20 inches are contained in 633600, the number of inches in 10 miles, which is 31680 seconds, which are equal to 8 hours 48 minutes, Am. 30. 4 grown persons and 3 children = 6 grown persons ; 3 grown persons and 8 children =^ 8^ grown persons. 184 MISCELLANEOUS EXAMPLES. Therefore, 6 : 8^ : : $150 : ( ); 31. (1) 20 bu. wheat + 15 bu. corn cost $36 ; (2) 15 bu. wheat + 25 bu. corn cost $32.50. Therefore, taking 3 times the quantities in (1) and 4 times the quantities in (2), we have (3) and (4). (3) 60 bu. wheat -f- 45 bu. corn would cost $108, and (4) 60 bu. wheat + 100 bu. corn would cost $130. Therefore, since the quantity of wheat is the same in (3) and (4), the difference in value must be caused by the difference in quantity of corn. Therefore, 55 bu. corn cost $22; 1 bu. corn cost ff, or | of a dollar, or $ .40, Ans, Since 20 bu. wheat + 15 bu. corn cost $36, and 15 bu. corn cost $6, 20 bu. wheat must cost $30; 1 bu. wheat must cost $1.50, Ans. 32. He gains 4 rods of every 30 he runs ; therefore, in order to gain 120 rods, he must run 30 rods 30 times, or 900 rods, Ans, Page 359. 33. A is 18 miles in advance of B. Since B gains 1 mile of every 4 miles he travels, to gain 18 miles he must travel 4 miles 18 times, or 72 miles, Ans, 34. Therefore the difference between ^ and ^ of the num- ber was equal to 10. \ — ^= 2V- Therefore -^ of the number was 10 ; the number was 20 times 10, or 200. MISCELLANEOUS EXAMPLES. 185 35. A number -f- -^ of it and ^ of it make If, or ^-, times the number. Since y- of the number = 105, -|^ of the number = 9^^ ; the number = 6 times 9y\, or 57^. 36. A number + -| of it = f of the number. Therefore, | of the number -f- 15 = 40. Since 15 has to be added to f of the number to make 40, f of the number must be 15 less than 40, or 25. Therefore, -f of the number = 25 ; "l^ of the number = 5 ; the number = 15, Ans, 37. Since 20 men need 45 days to do the work, 1 man would require 20 times 45 days, or 900 days; and 30 men would require -3^0- of 900 days, or 30 days, Ans. •38. f da. : fda. :: $| : (); $| X | -^ f = $ff , ^ns. 39. 4^ X If = -V^ X -V- = W s^- y^-» *^^ square contents of broadcloth ; ■^^ ~^^== ld^2 yd., the amount of silk, Am, 40 24- • 6^ 1 \^ : j|| :: 14 : ( ); YX-'#X|X| = 60A oz, 41. -I- mi. = 7920 in. ; 7920 -^ 14 = 565f ; that is, 566 tiles. I : : «300 : ( ) ; 1300X115X9^^^3 ^^^ 18 X 6 43. 1^ times the number lacks 20 of being double the num- ber; then, I of the number = 20; -J- of the number := 6f ; the number = 4 times 6f , or 26f , Ans. 42. fl8 : $115 6 mo. : 9 mo, 186 MISCELLANEOUS EXAMPLES. 44. At 5 o'clock the minute hand is 25 minute spaces be- hind the hour hand, and, therefore, if the hour hand were stationary the hands would be together at 25 minutes past 5. But the hour hand moves 1 space while the minute hand gains 11; therefore the hour hand has moved -^j of 25 minutes, or 2y\- minute spaces further, and therefore the whole number of minute spaces traversed by the minute hand before they are together must be 25 + ^^tj ^^ ^Tj^- spaces. That is, the hands will be together at 27^^^ minutes past 5 o'clock, Ans. 45. Since the time required to travel the whole distance was 18 days, the second soldier had traveled ^ the whole distance before the first turned back. And since the first turned back as far as the second had advanced at that time, he turned back ^ the dis- tance. This he walks back again, and, besides, com- pletes the whole distance, thereby going over the ground twice in 18 days, traveling 12 miles per day. The second soldier travels the whole distance once in 18 days, and, therefore, his rate per day is ^ of 12 miles, or 6 miles, Ajis, 46. Since 30 men did half of the work in 40 days, it would have required them 80 days to do the work, or it would have taken 1 man 30 times 80, or 2400 days. To do half of the work would require 1 man 1200 days, and since the work had to be finished in 20 days, it would require as many men as 20 is con- tained times in 1200, or 60 men. Inasmuch as he had 30 laborers, he was obliged to hire 60 — 30, or 30 laborers, Am, MISCELLANEOUS EXAMPLES. 187 Page 360. 47. f of time past noon = f of time to midnight ; ^ of time past noon = -§- of time to midnight ; The time past noon = | of time to midnight. Therefore, the whole time from noon to midnight was 1^ times the time to midnight. Therefore, 1|-, or f , of the time to midnight = 12 hr. ; ^ of the time to midnight = 1 J hr. ; The time to midnight - 6f hr. Therefore, The time past noon =12 — 6| =:: 5^ hr. That is, the time was 20 minutes past 5 o'clock, Arts, 48. $1 put at compound interest for 21 years at 6^, amounts to $3.399564. Therefore, it will take as many dollars to amount to $3000 in that time as $3.399564 are contained times in $3000. $3000 4-3.399564=. $882.46, Am, 49. August 11th was 21 days, or -^^ of a month, before the note was due. The use of any sum of money for that time at 6^ is equal to ^ifinj ^^ i^- The use of any sum for 60 days at 6% is equal to j^-^j of it. There- fore, since he was to pay such a sum that the use of it for 21 days was to equal the use of the sum unpaid for 60 days, yto" ^^ ^^^ ^^^ unpaid = y^Vo ^^ ^^^ ^"^ paid. The whole of the sum unpaid = -/oVo ^^ ^^^ sum paid. Therefore, |^^^ + ^\\\, or |^^, or |^ of the sum paid = $100; -^-^ of the sum paid = $-^7^* and the whole sum paid ==: 20 times $-i^<^/ = $74.07, the sum paid, Ans. 188 MISCELLANEOUS EXAMPLES. 50. His first annual saving draws interest from the time he is 22 years of age until he is 40, a period of 18 years ; his second saving draws interest for 17 years ; his third for 16 years, etc. Therefore his annual savings form a geometrical series, decreasing uniformly until the last saving, which does not draw interest, is reached. Since the last saving may be regarded as the first term, to find the value of an annual saving of $1 it is necessary to find the sum of a geometrical series in which the first term is 1, the ratio 1.06, and the number of terms 19. $1.06'^ — $1 ihe sum = $1.06 — $1 $1.06'^= the amount of $1 at compound interest for 19 years. By referring to the compound interest table (Prac. Arith., p. 227), it is found to be $3.0256. Then the sum = !?i^?5izi!l, or $33.76. $1.06 - $1 Since an annual saving of $1 would amount to $33.76 in the given time, it will require an annual saving of as many times $1 to secure $25000 as $33.76 are contained times in $25000. $25000 -^ $33.76 = $740.52. Therefore the annual saving is $740.52, Ans. 51. Since he sold f of the article for |- of the cost of it, he gained -|- of the cost. Therefore, he gained -I- on f, or ^, or 16f ^ of cost, Ans. 52. Since he sold ^ the quantity for f of the cost, he gained ^ — -I" or -j^ of the cost. Therefore, he gains -^^ on an investment of ^ the cost, or f, or 42f ^ , Ans» MISCELLANEOUS EXAMPLES. 189 53. Since f of cost of horse = |- of cost of carriage, \ of cost of horse = f of cost of carriage, and The cost of horse r=i f of cost of carriage. Since the cost of horse ^= f of cost of carriage, the gain by the sale of horse was 25% or ^ of f of cost of carriage, or f of cost of carriage. Therefore the selhng price of horse was ^^- of cost of carriage. Since the gain on the carriage was 10%, or y^^ of its cost, the selling price of carriage was \^ of its cost. Therefore the selling price of both was equal to ^ of cost of carriage + \^ of cost of carriage, or ^-^-^- of cost of carriage. Hence, ^^ of cost of carriage = $597, -^^ of cost of carriage = $3, and The cost of carriage = 90 times $3, or $270, Arts. The cost of horse = f of $270, or $240, Am, 54. If 300 cats kill 300 rats in 300 minutes, they kill a rat per minute, and therefore will kill 100 rats in 100 minutes. Therefore it will require 300 cats. Arts, 55. When 8 paid for the coach, each paid \ of the cost. If 12 had paid for it, each would have paid -^^ of the cost. Therefore, ^ of the cost — ^ of the cost = $1, or ^V of the cost of coach = $1 ; the cost of coach = $24, Ans, 56. If I had paid $250 for the goods more than I did, they would have cost me 100% of original cost -j- $250. Since, by buying at this price, I would have lost 20% by the sale, the selling price was 80% of what they 190 MISCELLANEOUS EXAMPLES. would have cost, or 80^ of (100% of original cost + $250), which was 80^ of original cost + $200. From the conditions first given, the selling price was 120% of original cost. Therefore, 80^ of original cost + $200 = 120^ of original cost ; 40 fo of original cost = $200 ; 1^ of original cost = $5; Original cost = $500, Ans. Page 361. 67. $ 1.25 X 25 = $31.25, am't he might have earned ; $31.25 — $19 = $12.25, am't lost in wages and board ; $ 1.25 + $.50 = $1.75, thedailyloss; $12.25 -^ 1.75 = 7, the number of days idle, Am. 58. A's age = 20 years ; B's age = 20 years -\- \ C's age ; C's age = 40 years -f \ C's age. Therefore, \ C's age = 40 years ; C's age = 80 years, Aiis. B's age = 20 years + 40 years = 60 years, Ans. 59. Since A received f of the profits, his capital must have been f of the entire capital. Therefore, f of capital = $4500 ; I of capital = $1500 ; entire capital = $7500 ; B's capital = f of $7500 = $3000, Am. 60. $2 -f- $6 = $8, half of his money before he bought the clothing. Therefore $16 was the amount of his money before he bought the clothing. $16 + 2 = $18, half of w^hat money he had after paying his traveling expenses. Therefore $36 was what he had after paying his travel- ing expenses. $36 + $4 = S40, half of his money. Therefore his money was 2 times $40, or $80, Ans. MISCELLANEOUS EXAMPLES. 191 61. Since 6 apples and 3 pears cost 21 cents, ( 1 ) 12 apples and 6 pears cost 42 cents; and since (2 ) 5 apples and 6 pears cost 28 cents, the difference between the cost of ( 1 ) and ( 2 ) must be the cost of the difference in the number of apples, since the number of pears is the same. Therefore 7 apples cost 14 cents, and 1 apple cost 2 cents, Ans. Since 6 apples cost 12 cents, and 6 apples and 3 pears cost 21 cents, 3 pears must cost the difference be- tween 12 cents and 21 cents, or 9 cents. There- fore, 1 pear cost 3 cents, Ans, 62. A and B can do ^ in 1 day. Since A does f as much as B, both do ^ as much as B ; and A does ^ of the work, and B ^ of the work. Therefore A does f of 2V> ^^ TTo> ^^ t^^ work in 1 day, and he would need as many days to do the whole as yf^ is con- tained times in H^, which is 46| days, Ans. B does ^- of -^Q, or yto, of the work in 1 day, and would need as many days to do the whole as j^ is contained times in ||^, which is 35 days, Ans, 63. By the conditions of the problem the payments are to include the interest accrued at the end of each year, together with a certain portion of the principal The principal for the first year will be S5000. The payment at the end of the first year will be equal to the interest, $300, and a portion of the principal. The principal for the second year will be less than the principal for the first year by the value of the first portion paid, and therefore the amount of interest 192 MISCELLANEOUS EXAMPLES. to be paid at the second payment will be less than the interest paid at the first payment by the interest for 1 year upon the first portioii of the principal paid, or 6^ of the portion of the principal previously paid. Hence, since the payments are to be equal, the j^ortion of the principal to be paid at the second payment must be as much more than the previous portion as the interest is less than the previous interest; that is, it must be 6^ more than the previous portion, or 1.06 of the previous portion. Reasoning in the same way regarding the subsequent payments, the third portion of the principal paid will be 1.06 of the second, the fourth 1.06 of the third, the fifth 1.06 of the fourth. Thus it is seen that the portions of the principal paid form a geometrical series in which the ratio is 1.06, the number of payments or terms 5, and the sum of payments $5000. We wish to find the first term, which will be the portion of the principal paid the first time. This sum plus the first interest, $300, will be the entire first payment; and since the payments are equal it will be the payment made each time. According to rule first, in Geometrical Progression, the last or fifth term = first term X 1.06*. Using this value for the last term in rule second, w^e have first term X 1.06^ — first term ^.r-AAn • r , 1^ = $5000, or simpli- 1.06 — 1 . . 1 . (1.06^—1) ,, . , , fymg the expression, ^ X first term = MISCELLANEOUS EXAMPLES. 198 1.06^ — 1 $5000. Dividiiio* by , the first term - - ^ ^ .06 $5000 X .06 , . , . , ^ .^_ .^ , which IS equal to $886.98; 1.06^—1 ^ 98, first portion of the principal paid, plus $300, the interest, = $1186.98, entire payment, Ans. Note. — 1.06^ is equal to the amount of $1 at compound inter- est for 5 years at 64, and may be found by reference to the compound interest table, Prac. Arith., p. 227. A general rule for the solution of this class of examples may be formed from the following : Let P represent the principal; Let p represent the payment ; Let r represent the rate per cent. ; FX 0- + f) = amount of principal for 1 year; P X (1 + ^) — i> = amount due at end of first year, or new principal for second year. Multiplying this new principal by (1 -f r), P (1 -f r)^ — p (1 -{- r) = amount of principal at end of second year; P (1 -|- r)^ — p (1 -\- r) — p = amount due at end of second year, or new principal for third year. Multi- plying this new principal by (1 -j- r), P(l -\- rf — _p (1 4- ry — p (1 -^r) =^ amount of prin- cipal at end of third year ; P(l + r/ — p (1 -j-ry — i> (1 + ^) — p = amount due at end of third year, or new principal for fourth year. By continuing the reasoning it may be shown that the amount due at end of fifth year may be expressed : 194 MISCKLLANEOXJS EXAMPLES. P (1 + ry -pa + ry - p (1 + r)'_- i> (l-f rf - j9 (1 -|- r) — p; and since the entire indebtedness was paid at that time, P (^1 + ry - p (1 + ry — p (^1 + ry —p (1 + ry ~ p (1 -\- r) — p=^0; therefore the sum of the sub- trahends must equal the minuend, and p(l+ry+p{l+ry + p(l+ ry + p (1 + r)+p = P(l+7')% and pi^a + ry+(l+ry+{l + ry+(l + r) + l^ = P(l+ry. Therefore, P(l+ry ^ - (1 +ry + a +ry + (1 +ry (1 +r) + 1 Substituting the numbers for the letters, the payment = 5000 X 1.06^ ^-.-.or. no , A = $1186.98+, Ans, 1.064 _|_ 1.063 4- 1.062 + 1.06 + 1 64, $300 + 1.25 = $240, the cost of one carriage ; $300 +- .75 = S400, the cost of other carriage ; $640, the cost of both carriages; $300 X 2 = $600, the selling price of both carriages ; $40, the loss by the sale. Am. The loss was -g*^, or -^, or 6^% of cost, Ans. 65. ■\/W+~S' = 40.79+ ft., AB, the length of the ladder. Therefore, in the right-angled triangle D C E, CE is 30 ft., and DE is 40.79+ ft. Therefore, 1/40.79^ — 30^ = 27.64+ ft., DC. MISCELLANEOUS EXAMPLES. 195 66. Since Mr. A. is 35, and his son 10, Mr. A. was 25 years of age when his son was born, and will be 2 times 25, or 50 years of age when his son's age is half of his. And since he is 35 years of age now, the son will be half the father's age in 50 years — 35 years, or 15 years, Ans, Page 362. 67. Since he lacked 30 cents of having money enough in the first instance, and had 15 cents left in the second instance, the difference in the cost was 45 cents. The difference per pound was ^ cent. Therefore 45 cents must have been the difference on 2 times 45, or 90 pounds, Ans. 68. Since | of No. in 1st field == | of No. in 2d field, ^ of No. in 1st field = I of No. in 2d field, and The entire No. in 1st field = f of No. in 2d field. Since | of No. in 3d field :r=| of No. in 2d field, I of No. in 3d field = 1 of No. in 2d field, and The entire No. in 3d field :=f of No. in 2d field. Therefore, since the number in the 1st field is f of the number in the 2d field, and the number in the 3d field is f of the number in the 2d field, the num- ber in the three fields will be equal to f + f + 1, or ^y^-y of the number in the 2d field ; and %7. of No. in 2d field = 434 ; yV of No. in 2d field = 2 ; Entire No. in 2d field = 72 times 2, or 144 ; The No. in 1st field = | of 144, or 162 ; The No. in 3d field = | of 144, or 128, Ans. Ift6 MISCELLANEOUS EXAMPLES. b'9. A and B can do j\ in 1 day ; B and C can do j\ in 1 day; A and C can do ^ in 1 day; A, B and C can do I of (yV + tV + tV). ^r I, in 1 day. Since A, B and C can do i, and A and B -^^^ in 1 day, C can do I — y^^ , or -^q, of the work per day, or he can do tlie whole in 40 days, Am. Since A, B and C can do ^, and B and C -jL^ in 1 day, A can do I — ^j, or J^? P^^ ^^7> or he can do the whole work in 24 days, Aris, Since A, B and C can do ^, and A and C can do -^^ in 1 day, B can do i — -^, or y|-g-, in 1 day, or he will require as many days to do the whole as j^-^ is contained times in -J-|^, which is 17| days, Ans, 70. Since the field contains 24 acres, each man's, cattle would eat ^ of the 24 acres, or 8 acres. Since A owned only 9 acres, and his cattle ate the grass upon 8 acres, he really furnished only 1 acre for the pasturage of C's cattle. B furnished 15 — 8, or 7 acres. Therefore, since A furnished 1 acre and B 7 acres, A is entitled to I of the sum paid by C, or ^ of $24, which is S3 ; and B is entitled to -J of $24, or $21, Am, 71, Since the boards are 11 ft. long, and fence 4 boards high, the number of boards in a rod would be equal to 6. Since the field is a square field, and the number of boards per rod is 6, the number of boards needed to inclose the field will be 24 times the number of rods in the length of one side. The number of acres is equal to the square of the num- ber of rods in length divided by 160. MISCELLANEOUS EXAMPLES. 197 And since the number of acres and the number of boards are equal, we have the following equation : Rods in length X ^4 == C ^^ ^ ^^^ ^^g _ ; therefore, ^ 160 „ . rods in length , 24 = ^— ; and 160 24 X 160 or 3840 = rods in length; 24 times 3840 r= 92160 acres, Am. 72. Since the numbers have a common factor plus the same remainder, if. the numbers are subtracted from each other the results will contain the common factor without the remainder. Thus : 1st set: 2d set: 21 63 147 The greatest common divisor of these numbers is 21. 73. A's present stock is 1^ times the original amount , B's stock was |- of A's present stock, or f of A's or B's original stock. Therefore, B's loss was equal to f of A's or B's original stock; f of As or B's original stock == $225 ; \ of A's or B's original stock :^ $75 ; A's or B's original stock = $600, Ans. 27 48 90 174 27 48 90 21 42 84 48 90 174 ^ 27 27 27 198 MISCELLANEOUS EXAMPLES. 74. Since, if he left only a daughter, the wife was to have f and the daughter ^, the wife's share was to be 8 times the daughter's; and since, if he left only a son, the wife was to receive \ and the son f , the son's share was to be 3 times the wife's; when the daughter gets $1, the wife gets $3, and the son gets $9. Therefore, the daughter gets ^ of the sum, the wife yS^, the son ^. ^ of $6591 = $507, the daughter's share ; ■^ of $6591 = $1521, the wife's share; ■^ of $6591 = $4563, the son's share, Ans. M577039 M54 key Educ . Lib. ^^