University of California • Berkeley The Theodore P, Hill Collection of Early American Mathematics Books KEY TO THE PROGRESSIVE PEACTICAL ARITHMETIC. INCLUDINa ANALYSES OF THE MISCELLANEOUS EXAMPLES PROGRESSIVE INTELLECTUAL ARITHMETIC. FOR TEACHERS ONLY. IVISON, BLAKEMAN, TAYLOR & CO.. NEW YORK AND CHICAGO. 1877. ROBINSON^S Mathematical Series. Graded to tlie wants of Primary, Intermediate, Grammar, Normal, and High Schools, Academies, and Colleges. Progressire Table Boob. Progressive Primary Aritliraetic. ProgressiYO Intellectual Arithmetic, Rudiments of Written Arithmetic. JUNIOR-CLiSS ARITHMETIC, Oral and Written. NEW. Progressive Practical Aritlimetic. Key to Practical Arithmetic. Progressive Higher Arithmetic. Key to Higher Arithmetic. Nevr Elementary Algebra. Key to New Elementary Algebra. New University Algebra. Key to New University Algebra. New Geometry and Trigonometry, In one vol. Geometry, Plane and Solid. In separate vol. Trigonometry, Plane and Spherical. In separate vol. New Analytical Geometry and Conic Sections. New Surveying and Navigation. New Differential and Integral Calculus. University Astronomy— Descriptive and Physical. Key to Geometry and Trigonometry, Analytical Geometry and Conie Sections, Surveying and Navigation. Entered, according to Act of Congress, in the year 1860, by HORATIO N. ROBINSON, LL.D., in the Clerk's Office of the District Court of the United States for the Northern District of New York. P R E I^ 1 1 . A Key to any Matliematical work is not intended to sa- persede labor and study, but to give direction to the latter and make it more effective and useful. In many examples and problems tbe same results may be obtained by different processes, but the shortest and most simple method is to be desired ; hence the object of a Key should be to give not results only, but the explanation of methods, and a full analysis of such questions as contain a peculiar application of principles involved. It is supposed, of course, that every teacher is fully com- petent to solve all the questions, but with the multiplicity of duties ordinarily put upon the teacher, time cannot always be had to answer or solve all the questions presented by the pupil. Therefore the Key is intended to lessen the labor and save the time of the teacher by presenting the shortest solution, and the best form of analysis as a standard to which the pupil should be required to conform. IV PREFACE. In compliance with the wishes of many teachers, brief analyses of the Miscellaneous Examples in the Intellectual Arithmetic have been added to the latter part of this work. Much labor has been bestowed upon the present work to give a full, complete, and logical analysis of all difficult rxamples^ and of g'jch questions as contain the application of a new principle, The arrangemeut i& such as to be easily understood. KEY. NOTATION. ROMAN NOTATION . (17, page 9.) Ex. Ex. Ex. Ex. 1. Ans XL S, Ans, XXV. 5. Ans. XLVm. Y. Ans. CLIX. 9. Ans. MDXXXYIII. Ex. 2. Ans. XV. Ex. 4. Ans. XXXIX. Ex. 6. Ans. LXXVII. Ex. 8. Ans. DXCIV. Ex. 10. Ans. MDCCCCX, I Ex. Ex. Ex. 1. Ans. 125. 4. Ans, 900. 1. Ans. 505. ARABIC NOTATION. (26, page 12.) Ex. 2. Ans. 483. Ex.5. Ex. 8. Ans. 290. w4?i5, 557. Ex.3. Ans. 716. Ex. 6. ^/i5. 809. (28, page 13.) Ex. 2. ^W5. 5160. Ex. 5. ^715. 2090. Ex. 8. Ans. 9427. Ex. 3. Ans. 3741. Ex. 6. ^715. 7009. Ex. 9. Ans. 4035. Ex. 1. Ans. 1200. Ex. 4. ^715. 8056. Ex. 7. Ans. 1001. Ex. 10. ^»5. 1904. Ex. 11. Ans. Seventy-six; one hundred twenty-eight ; four hundred five ; nine hundred ten ; one hundred sixteen ; three thousand four hundred sixteen ; one thousand twenty-five. Ex. 12. Ans. Two thousand one hundred; five thousand forty-seven ; seven thousand nine ; four thousand six hundred 6 SIMPLE NUMBERS. seventy; three thousand nine hundred ninety seven; one thousand one. (29, page 14.) Ex. 1. Ans. 20000. Ex. 2. Ans. 47000. Ex. 3. Ajis. 18100. Ex. 4. Ans. 12350. Ex. 5. Ans. 39522. Ex. 6. Ans. 15206. Ex. 7» Ans. 11024. Ex. 8. Ans. 40010. Ex. 9. Ans. 60600. Ex. 10. Ans. 220000. Ex. 11. Ans. 156000. Ex. 12. Ans. 840300. Ex. 13. Ans. 501964. Ex. 14. Ans. 100100. Ex. 15. Ans. 313313. Ex. 16. Ans. 718004. Ex. 17. Ans. 100010. Ex. 18. Ans. Five thousand six; twelve thousand three hundred four ; ninety-six thousand seventy-one ; five thousand four hundred seventy ; two hundred three thousand four hun- dred ten. Ex. 19. Ans. Thirty-six thousand seven hundred forty-one ; four hundred thousand five hundred sixty ; thirteen thousand sixty-one ; forty -nine thousand ; one hundred thousand ten. Ex. 20. Ans. Two hundred thousand two hundred ; seventy five thousand six hundred twenty ; ninety thousand four hun- dred two ; two hundred eighteen thousand ninety-four ; one hundred thousand one hundred one. (31 5 page 16.) Ex 1. Ans. 140. Ex. 2. Ans. 30201. Ex. 3. Ans. 8050. Ex 4. Ans. 2900417. Ex. 5. Ans. 300040. Ex. 6. Ans. 96037009. Ex. 7. Ans. 4064200150, Ex. 8. Ans. 846009350208. (34, p. 19.) Ex. 1. Ans. 436. Ex. 2. Ans. 7164. Ex. 3. Ans. 26026. Ex. 4. Ans. 14280. Ex„ 5. Ans. 176000. Ex. 6. Ans. 450039. Ex. 7. Ans, 95000000. Ex. 8. Ans. 433816149. NOTATION AND NUMERATION. 7 Ex.9. Ans, 900090. Ex. 10. Ans. 10011010. Ex. 11. Ans. 61005000000. Ex. 12. Ans, 5080009000001. Ex. 13. Ans, Eight thousand two hundred forty. Ex. 14. Ans, Four hundred thousand nine hundred. Ex. 15. Ans. Three hundred eight. Ex. 16. Ans, Sixty thousand seven hundred twenty, Ex. 17. Ans, One thousand ton. Ex. 18. Ans, Fifty-seven million four hundred sixty-eight thousand one hundred thirty-nine. Ex. 19. Ans, Five thousand six hundred twenty- eight Ex. 20. Ans, Eight hundred fifty million twenty-six thou- eand eight hundred. Ex. 21. Ans, Three hundred seventy thousand five. Ex. 22. A71S, Nine billion four hundred million seven hun- dred six thousand three hundred forty-two. Ex. 23. Ans, Thirty-eight million four hundred twenty-nine thousand five hundred twenty-six. Ex. 24. Ans, Seventy-four billion two hundred sixty-eight million one hundred thirteen thousand seven hundred fifty- nine. Ex. 25. Ans, 7000036. Ex. 26. Ans, 563004. Ex.27. Ans, 1096000. Ex. 28. Ans, Nine billion four million eighty-two thousand five hundred one. Ex. 29. Ans, Two trillion five hundred eighty-four billion five hundred three million nine hundred sixty-two thousand forty-seven, Ex. 30. Ans. 3064159, Ex, 31. A71S. Two of the sixth order, 9 of the fifth, 6 of the third, 4 of the second, and 8 of the first. Ex. 32. Ans. One of the seventh order, 3 of the fifth order, 7 of the fourth order, and 5 of the second order. Ex. 3 Ex.6. Ans. 6i>8. Ans. 898. SIMPLE NUMBElia ADDITION. (40, page 21.) Ex. 4. Ans. 967. (43, page 24.) Ex. 7. ^7i5. 1807. Ex. 8. Ans. 27246. Ex. 9. Ans. 4945. Ex. 10. ^7Z5. 78313. Ex. 19. Ans. 145. Ex. 22. Ans. 69585. Ex. 23. Ans. 566. Ex. 24. Ans. 3746. Ex. 27. ^^^5. 4619. Ex. 28. Ans. 4915. Ex. 29. Ans. 4320. Ex. 30. Ans. 4623. Ex. 31. Ans. 3871. Ex. 35. ^?Z5. 101500. Ex.37, ^ns. 50000000. Ex.40, ^w*. 1194. Ex. 44. Ans. 2773820. Ex. 45. Ans. 4403241. Ex. e. Ex. 8. Ex. 19. Ex. 24. Ex. 26. SUBTKACTION. (49, page 30.) Ans. 353. Ans. 205. Ans. 123. Ans. 4202. Ans. 16348755. Ex. 7. Ex. 9. Ex. 22. Ex. 25. Ex. 27. Ans. 210. Ans. 320. Ans. 2113. Ans. 11425, Ans. 4014580. (51, page 32.) Ex. 3. Ans. 721. Ex. 4. Ans. 561. Ex. 5. ^ws. 3769. Ex. 6. Ans. 269, SUBTRACTION. Ex. 7. Arts. 4509. Ex. 8. Ans. 3449. Ex. 9 Arts. 1288. Ex. 10. Ans. 30616. Ex. 11. Ans. 21078. Ex. 12. Ans. 142. Ex. 13. Ans. 762301. Ex. 19. Ans. 224130. Ex 20. Ans. 220874. Ex. 25. Ans. 181972. Ex. 31. Ans. 529509693. Ex. 34. Ans. 1902001 EXAMPLES COMBINING ADDITION AND SUBTRACTION, (52, page 33.) Ex. 1. 2500+ 175 = 2675 6200—2675 = 2525 dollars, Ans. Ex. 2. 235 + 275 + 325 + 280 = 1115; 1300 — 1115 = 185 miles, Ans. Ex. 3. 4234 + 1700 + 962 + 49 = 6945, 87,^^-6945 = 1807 dollars, Ans. Ex. 4. 47(^5 + 750=5515; 5515 — 384 = 5131 dollars, Ans. Ex. 6. 1224 + 1500 + 1805 = 4529; 7520 — 4529 = 2991 barrels, Ans. Ex. 6. 450 + 175= 625, B's; 450 + 625 = 1075, A'sandB's; 1075 — 114= 961, C's sheep, Ans. Ex. 7. 1575 — 807 = 768, bushels of wheat, 900 — 391 = 509, " " corn, Ex. 8. 2324 + 1570 + 450 + 175=4519; 6784—4519 = 2265 miles, Ans. at, ) Ex. 9. 7375, first paid; 7375+ 7375 = 14750, second paid; 7375 + 14750 = 22125; 36680 — 22125 = 13555, dollars, Ans. Ann, 10 SIMPLE NUMBERS. Ex. 10. '750-{-3'79 + 450 = 1579; 1579 — 1000=579, dollars, Ans. Ex. 11. 6709 + 3000=9709; 9709—4379 = 5330 dollars, Ans, Ex.12. 10026402+ 9526666 = 19553068, total ; 8786968+ 8525565 = 17312533, native; 19553068 — 17312533= 2240535, foreign, Ans. MULTIPLICATION. (61, page 38.) Ex. 5 Ans. 247368. Ex. 6. Ans. 648998. Ex. 7. Ans. 224744. (64, Ex. 8. page 41.) Ans. 416223. Ex. 5. Ans. 2508544. Ex. 6. Ans. 15731848, Ex. 7. Ans. 16173942. Ex. 9. Ans. 78798. Ex. 13. Ans. 937456. CONTRACTIONS. Ex. 2. Ex. 3. Ex. 4. Ex. 5. (675 page 43.) 3472x6 = 20832; 20832x8 = 166656, Ans. 14761 X 8 = 118088 ; 118088 x 8=944704, Ans. 87034 X 3 = 261102 ; 261102 x 3 = 783306 ; 783306x9 = 7049754, Ans. 47326 X 6 = 283956 ; 283956 x 5 = 1419780 • 1419780x4 = 5679120, Aiis. MULTIPLICATION, II Ex. 6. 60315x8X3x4 = 5790240, Ans, Ex. 7. 291042x5x5x5 = 36380250, Ans. Ex. 8. 430 X 7 X 8 = 24416 miles, Ans. Ex. 9. 124 X 6 X 3 X 4 = 8928 dollai-s, Ans. Ex.13. 5280x7x3x4 = 443520 feet, Ans. Ex. 11. 120x5x5x5 = 15000 dollars, Ans. {61>5 page 44.) Ex. 3. Ans. 13071000. Ex. 4. Ans. 890170000. (70, page 45.) Ex. 12. 296 3000 888000 dollars, Ans. EXAMPLES COMBINING ADDITION, SUBTRACTION, AHD MULTIPLICATION. Ex. 1. 4x45 = 180; 13x9 = 117; 180 + 117=297 dollars, Ans. Ex. 2. 31x6 = 186; 39x6 = 234; 234 — 186 = 48 dollars, Ans, Ex. 3. 288x9 = 2592; 2592 — 1875 = 717 dollars, Ans. Ex. 4. 240 + 125 + 75 + 50 = 490; 500 — 490 = 10 dollars, Ans, Ex. 5. 184x2=368; 67x4 = 268; 368 — 268 = 100 dollars, Aiis Ex. 6. 36 X 320 = 11520, A received ; 48 X 244 = 11712, B received ; 11712 — 11520 = 192 dollars, Ans. Ex, 7 35 + 29 = 64 miles, in one day ; 64 X 16 = 1024 miles, ^^. 12 SIMPLE NUMBERS. E%. 8. 14 X 26 X 43 = 15652 yards, Am* Ex. 9. 4 X 365=: 1400, yearly expenses ; 3700—1460 = 2240 dollars, Ans, Ex.10. 2475, first; 2475— 840 = 1635, second; 2475 + 1635 = 4110, third; 8220 dollars, Ang. Ex. II. 336 — (28 X 10) = 56 miles. Am. Ex.12. 23 X 14= ^22, cost of cows; 96 X 7= 672, " " horses; 57 X 34 = 1938, " " oxen; 2x300= 600, " " sheep; 3532, " " whole. 3842 — 3532 = 310 dollars, Ans. Ex. 13. 36X164 = 5904 Sx 850 = 2550 8454 dollars, Ans. Ex. 14. 14760— (1575 x 5)=6885 dollars, An9» Ex. 15. 936 X 9=8424, cost ; 480 x 10 = 4800 456 X 8=3648 Flour sells for, 8448. 8448 — 8424 = 24 dollars, Ans. DIVISION. (77, page 50.) Ex. 2. Ans, 16358. Ex. 3. Ans, 17827. Ex. 4. Ans. 29822. Ex. 5. Ans. 672705. Ex. 6. Ans. 182797. Ex. 7. Ans. 829838. DIVISION. 13 Ex. 13. Ans, 1048795f. Ex. 14. Ans, 635926f . Ex. 15. Ans. 2379839I-. Ex. 16. Ans. 9355Y51f Ex. 17. Ans, 245640}-!. Ex. 18. Ans. 70141321. Ex. 19. 47645-^5 = 9529 dollars, Ans. • Ex. 20. 17675-^7:=:2525 weeks, Ans. Ex. 21. 6756^6 = 1126 barrels, Ans. Ex. 22. 46216464^12:^3851372 dozen, Ans. Ex.23. 347560-^5 = 69512 barrels, Ans. Ex.24. 3240622-7-11 = 294602 acres, Ans. Ex. 25. 38470-T-5=7694 dollars, Ans. (80, page 54.) Ex. 5. Ans. 212//4-. Ex. 14. Ans. 1489f|. Ex. 15. Ans. 121522%. Ex. 16. Ans. 508301yV2- Ex. 17. Ans. 12109001$ Jf Ex. 18. Ans. 9974^9-. Ex. 19. Ans. 1343if Jf. Ex. 20. Ans. 5473|fff Ex. 21. Ans. 7500yVTTV Ex.27. 1850400-^18504 = 100 dollars, Ans. Ex. 28. 72320060-^10735 = 6736yVT3J . 3 2x20 4x10 5x8 2x2x10 2x4x5 2x2x2x5 ^2x36 3x24 4x18 6x12 8x9 2x2x18 - 2x3x12 12= < 2x4x9 3x4x6 2x6x6 3x3x8 2x2x2x9 2x2x3x6 2x3x3x4 *>x2x2x3xa 20 PROPERTIES OF NUMBERS. CANCELLATION. Ex. 3. u (95, pa n u ige 72.) Ex. 4. 1$ n 8 11 ^0 14, Ans. 33, Ans. Ex. 5. 1 $4 X$ H 7 £$ 16 .%$ 4 $ 64 Ex. 6. 30 $ n n W0 13 t 3 13 9|, Jns. 41, -4ms. Ex. 7. 8 8 00 3 9 X$ 3 81 Ex, 8. tn 1$ u 200 ?0 2 $0 2, Ana. lOi, Ans. Ex. 9. n 00 2 Ex. 10. 4 $ t i?2 *< S,Ans. ^^2 4, u4»». Ex. 11. it 1 t.nns_ AnR ka. 12 00 4 xn 6 . >, ww--^, ( i firkms, A7if El. 13. GREATEST COMMON DIVISOR ^ Ex. 14. 3 U t 21 20 suits, Ans. 3 115 00 2 230 76|days,-4w«. GREATEST COMMON DIYISOR. (98, page 74.) Ex. 1, 3 X4=:12, ^?25. Ex. 3. 4x6 = 24, Ans, Ex. 5. 2x7=1 14, Ans. Ex. 7. 7x10 = 70, ^?Z5. Ex.9. 2x7x9 = 126,^715. Ex. 12. 5x5 = 25, Ans, Ex.2. Ex. 4. Ex. 6. Ex.8. 2x3 = 6, Am. 3 X 6 = 18, Ans. 4 X 4 = 16, Am. 3 x5 x5 = 75. Am, Ex. 10.4x8 = 32, ^/i5. (OOj page 78.) Ex. 8. To arrange a nnmber of things in equal parcels, the parcel must be a divisor of the number ; and to arrange two numbers in equal parcels, the parcel must be a common divisor of the two numbers. Ans. 5 in a parcel. Ex. 9. The lots, to be equal, must be a common divisor of the three fields, and to be the greatest possible, must be the greatest common divisor. Ans. 2 acres. Ex. 10. To avoid mixing, the capacity of a bin must be a common divisor of the two numbers of bushels ; and to have the least number of bins, will require the yreatest common di- visor of the two numbers of bushels. Ans. 21 bushels. Ex. 11. Tlie greatest common divisor of 124, 116, and 92 rods, the three fronts. Ans. 4 rods. 22 PROPERTIES OF NUMBERS. Ex 12. The greatest common divisor of the three lengths, 3013, 2231, and 2047 feet. Ans. 23 feet. Ex. 13. The greatest common divisor of the three num- bers of bushels is 2, which must be the capacity of the bag. Now there are to be forwarded 2722-1-1822 + 1226 = 5770 bushels; and 5770-f-2zii:2885. Ans, 2885 bags. Ex. 14. The greatest common divisor of $120, $240, and $384, is $24, the price of the cows; and $120-t-$24=:5, A's number; $240-^$24 = 10 B's number; and $384-r-$24 = 16, C's number. MULTIPLES. (104, page 81.) Ex. 2. 2 X 5 X 7 X 7=490, Ans. Ex. 3. 2x2x2x3x7x 17 = 2856, Ans. Ex. 4. 2x2x2x3 x 3 = 72, Ans. Ex. 5. 2x2x2x3x5x7x11 = 9240, Am. Ex.6. 2x3x3x5X5 = 450, ^7Z5. Ex. 7. 2x2x3x3x5x 7 = 1260, Ans. Ex. 4. 2, 3 2,5,7 (10^5 page 83.) 42 .. 60 7 .. 10 2x2x3x5x 7=420, Ans, Ex. 5. 3, 7 2,5 21 .. 35 .. 42 5 ., 8x7x2 X 5 = 210, Ans, Ex. 6. Ex. 7. Ex. S. Ex. 9 Ex. )'>. LEAST COMMON MULTIPLE. 28 2,5 3, 2, 5, 5 60 .. 100 .. 125 6 .. 10 25 2x5x3x2x5x 5==1500, Ans. 2,2,5 40 . . 96 .. 105 2, 2, 2, 3 2 . .24.. 21 1 1 2x2x5x2x2x2x3x '7 = 3360, An^ 2,2,3 48 . .60. .12 2, 2, 3, 5 4 : . 5 . . 6 2x2x3x2x2x3x5 = 720, Ans. 2,2,3 84 . . 224 . . 300 7,5 7 . . 56 . . 25 2, 2, 2, 5 8 . . 5 2x2x3x7x5x2x2x2x 5 = 16800, Ans. 3,3 270 .. 189 . 297 . . 243 3, 3,7 30.. 21. . 33 . . 27 2,5,11,3 10 .. 11 . . 3 3x3x3x3x7x2x5x11 x 3 = 187110, Ans. Ex. 11. 2,3,5 5 . .6 . .7 . . 8 . .9 2, 2, 1, 3 , . 7. .4 . .3 2x3x5x2x2x7x 3 = 2520, Ans. Ex. 12. To purchase books at 5 dollars, or 3 dollars, or 4 dollars, or 6 dollars, the sum of money must be some common mvliiple of 5, 3, 4 and 6 ; and the least sum will be the teasi iommor, multiple^ which is 60 dollars, Ans, E.c. 13. The least common multiple of 12, 15, and ly barrels, whicb is 180 barrels, Ans. Ex. 14. The least common multiple of the prices, $30, $o5, and $105, which is $2310, Ans. 24 FRACTIONS. Ex. 15. The least common multiple of 41, 63, and 64 sheep, which is 15498 sheep, Ans, Ex. 16. He must spend in the purchase of each kind of fowls a sum equal to the least common multiple of the prices paid. Suppose he takes the cheaper turkeys ; the least com- mon multiple of 12, 30, and 75 is 300; and 300^12=25, number of chickens; 300-^301=10, number of ducks; 300 -i-75 = 4, number of turkeys; and 25 -flO + 4 = 39, the whole number of fowls purchased. Next suppose he takes the tur- keys at the higher price; the least common multiple of 12, 30, and 90 is 180; and 1 80-^-12 ==15, number of chickens ; 180-f-30 = 6, number of ducks; 180-^90 = 2, number of tur- keys; and 154-6 + 2 = 23, whole number of fowls. But 39 — 23 = 16, number of fowls purchased more than was n^ cessary; and 16x5 = 80 cents, Ans. FRACTIONS. (ISO, page 88.) Ex. 2. Ans, /j, Ex. 4. Ans, ij. Ex. 6. Ans, JJ. Ex. 8. Ans. p^fy. Ey 10 An -^'*^« Ex. 4. 3, 3, 2 7,2 9 ..21 ..4 3x3x2x7x 2 = 252, least com. denom. a. _1_ 3 3. — _5JB_ Jl_2_ J.19. _1_5 1.5. J770 ¥) 21) 4) 1 — 252) 252) 252) 252 » -^'*''»» ?JL SL JJ.— AA j_a. j_i. Ans T)4) 8 — 8) 8) 8) ■^'t'^' 3 1 2 i_7 _5_ — 2.3. A _aJ_ JL8_ ±l.ft JLO yl-w* 4) 8') T) 6 ) 14 — T6 8) 16 8) 16 T) 16 8) 168) ^^^» 4 7 115. 1 — 3i. 31 1_6_5 4JL5 35. y|^o 5) TJ) 3) 1) 9—45) 4 5) 45 ) 45 ) iJ) ■^'*^« 2 1 J 3. 1 11 6. — 4.2. i_o_4 45. 4 A Vie Ans 2" 8") T) 8) 14)1 56) 56)56»56) 56) -^'*'^« 2 1 31 A 5. 11 5 32.1 111 A 8.0 2Cr 4 4_ 7J ^yij 10)40) 1)3)30)8 120) 120) 120)1^^*120) I t^l ^"'' A 2 1 _7_- 9) 3) ¥) 12- 7 5 1_1 1- fi) 7) 4)2" -1^ 2 A _ft_ ' 36) 36) 36)36 "56) 56) 16 ) 5 , Ans. ADDITION. 27 iliX. Id. ^f, fo, 3, -'j— fQ, l^o, 3^, f 0", ^7i6. Fv Ifi 7 2 5 _$ 13 3 7 J2_5.18 140l2iI0 AnS^ UiX. ID. 2 0> 4 > 10 » 1> S> 2 — 2 0> 20 » 2 0") "2 0-? 2 0» 2 0» ^'^* ADDITION. (133, page 96.) 7 + 3+1+54-9 25 ^5 ^1 ^ Ex.2. --±-^ =-=2-=2-,^.. 4 + 5 + 7 + 1 + 3 + 11 31 7 ^''•^' 12 -12^^2' "^'''• ^ , 7 + 9 + 2 + 13 + 16 + 21 68 ^18 ^ Ex.4. ^^ =.-=2-,Ans. 41+63 + 71+89 + 109 373 13 , Ex. 5. = =:3 , Ans, 120 120 120' ^ ^ 13 + 76 + 140 + 181+223 683 61 ^ Ex.6. ■ ■ ■ ■ = — =2 — , Ans. 225 225 75' (134, page 97.) „ „ 3 2 27 + 8 35 . Ex. 2. T + - = ■ — = — » -4rwr. 4 9 36 36' ^ „ 4 11 56+55 111 41 . ^,3125 ^^•'^ 4 + 8 + 7 + 12= 126+214-48+70 _265_, 97 168 -168-^168' "***'* 18 FRACTIONS. 14 . 9 2 1274 + 2835 + 390 4499 404 , = ~1- , Ans, 4095 4095 4095* 42 9 Y 1 ^'- li0+70 + 28+n= 42 + 18 + 35 + 10 105 3 . --, Ans 140 140 4' ^ ^ bl 131 24 1 2 ^^•'' Y5 + 160 +2^ + 2 + 3 = 102 + 131 + 144 + 75 + 100 _ 17 150 "" 25' „« 3124.56 78 9 '^•«- 4 + 2 + 3 + 5 + 6 + 7 + 8 + 9 + 10 = 1^90 + 12 60 + 1680 + 2016 + 2100 + 2160 + 2205 + 2240 + 2268 2520 17819 ^179 , 2520 2520' T. ,^ 4 9 2 19 ^19 ^^•^^- 5-^10 + 3 + 20=^60 14 + 3+ 1 = 18 21U, Ans. Ex.11. f + T\ + f= 2/j 1 + 10 + 5 = 16 18^, -4/i«. Ex. 12. f + A + ,-V = lA 17 + 18 + 26 = 61 62/^, Ans. Ex. 13. /8+H + i + H = HJ 1 + 3 = 4 5^J, -4««. SUBTRACTION. 2^ Ex. 14. ;}+ t\+ 1 = l^T 125 + 327 + 25— 477 4782V, -^^*- Ex.15. in + U+H + ii-^\n=^Hih^'^' Ex. 16. /o+H+T + To= 4 3 + 2 + 40 + 10^:55 55f , ^ws. Ex. 17. f + f + 1 = 2/:, 125 + 96+48=:269_ 27I2V yards, Arts. Ex. 18. | + ^ + J=zlJ^ 5 + 3 — 8 9jV yards, Ans, Ex. 19. ^V + i^ + 1 + 3 + 2 ^ 311 26+40 + 51+59 + 62 = 238 241^f acres, Ans, Ex. 20. | + 4+±i + _7_ := 23«, 175 + 325 + 270 + 437 = 1207 35 1209/5 bushels. 205 + 296 + 200 + 156 = 857 $859||, dollare. SUBTRACTION. (135, page 99.) Ex. 2. —-—=-=-, Ans, 9 93' 14-11 3 1 . Ex.3. -^-=^^=-,Ans. 80 FRACTIONS. 20 — 6 14 , Ex.5. ^±Z?lJ^^,Ans. ^ ^ YS-ll 64 1 . Ex. 6. -— ^^~ ^7i^. 128 128 2' » ^ ^ 182-110 72 6 ^ Ex. 7. = — =r — , Ans. 348 348 29' (I365 page 99.) Ex. 2. -— -=-^^= — , Ans, 2 9 18 18' 15 2 75-48 27 9 ^'''^' 24-5=~12Cr=T20 = 40'"^'**- ^,34 51 — 32 19 . Ex. 4. -:=z = — , Ans, 8 17 136 136' ^ ^ 84 4 49-8 41 . Ex. 5. ■ = -i: — =ir-, Ans, 120 35 70 70' 1500 50 125 — 100 25 . ^^•'- 1728-7^==— 4-4-=U4'^"^- ^ ^ 60 332 720 — 83 637 , Ex. 7. — -= = , Ans, 89 4272 1068 1068' Ex. 9. 8i = 8y\ Ex. 10. 25| = 25^^ 4|f, ^?Z5. 163^ = 16^3,^/1*. Ex. 11. 4f =4|| Ex. 12. 6 3 jj, ^725. 4-f, ^715. Ex. 13. 450J =450^1 Ex. 14. ^^J = Z^^ i20Jv=i2oii tVs= m 330jf, Ans, 33Y5, ^n». MULTIPLICATION. 81 Ex. 15. 751 Ex. 16. 227f 49 1961=1961 26i; Ans. 30f , Ans. Ex. 19. $Y|-~$6i=$lf^2, Arts. Ex. 20. 4 + 31 =4V\ Ex. 21. 6i + 2i+f= 9^^ 5i-4i =- H $255-$92«o =$16Ht 4_3_ - |i = 3|f , ^n^. -^^^*" Ex. 22. n-2f =4|f, ^7^5. Ex. 23. ^f-H=^» ^^^«- Ex. 24. 9121 + 5451 =145^^ |2000-$145YyV=$542^^, Ans. Ex. 25. $136yV + $350|=$487|i cost. $184i + $416J = $6011 receipts. $601i-$487ii=$llV8» ^^^• MULTIPLICATION. (I375 page 101.) Ex. 4. tV X '^=iT=lTf» ^^^• Ex. 5. 1^4 X 12 = VV='7f» ^^^• Ex. 6. /t X 63=5 X 3 = 15, ^W5. Ex.8. '7|xl2 = *|«=91i, ^^5. Ex. 9. tVt X 8=f It^SyVr, ^^^• Ex.10, ^l^x51=f = 2,^r^5. Ex.11. 15fx 16 = 125 X 2=250, ^w«. Ex.12. mx22 = 'i'=lH,Ans. Ex.13. $8Axl2=$Hf'=^106i, ^n#. Ex. 14. |iix9 = $H=^S-I' ^''^' Ex. 15. $J X 2'7=$H^=$23f, ^w«. 32 Ex. 2. FRACTIONS. (138, page 103.) Ex. 3. 14 100 9 450 Ex. 4. 21 li, Ans. 105 17 Ex. 5. 47 85, Ans, 47 64f, Ans, 19 13 247 Ex. 7. 42 39 Ex. 8. 16 5if, Ans. 80 233 819 '1165, Ans. 409^, Ans. Ex. 9. 39 156 27 Ex. 10. $8xf = 6| dollars, ^/MT. Ex. 11, 36 X 10|=384 miles. Ana. 108, Ans. Ex. 12. $450 X ^2 =$262|, Ans. Ex. 13. $16 X 2J=$44^, Ans. (139, page 104.) I Ex. 3. 8 Ex. 2. 10 Ex. 4. 24 65 10 ^, Ans. 11 36 Ex. 5. 6 7 y\, Ans. 21 6 18 3|, Ans. MULTIPLICATION. 33 Ex. 6. 10 9 1 2 9 5 4 1 28 1 aV, ^ns. E^. 9 Ex. 11. Ex. 13. 15 8 4 9 5 1 3 22 25 44 lif, Arts. 3 2 1 n 4 5 4 4 13 35 78 2A, -4^5. 8 7 2 1 9 11 2 3 1 8 12 77 6y5_ Arts. Ex. 15 $ixf=$i u4w5. Ex.17. ix^=^J Ans, :. 7. 6 11 6 3 2 3 16 15 176 1111, Ans. Ex. 10. 7 2 1 16 10 7 3 80 3 256 851, Am Ex. 12. 2 6 4 3 5 4 3 4 2, Ans. 2 25 2 11 4 27 16 7425 Ex. 14. 464i-V» -^^** Ex. 16. 4x|=:J^, ^W5. Ex.18. V x$|=$ff, ^'/wj 84 Ex.20. 4 2 51 17 8 867 FRACTIONS. Ex. 21. Ans., 108f. 8 6 51 14 20 357 Ans,^ 11^1 dollai*. Ex. 22. Ex. 23. Ex. 24. Ex. 25. Ex. Ex. Ex. 26. 27. 28. f xf x$VV=$21y\-, Ans. V x$V-=^22iJ, Ans. f X V- X I X P^^-=$25^j, Ans. Xx|=i4, Ans. $A.^xi=$25-^\,Ans. ■^^ acres x | x 1=49-,-^ acres, Ans^ f-f X f barrels =6| barrels, An>s. DIVISION. (I4O5 page 107.). Ex. 6. j\^j---25z=^.l-^,Ans. Ex. 9. $|-r-6=$i, Ans. Ex. 10. J-v-7=|-, ^7i5. Ex. 11. -f -v-5 = 3«j, -47^5. Ex. 12. $^-^-9=$^, Ans. Ex. 14. V-v-3 = 5yV ^^5- Ex. 15. ;2 x Y-^^ -|,^w*, Ex. 16. $2.±|iL_^4 = $\2_4._$24||, ^7^s. (141 5 page 109.) Ex, 7. I X 9-^1=15, ^715. Ex. 8. 121-x-^— $49, u4w5. Ex. 9. 16xf = 10; 10-^|-=:22i, .^WA. Ex. 11. 75-i-Y-=^H» •^^^• Ex. 12. 149-^J4i=6yVj, ^w«- Ex. 18. 15^-f =9, Ans. Ill VISION. B6 Ev 14. f X 320=200 ; f x Y=¥ ; 200— V =254, Ans. Ex.15. $32x1=8; Yxi=f; Ex. 16. 183~J^i=4, Ans. Ex. 2. Ex. 4. Ex. 6. Ex. 8. Ex. 10. (143, page 110.] ) 8 7 Ex.3. 9 6 3 4 1 6 6 r" 3 10 1 1^, Ans. 31, Ans. 1 4 Ex. 5. 2 1 9 10 7 14 13 63 40 13 ih ^^«- \h -4^*' 3 2 Ex.7. 6 5 27 28 4 5 81 56 • 24 25 ih ^ns. l^V, ^^*' 3 5 Ex. 9. 19 17 8 7 7 19 9 35 7 17 3f , Ans. 2f , ^»«. 20 13 Ex. 11. 7 2 5 16 2 6 25 52 1 3 2 4 2^, ^»5. 21 40 l^f, Ans. FRACTIOOTL Ex. 12. 10 9 4 5 6 13 4 325 432 6f 56 3 28 . Ex. 15. -^=— - X TTTT^rrr, Arts. 81 9 26 39' Ex. 16. Ex. 17. Ex. 18. Ex. 19. "'a 80 1 — ^=— -x-=20, Ans. 4 Y 4 ' ^5 5 5 25 . ii= — X — = — , Ans* 4J 11 22 242' _-x-x--l,^n., 2 5 9 2 1. — - X - X - X -=-, Ans. |xf5629 3' 2 X A Ex. 20. V x|=14, Ans. Ex. 21. 3_3 X f = Y=6|, ^M5. Ex. 22. 8 35 • Ex. 23. 8 35 1 2 6 6 2 21 3 14 98 6 35 'uT^Ans. Ex.24. $J43.xixf=$14f ; $17— $141=$ 2^, Ans. Ex. 25 10 2 20 $/^, Ans. Ex. 26. 3 10 PROMISCUOUS EXAMPLES. Ex. 27. 1 87 10 2 3 2 bu., Arts. 10 1905 8 127 12j-V, Ans PROMISCUOUS EXAMPLES, (Page 112.) Ex.2. 91^7 = 13; 4^; 1 3=f^, ^w*. Ex. 3. 3, 40 I 3 :: 40 3x40 = 120, Ans, Ex. 4. 4 + 3 =7 i + J + f of f = 2|f|_ Ex.6. |xn=fi=4J, li|}, Ans. Ex. 6. 47561- + 1281=4885 3V, ^w«. Ex.7, f xix|xY-=H=VTr Yxfxfx |=H=3li_ 31J, ^ris. Ex. 8. f X f =20, ^W5. ; f x f = V =1t» ^^*- Ex. 9. 18251=1^1-^; iAjiLixf=^4|i^=3043i, .4rw. Ex. 10. i+i=/o ; 1-2V=H; ^V-i^ = 140, ^715. Ex. 11. ^ X $ V X V =^24^1, ^ws. Ex. 12. $V X |-=$23i, ^715. 88 Ex 13. Ex. 15. Ex. 16. Ex. 17. Ex. 18. Ex. 19. Ex. 20. Ex. 21. Ex. 22. Ex. 23. Ex. 24. Ex. 25. Ex. 26. Ex. 27. FRACTIONS. 8 14701 14701 Ex. 14. 8 2 471 37803 2 4 1 628 $12601, Ans, $1, Ans. P-Y^ X ^ Xf =$40551, ^ws. 1 42 10 2,22,, ^*1 ( y x|xf=27, Ans. V X V X l=¥/=24-i-V, ^ns. V xf x|=34i, ^715. lx\^=^=1j\,Ans. 3_3 X _3_ xifi=iy-a=589|, ^W5, H^ X yaVo X ^V^=2500, Ans. $204- f =$50, ^715. 4 5 — 20J-'- 20 — 20? ^^'20 — ^^) -n/«^, $i7^j.Xf=$4608, Ans. 320 X $21=$ 720 755x|X$lf= 528^ 435x$lJ=:$ 815f 755x|x$2i-: 755 $1535| $15351— $1491^ 962f $14911 l:^^, Ans. ^ ^^ 14-5 12 12 7 5 , £x. 28. - -= — : = — , Ans. 8 + 5 13' 13 8 104' _ ^^ 8 + 5 13 8 13 6 ^ Ex. 29. J- ,=— -; 71— — =ri, Ans. 7 + 5 12' 7 12 84' Ex. 30. f X f X f X |=7-J, Ans. Ex. 31. J X $V X f X ^= V =$3J, Ans. DIVISION. 39 Ex 32. 16f-3J = 12l=i?/ H^x'V X 2%=^ W^ =95311, Ans. Ex.33. -Vxy = 'LV = 12ff, -4w5. Ex. 34. 6-^-1846=^1^, Ans. Ex 35. $V X f X 3 J =$3, ^W5. Ex 36. $1-6 X f X i X J X f-=$5, ^W5. Ex SY 5 4-J'- — 17. 1 11 — _3_- _3 1—1 20-^^V = 800, ^715. Ex. 38. Jf cents xy xf = 100 cents, Ans. Ex. 39. i+f =H ; 1- H = 2T, remainder. ^f i X 2^ X $2.1^=^1 i-^=$45YYW, ^^^• Ex. 40. If the horse cost 1\ times as much as the wagon, ihe horse and wagon must cost 2 J times the wagon. Hence, $270-T-2i=$120, Ans. Ex.41. Y^xf=32; 32-20f = lli, ^W5. Ex. 42. P^-i^ X aV X $1=126, Ans, Ex. 43. If A can do f as much as B, he can do the work in 4. of the time that B will require, and in 1+|=|- of the time be Ah will require. Hence 14 days x |=32| days, A's time ; ) 32| days x J=24i days, B's time ; ) ^* Ex. > 4. -V^ X } X f =11 J, Ans. Ex. 45. A, B, and C can do J of the work in a day ; B and C can do \ of the work in a day ; hence A alone can do \--\z=:^^ of it in a day ; and be wiP therefore require ^^=:1Z\ days, Ans. fix. 46. 1 + 1+1=/^; 1-^^=J^, remainder; tV-tV=3V; $24~3V=$720, Ans. Ex 11 -V^ X aV X V =Hh ^^5- 10 DECIMALS. Ex. 48. Ex. 49. Ex. 50. Ex. 51. Ex.52. i-i=j\ ; 30 feet-^y3__ioo feet, Ans, i+ i =T2> fraction of the post below water, 1-t'2=tV " " " (^^ove " 21-h/^=36 feet, Ans. ^= eldest son's fraction; ^x^=l^= youngest son's fraction; 1 — (-? + If) == Jf = daughter's fraction ; n-if = 4V; ll'723f--/^=$21}U;f ^Twr. DECIMAL FRACTIONS. (145, page 118.) Ex. 1. Ans, .38. Ex. 2. Ans (. Ex. 3. Ans. .325. Ex. 4. Ans 04. Ex. 5. Ans. .016. Ex. 6. Ans. .00074. Ex. 7. Ans. .000745. Ex. 8. Ans. .4232. Ex.9. Ans. .500000. Ex. 10 . Five hundredths; twenty-four hundredths; six hundred seventy-two thousandths; six hundred eighty-one thousandths ; twenty-four thousandths; eight thousand four and seventy-one ten-thousandths; nine thousand thirty-four ten- thousandths; five ten-thousands ; one hundred thousand two hundred forty-eight millionths ; nineteen thousand two hun- dred forty-eight hundred- thousandths ; opc thou&dnd three hundred eighty-five millionths ; one million eighty-pere^ t^D- millionths. NOTATION AND NUMERATION. 41 (I465 page 118.) Ex. 1. Ans, 18.027. Ex. 2. Ans. 400.0000019. Ex. 3. Ans. 54.000054. Ex. 4. Ans. 81.0001. Ex. 5. Ans» 100.0067. Ex. 6. Eighteen, and twenty-seven thousandths; eighty* one, and one ten-thousandth ; seventy-five, and seventy-fiv<, thousandths ; one hundred, and sixty-seven ten-thousandths ; fifty-four, and fifty-four millionths ; nine, and two thousand eight hundred six ten-thousandths ; four hundred, and nine- teen ten-millionths ; three, and three hundredths ; forty, and forty thousand four hundred four hundred-thousandths. (148, page 120.) Ex. 1. Ans. ,000^25, Ex. 2. Ans. .6000. Ex. 3. Ans. .01859. Ex. 4. Ans. .000260008. Ex. 5. Six thousand three hundred twenty-one ten thous- anths ; five million four hundred thousand twenty-seven ten- millionths ; seven hundred forty-eight thousand two hundred forty-three millionths ; sixty million hundred-millionths ; two million nine hundred sixty-two thousand nine hundred ninety- nine ten-millionths ; six hundred-millionths. Ex. 6. Ans. 502.001006. Ex. 7. Ans. 31.0000002 Ex. 8. Ans. 11000.00011. Ex. 9. Ans. 9000000.000000009. Ex. 10. Ans. 10.2. Ex. 11. Ans. 124.315. Ex. 12. Ans. .700. Ex. 13. Ans. .00007. Ex. 14. Twelve, and thirty-six hundredths ; one Lundred forty-two, and eight hundred forty-seven thousanths ; one, and two hundredths ; nine, and fifty-two thousandths ; thirty-two, and four thousandths ; four, and five ten-thousandths ; six- ty-two and nine thousand nine hundred ninety-nine ten- thousandths ; one thousand eight hundred fifty-eight, and four ^2 DECIMALS. thousand five huadred eighty-three ten-thousandths ; twenty seven, and forty-five hundred-thousandths. REDUCTION. (149, page 121 Ex. 2. .I'ZOOOOO Ex.3. .'TOOOOO 24.6000000 .024000 .0003000 .000187 84.0000000 .000500 721.80002'71 108.450000 Ex. 4. 1000.001000 841.'780000 2.600400 90.000009 6000.000000 (150, page 122.) Ex. 2. TV¥o=i» ^^«- Ex. 3. tVo = 2V» ^^«- Ex.4. jWo=m^^^^' Ex.5. y»JV¥o =H» -4^- Ex.6. j^i^^=^^\^^ Arts. (151, page 123.) Ex. 4. Arts. .4. Ex. 6. Ans. .875. Ex. 9. Ans. .375. Ex. 10. Ans. .0375. ADDITION. (152, page 124.) Ex, 6. 26.26 Ex. 7. 36.015 .7 300.0605 6.083 5.000003 4.004 60.0000087 87 047, Ans. 401.0755117, Ans, ADDITION. 43 Ex. 8. 64.34 1.0009 3.000207 .023 8.9 4.135 71.399107; Ans. Ex. 9. 18.375 41.625 35.5 95.500, Ans. Ex. 10 61.843 143.75 218.4375 21.9 Ex. 11. 445.9305, Ans. Ex. 12. ^= 2.5 5f = 5.75 H= 3.625 3.0642 8.925 23.8642 barrels, Ans. 12J =12.75 18f =18.4 9=9 241 =24.125 4|f= 4.8125 8yV= 8.9 15iJ=15.65 93.6375, Ans, Broadcloth. Ex. 13. First suit, 2.125 Second " 2.25 Third " 5.0625 Sums Total Cassimere. Satin. 3.0625 .875 2.875 1.000 1.125 9.4375 5.9375 3.000 9.4375 + 5.9375+3 = 18.375, Ans, 44 DECIMALS. SUBTRACTION. (I535 Pag^ 126.) Ex. 4. 6. 8. 10. 12. 714.000 .916 Ex. 5. Ex. 7. Ex. 9. Ex. 11. 2.000 .298 Ex. 713.084, Ans. 21.004 .75 1.702, Ans. 10.0302 .0002 Ex. 20.254, Ans, 900. .009 10.03, Ans 2000. .002 Ex. 899.991, Ans. 1. .000001 1999.998, Ans. .427 .000427 Ex. .999999, Ans. .34 .034 .426573, Ans. .306, Ans. MULTIPLICATION. \ (1*3^4:5 page 127.) Ex. 4. 274.855, Ans. Ex. 8. 243.6, Ans. Ex. 12. .000030624, Ans. NOTATION AND NUMBKATION. 46 \ DIVISION. (15>5, page 129.) Ex. 5. .111, Ans. Ex. 6. 11.1, Ans. Ex. 8. 15,21 -\-, Ans. Ex. 9. 1; 10; 100; 1000, ^ws, Ex.10. 5,68Ui-,Ans. Ex.12. 3020,^^5. Ex. 17. 3.65, Ans. PROMISCUOUS EXAMPLES. (Page 130.) Ex. 2. 6188.311478, Ans. Ex. 3. 86.913, Ans. Ex. 6. .00012, Ans. Ex. 9. 4, Ans. Ex. 11. 70.6755-T-6.35r^ll.l3, Ans. Ex. 12. tWo -f, ^ns. Ex. 13. 26yVVo=26i, Am, ^■'''^^=-''^^- ^ ,^ .25x17.5 ^ . Ex. 16. —-—=5, Ans. .5x1.75 Ex. 17. 3.625 X 36.75 x $.85=$113.2359375, Ans. Ex.18. 56.925-^4.6 = 12.375 = 12f, ^n«. DECIMAL CURRENCY. NOTATION AND NUMERATION. (I6O5 page 132.) Ex. 2. Ans. $2.09. Ex. 3. Ans. $10.10. Ex. 6. Ans. $.032. Ex. 7. Ans. $100,011 46 DECIMAL CURRENCY. Ex. 8. Seven dollars ninety-three cents/; eight dollars two cents ; six dollars fifty-four cents two mills. Ex. 9. Five dollars twenty-seven cents two mills ; ono hundred dollars two cents five mills; seventeen dollars five mills. Ex. 10. Sixteen dollars twenty cents five mills ; two hun- dred fifteen dollars eight cents one mill ; one thousand dol- hrs one cent one mill ; four dollars two mills. REDUCTION. (161, page 133.) Ex. 2. Ans. 3600 cents. Ex. 3. Ans. 524800 cents. Ex. 6. Ans. 160 mills. Ex. 1. Ans. 3008 mills. Ex. 8. Ans. 890 mills. (162, page 134.) Ex. 2. Ans. $15.04. Ex. 3. Ans. $138.75. Ex. 4. Ans. $16,525. Ex. 5. Ans. 52.4 cents. Ex. 6. Ans. $6,524. ADDITION. (163, page 134 .) Ex. 2. $ 50.07 Ex. 3. $ 364.541 1000.75 486.06 60.003 93.009 .184 1742.80 1.01 Ans. 3.276 25.458 $2689.686, - $1137.475, ADDITION. Ex. 1 $ .92 Ex. 5. $89.74 .104 13.03 .357 6.375 .186 19.625 .125 .99 $ai2e Ans. Ex. 6. $ 9.17 Ex. 7. $2175.75 .875 240.375 .0625 605.40 .04 140.125 .08 .11 $3161.65, Ans. $10.3375, Ans. Ex. 8. $ 6.08 Ex. 9. $7425.50 26.625 253.96 16.000 170.09 7.40 156.105, Ans. $7849.55, Ans. Ex. 10. $3,625 1.75 1.375 .625 .875 $8.25, Ans. 4^ DECIMAL CURRENCY. SUBTRACTION. (164 5 page 136.) Ex. 2. $365,005 Ex. 3. $50. 267.018 .60 $97,987, Ans. $49.50, Ans. Ex. 4. $100. Ex. 5. $1000. .001 .037 $ 99.999, Ans. $ 999.963, Ans. Ex. 6. $1834.16 Ex. 7. $145.27 1575.24 37.69 $ 258.92, Ans. $107.58, Ans. Ex. 8. $6.84 Ex. 9. $14725 5.625 $3560-|-$'3^015.875=:10575.876 $1,215, Ans. $4149.125, Ans Ex. 10. $13.75 5.25 1.375 .875 $25-$21.25=: $3.75, Ans. Ex. 11. $480 80.50 $560.50- -$200=$360.50, Ans. MULTIPLICATIOK (16^5 page 137.) Ex. 2. $4,275 X 300=:$1282.50, Ans. Ex. 3. $2.45 X 176=:$428.75, Ans. DIVISION. 49 Ex. 4. 11.28 X 800:=$1024, Ans. Ex.5. $.15 x3Y2=:$55.80 .125x434= 54.25 .33 X 16= 5.28 $115.33, Ans. Ex. 6. $.56 X 3=:$1.68 .07x15= 1.05 .08x27= 2.16 $5--$4.89 = $.ll, Ans. Ex. 7. $.375 X 125 =$46,875 .09 X 75 = $6.75 .60 X 12= 7.20 = 13.95 $32,925, Ans. Ex.8. $32.50 X 80 r 34.25x70= 2397.50 $4997.50 3975 $1022.50, Ans. DIVISION. (166, page 138.) Ex. 2. $41.25-^33=$1.25, Ans. Ex. 3. $94.50-^27=$3.50, ^715. Ex. 4. $136-h64 = $2.125, Ans. Ex. 5. $1.32-^$.12 = 11, Ans. Ex. 6. $405-f-$.54 = 750, Ans. Ex. 7. $180-^12=$15, Ans. Ex. 8. $2847.50 -M00=$28.475, -4715. Ex 9. $80.46-j-894=$.09. Arts. Ex. 10. $1,125 x 120=$135 ; $135-4-27=$5, Ans. K.P, 3 so DECIMAL CUiaiENCY. Ex. 11. $3.20 X 4=:$12.80 .08x37= 2.96 $15.76 6.80 $8.96-^$.16=56, Ans. Ex. 12. $4.50 4- $2.75 =$7.25 ; $166.75-^$7.25 = 23, Ans, Ex. 13. $18.48-M54=:$.12, Ans. Ex. 14. $560 106.75 $453.25-M4=$32.37i, Ans. ADDITIONAL APPLICATIONa (I685 page 139.) Ex.2. 693x$i=$321, ^7i6\ Ex.3. 478x^=$2S9, Ans. Ex. 4. 4266 X $yV=$355.50, Ans, Ex. 5. 1250 X $i=$156.25, Ans, Ex. 6. 3126 X $yV=$195.375, Ans. Ex. 7. 1935 X $i=$322.50, Ans. Ex. 8. 56480 x $^=$7060, Ans. Ex. 9. 1275 X $i=$255, Ans, (169, page 140.) Ex. 2. $.09 X 864=177.76, Ans. Ex. 3. $1.25 X 87=$108.75, Ans. Ex. 4. $1.45 X 400 = 1580, Ans, Ex. 5. $.44 X 52 X 16=$366.08, Ans. (170, page 141.) Ex. 2. $l75-4-25=$7, Ans, Ex. 3. $200-^48=$4.16|, ^/^^. ADDITIONAL APPLICATIONS. 51 Ex. 4. $1200-f-96 = $12.50, Ans. Ex. 5. $56.25-f-10 = $5.62J, Arts. Ex. 6. m.^O-MS^r^-GS, Ans. Ex. V. llO.O'Z-f-SS^S.lQ, Ans. Ex. 8. $1016-T-800 = $1.2Y, Ans. Ex. 9. $8'74.65-^343=$2.55, Ans. Ex. 10. $684.3'75-^36o=:$1.8'75, Ans. (in, page 142.) Ex. 2. $5.55-f-$.15=37, ^lw5. Ex. 3. $2 16 ^$12 — 18, Ans. Ex. 4. $21Y8.'75-^$1.25 = 1'743, ^W5. Ex. 5. $643.50^$19.5=ir33, ^?^5. Ex. 6. $52.65-f-$.45 = ll'7, ^?i5. Ex. Y. $6336-f-$132=:48, ^n5. ^ Ex. 8. $117715-^165 = 1811, ^w«. (ITS, page 143.) Ex. 2. $4.50 X 42.65 = $191.925, Ans. Ex. 3. $.85 X 24.89 = $21.156.+, Ans. Ex. 4. $17.25 X 7.842=$135.274+, ^w«. Ex. 5. $12.50 X 23.48 =$293.50, Ans. Ex. 6. $3 X 1.728=$5.184, Aiis. Ex. 7. $7 X 2.40 =$16.80 6.40 X .865= 4.671 .80X12.56 = 10.048 $31,519, Alls. Ex. 8. $4,375 X 14.76 =$64,575, Ans. (173, page 144.) Ex.2. $7-4-2 = $3.50; $3.50 X 1.495=$5.2826, Ans. 52 DECIMAL CURRENCY. Ex. 3. $S.75~-2=z$4.Sl5; $4,375 X .325 = $1.421 -f , Ans. Ex. i. $3.84-^2=$1.92; $1.92 X 3.142=$6.032 +, An8. Ex. 5. $5.60-^2=$2.80; $2.80 X 1.848 = $5.1744, Ans. Ex. e $18-^2=$9; $9 X 125 X .148=r:$33.30, Ans. Ex. 7 $3.054-2r:r$1.525; $1,525 X 31.640=$48.251, Ana, (174.5 page 145.) Ex. 1. $3.60 X '7=r$25.20 1.125 X 9= 10.125 •.90 xl2= 10.80 1.375x24= 33.00 .65 x32=: 20.80 Ex. 2. Ex. 3. $99,925, Ans. $3.75 X 67 = :$251.25 2.62 xl08 = : 282.96 1.12 X 75 = : 84.00 .86 X 27 = : 23.22 .70 X 35 = : 24.50 1.04 X 50 = : 52.00 $717.93, Am. $.07 X325: =$22.75 .0625x148: = 9.25 .05 X286: = 14.30 .125 X 95 = = 11.875 2.75 X 50: = 1-37.60 3.625 X 75: = 271.875 2.85 X 12r = 34.20 $501.75, Ans. PROMISCUOUS EXxiMPLES. ti Ex. 4. $15 X 20 =8300. 9.50 X 7.5 = 71.25 C.25 X 10.75 = C7.1875 2.625 X 3.96 = 10.395 3.00 X 5.287 = 15.801 $464.6935, Atu Ex.6. $.11 X 25=12.75 .625 X 5= 3.125 .0625x26=: 1.625 .42 X 4= 1.68 .09 x46= 4.14 .14 x30= 4.20 .04 X 6= .24 .12 X 4= .48 $18.24, Ans. PEOMISCUOUS EXAMPLES. (Page 146.) Ex. 1 $124.35 X 62.75=$7802.9625, Am. Ex. 2. $.17x15=82.55, ^?i5. Ex. 3. $1406.25-v-2250=$f, Arts. Ex. 4. $48.96-M2=$4.08, Ans. ^ Ex. 5. 325 miles x .45 = 146.25 miles, Ans, Ex. 6. 657-^36.5 = 18,^725. Ex. 7 $105+$125 + ($35x4)=$370 $400 — $370 =$30, Ans. Ex. 8. $19— $15 = $4; $4x28=$112, ^?i«. Ex. 9. ■V-X2\ = ¥-='^A •^^^• Ex. 10. $9-^$.3125 = 28.8, Ans. Ex. 11. $3.50 X 365=$1277.50 $2000-1277.50 = $722.50, ^/w?. M DECIMALS. Ex. 12. $687.25 + $943.64=:$1G30.89 $1630.89 — $875.29 = $755.60, Ans. Ex. 13 $l728-7-2=:$864 1st half sold for; 144x8 = 1152 2d " " " $2016,^725. Ex. 14. $3.75 X .875 = 13.281 +, Ans. Ex. 15. $65.42 — $46.56 = 118.86, gain per head; $3526.82-^-$18.86 = 187, Ans, Ex. 16. $54.72-^36.48 = $1.50 ; $1.50 X 14.25 = 821.375, Ans. Ex. 17. $3548-^4 = $887, Ans. Ex.18. 112.34-^$.82 = 137, ^n5. Ex. 19. $3461.50-^46 = $75.25 ; $75.25 X 5 = $386.25, A718. Ex. 20. $24000 X | x j = $3200, Ans. Ex.21. $1.25x160= $200 $5 X 80= 400 $600 $2.50x240= 600 Loss 000, Ans, Ex. 22. $1.70x48 = 881.60 72.90 $ 8.70, Ans. Ex.23. 122|4-75i = 197|; 197f — 60 = 137f ; $.9375 — $.8125 = $.125, loss per bushel; $.125 xl37f = $17.218 -f loss; 12.50 gain; $4,718 + , loss, Ans. Ex 24. $1.40 X e =$8.40 wages ; $ .75x7= 5.25 expenses, $3.15 savings, Ans. PROMISCUOUS EXAMPLES. 55 $.08 X 39 Ex.25, -jj^ =19^, Ans. Ex. 26. $4.50 X 23.487=:$105.6915, Ans. Ex. 27 $1200-^365:=z$3.2874f, Ans. Ek 28. $.17 X 56 X 28 = 1266.56, ^W5. Ex 29. $.07 X 26 X 13 X 16=$378.56, Ans, Ex. 30. $4.75 X 4.868=$23.123, Ans. Ex. 31. $.33ix27 = $9.00 .25 x28= 7.00 .50 xl9= 9.50 $25.50, Ans. Ex.32. 44—32 = 12; — - — -=21^ minutes, Ans, 12 ^ ' $32.3 4 15 ^ Ex. 33. — -- X — X —=$51, Ans. J It/ z Ex 34. $5.d35-f-.875=$6.44; $6.44 x 9^ =$59.57, Ans. Ex. 35. $5000 $1200.25 x3 = $3600.75 1800.62x8= 5401.86 $14002.61 $950.87x2= 1901.74 $12100.87, Ans. Ex 36. $4.50xl86.40=$838.80, J7i5. Ex. SS. $96.40-v-2=$48.20 ; $48.20 X 1.375=$66.275, Ans. Ex.39. ,Vo¥t=JL, ^W5 5Q COMPOUND NUMBERS. Ex.40. 3\=.09375; .62i=.625; .37y^ ::^,370625 ; |=::.375 ; .09375 + .625 + .370625 + .375 = 1.464375, Ans. Dr. Cr. Ex. 41. $4,745 $2,765 2.625 1.245 1.27 .625 .45 3.45 5.285 1.875 Ex. 42. $14,375 - $9.9C $.125 X 120=115.00 .625 X 18= 11.25 .07 X 47= 3.29 .18 X 6= 1.08 J=$4.415, Ans. $1.50 1.27 1.87 2.30 $30.62 - $6.94 =$23.68, • REDUCTION. (183, page 152.) Ex. 1. 14194 far.-^4 = 3548 d. 2 far. ; 3548 d.-^12 = 295 s. 8 d. ; 295 s.-f-20=14 £ 15 s. Ans, 14 £ 15 s. 8 d. 2 far. Ex. 2. 14£x 20 + 15 s.=295s.; 295s. xl2 + 8d.=3548d,: 3548d.x4 + 2 far. = 14194 far., Ans. Ex.3. 15 £x 20 + 19 s. = 319 s.; 319 s.xl2 + lld. = 3839 d. ; 3839 d.x 44-3 far.= 15359 'far., Ans Ex. 4 15395 far.-~4 = 3839 d. 3 far.; 3839 d.-7-12 = 319 s. 11 d. ; 319 s.-v-20 = 15£ 19 s. Ans. 15 £19 s. 11 d. 3 far. Ex.5. 46 sov.x 20 + 12 s. = 932s.; 932s.xl2+2d. = 11186 d., Ans REDUCTION. 57 Ex. 6. 11186 d.-M2 = 932 s. 2d.; 932 s.-f-20==:46 sov 12 s. Ans. 46 sov. 12 s. 2 d (IStJ, page 153.) Ex.3. 5lb. X12 + 7 oz.=:67 oz. ; 67 oz.x 20 + 12 pwt.^ 1352 pwt.; 1352 pwt. x 24 + 9 gT.=: 32457 gr., Aiis Ex.4 43457 gr.-^24=rl810 pwt. 17 gr. ; 1810 pwt.^ 20 = 90 oz. 10 pwt.; 90 oz. +-12 = 7 lb. 6 oz. Ans, 7 lb. 6 oz. 10 pwt. 17 gr. Ex. 5. 41760 gr. -^24 = 1740 pwt. ;l740pwt.-^20=::87oz.; 87 oz.-r-12i=7 lb. + 3 oz., Ans, Ex. 6. 14 lb. 10 oz. 18 pwt. = 3578 pwt. ; 3578 X ^$.75=$2683.50, Ans, Ex. 7. 5 lb. 6 oz.=1320 pwt. ; 2 oz. 15 pwt.=:55 pwt. ; 1320-^55 = 24, Ans, Ex. 8. 1 lb. 1 pwt. 16 gr.=5800 gr. ; 4 pwt. 20 gr.= 116 gr. ; 5800-M16=:50 ; $1.25 X 50 ==$62.50, Ans, (I865 page 155.) Ex.3. 16 lb. X 12 + 11 oz. = 203 oz. ; 203 oz. x 8 + 7 dr. = 1631 dr.; 1631 dr.x3 + 2 sc. =4895 sc; 4895 sex 20+19 gr. = 97919 gr., Ans, Ex.4. 471b. X 12 + 6 | =570 3 ; 570 | x8 + 4 3 = 4564 3 ; 4564 3 x 3 = 13692 3, Ans. Ex. 5. 20 gr.x 5x365 = 36500 gr. ; 36500 gr.-^20 = 1825^; 1825 3-^-3 = 608 31 3; 608 3 +-8 = 76 5 ; 76 | -M2 = 6 lb. 4 ^ . Ans, 6 lb. 4 3 13. (187 5 page 156.) Ex.3. 3 T.X20 + 14 cwt.=74 cwt. ; 74 cwt.xlOO + 74 lb.=7474 lb.; 7474 lb. x 16 + 12 oz.= 11959e oz.; 119596 oz.x 16 + 15 dr.= 1913551 dr., Ans. 68 COMPOUND NUMBERS. Ex. 4. 1913551 dr.~16 = 119596 oz. 15 dr.; 11959G 02. -M6 = 7474 lb. 12 oz. ; 7474 lb. -^100=74 cwt. 74 lb.; 74 cwt.-^20=:3 T. 14 cwt. Ans, 3 T. 14 cwt. 74 lb. 12 oz. 15 dr. Ex 5. 3 T. 15 cwt. 20 ]b.=:7520 lb. Ans. $.22 X 7520=:$1654.40. Ex. 6, 115 lb.-=-2000=.0575 T. ; $10 x .0575=i$.575, Am Ex. 7 217 lb. X 10 = 2170 lb. @ $.06 =$130.20 306 1b. X 5=:1530 lb. @ $.07A = 114.75 3700 lb. @ $.08 =$296.00, which — $244.95 =$51.05, Ans. Ex. 8. 2 T. X 2000 = 4000 lb. ; 4000 x $.12i-=$500 ; $500 — $360 = $140, Ans. Ex. 9. 10 T. X20 + 6 cwt. = 206 cwt.; 206 cwt. x 4 + 3 qr = 827 qr. ; 827 qr.x 28 + 14 lb.=23l70 lb. $.06 buying price. $130-r-2000 = .065 selling price. $.005 gain per pound. $.005x23l70 = $115.85, Ans. Ex. 10. 2352 lb.-v-56=42 bu.; $.90 X 42 X 2 = $75.60, Ans. Ex. 11. 300 bbl. X 196 = 58800 lb., Ans. Ex. 12. $1.25 X 3 = $3.75 cost^ $.0075x280x3= 6.30 $2.55, Ans. REDUCTION. 59 (191, page 157.) Ex. 1. 6 lb. 10 oz.=90 oz. ; $.50 x 90 = $45.0C .ost. $J2_x_8_><_437^ x^^^^8.Y5 sold lor. 480 $33. 75, Ans. Ex. 2. 424 dr. -^8 =53 oz. ; 53 oz.-t-12 = 4 lb. 5 oz., Jps Ex. 3. 20 lb. 8 oz. 12 pwt. = 119328 gr. 119328 gr.^7000=:l7gV3 lb., Ans. Ex.4. $.40x16x20 =$128 cost $.50x320x437.5 480 .=1 145.83^ $ 17.831, Ana, (193, page 159.) Ex. 3. 7912 mi. x 63360 = 501304320 ia., Ans. Ex.4. 168i74 ft.-^3 = 56158 yd.; 56158 yd. -^5^ = 10210 id. 3 yd.; 10210 rd. -4-40 = 255 fur. 10 rd. ; 255 fur.-^-8 = 31 mi. 7 fur. Ans. 31 mi. 7 fur. 10 rd. 3 yd. Ex.5. 31 mi.x84-7 fur.=255 fur.; 255 fur.x40-fl0 rd.=10210 rd.; 10210 rd. x 5i + 3 yd.=56158 yd.; 56158 yd. X 3 = 168474 ft., Ans. Ex.6. 2500 fathoms X 6 = 15000 ft.; 15000 ft. -^161 = 909 rd. l|ft. ; 909 rd.-4-40 = 22 fur. 29 rd. ; 22 fur.-f-8 = 2 mi. 6 fur. Ans. 2 mi. 6 fur. 29 rd. 1| ft. Ex. 7. 2200 mi. X 5280 = 11616000 ft.; $.10 X 11616000 = $1161600, Ans. Ex.8. 4 fathoms X 6 + 3 fl.=27 ft.; 27 ft. xl2+8in.= 832 in Ans. Ex. 9 200 mi. = 12672000 in. ; 18 ft 4 in = 220 in. ; 12672000-4-220 = 57600 times, Ans. Ex. 10. 120 lea. x 3 = 360 geo. mi ; 360 geo. mi x 1.15 - 4 14 Eng. mi., Ans, Ex. 11. 141 hands x 4=58 in., Ans. 60 COMPOUND NUMBERS. (1045 page 160.) Ex.1. 3 mi. X 80 + 51 d\.^291 cb. ; 201 ckxlOO-i- 1S l. = 291'73 1., Ans. Ex, 2. 291'73 l.-^100--291 ch. 73 1. ; 291 ch,-v-80 = 3 ml 51 ch. Ans. 3 mi. 51 ch. 13 L Ex 3. 17 cL 31 1. = 17.31 ch. 12 ch. 87 l. = 12.87 ch. 30.18 ch. half romid the field. 30.18 ch. X 2 X 66 = 3983.76 ft., Ans, (lOGj page 163.) Ex.3. 87 A.X4 + 2 R. = 350 R. ; 350 R. x 40 -f-38 sq. rd = 14038 sq. id. ; 14038 sq. rd. x 30^ + 7 sq. yd.==424656i sq yd.; 4246561 sq. yd. x 9 + 1 sq. ft. = 3821909i sq. ft.; 38219091 sq. ft. X 144 + 100 sq. in=:550355068 sq. in., Ans, Ex. 4. 550355068 sq. in. +-144 = 3821910 sq.ft. 28sq.in.; 3821910 sq. ft. -+9 = 424656 sq. yd. 6 sq. ft. ; 424656 sq. yd.-+ 301 = 14038 sq. rd. 61 sq. yd.; 14038 sq. rd.+-40 = 350 R. 38 sq. rd.; 350 R.+-4 = 87 A. 2 R. Ans, 87 A. 2 R. 38 sq. rd. 6i sq. yd. 6 sq. ft. 28 sq. in. But (-J- sq. rd.)=:4 sq. ft. 72 sq. in. Hence, Ans, 87 A. 2 R. 38 sq. rd. 7 sq. yd. 1 sq. ft. 100 sq in. Ex. 5. 100 X 30 = 3000 sq. rd.=:18 A. 3 R., Ans, Ex. 6. 4 mi. x 320 = 1280 rd., Ans, Ex. 7. 2 mi. x 320 = 640 rd., Ans, Ex. 8. 100000 sq. ft. +- 9 =11111 sq. yd. 1 sq. ft.; IIIU sq. yd.+-30i = 367 sq. rd. 91 sq. yd 367 sq. rd.+-40 = 9 R. 7 sq. rd. 9 R.+-4 = 2 A. 1 R. Ans. 2 A. 1 R. 7 sq. rd. 9i sq. yd. 1 sq. ft. ; or 2 A 1 R. 7 sq. rd. 9 sq. yd. 3i sq. ft. REDUCTION. 61 Ex. 9. 181x16 = 296 sq.ft.; 296 sq. ft.-f-9=:32f sq. yd., Ans. Ex. 10. (18 4- 161) X 2 = 69 ft., distance round the room ; 69x9 -69 sq. yd., in tlie walls; 9 18x161 = 83 sq. yd., in ceiling; 69 sq. yd. + 33 sq. yd. = 102 sq. yd. $.22xl02=:$22.44, Ans, Ex. 11. 40x20x2 = 1600 sq. ft. = 16 squares; $10xl6=$160, Ans. (lOr, page 164.) Ex. 2. 3686400 P.-M02400 = 36 sq. mi., Ans. Ex. 3. 94 A. X 10 + 7 sq. ch. = 94'7 sq. ch. ; 947 sq. ch.xl6 + 12 P.= 15164 P.; 15164 P. X 625 4-118 sq. 1 = 9477618 sq. l, A^^s. Ex. 4. 4550000 sq. l.-M0000 = 455 A. $50x455=$22750, Ans, (lOO, page 166.) Ex. 1. 125 cu. ft. X 1728 + 840 cu. in. = 216840 cu. in., ^ai5. Ex.2. 5224 cu. ft.-M28=40if Cd., ^715. Ex. 3. 3ft.2in.=38in.; 2ft. 2 in.=26 in.; 1 ft. 8 in. =20 in.; 88 X 26 X 20 = 19760 cu. in., Ans. Ex. 4. 6x6x6 = 216 cu. ft.; 216 cu. ft. X 1728 = 373248 cu. in., Ans. Ex. 5. 60 X 20 X 15 = 18000 cu. ft. ; 18000 cu. ft.-^128 = 140| Cd., Ans. Ex. 6. 10 X 31 X 3i = 113f cu. ft., Ans, Ex. 7. 128^(3 X 12) = 3f ft. high, Ans 62 COMPOUND NUMBERS. Ex. 8. 21x115 lb.=4725 lb.=2 T. 1 cwt. 25 lb., Ann Ex. 9. 32 ft. + 24 ft.=i56 ft.; 56 ft. x 2=^112 ft. girt; 112 X 1| X 6 = 1008 cu. ft.; 1008 cu. ft.-^24^—A0j\ Pch. $1.25 X 40y\=:$50.909 -f , Arts. 32 X 24 X 6 Ex.10. $.15 X — =$25.60,^715. Ex. 11. 10x9x8 = 720 cu. ft.; '720+-10 = Y2 minutes, Ana. Ei. 12. 30 X 20 X 10=6000 cu. ft. ; 6000 ,, . , -=12 minutes, Arts, 50x10 ' (SOO, page 168.) Ex. 3. 3 hhd. X 2016 = 6048 gi., A7is, Ex. 4. 6048 gi.+2016 = 3 hhd., Ans. Ex.5. 13 hhd. X 63 + 15 ga]. = 834 gal.; 834 gal. x 4 x 1 qt. = 3337 qt. ; 3337 qt. x 2 = 6674 pt., Ans. Ex. 6. 6674 pt.-^2 = 3337 qt. ; 3337 qt.-v-4 = 834 gal. 1 qt.; 834 gal.-v-63 = 13 hhd. 15 gal. 13 hhd. 15 gal. 1 qt. Ans. Ex. 7. 1 hhd.=2016 gi. ; $.06 x 2016=$120.96, Ans, Ex. 8. $2 X 10 = $20 cost ; $.05x 4 x 31^ x 10=$63 reed. $63 — $20 = $43 gain, Ans. Ex. 9. $3.84-T-$.06 = 64 pt.=8 gal., Ans. Ex. 10. 2 gal. 2 qt. 1 pt.=21 pt.; 1 hhd.=504 pt. ; 604-4-21 = 24, Ans. (SOI, page 169.) Ex.1. 49 bu. x4 + 3 pk.=199 pk. ; 199 pk. x8 + 7 qt,= 1599 qt. ; 1599 qt. X 2 + 1 pt.=3199 pt., Ans. REDUCTION. 68 Kx. 2. 3199 pt.-^23:=1599 qt. 1 pt. ; 1599 qt.-^8 = l99 pk. 7 qt; 199 pk.^4r=:49 bu. 3 pk. Ans. 49 bu. 3 pk. 7 qt. 1 pt. Ex. 3. 1 bu. x4 + l pk.=5 peck; 5pk. xS + l qt.— 41 qt, ; 41 qt.x2 + l pt.=83 pt., Ans. Ex. 4. 83 pt.-^2rr41 qt. 1 pt.; 41 qt.-f-8=:5 pk. 1 qt. 5 pk.-^4 = l bu. 1 pk. Ans, 1 bu. 1 pk. 1 qt. 1 pt. Ex. 5. $.Q5 X 60=$32.50 cost ; $.25 X 4 X 50 = $50.00 sold for ; $17.60 Ans. (205, page 170.) Ex. 1. 1 bu. (Dry Measure) =21501 cu. in.; 2150| cu. in.-^57j = 37;^J wine quarts; 37±| qts.— 32:=5i| qts., Ans, Ex. 2. 40 qt.-f-4=rl0 gal. ; 10 gal. x 282 = 2820 cu. in. , 2820 cu. in.-^57f = 48|| qts. Wine Measure : 48^ qts.~40 qts. = 8f| qts., Ans, Ex. 3. 1 bu. Dry Measure =2150f cu. in. 32 qt. Wine Measures 1848 cu. in. 302| cu. in., Ans. (206, page 171.) Ex. 1. 365 da. X 24 4-5 h.=8765 b. ; 8765 h. x60f48 min.=525948 mi»;t 625948 min.x 60+46 se us=s 31556926 sec, Ans, Ex. 2 31556926 sec. -^60 = 525948 min. 46 sec; 525948 min-v-60 = 8765 h. 48 min.; 8765 h.-^24 =365 da. 5 b. Ans. 365 da. 5 b. 48 min. 46 sea 64 COMPOUND NUMBERS. Ex.3. 5 wk. x7 + l da. = 36 da.; 36 da. x 24+1 h,= 865 h. ; 865 li. x60 + l niin.:=51901 mm.; 51901 min. x60 + l scc. = 31 14061 sec, Ans, Ex. 4. 3114061 sec. -^60 = 51901 mm. 1 sec; 51901 mm. -^60:=865 h. 1 min.; 865 h.-^24 = 36 da. 1 h. ; 36 da.-^-7 = 5 \vk. 1 da. Ans, 5 wk. 1 da. 1 h. 1 min. 1 sec Ex. 6. 10 mi.=:l'7600 yd.; 17600 sec-^60==293 min. 20 sec ; 293 min.-^60 =4 h. 53 min. Ans, 4 h. 53 min. 20 sec Ex.7. 29 da.x24+12 li.=708 h. ; 708 h. x 60 + 44 min, =42524 min. ; 42524 min. x 60 + 3 sec = 2551443 sec, Ans, Ex.8. 40 yr.x365^z=:14610 da.; 14610 da. x45 = 657450 min. gained. 657450 min.-f-60=rl0957 h. 30 min.; 10957 h.-^ 24=456 da. 13 li. Ans, 456 da. 13 h. 30 min. (207, page 173.) Ex.1. 10 S. x30 + 10°=310°; 310° x 60 + 10'=18610' ; 18610'x60 + 10''=1116610^ Ans, Ex. 2. 1116610'' -^60 = 18610' 10''; 18610'-^60 = 310° 10'; 310°-^30=:10 S. 10°. Ans. 10 S. 10° 10' 10". Ex. 3. 11400'-^60 = 190°, Ans, Ex. 4. 190° X 691 = 13148 miles, Ans. Ex. 5. 360° X 60 = 21600', Ans, Ex. 6. 397'-^60=6° 37', Ans, (210, page 174.) Ex.1. 150000000-^12 = 12500000 doz.; 12500000 doz. -^12 = 1041666 gross 8 doz.; 1041666 gross-^12 = 86805 great gross+6 gross, Ans, 86805 great gross 6 gross 8 doz. REDUCTION. 65 Ex. 2. 100000 sheets-^24r=4166 quires 16 sheets; 4166 quires-^20 = 208 reams 6 quires; 208 reams-^2r=104 bundles ; 104 bundles-4-5 — 20 bales 4 bundles. Ans. 20 bales 4 bundles 6 quires 16 sheets. Ex, 3. 20 years x 4 + 10 years =90 years, Ans. Et 4. 8 sheets x 8=64 leaves ; 64 leaves x 2 = 128 pages, Ans, Ex. 5. 32 pages xl0x2 = 640 pages, Ans, PROMISCUOUS EXAMPLES IN REDUCTION. Ex. 1. 6 yd. 3f qr. = 27J qr.; 333 yd. = 1332 qr. ; 1332-^273 =48 suits, Ans, Ex. 2. 1 oz. 15 pwt.i=:35 pwt. ; I.YOx 35 =$24.50, Arts. Ex. 3. 2 lb. 3 3 5 3 13 10 gr.=13290 gr.; 13290-^15 = 886, Ans, Ex. 4. 1 T. 11 cwt. 12 lb. = 3112 lb.; 3112x$.01i=$38.90, Ans, Ex. 5. 1456 lb.-^32 = 45.5 bu. ; $.375 X 45.5=$17.0625, Aiis. Ex. 6. 45 lb. X 1000 = 45000 lb. 45000 lb.-M96 = 229 bbl. 116 lb., Ans. Ex. 1, 2430 lb.-T-60c±40.5 bu. ; $1.20 X 40.5=$48.60, Ans, Ex. 8. $12.50-^200=$.06i, Ans, Ex. 9 360° X 69.15=24897.6 stat. mi. ; 24897.6 X 63360=1577511936, Ans. Ex. 10. 10 mi. X 80 + 7 ch. + l eh. (4 rd.)=808 ch. ; 808 nh. X 100 + 20 1.= 80820 L, Ans. 1j6 compound numbers. Ex. 11. 25 X 100 X 144 = 360000 sq. in. ; $.01 X 360000 = 13600, Ans. Ex. 12. 50 X 25 X 10 = 12500 cu. ft. ; 12500 cu. ft. -^1 6 = 781 cd. ft. 4 cu. ft.; '781 cd. ft.^8=97 Cd. 5 cd ft. ; Ans, 97 Cd. 5 cd. ft. 4 cu. ft Ex 13, 10 X 10 X 10 X 1728 = 1728000 cu. in. ; 1728000 cu. in.-v-231 = 74804^ gal.; 1i80^ gal.-^63 = 118 hhd. 464-f gal., ^W5. Ex. 14. 8 X 5 X 41=180 cu. ft.=311040 cu. in. ; 311040 cu. in.-T-2150.4 = 144y\ bu., Ans. Ex. 15. Mar. 31 da. June 30 da. Sept. 30 da, Apr. 30 da. July 31 da. Oct. 31 da. May 31 da. Aug. 31 da. Nov. 30 da. Spring, 92 da. Summer, 92 da. Autumn 91 da. 92 da.— 91 da. = l da. = 86400 sec, Aiis. Ex. 16. 1296000 sec.-=- 86400 = 15 da., Ans. ^ ,^ 20x13 ,^ , . Ex. 17. =40 yd., Am. Ex 18. 4 reams X 20 + 10 quires= 90 quires; 90 quires x 24 + 10 sheets=2l70 sheets, Ans. Ex.19. 16 ft. 6 in.=l rd. ; 1 mi. = 320 rd. ; 320^1 = Ifto times in 1 mi. 320 x 42 = 13440 times, Ans. )|Ex. 20. 1000000 sec.-^60 = 16666 min. 40 sec. ; 16666 min.-f-60 = 277 h. 46 min. ; 277 h.-f-10 = 27 da. 7 h. ; Ans. 27 da. 7 h. 46 min. 40 sec. Ex. 21. 6x4^ =27 sq. mi ; — — — =216 farms, Ans. Ex. 22. 10 mi. 176 rd.=3376 rd. $21.75 x 3376 ^$73428, Ans. REDUCTION. til (21 1 5 pagelTG.) Ex. 2. j-oV f iJ X V X -V = 2«5 d., A71S, Ex. 3. y^ ^ Wk. X I X V- X -¥- = f ^^^-j ^'^*- Ex. 4. y-,V2 ^^^^' X r- X f X f- X f - 1 gi., ^ws. Ex 5. ^J-o oz. X -/ X V — i gr., J.71S. Rv fi 1 mi X ^ X ^-" X -3-3- X J^= J-9-®-, in.. Ans Ex. 7. f X i X f lb. X Y =1 oz-j ^^^5« Ex. 8. elo hhd. X V- X f X 2. = f J pt., ^?^5. Ex. 9. yyVo A. X f X V-=i- rd., ^n5. (SIS, page 177.) Ex. 2. i ft. X 3=3 = ri2 r^- ■^^^^• Ex. 3. f dr. X j\ X tV^tAo lb., ^7^5. Ex. 4. J ct. X y^Vo = 2 oVo E., ^?^5. Ex. 5. i ft. X 3 J-^=y^i^^ mi., Ans. Ex. 6. -f X t pwt. X ^V X T2 = 2 « lt>., Ans. Ex.7, f pt.x-Lxix^V=8io l^bd. ^ hhd.—- gJ liiid.=:f J-f hhd., Ans. Ex. 8. I in. X c 3 k =To jVo i^i-, ^'^s- Ex. 9. f oz. X tV=:2\ lb— 2T <^^' 2 lb. ; and ^V of 2 lb. is I of f of 2V of 2 lb., or I of ^ of 2 lb., Ans. Ex. 10. f oz. X tV=2"t lb. = 2T of 2 lb. ; and ^\ of 2 lb. ia J. of f of J, of 2 lb., or I of f of 2 lb., Ans. (2185 page 178.) Ex.2. 4mo.x30 = l7| da.; -Ida. x24 = 3f h. ; ^ h. x 60 = 254 rain. ; 4 min. x 00 = 424 sec. A71S. 17 da. 3 b. 25 min. 42^ sec. Ex. 3 |£ X 20 = 84 s. ; 4 s. X 12 = 64 d. ; 4 d. x 4 = 34 ^ A71S. 8 s. 6 d. 34 far. 68 COMPOUND NUMBERS. Ex. 4. § bu. x4 = lj pk. ; J pk.x8 = 4f qt.; f qt x2-= 1| pt. Ans, 1 pk. 4 qt. If pt. Ex.5. 4 of 15 cwt. = 124 cwt. ; f cwt. xl00=:85f lb.; f lb. X 16 = llf oz. ; f oz. x 16 = 6^ dr. Ans. 12 cwt. 85 lb. 11 oz. Of dr. Ex.6. |x-}x V^3=4|f oz. ; |^ oz. x IGnzllf^ di. Ans. 4 oz. 11|^ dr. Ex. 7. I A. X 4 = 3i R. ; ^ R. x 40 = 131 P. ^m\ 3 R. 131 p. Ex. 8. If da. X 24r=16/3 h. ; j\ b. x 60 = 36|| min. ; \i inin. X 60 = 55 j^^ sec. Ans, 16 b. 36 min. 55-^^^ sec. Ex. 9. I lb. X 12 = '7i oz. ; i oz. x 20 = 4 pwt. Ans, 7 oz. 4 pwt, Ex. 10. i ofVT.=4/_ T. ; /^ T.x20=5f cwt; f cwt X 100 = 55f lb. Ans. 4 T. 5 cwt 55| lb. Ex. 11. I ofY A. = lf A.; f A. X 4 = 11 R. ; i Rx40 = 20 P. Ans. 1 A. 3 R. 20 P. (214, page 179.) Ex. 2. 6 fur. 26 rd. 3 yd. 2 ft=4400 ft ; 1 mi.=5280 ft; UU mi.=f mi., Ans. Ex. 3. 13 s. 7 d. 3 far.=655 far. ; l£ = 960 far. ; 960 ^ — 192 "^i -^rx/Cis, Ex. 4. 10 oz. 10 pwt 10 gr. = 5050 gr.; 1 lb. = 5760 gr.; HU Ib.-Hf lb., Ans. Ex. 5. 2 cd. ft 8 cu. ft =40 cu. ft ; 1 Cd. = 128 cu. ft ; tVV Cd.=tV Cd., Ans. Ex.6. lbbl.lgal.lqtlptlgi. = 1053gi.; lLbd.=2016gi»^ i^fl bbd.=iif hbd., Ans. Ex. 7. 4 yd. li ft = 27 half-feet; 2 rd.=66 half-feet; REDUCTION. 89 Ex. 8. -p bu- =—-=:- bu., Ans. 4 20 5 Ex. 9. -^j = ij Ans. Ex. 10. 2 yd. 2 qr.=10 qr ; 8 yd. 3 qr.=:35 qr. ; H yd.==f yd., Ans. (215>5 page 180.) Ex.2. .217° X 60=13.02'; .02'x 60 = 1.2^ Ans, 13' 1.2^ Ex.3. .659wk. X 7=4.613 da. ; .613 da. x 24 = 14.712 h.; •712 h. X 60 = 42.72 min. ; .72 min. x 60 = 43.2 sec. Ans, 4 da. 14 h. 42 min. 43.2 sec. Kx. 4. .578125 bu. X 4 = 2.3125 pk.; .3125 pk. x 8=2.5 qt.; .5 qt. X 2 = 1 pt. Ans, 2 pk. 2 qt. 1 pt. Ex. 5. .125 bbl. X 31.5 = 3.9375 gal. ; .9375 gal. x 4 = 3.75 qt. ; .75 qt. x 2 = 1.5 pt. ; .5 pt. x 4 = 2 gi. Ans. 3 gal. 3 qt. 1 pt. 2 gi. Ex. 6. .628125 £ x 20 = 12.5625 s. ; .5625 s. x 12 = 6.75 d.; .75 d. X 4 = 3 far. A7is, 12 s. 6 d. 3 far. Ex. 7. .22 bhd. x 63 = 13.86 gal. ; .86 gal. x 4 = 3.44 qt. , .44 qt. X 2 = .88 pt. ; .88 pt. x 4 = 3.52 gi. Ans, 13 gal. 3 qt. 3.52 gi. Ex. 8. .67x3.45 = 2.3115; .3115x320 = 99.68; .68x16^=11.22; .22x12 = 2.64. Ans. 2 rai. 99 rd. 11 ft. 2.64 in. Ex. 9. .42857 mo. x 30 = 12.8571 da.; .8571 da. x 24 = 20.5704 h. ; .5704 h. x 60 = 34.224 min. ; .224 min. x 60^13.44 sec. Ans. 12 da. 20 h. 34 min. 13^} sec. Te COMPOUND NUMBERS. Ex. 10. .78875 T. x 20 = 15.775 cwt.; ,775 cwt. x 4=3.1 qr.; .1 qr. X 28 = 2.8 lb. ; .8 lb. x 16 = 12.8 oz. Ans. 15 cwt. 3 qr. 2 lb. 12.8 oz. Ex. 11. .88125 A. X 4 = 3.525 R. ; .525 E.x 40 = 21 P.; Ans, 5 A. 3 R, 21 P. Ex 12. .0055 T.x 2000 = 11 lb., Ans. Ex. 13. .034375 bundles x 40 = 1.375 quires; .-^>75 quires x 24=9 sheets; Ans, 1 quiie 9 sheets. (916, page 181.) Ex. 1.00 gi. 1.250 pt. 3.625 qt. Ans. .90625 cral. Ex. 3. 24 20 12 9.000 gr. 13.375 pwt. 10.66875 oz. Ans. .8890625 ib Ex.4. 2 4 1.2 pt. .6 qt. 63 .150 gal. Ans. .00238 + hhd. Ex. 5. 1.12 qt. 3.14 pk. Ans. .785 bu. Ex. I 40 4 12.56 P. 3.314 R. Ans. .8285 A. Ex. 7. 12 3 5,5 40 8 6 in. 1.5 ft. iT^yd. 3.1818181+ rd. .07954545 + fur Ans. .00994318 + mi. Ex. 8. Ex. 9. Ex. 10. Ex. 11. Ex. 12. .32 pt.-h64=.005 bu., Ans. 4.875 ft.+6 = .8125 fathoms, Ans. 150 sheets-^480 = .3125 Rm., Ans. 47.04 lb.-f-196=.24 bbl flour., An^ .33 ft.-^5280 = .0000625 mi., Ans, ADDITION. n Ex. 13. 60 51.6 sec. 60 36.96 min. 24 5.616 h. Ans. .234 da. ADDITION. (Sir, page 183.) Ea 3. 43 10. 1 3 2 ^ 16 gr., Ans. Ex. 5. 68 bu. 3 pk. 1 qt. 1 pt., Ans, Ex. 6. 21 mi. 5 fur. 23 rd. 1 yd. 4 in., Ans. Ex. 10. 627 hbd. 1 gal. 1 qt. 1 pt., Ans. Ex. 11. 187 bu. 3 pk. 1 pt., Ans. Ex. 16. 152 cu. yd. 9 cu. ft.=:152i- cu. yd. ) $.16xl52J- ===$24,371. ) Ex. 17. 2564 lbs. 2713 " 3000 " 3109 " ~" =203.3214+ bu. 2.657 + , At^. Ex. An3» 11386 lb«i,- -56 =: $.80 X 203.3214: bbls. gal. qt. pt. gi. 18. 1 4 1 30 2 1 2 15 3 49 2 1 1=:4613 gi. $.09x4613=:$415.l7, Ans. (21 85 page 185.) Ex. 2 i rd. = 12 ft. 4i in. I ft. = 9_ " 13 ft. 11 in., Ans. T2 COMPOUND NUxMBERS. Ex, 3. 1 mi. = 7 fur. I fur.= 26 rd. 11 ft. I rd. = 13 " 9 in. 7 fur. 27 rd. 1^ ft. 9 in. ; or 1 fur. 27 rd. 8 ft. 3 in., Am. Ex.4. |£=13s. 4 d. ^ s. = 6 " 2| far. 13 s. 10 d. 2| far., A7is, Ex. 6. f T. =12 cwt. ^ cwt.= 42 lb. 134 oz. 12 cwt. 42 lb. 13f oz., An4. Ex. 6. f da. = 9 h. i h. = 30 min. 9 h. 30 min., Ans, Ex. 7. 1 wk. = l da. 4 h. f da. = 18 " ^ h. = 15 min. 1 da. 22 h. 15 min., Arts, Ex. 8. -f lihd. = 54 gal. f gal. = ^qt. 64 gal. 3 qt., Ans. Ex. 9. 4 cwt. =64 lb. 8f lb. = 8 " 13 oz. 5i dr. 3yV oz. = 3 " 14| " 73 lb. 1 oz. 3|i dr., Ana. Ex. 10. I mi. = 3 fur. |yd.= 2 ft. J ft. = 9 in. 3 fur. 2 ft. 9 in., An8. SUBTRACTION. T8 Ex. 11. 1 of } A.=:i A. - 26 P. 181 ^ sq. ft. 60f rd. = l K. 20 " 204^^ « f A.=l " 5 " 194 If " Y^ A. == 13 " 90 f " 3 K. 26 P. 126yV2 sq. ft., Am, Ex. 12. 11 T. =1 T. 3 cwt. 33 lb. 5^ oz. If^ T. =1 " 3 " 75 " 18f cwt.= 18 " 83 " 6i '•• 3 T. 5 cwt. 91 lb. 10| oz., Ans. SUBTRACTION. (2195 page 187.) Ex 4. 3 T. 18 cwt. 70f lb., Ans. Ex. 6. 2953 £ 2 s. 7f d., Ans. Ex. 11. 365 da. X 5 4-2 da.=1827 da. 1 bhd.=63 gal. 1827 gi. =57 " qt. pt. 3 gi. 6 gal. 3 qt. 1 pt. 1 gi., Ans. Ex. 12. 196 A. 2 R. 16.25 P. 200 " 1 " 20 " 177 " " 36 " 1st, 2d, and 3d own 574 A. R. 32.25 P. 640 " 4th owns 65 A. 3 R. 7.75 P., A71S. Ex. 13. 16 Cd. 5 cd. ft. 75 Cd. 6 cd. ft. 24 " 6 " 12cii.ft. 69 " 2 " 12 cu. ft. ^^ * 6 Cd. 3 cd. ft. 4 cu. ft. 69 Cd. 2 cd.ft. 12cu.ft. Ans. K. P. ^ COMPOUND NUMBERS. Ex. 14. 10 gal. 1 qt. 1 pt. 63 gal 15 " 1 pt. 40 " 1 qt. 14. « Q « 22 gal. 3 qt., Ans, 40 gal. 1 qt., Ans. (32O5 page 189.) yr. mo, da. yr. mo. da. Ex. 2. 1799 12 14 Ex. 5. 1815 G 18 1732 2 22 1775 6 17 67 9 22, Ans. 40 1, Ana jT. mo. da. h. min. Ex. 6. 1861 1 3 8 50 1856 6 24 14 20 4 6 8 18 30, Ans, Ex. 7. 122 da. ; 244 da. ; 306 da., Ans. Ex. 8. Erom Nov. 6 to April 6, 151 da. From Apr. 6 to Apr. 15, 9 " 160 da., Am, Ex. 9. From Aug. 20 to June 20, 304 da. Subtract 5 " 299 da., Aru (221 5 page 190.) Ex. 2. i rd.=:8 ft. 3 in. |ft. = 9 « 7 ft. 6 in., Ans. Ex. 3. f £=11 s. 1 d. li far. 4s.= 6" 10 8. 7 d. 11 far., Aiu. MULTIPLICATION^. 75 Ex. 4. I of 3.45 mi.=2.3 mi ; 2.3 mi.— .7 mi.=:1.6 mi. 1.6 mi. = l mi. 4 fur. 32 rd., Ans. Ex. 5. 8yV cwt. =8 cwt. 3 qr. 16 lb. 12 oz. 12f dr 1 qr. 2^ lb. = 1 qr. 2 " 6 " 13 4 " Ans. 8 cwt. 2 qr. 14 lb. 5 oz. 15/j dr Ex. 6. 1 wk.=l da. 9 h. 36 min. 1 da. = 4 " 48 " 1 da. 4 li. 48 min., Ans. f I of 120 mi.=41 mi. 7 fur. 9 rd. 8 ft. 7^ in.. Am Ex.8. l-i=J; ioff=/^; y^ of 96 gal.=25 gal. 2 qt. 3i gi., Ans MULTIPLICATION. (222, page 192.) Ex. 4. Ans. 23 £ 13 s. 4 d. Ex. 5. Ans. 23 lb. 4 oz. 6 pwt. 10 gr. Ex. 6. ^7i5. 163 T. 1 cwt. 36 lb. 4 oz. Ex. 7. ^W5. 128° 35' 15"^. Ex. 9. Ans. 20fi) 1 | 3 3 1 3 16 gr. Ex. 10. Ans. 235 mi. 6 far. 7 rd. 4^ ft. Ex. 13 bu. pk. qt pt. 45 3 6 1 8 Ex. 14. 1 a. 9 d. 12 367 2 4 8 17 16 6 10 2941, Ans 178 5, ^w«. 76 COMPOUND NUMBERS. Ex. 15. U800-T-$80 = 60 — 6 x 10. A. R- P. 8q. yd. 8a^ $m.74x^^ ^^3^^ ^^^^ 154.4375 30.8875 Ex. 30. .0125 T.x 2000 = 25 lb., Ans. Ex. 31. j\ of 2 bu. 3 pk.= Jl pk. ; 3 bu.=12 pk. ; ^1^12 =^^\, ^715. Ex. 3^ 4- X I X I X -i-V-^ X 2 K=H?^^=1'747V, Ans. Ex. 33. V- X J/ x^3Q-xi-V-^x^yio,T = Vr32'= 301.339 f , Ans „ ^^ $.26x36x20 ^^^^^ . Ex. 34. =$20.80, Ans. y ^ ^ 46x20x2x144 Ex 35. = 13248,^715. 4x5 26» 200) 800) tof) .0025 .0075 .00666 + .008 .00625 .0125 .028 .04333 + .0575 .07125 .122 .25375 11 3 .* 4 0) ^8 . 26 ) (S355 page 207.) Ex. 9. ^W5. 63 sheep. Ex. 10. Ans, 620 men. Ex. 12. Ans, $22692.25. Ex. 20 1.00 — .25=:.75; 760 sheep x .75=570 sheep, ^w« Ex.21. .18+.30=:.48; 1.00-.48=.52; $24500 X .52=$12740, Atis. 1576 barrels x.l 2 5 = 197 barrels, Ans. Ex, 22 Ex.2? .75=1; .33i=i; ^ — 1= 1^) $2760 X y\=$1150, A71S. Ex.24. jXyVo=/5 sold; |-^\=i| left, ^n«. 86 PERCBNTAGB. Ex. 25. I, owed after the 1st payment, fxf, " " *' 2d I X f X I, " " " 3d " $^f ^ X f X f X ■i=$226.5G|, Ans. (S36, page 208.) Ex. 2. 90-^450 = .20=:20 per cent., Ans, Ex 3 175^1400=.125 = 12i per cent., Atv*. Ex. 4. 165-v-'750r=.22 = 22 per cent., Ans. Ex. 5. 13.20-^240=. 055 = 51 per cent, Ans, Ex. 6. .15-f-2 = .075 = 7^ per cent., Ans, Ex. 1. 6 bu. 1 pk.=200 qt; 4 bu. 2 pk. 6 qt.= 150 qt. 150^200 = .'75 = 75 per cent., Ans, Ex. 8. 15 lb. = 240 oz. ; 5 lb. 10 oz.=90 oz. ; 90-r-240 = . 375 = 37^ per cent., Ans, Ex. 9. 40-T-250=.16 = 16 per cent., ^?i5. Ex. 10. 100 + 90 = 190; 190-f-760=.25 = 25 per cent., Ans, Ex. 11. I of f =|=.50=50 per cent., Ans (237, page 209 ) Ex. 2. 16^.08 = 200, Ans, Ex. 3. 42-r- .07=600, Ans, Ex. 4. 75-^.125 = 600, ^?i5. Ex. 5. 33^.0275 = 1200, Ans. Ex. 6. $281.25-^.375 = $750, Ans. Ek, 7. 50-^.20 = 250, Ans, Ex.8. $59.75-T-.125 = $478, ^W5. Ex. 9. $975^.15=16500, Ans. Ex. IC .40x.25=.l $1246.50-^.l =$12465, Ans, Ex.11. 2000-r- .40 = 5000; 5000-2000 = 3000, Jn. premium ; $3200— $36==$3164, saved by insuring; $4800 — $3164=$1636, owner's loss, Ans, Ex. 10. $21000 X 5^0=^^168; $15400x^^0=^ 9^-25; $264.25, Ans, TAXES. (289, page 23G.) Ex. 3. Property tax ==$26.95 1 poll .75 $27.70,^^15. Ex. 4. $.50 X 2981 =$1490.50, poll tax ; $9190.50— $1490. 50=$7700, properly tax ; $7700-^-$l,400,000 = .0055, rate of taxation. $12450 x.00553=$68.475 2 polls =$ 1.00 C's tax $69,475, Ans, Ex. 5 $5375 X .0055 =$29.5625, Ans. K. p. 5 M PERCENTAGE. Ex. 6. $.625 X 30 = ^18.75, poll tax ; $4342.75— $18.75 = $4324, property tax; $4324-^$188000=:.023, rate of taxation. $2500 X. 023 + $.625 =$58,125, Ans. Ex. 7. $.30 X 25482 = $7644.60, poll Jpx ; $103294.60 — $7644.60=$95650, property tax ; $95650-J-$38260000=.0025, rate of taxation. $9470 X. 0025 -f-$.90 = $24.575, Ans. Ex. 8. $10000 X 1.025 = $10250, whole tax; $1.25x225=:$281.25, poll tax ; $10250-$281.25=$9968.75, property tax; $9968.75-^$1246093.75=:.008, rate of taxatioL . $11500 X .008+ $1.25 = $93.25, E's tax, Am. Ex. 9 $275.57 — $98=$177.57, tax ; $l77.57-^3946=$.045, rate per day; $.045 X 118 X 2=$10.62, Ans. CUSTOM HOUSE BUSINESS. 95 CUSTOM HOUSE BUSINESS. (303, page 239.) Ex. 2. $-95 X 224=$212.80, value of the silk , $212.80 X.19=$40.432, Ans. Ex. 3. $.54 X 31.6 X 50=$850.50, gross value ; deduct $850.50 x .02=5 17.01 for leakage ; $833.49, net value; $833.49 X .24 =$200.0376, Ans. Ex. 4. $.15 X 115 X 175=$3018.75, value of the coflfU? $3018.75 X .15=$452.81J, Ans. Ex. 5. $.36 X 63 X 25 =$567, gross value ; deduct $567 x .005= 2.835, for leakage ; $564,165, net value; -^ $564,165 X .24=$135.3906, Ans. PERCENTAGB. SIMPLE INTEREST. (311, page 241.) Ex. 3. $45.92, Ans, Ex. 8. $093.83+, Ara Ex. 11. $607.50, Ans, Ex. 15. $440,625, Ans. Ex. 17. $605.70 + liit. for 3 yr.=$751.068, Ans. (313, page 245.) Ex. 7. $106,855, Ans. Ex. 8. $1.72+, Ans. Ex. 11. $91.85 + , Ans. Ex. 15. $2,138+, Ans. ' Ex. 18. $24.87+, Ans. Ex. 19. $282.75+, Ans. Ex. 22. $82.36+, Ans. Ex. 23. Time, 7 yr. 7 mo. 2 da.; $51.98 + , Ans. Ex. 24. Time, 2 yr. 1 mo. 4 da. ; $4,474, Ans. Ex. 25. Time, 11 yr.3mo.27 da.; $19.818f + , Ans. Ex. 26. Time, 9 mo. 19 da.; $408,957 + , Ans. Ex. 27. First payment, $2000 Second payment, $3157.50 Third payment, $1105. $6262.50, Ans. Ex. 28. $350. + int. for 11 mo. 21 da.= $373,887 + $150. +int. for 8 mo. 16 da.= 157.466 + $550.50 + int. for 3 mo. 11 da.= 561.310 + Total = $1092.663 + , Ansi PARTIAL PAYMENTS. 97 PARTIAL PAYMENTS. (314, page 249.) Kx 3. Amt.ofnotetoNov.l2, 1858, (4mo. 22da.)$535,27-f" Payment, 105.50 New Principal, $429.77-+- Amt., Mar. 20, 1860, (16 mo. 8 da.) 488.03 + Payment, 200 New Principal, $288.03 + Amt., July 10, 1860, (3 mo. 20 da.) 296. 83 -j Payment, 75.60 New Principal, ..$221.23 + Amt., June 20, 1861, (11 mo. 10 da.) $242.12+, Ann Ex. 4. Amt. of note, Jan. 1, 1860, (7 mo. 24 da.) $3136.50 Sum of payments to tliis date, 525.00 New Principal, $2611.50 Amt., April 4, 1861, (15 mo. 3 da.) 2841.53 + Sum of payments, 1575.00 ' New Principal, $1266.53 + Amt., Feb. 20, 1862, (10 mo. 16 da.) $1344.35+, Ex. 6. Amt. of note, Jar.. 1, 1852, (16 mo. 28 da.)$977.15 + Payment, 250.00 New Principal, $727.15 - Amt. May 4, 1853, (16 mo. 3 da.) 775.93 + Payment, 316.75 New Principal $459,18 4 Amt. Sept. 15, 1853, (4 mo. 11 da.) $467.53+, 5 Ans 98 PERCENTAGE. Ex. 6. Interest commence 1 Aug. 2, 1860. Amt. of note, May 6, 1861, (9 mo. 4 da.),. $192.988 -f Payment, 50 New Principal, $142,988 + Amt. Aug. 26, 1862, (15 mo. 20 da.) $154,188+. Ex. 1. Amt. of mortgage, Jan. 1, 1852, (3 mo.). .$6120.00 Payment, 500 New Principal, $5620.00 Amt. Sept. 10, 1852, (8 mo. 9 da.) 5930.98 + Payment, 1126.00 New Principal, $4804.98 + Amt. March 31, 1854, (18 mo. 21 da.) . . . 5404.00 -f- Payment, 2000.00 New Principal, $3404.00 + Amt Aug. 10, 1854, (4 mo. 9 da.) 3501.57 + Payment, 876.50 New Principal, .$2625.07 + Amt. Oct. 1, 1857, (37 mo. 21 da.) $3284.84 + (315, page 251.) Ex. 1, Amt. from Jan. 1, 1858, to Jan. 1, 1859, (1 yr.) $487.60 Amt. of 1st pay't from Apr. 16, 1858, to Jan. 1, 1859, (8 mo. 15 da.) 154.29 New Principal, $333.31 Amt. from Jan. 1, 1859, to Mar. 11, 1860, (14 mo. 10 da.) 357.19 + Payment, 75.00 New Principal, $282.19 + PARTIAL PAYMENTS. H Ami. from Mar. 1 1, 1860, to Dec. 11, 1860, (9 mo.) $294. 89 + Amt. of 3d pay't from Sept. 21, 1860, to Dec. 11,1860, (2 mo. 20 da.) 56.74 + Ans, $238.15 -h (316, page 251.) Ex. 1. Amt. of Principal, Jan. 1, 1859, (2 yr. 8 mo. 20 da.) $698.00 A.mt. of 1st endorsement, (for 2 yr. 4 mo. 21 da.) $178,386 Amt. of 2d endorsement, (for 1 yr. 10 mo. 19 da.) 222.633 Amt,of3dendorsement,(for7mo.) 191.475 592.494 Ans. $105.50 + , (SIT, page 252.) Ex 1. Amt. of note, Aug. 4, 1859, (1 yr.) $609.50 Amt. of 1st pay't, Aug. 4, 1859, (9 mo.) . . 66.88 New Principal, $542.62 Amt. of new Principal, Aug. 4, 1860, (1 yr.) 575.17 Amt. of 2d pay't, Aug. 4, 18G0, (7 mo. 21 da.) $49.85 Amt. of 3d pay't, Aug. 4, 1860, (4 mo. 18 da.) 253.70 303.55 New Principal, $271.62 Amt. of new Prin., Nov. 4, 1860, (3 mo.). 275.69 Amt. of 4tli pay't, Nov. 4, 1860, (1 mo. 6 da.) 60.30 Ans, $215.33. 100 PERCENTAGE. (31 85 page 253) Ex. 1. 1st installment of interest, due Feb. 2, 1856, ^30 2d " " " '' " " 1857, 30 3d " " " " " « 1858, 30 4th « " " " " " 1859, 30 5th " " . " " Aug. 2, 1859 15 4135 1st installment draws int. 3 jr. 6 mo. 2d " " " 2 jr. 6 mo. 3d " " " 1 jY, 6 mo. 4th " " " 6 mo. Int. of $30 for 8 yr. mo $14.40 Principal, $500.00 Ans, $649.40 PROBLEMS IN INTEREST. (320., page 253.) Ex. 2. Int. of $1 for 6 yr. 3 mo. at 6 per cent., is $.395 ; $28.125^.375 = $75, Ans. Ex. 3. Int. of $1 for 4 mo. 18 da. at 4 per cent., $.015^ ; $9.20-^.015i=z$600, Ans. Ex. 4. $1260^.07 = $18000, Ans. Ex. 5. $33'70-^.10=z:$33'700, Ans. (321, page 254.) Ex. 2. $1 for 8 mo. at 6 per cent., amounts to $1.04} $655.20-^1.04=:$630, Ans. Ex. 3. Amt. of $1 for 5 yr. 5 mo. 9 da. at 5 per cent., 1.27201 ; $106.855 4-1.2720f =$84, Ans. .^tyh PROBLEMS IN INTEREST. 101 Ex. 4. Amt. of $1 for 8 yr. 5 mo. at 5 J per cent., $1.462916+ ; $1897.545^1.462916 + =$1297.09+, Ans. Ex. 6 Amt. of $1 for 3 yr. 4 mo. at 7 per cent., $1.23^ ; $221.075-+1.23i = $l79.25, Ans. Ex. 6. Amt. of $1 for 11 yr. 8 da. at IQi per cent., $2.1 57 -J ; $857.54 + 2.157iirr$397.50, principal ; Int. of $397.50 for 11 yr. 8 da., at 10^ per cent., = $460.04, Ans. (322, page 255.) Ex. 2. Int. of $500 for 3 yr. at 1 per cent., $15 ; $45 -+$15 = 3 per cent., Ans. Ex. 3. Int. of $180 for 1 yr. 2 mo. 6 da. at 1 per cent., $2.13 ; $12.78-+$2.13=:6 per cent., Ans. Ex. 4. Int. of $2000 for 6 mo. at 1 per cent., $10 ; $75-^$10 = 7^ per cent, per annum, Ans. Ex. 5. Int. of $1000 for 3 yr. 3 mo. 29 da. at 1 per cent, $33,305+ ; $183.18-+$33,305=:5.5 per cent., Ans. Ex. 6. Int. of $21640 for 1 year at 1 per cent, $216.40 ; $2596.80-^$216.40=rl2 per cent, Ans. (323, page 256.) Ex. 2. $325 X .06 =$19.50, int for 1 yr. ; $58.50-+$19.50 = 3 yr., Ans. Ex.3. $1600x.06 = $96; $2000-$1600=$400; $400+-$96 = 4J yr-=4 yr. 2 mo., Ans. Ex.4. $204x.07--$14.28; $217.09 — $204=$13.09; $13.09 -; $14.28 = f J- yr. = ll mo.. Am, 102 PERCENTAGE. Ex. 5. $750 X .Oe = $45 ; $942 — S750=$192 ; $192-v-$45 = 4y4j yr. = 4 yr. 3 mo. 6 da., Am. Ex.6. $200 X. 06 =$12; $200^$12=:16 I yr.=16 yr. 8 mo., An8. Ex.7. $675x.05=r$33.75; $675-^$33.75 = 20 years, Ans. COMPOUND INTEREST. (324, page 257.) Ex. 2. $500.00 Prin. for 1st year. 35.00 Int. " " " $535.00 Prin. " 2d ** 37.45 Int. " " ** $572.45 Amt. " 2 years. 600. Given Prin. Ajis, $ 72.45 Compound interest. fix. 3. $312.00 Prin. for 1st year. 18.72 Int. " " " $330.72 Prin. " 2d " 19.84 Int. " " " $350.56 Prin. " 3d " 21.03 Int. Ans. $371.59 + , Arat. " 3 year^ COMPOUND INTEREST. 103 Ex. 4. $250.00 Prin. for 1st. half year. 7.50 Int. " " " $257.50 7.72 $265.22 7.96 Prin. Int. Prin. Int. Prin. Int. Amt u u u u 2d 3d $273.18 8.19 4tli u $281.37 250.00 2 years. Ans, $31.37 + Compound interest Ex. 5. $450.00 Prin. for 1st quarter. 7.87 Int. " " " $457.87 Prin. " 2d 8.01 Int. " Prin. " u $465.88 3d 8.15 Int. " Prin. " (( $474.03 4th 8.30 Int. " (( ^?6S. $482.33+ Amt. " 1 year. Ex. 6. $236.00 Prin. for 1st year. 14.16 Int. " " " $250.16 Prin. " 2d 15.01 Int. " " $265.17 Prin. " 3d 15.91 Int. " " $281.08 Prin. " 4th •* 16.86 Int. « « - IGil PERCENTAGE. 1297.94 Prin. " 1 mo. G d*. 10.72 Int. " 7 " 6 " $308.66 Arat. " 4 yr. 7 mo. 6 dfi, 230.00 Given principal. Ans, $72.66+ Int. 4 yr. 7 mo. 6 da. Ex. 7 $700.00 Prin. for 1st year. 49.00 Int. " " " $749.00 Prin. " 2d " 52.43 Int. " " " $801.43 Prin. " 3d " 56.10 Int. " " *' $857.53 Prin. " 9 mo. 24 da. 49.02 Int. " 9 " 24 " Ans, $906.55 -f,Amt." 4 yr. 9 mo. 24 da. Ex. 9. $120 X 2.078928==$129.47 + , ^/2S. Ex.10. $.10x3.86968=$.386968, ^^5. DISCOUNT. (326, page 259.) Ex. 2. $180-M.20r=:$150, Ans, Ex. 3. $1315.389^1.175 = $1119.48, Ans, Ex. 4. $866.0384-1.281i = $675.888-f,pref. worth. [ $866.038— $675.888 + =$190.15+. viuv-ft ] '** ^x. 5. $1005 — $475=$530 $475-^1.05 =$452.38 + $530-^1.075 = 493.02 + $945.40 f , Ans. PROMISCUOUS EXAMPLES, 10t5 Ex. 6. Term of discount, 6 mo. 24 da. $529.925-^1.034 = $512.50 present worth. $529.925 — $512.50 = $17.425 discount, Ans. Ex. 7. $3675 cash offer. $4235-^1.21=$3500 cash value of note. Am, $ 175, loss. Ex. 8 $550-^1.10 = $500, present value of note ; $480, cash offer ; Ans, $ 20, gain. Ex. 9. $517.50-^1.035:=$ 500 $793.75-M.05f=:$ 750 $1326.47-f-1.105 = $1200.426-f- $2450.426 + , entire pres. worth $2637.72-$2450.426 + =:$187,29-f,^/^s. Ex. 10. $.10 X If r=$y^^,int. of $1 for 10 mo. at 10 per cent $130^1y'2 =::$120 ; $130 — $120 = $10.00, discount. $130 X yV =$10,831, interest. Ans, $.83^. PROMISCUOUS EXAMPLES IN PERCENTAGE. (Pag^ 260.) Ex. 1. .02 4- .25 = . 27 gain per cent, on cost. V' ^ fo =^tV <^ents, selling price of what remains of every pound, after transportation ; Ex. 2. $200 X .40 = $80 gain on one ; $200 X .20 $40 loss on the other; Ans, $40. 106 PEKCENTAGB. Ex. 3, $425-^1.03=$412.62+ cash value of sale; $425 — $25 =$400.00 cost ; Ans, $1^.624, profit. Ex.4 $.13-f-1.04 = $.125; $.13— $.125=$.005 ; $.005-r-$.125=.04, ^?i5. Ex. 5. $150-^-1.25=$120 costof one; $150^ 15= 200 cost of the other; $320 cost of both; $320 — $300 = $20, Ans. Ex. 6. Amt. of $1 for 2 yr. 8 mo. at 9 per cent., $1.24 ; jiyji X |« J X f =$3750, Ans. Ex. 1. 1.00 — .07 = 14f years, Ans. Ex. 8. 3 yr. 4 mo.=r3i yr. ; .121 x 3i=:.4is., whole nito of gain; $5000-r-.41| = $12000, capital ; $12000 X f = $7500, ^'s, ) $12000 X f — $4500, B's, J Ex. 9. $800 X .15 = $120, gain on groceries ; 500 X .20:= 100, loss on dry goods ; Whole gain $20, Ans. Ex.10. 1.00-.08i=.91|; $1100h-.91| = $1200, Ans. Ex. II. $667-f-1.04=$C41.346 +, cash value of goods ; 600x1.06= 636.^ pres. valuation of goods , True gain, $5,346 + , Ans. Ex. 12 $18xi=$6, profits; $18— $6=$12, cost; $6-7-12 = 50 per cent., profit, Ans, Ex. 13. If f sell for f of cost, the whole would sell for f x | >f the cost, which is 1]| times cost. Hence JJ = .40f, is the S|;ain per cent. PROMISCUOUS EXAMPLE? 107 Ex. 14. $1.30, received for lumber originally wortli $1.00 ; $1.06|, valuation of ditto, after 16 mo. int. accrues ; ■ $.231 gain on $1.06|; $.23iH-$1.0G|=.21|, Ans. Ex.15. 1-1 = 1; ix^^i,Ans. Ex. 16. $1,121 x728 = $819, expended in wheat; .60 X .30 X .50=1.09 ; $819^.09 =:$9100,in bank; 1.00 — .60^.40; $9100x.40=:$3640, Ans. Ex. 17. 6x5=i30sq. yd., area to be covered ; 4 per cent. IS 2^, and 5 per cent, is ^V ; hence every yard purchased will be, after shrinking, || of 1 yd. long, and if off yd.==|-J yd. wide, and will contain || x f J sq. yd. Therefore, as many yards must be purchased as |i x f i is contained times in 30. Ex. 18. 1.00 per cent.=B's money; 1.28 " =A's " .28-4-1.28-. 211, Ans, Kx. 19. $1200-T-.12:=$10000, f of his capital; $10000-4-|=$25000, whole capital; $25000 X f X yi^=$750, loss on | of the capital ; $1200— $750 = $450, gain. Ex. 20. Amt. of $1 for 3 yr. 9 mo. at 10 per cent., $1,375 J $1933.25^-1. 375=:$1406=i of C's money. $1406x2=±$2812, C's, $2812 X2=±$2812, C's, ) x#=$4218,D's, ) • 108 PERCENTAGE. BANKING. (336^ page 264). Ex. 1. Int. of $450 at 6 per cent., for 63 da.=$4.725 lisc't : $450— $4,725=1445.275, proceeds. Ex. 2. Int. of$368at7percent.,for93da. = $6.654+,disc't; $368 — $6,654+ =$361,345 + , proceeds, Ans, Ex.3. Int. of $475.50 at 5 per cent., for 63 da.=$4.16 + , discount; $475.50— $4.16 + =$471.33 + , proceeds. Ex. 4. Int. of $10000 at 6 per cent., for 93 da.=$155, disc't; $10000— $155 = $9845, proceeds, Ans. Ex. 5. Proceds of the note, disc'ted at 6 per cent., $247,375 ; $247.375— $240=$7.375, Ans, Ex. 6. Int. of $360.76 at 6 per cent., for 93 da. =$5,591 +, disc't; $360.76— $5.591 + =$355,168+, proceeds. Ex. 7. Proceeds of the note, $529.2355 ; $530 — $529.2355 = $.7645, Ans. Ex. 8. From Mar. 2 to Apr. 7 is 36 da., term of discount; Int. of $500, at 6 per cent., for 36 di., is $3.00, disc't ; $500-$3.00 = $497, proceeds. Ex. 9. From Nov. 15 to Dec. 15 is 30 da term of discount ; Amt. of $750 on interest for 6 no. 3 da. at pel cent, is $772,875 ; Bank discount of $772,875 for 30 aa., x\ iO km^ g^^av is $6,440+ ; $772.875— $r.44U^ ^ -^ $766,434+, proceeds EXCHANGE. 109 (337, page 2G6.) Ex. 2. $680-^.9895 — $687,215+, Ans. Ex. 3. $1000-^.9870| = $1013.085+, Ans, Ex. 4. $o00-+.9935f =:$503.22+, Ans, Ex. 5 $1256+- .96441=11302.341 +, Ans. EXCHANGE. (350, page 269.) Ex. 3. $1000 X 1.03 = $1030, Ans. Ex. 4. $400 X 1.0075 :=:r$403, Ans. Ex. 5. 8530 X 1.0275rrr$544.575, cost of draft ; 20. transportation ; Ans. $564,575. Ex. 6. $1— $.01225 =$.98775, proceeds of $1 at b'k disc'tj. Add .02 premium ; $1.00775, cost of exchange for $1 ; $800 X 1.00775 =$806.20, Ans. Fa. 7. $1— $.0055 = $.9945 Subtract .015 discount; $.9795, cost of exchange for $1 ; $420 X. 9795 =$411.39, Ans. Ex. 8. $1 — $.0225=$.9775; 320 x $10=$3200; $3200x.9775 = $3128, draft; 312, transportation ; 400, gain ; To be sold for $3840 $3840^320=$12, Ans. 110 PERCENTAGE. (351, page 271.) Ex. 2. $243.60^1.015=1240, Ans. Ex. 3. $79.20-^.99=:$80, Ans. Ex. 4. $1— $.0105 = $ .9895 Add .02 $1.0095, draft for $1; $282.66^1.0095 =$280, An€. Ex. 5. $1— .0055=$.9945 Subtract .0125 $.982, draft for $1 ; $240-^.982 = $244.399 + , draft hot. for $240; 1240 — ($240 X. 005)= 238.80 current money for $240 ; Ans. $ 5.599 + , saved. Ex. 6. $1.00000 .01225, bank disc't at 7 per cent. (63 da^ $ .98775 .0075 premium. $3C00-h .99525 = $3617.181 +, draft required. $3600^ 1.0075 =$3573.200 + , draft sent. Add int. for 60 da. $ 35.732+ (at 6 per cent.) Amt. at time req'd, $3608.932 + $3617.181 -$3608.932 =$8.24+, loss, Ans. EQUATION OF PAYMENTS. Ill EQUATION OF PAYMENTS. (356, page 273.) Ex. 2. $700x20 = 114000 400x30= 12000 700x40= 28000 $1800 $54000 54000^1800 = 30 da. Average credit Sept. 25-f30 da. = Oct. 25, equated time. Ex. 3. $250x4 = $1000 750x2= 1500 500x7= 3500 $1500 $6000 6000 -T- 15 00 =4 mo. average credit, July 1+4 mo. = Nov. 1, Ans, Ex. 4. $1x0=$ 2x1= 2 3x2= 6 4x3= 12 5x4= 20 6x5= 30 7x6= 42 $28 $112 112-r-28=4 da. Monday -I- 4 da. = Friday, Ans, Ex. 5 $650 X 4=$2600 725 X 8= 5800 500x12= 6000 $1875 $14400 14400^1875 = 7.68 mo.=7 mo. 20 da. Jan. 1 + 7 mo. 20 da. = Aug. 21, Ans, 112 AVERAGINa ACCOUNTS. Ex. 2. £x. 3. (362, page 276. Due. da. Items. Prod. Jan. 1 16 Feb. 4 March 3 15 34 61 150 200 100 160 3000 3400 9760 610 16160 16160^610 = 26 da. Jan. l-}-26 da. = Jan. 27, Ans. Due. da. Items. Prod. March 1 April 4 Aug. 18 Aug. 8 34 170 160 300 240 100 400 8160 17000 64000 1040 89160 89160 — 1040 = 86 da. March 1 + 86 da.=May 26, Ans. Ex. 4. Due. da. Items. Prod. June 1 600 " 12 11 300 3300 " 15 14 832 11648 " 25 24 760 18240 30 29 750 21750 8242 , 54938 54938-^3242 = 17 da. June 1 + 17 da.=June 18, Ans. AVERAGING ACCOUNTS. 118 Ex. 5. Due. da. Items. Prod. Jan. 16 Feb. 20 March 4 April 24 35 48 99 536.78 425.36 259.25 786.36 14887.60 12444.00 77849.64 2007.75 105181.24 j 105181.24-^-2007.75 = 52 da. ^^' \ Jan. 16, 1856 + 52 da.=March 8, 1856. Ex.6. Due. da. Items. Prod. April 1 " 28 June 15 27 75 420 135 1800 3645 135000 2355 138645 A71S. ( 139650-^2355 = 59 da. ( Apr. 1 + 59 da.=May 30. Ex. 2. (36 3, page 278.) Br. Or, Due. da. Items. Prod. Due. da. Items. Prod. Jan. 1 Feb. 4 *' 20 34 50 448 364 232 12376 11600 Jan. 20 Feb. 16 " 25 19 46 55 560 264 900 10640 12144 49500 104423976| 172472284 1044'23976| Balances 68048308 Ans, 48308-^-680 = 71 da. Jan. 1+71 da.=March 13 114 RATIO. Ex.3. Dr. Cr. DnSf da. Items. Prod. Due. da. Items. Prod. Apr. 1 June 12 Sept. 3 Oct. 4 72 155 186 145.86 37.48 12.25 66.48 2698.56 1898.75 12365.28 May 11 July 12 Oct. 12 40 102 194 11.00 15.00 82.00 440.001 1530.00 15908.00 262.07 108.00 154.07 — 16962.59 108.00 17878.00 16962.55 Bal acct. Bal. Prod. 915.41 ( 915.41-f-154.07 = 6 da. ' I Apr. 1 — 6 da.=:Marcli 26, 1858. RATIO. Ex. 3. 3V = 1, Ans. Ex. 5. ^ = 5l,Ans. Ex. 7. \*~Xj% = l2, Ans, Ex.9. ixi=i,Ans. Ex. 11. 3-0x-^-z=5, Ans. (379^ page 281.) Ex. 2. 2\={, Ans, Ex.4. ^f^=1,Ans. Ex. 6. 2J.x}={i,Ans, Ex.8, ^f = 41, Ans. Ex. 10. -j4_x|=:|, ,Ans, Ex. 12. 3 gal. = 24 pt. ; 2 qt. 1 pt. = 5 pt. ; 5-^24 = 2^7 ■^^^' Ex. 13. 8 s. 6 d.=:8.5 s. ; |;f =|f = 1||, Ans. Ex. 14. -/I^/Zo =tV, ^ris. Ex. 15. 19 lb. 5 oz. 8 pwt.=4668 pwt. ; 25 lbs. 11 oz. 4 pwt. = 6224 pwt. |||i = li, Ans. Ex. 16. ||=f, Ans. Ex. 17. f x l = j\y Ans. Ex. 18. 2^3 x^^rzz^, Ans. Ex. 19. 16-r-2f ==7, Ans. SIMPLE PROPORTION. 115 Ex. 20. 14.5 X 3 = 43.5, Ans. Ex. 21. J X f =| = li, Ans. Ex. 22. ^xiz=j\,Ans. PROPORTION. (388, page 283.) „ , 48x50 , ^ . ^ ^ 70x3 ^ . Ex. 1. =120, Ans, Ex. 2. ■— — = 5, Ans, 20 ' 42 ' Ex. 3. =6, Ans. Ex. 4. -^=12, Ans. _ ^ 201.75x48 yd. ^^^ , . Ex. 6. 3 lb. 12 oz. = 60 oz. 10.50x60 oz. 3.50 = 180 oz. = ll lb. 4 oz., ^n5. Ex. Y. 8 bu. 2 pk.=34 pL; 1Q bu. 2 pk.: ?06 pk. $38.25x34 ^, , , =$4.25, ^n5. 306 Ex. 8. V X ^^ X ^1 ^= Y-=8i, ^Tis. Ex. 9. V X I X 5 = 7, ^?25. Ex. 10.^ X I X f =|, ^ns. SIMPLE PROPORTION. (3995 page 287.) Ex 1. 48 Cd. : 20 Cd. : : $120 : ( ) , , $120x20 ,^ . ( )=^-jg =$50, ^/w Or. $120xf? = $50, Ans, 116 PKOPORTION. Ex. 2. 6 bu. ; 75 bu. : : $4.75 : ( ) , , $4.75x75 ^ ^ ^, , ( ) = =$59,371, ^W5. Or, $4.75 X V =$59.37^, Ans. Ex. 3. $3i : $50 : : 8 yd. : ( ) , , 50x8 yd. ^^^, , . Ex. 4. 12 : 20 : : 42 bu. : ( ) , , 42 bu. X 20 . ^ , . ( )=z -^ :^10hu., Ans. Or, 42 bu. X ^:=^10 t>u., Ans. Ex. 5. $.75 : $9.00 : : 7 lb. : ( ) , , 900 X 7 lb. ^^ ,^ , ( )=-Y^ ^84lb., ^7Z5. Ex. 6. 3 lb. 12 07. : 11 lb. 4 oz. : : $3.50 : ( ) 60 oz. : 180 oz. :: $3.50 : ( ) , , $3.50x180 ^ -^ , ( ) = - =$10.50, Ans. Ex. 7. 1 ft. 6 in. : 75 ft. : : 3 ft. 8 in. : ( ) lift. : 75 ft. :: 3| : ( ) ( ) = Y x-y ft. x|==183J ft. = 183 ft. 4 in., ^n*. Ex. 8. $2.75x VV =$19-^^7 ) ^^^^• Ex. 9. $13.32 : $51.06 :: 12 bu. : ( ) , , 51.06 x 12 bu. ^^, ( ) — — 46 bu., Ans. ^ ' 13.32 ' Ex. 10. 15hhd.i=945 gal. 945 gal : 28.5 gal. : : $236.25 : ( ) , , $236.25x28.5 . ^^, . ( ^" ^45" -'^.ISi, ^/^s. SIMPLE PROPORTION. ' 117 Ex. 11. 6 mo. : 11 mo. : : 1 bbl. : ( ) , . 11 xY bbl. ,^, ,, , . ( )=~ =l2^hhl,Ans. Ex. 12. 5 £ 12 s. : 44 £ 16 s. : : 9 yd. : ( ) , . 9 yd. X 896 .^ , . ( )=-^-u^ = 72 yd., ^715. Ex. 13. $3100 X J7//^=$310, Ans. Ex. 14. 100 lbs. cofFee=100 X 1 = 160 lbs. sugar; 2 : 160 : : $.25 : ( ) , , $.25x160 ^^^ , ( )= =$20,^715. Ex. 15. 13° 10' SO' : 360° : : 1 da. : { ) 47435'' : 1296000" : : 1 da. : ( ) ( )r=J-f i| jiiL da. = 27 da. 7 h. 43 min. 6.06 + sec, Ans. Ex. 16. 8J : 13^ : : $4.20 : ( ) ( ) =$4-f-o X V- X 3V=$6.48, Ans. Ex. 17. 6J d. : 10 £ 6 s. 8 d. : : If yd. : ( ) ( )=2.4_8jL X J yd. X 2T = 694| yds., Ans. Ex. 18. 121 cwt. : 48f cwt. : : $421 : ( ) ( ) = $i|5.xi|ix 22j=$163.50 + , Ans. Ex.19. $lf : $317.23 :: 8| lb. : ( ) ( )=:317.23 X 8.4 lb. X 4 = 1522.7+ lb.= 15 cwt. 22.7+ lb., Ans. Ex. 20. $1561 : $95.75 : : 15f bu. : ( ) ( ) = ^J^■LS-x^^ bu. 615=9.575 bu.= 9 bu. 2 pk. 2 1 qt., Ans. HJx. 21. I bar. : 1 bar. : : $^| : ( ) ( )=$^ xixi = $j^j, Ans. K. P 6 118 Ex. 22. Ex. 23. Ex. 24. Ex. 25. PROPORTION 4 rd. : llf rd. :: f A. : ( ) ( )=:} A. X V- X i = 2^\ A.=2 A. 28 rd., ins 13cwt. : 12 cwt. :: $421 : ( ) 16 oz. : 12 oz. : 3x12 ( y- 16 : $28 : ( ) =$21, ^/i5. 16 oz.— 1411 Q2.=ly5_ oz., cheat in 16 oz, 16 oz. : l/g oz. : : $30 : ( ) ( )=:$3jP X 2.1 XyV=$HI ==$2.46 + , Ant Ex. 26. 1 yr. 6 mo. : 3 yr. 9 mo. : : $750 : ( ) , , $750x45 ^ ^ 18 Ex. 27. 10 mo. x V/r-^^^ ^o-> ^^^• Ex. 28. $25 : $30^ : : $28 : ( ) ( )=$2J- X V- X 2V = $34.16, Ans. Ex. 29. 1 yr. 4 mo. = li = | yr. ; i yr. X|J| = 2J yr. = 2 yr. 9 mo. 10 da. Am COMPOUND PROPORTION. \ (401 5 page 292.) . 1. 16 50 J /o[-^^« = ( ) - 128 5 ( ) 90 ( ) = 72 bii., Am. COMPOUND PROPORTION. 119 Ex. 2. 3 12 h<4 120 ; 360 120 10 ( ) 300 12 54 ( ) = 10f days, Ans. Ex.3. 6 10 ( ) 34 20 15 ( ) = 170 yards, Ans. Ex. 4 450 ( ) 12 V : 9 12 ) 8 1 : 1 8 9 ( ) 12 12 450 ( )=900, Jtis. Ex. 5. 1200 ; Ex. 6. Ex.7. wi 500 : 960 m-- 4 ^ 8J ■■v\-- 48 6|:( ) 500 960 1 8 4 5 ( ) 1200 1 23040 ( )=3291^, An9 36 12 12 ( ) 48 9 9 8 ( )=:6 men, Ans, V-xVx*xYxixi>< 7V=¥i^ = 40tt» ^^*' 120 PROPORTIDN. Ex.8. 41 ) 4| } 6 ^ : 9 V : : 540 : GOO 20 ) ( ) ) Ex. 9. 2^ If [ :^;| j ::$3.27i:( ) .6.75 73 3 2 5 J478.25 ^_ ^^ Ex. 10. 5 6 [ • ^ 12 [ • • ^^-^ • ^^'^•^ 417.6x5x6 ^^ — ^ttt: — 7^ — = 20 men, Ans, 52.2x12 ' Ex. 11. 6 ) 9 ) 22.5 ) 45 2.5 y : ( ) [ :: 17.3 [ : 34.6 12.3 ) 8.2 ) 10.25 ) 12.3 6x2^x12^3x^5 X 34.6 X 12.3 _ 9x8.2x22.5x17.3x10.25 *"^^ ^^^^' ^'''' Ex. 12. 54 J 75 24J f : ( ) 121 ) lOJ V- X V X V X T J X /t=21 days, Ans. Ex. 13. 24 ^ 217 J 33J ) 23i 189 [ : 51 V : : 5| U) { ) ) 31) 21 Vx^l^XVxVxy XlX2lTXi^TXHjX^YXf = 16, ^7lS. PARTNERSHIP. 121 Ex.2, $ 8000 12000 20000 $40000 PARTNERSHIP. (407, page 295.) TVoW=i» A's fraction. ±2 0.0.0. _3_ TJ'q U 40000 10> -^^ 2.00.00. — i n\ a -$336, A's; $1680 x tV=$504, B's $1680xi==:$840, C's. Ex. 3. $1200+$1000 + $600=::$2800; $2800 : $1200 : ' $224 : ( )=:$96, A's share; $2800 : $1000 : : $224 '{ ) = $80, B's " $2800 : $ 600 : , $224 ( )=$48, C's " Kx. 4. $20000 : $13654 :: $3060 : ( )=$2089.062 $20000 : $13654 :: $1530 : ( )=$1044.531 Ex. 5. 16 + 24 + 28 + 36 = 104 $13XyV4=^2, A pays; 'x/oV=^3, B pays; $13 X tVt ==^3.50, C pays ; $13 x j\\=$^,50, D pays. Ex.6. 14 + 6 + 12 = 32 shares. $2240 X i|=:$980. Captain's share ; $2240 X 3\=:$420, Mate's share ; $2240 X i|=$840, divided among the sailors ^ $840-r-12=$'70, each sailor's share. fix.) $3475.60— $2512=$963.60, lost to the owner* ; $963.60 X i=$120.45, A's \ $963.60 X 1 = $240.90, B's i Jns. $963.60 X f = $602.25, C's ) Proof 1 =$963.60 122 PARTNERSHIP. • Ex 8. 6, C's proportional share. 6 f 4=10; 10x1= 2, E's ** ** 6 + 4 + 2 = 12 $2571.24 X ^2 =$1285.62, C's ; \ $2571.24 X yV=$ 857.08, D's ; V Ans. $2571.24 Xy2-=$ 428.54, E's;) Ex. 9 $7500-($2000 + $2800.75 + $1685.25) = $1014, D's gain ; gain. cap. gain. cap. $1014 : $3042 :: $2000 : ( )=$6000 Am $1014 : $3042 :: $2800.75 , ( )=$8402.25, B; i Am, $1014 : $3042 :: $1685.25 : ( ) = $5055.75, C ; ) (408, page 297.) Ex. 2. $250 X 6 =$1500, B's product ; 275x8= 2200, C's " 450x4= 1800, D's " $5500 $825 X If =$225, B's share of gain ; 825 X 11= 330, C's " " " 825X;?-|= 270, D's " (( $1000 X 8=$ 8000 1600x10= 16000 Ex.3. $1000 X 8=$ 8000 $1500 x 4 = $ 600C 1200x14= 16800 A's product, $24000 B's product, $22800 $2l000 + $22800 = $46800, sum of products. $46800 : $24000 :: $1394.64 : ( ) =$715.20, A's gain • $46800 : $22800 :: $1394.64 : ( )=$679.44, B's " PARTNERSHIP. 123 Ex. 4. 4x5 days =20 days' work A furnished ; 3x6 " =18 " " B " 6x4 " =24 " . " C " 62 " " all " 372 buslie]s-f-4 = 93 bushels to be divided. 62 : 20 :: 93 : ( )=30 bu., A's ; j 62 : 18 :: 93 : ( ) = 27 " B's ; [ Ans. 62 : 24 :: 93 : ( )=36 " C's; J Ex. 5. From Jan. 1, 1856, to Apr. 1, 1858, is 27 mo., Gallup's time ; " Mar. 1, 1856, " Apr. 1, 1858, " 25 " Decker's " " July 1, 1856, " Apr. 1, 1858, " 21 " Newman's" $3000 X 27 = $81000, Gallup's product ; 2000x25= 50000, Decker's " 1800x21= 37800, Newman's " $168800, sum of products. $4388.80 : ( ) = $2106, Gallup's gain; $4388.80 : ( ) =$1300, Decker's '' $4388.80 : ( ) =$982.80, Newman's ". $168800 $168800 $168800 $81000 $50000 $37800 Ex. 6. $560— 8=$70, A's monthly profit; $800^10=$80, B's " " $150 Since the gains of the partners are proportional to their amounts of capital when the times are equal, we have $150 : $70 : : $5600 : ( ) =$2613.331-, A's gain; $150 : $80 : : $5600 : ( ) = $2986.6G|, B's gain. Ex. 7. If we allow 2 parts of the gain to A, 3 parts to B, and 4 parts to C, | of A's gain will be equal to I of B's, and to J of C's, and the proportion of the shares will corres- Dond to the conditions. 124 ANALYSIS. 2+3+4=9 $117 X |=:$26, A's gain . $ll7xf=$39, B's " |ll7x^riz$52, C's " If we now divide the proportional shares of the gain, 2, 3, 4, by the respective times, 3, 5, 7, we shall obtain the piopor- tional monthly shares of the gain, which must be in the sa.i:*e proportion as the respective shares of the capital ; 2-f-3 = |, A's proportional share of capital • 3-+5 = f, B's " " " 4-h7=4, ^'s " " " " l"^! + 7=Tol» ^^™ of proportional shares. : ( ) = $700, A's capital ; :( )=$630, B's " :( )=$600, C's tf f : I : : $1930 19 3 . 3 105*5 HI: 4 $1930 $1930 ANALTSia (412, page 300.) Ex. 5. We multiply the number of casks by the number of pounds per cask, and this product by the number of pence per pound, and obtain the cost in pence; which, divided by 56, the number of pence in a dollar, gives $27, answer. 56 3 126 4 $27, Ans, Ex. 6. We multiply 19 (pence) by 28 7 for the cost of the butter (in pence), and j_^_ livide by 1 times 12 (pence) the price of _£_ the tea. 28 19 "l9 61, Ans. ANALYSIS. 125 Ex 7. 10 s. 6d=126d. The product of 2, 72, and 4 is the num- ber of quarts. Multiply this oy 126, the selling price per quart, and divide by 96, to reduce the result to Decimal currency, and we $756 — obtain $756. Subtracting the cost we obtain the profit. 96 2 72 4 126 $756 = $108, An9. Ex.8. 2 s. 6d.=:30d. Then 2 x 3 x 7 X 30= cost in pence. Divide by 60 to reduce to Decimal currency. 60 2 3 7 30 $21, Ans. Ex. 9. 20 X 3 X 12r=value of the ap- ples in pence. Divide by 6 s. 3 d. (=75 d.) to find the number of days' work to be given in exchange. 20 3 75 12 5 48 9|, Ans. Ex. 10. 96 160 18 Cost, $30 $42.66f- 90 160 24 sold for $42.66|. )=$12.66|, ^/i5. Ex. 11. 431= V; 10 s. 6 d.= 8 s. 3 d. = 99 d. :126d. Ex. 12. 9 s. 4 d. = 112 d. ; $1=96 d., Mich, currency 2 99 87 126 11 609 65y*i- A718, 96 300 112 $350, Am 126 ANALYSIS. 128 5 90 Ex.16. Dividing 128 by 16, we ob- tain what 1 horse Avill consume in 50 iq days ; dividing this result by 50, we ob- 50 tain what 1 horse will consume in 1 day. 72, Ans, Multiplying by 5, we find what 5 horses will consume in 1 day ; and multiplying this result by 90, we find what 5 horses will consume in 90 days. Ex. 11. Divide 4| (rr:Y) ^7 l^i (=%') to find what amount of wood $1 will buy ; then multiply by 24:^[=:\^) to find how much 824J will buy. Ex. 18. 52 X 3 X ^^= the money given for the cloth. Divide this result by 65, the number of yards, to find the price per yard. Ex. 19. A shadow of 1 foot will require an object J of 3 feet in length ; and a shadow of 46| feet will require an object 46 J times ^ of 3 feet in length ; hence ^xlx^^=28ieet,Ans. s 14 21 2 4 99 U.Ans. 52 3 3 100 65 Ans . 80 cents. Ex. 20. 8 sheep x Y|= 60 sheep for 1 mo., A's use of the pasture ; 12 15 " vfi2_inn « " " P.'a " " « X4i=: 50 X6|=100 u u " ,B's " " ,C's " " , total " 210 (C Each man should pay such part of the whole cost as his use of the pasture is part of the total use ; hence $63 X 2^=118, A must pay ; $63x^y=:$15, B " " $63xif==|30, C " " ANALYSIS. 127 Ex. 21. 1 bu. oats =1 dollars ; 1 bu. rye =-y bu. oats = Y ^ i rom tlve greater to find the less. The greater diminished by i of the greater equals the less, which must be | of the greater. And if the less be | of the greater, their sum, 20, is 1| times tb<5^2><3><3-225=-^^ Ex. 54. $3500— $2100=$1400, what B owns now ; $1400^ 1.40=$1000, B put in ; $2100-T- 1.40 = $1500, C put in. Ex. 65. i/S'-f 16^ = 17.88 +ft., Ans. Ex. 56. 12 ) 10 ) ^. ^ . V J Ex. 57. $12-v-1.09=$11.0091-f, worth of sugar; $12— $11.0091=$.9909 + , grocer's profit; $6^1.10=$5.4545+, worth of beef; $6— $5.4545=$.5455+, farmer's profit; $.9909— $.5455 =$.445+, grocer gains more, Ex. 58. $336.42 — $311.50 = $24.92, interest; $4.15i int. of $311.50 for 1 yr. 4 mo. at 1 per cent $24.92-i-$4.15|^=6 per cent, Ans. 148 PROMISCUOUS EXAMPLES. Ex 50 A 2 «,-,/] 2 — 12 10 anri -8- r^A. OJ. 513? '*"'^ 5 — 15) 1 5> ^l^'-i 1 5» These fractions are to each other as their nuiaerators, 12, 10, and 6 ; and these numerators are to each other as 6, 5, and 3, Hence, we have 6 shares for A and B ; 5 shares for A and G ; 3 shares for B and C. 14, twice the number of shares for A, B, an J C. 14 -7- 2 ==7 shares for A, B, and C. 7 — 3 = 4 shares for A ; 7 — 5 = 2 shares for B ; 7 — 6 = 1 share for C. Hence $20 x -f =$llf , A's ; $20xf = $54,B's; $20x|=$24, C's. Ex. .60. $375-^.025 = 815000, Ans. Ex. 61. Interest commenced Apr. 1, 1857. Amt. of note July 1, 1857, (3 mo.) $1015 Payment, 560 New Principal, $ 455 Amt., Dec. 1, 1857, (5 mo.) 466.37 -f Payment, 406 New Principal, $ 60.37 4- Amt., Aug. 23, 1859, (1 yr. 8 mo. 22 da.). .$ 66.63+, Ans. Ex. 62. B has | x ^ of A's =f of A's ; C has 4 X f of B's=-^ x f x f of A's =|f of A's ; D has I X f of C's= J x f x |? of A's=-; f of A's And since D has $45 more than C, If of A's— 1^ of A's=$45; or ^% of xVs -$45. Hencc; $45-^/^ =$378, A's;] $378 X I =$336, B's; $378x|^=$360, C's; ^^^^• $378 X If =$405, D's. PROMISCUOUS EXAMPLES. 149 Ex. 63. B had the use of $300 for 2Y months before it waa dne, which was equivalent to the use of $1 for 27 x 800=8100 months. But the use of $1 for 8100 months is equal to the use of $600 for VoV-=13i months, the time he should wait. Ex. 64. His savings and expenses together, or his sal&ry is If + A = Tf ^^ what he saves ; hence $800 = if rf That ^ie aves; and $800-^|f =$550, ^/i5. Ex. 65. $.87i=$J ; $1.00 H- 1.1 0=:$|^, cost per yd.; $-;^— $1 — $JL., loss at $.871 per yd. ; Tj^TT^/o^.OSf = 3^ per cent., ^;25. Ex. 66. 9 1 JL 2 4 ^^ 1 I 11 . 240 '320 3 /3\* 27 , ^ ^^ 63 189 27 /27 3 , ^"'^- 1491=448=64' / 64=? ^"^• Ex. 68. 50 - 20 tV 20 2 2 20 35 tV 10 2 2 20 60 To^ 15 3 3 30 70 ^V 30 3 3 "lo" 30 100 Ans, 20 of oats and corn, and 30 of rye and wheat. Ex. 69. 40 + 500 = 270, half the sum of the extremes. And since the sum of all the terms is equal to half the sum of the extremes multiplied by the number of terms, 6480-7-270=24, the number of creditors. 500 — 40=460, difference of extremes. $4604-23 =$20, common difference. Ex. 70. Vl28''-f 72' = 146.86-h, Ans. 150 PROMISCUOUS EXAMPLES. 10 11 14 4 10 550, Ans. Ex. 71. If 7 pounds of butter are equal to 10 pouuds of cheese, 11 pounds of but- ter are equal to y of 10 pounds of cheese ; and if 2 bushels of corn are equal to 11 pounds of butter, 14 bushels of corn are eqial to 'g*- of 11 pounds of butter, or y- of y of 10 pounds of cheese ; and if 8 bushels of rye are equal to 14 bushels of corn, 4 bushels of rye arc equal to f of 14 bushels of corn, or | of Jg^ of Jy of 10 pounds of cheese ; and, finally, if 1 cord of wood is equal to 4 bushels of rye, 10 cords of wood are equal to 10 times 4 bushels of rye, or \^ of f of V ^^ V ^^ ^^ pounds of cheese=550 pounds of cheese, Ans, Note. — ^Instead of the fractional form the vertical lino may bo iSQd^ as above. Ex. 72. $18-^f = $45 A's gain. f : ^ :: $45 : ( ) =$37.50, B's gain. $45-^.06 =$750, A's stock ; $37.50-^.06 = $625, B's stock '. 73. 20 ) ( ) ) 30 ' 45^ 21 V : 25 V :: 15 16 10) 8) 12 ^- 18 >■ 3^ 5 Reducing the statement, we have ( )-=84 , Ans, Ex. 74. 2| : 27^ :: 10 ft. : ( ) = 103i it., Ans. Ex. 75. A can do ^ of the work in 1 day ; J^ u ^ a u a -j^ u C " ^^ " " " 1 " They all " i + 1 + 7*2 = f of the piece in 1 day. Hence it will require 1-^f =f of a day. PROMISCUOUS EXAMPLES. 151 Ex. Y6. $1890-^1.25=:$1512, true value of 1st. $1890— .75=$2520, " " "2d. $4032, « " " both $1890 X 2 =$3780, received for both. Ans, $ 252, lost. Ex. 77. If C paid one half the cost, A and B together paid $50. Since C cows eat as much as 4 horses, 12 cows eat as much as 8 horses. Therefore, A put in 9 horses for 1 unit of time, and B put in the same as 8 horses for 2 units of time. Hence, A's use of the pasture was 9x1= 9 horses for 1 unit of time ; B's " " " " 8x2 = 16 " •« 1 " « " A andB's use of the pasture was =25 " " 1 " " " $50x^5 =$18, A paid ; $50 X if =$32, B paid. Again, C's time was 2i times B's time, or 2 x 2^ = 5 units of time. And since C paid half the cost, his use of the pasture must have been as much as A's and B's together, which is 25 horses for 1 unit of time ; and this is equal to 5 horses for C's 5 units of time, Now the pasturage of 1 horse is f times the pasturage of 1 cow, and the pasturage of 1 cow is J/- times the pasturage of one sheep ; hence the pasturage of 5 horses (for which C paid) is '/xf x5=25 times the pasturage of 1 sheep. Therefore C put in 25 sheep. Ex. 78 $350C-4-1.0175=$3439.803-f , pres. worth of 1st installment; 13500 -M.02i =$3420.195+, " " 2d ** |3500-^l.04| =$3343.949+, " " 3d ** $10203.947 -f, Ans. Ex. 79. Had the farmer sold both geese and turkeys at $.75 %piece,thc50 fowls would have brought $.75 x 50 =$37.50, 152 PROMISCUOUS EXAMPLES. which is $52.50 — $87.50 = $15 less than they really brought ; consequently the difference between the two estimates of the turkeys, reckoned at $.75 and $1.25 apiece, is $15. Hence. $].25-$.75 = S.50; $15-^$.50=:30, the number of turkeys; and 50 — 30=::20, the number of geese. Ex. 80. B gains of A 3 miles an hour, and C 5 mih:a an hour. Hence B will pass A every ^ hours=24 h. ?0 inin.=:1460 min. ; and C will pass A every y hours=14 h, 36 min. rr: 876 min. Now the least common multiple of 1460 and 876 will express the number of minutes in which B and C will first pass A together. 2,2 1460 . . 876 73 865 . . 219 5,3 5 . . 3 2x2x73x5x 3 = 4380 min.=:6 da. 1 h., Ans. Ex. 81. i, -} and i^f f, ii and i|. And since fractions having a common denominator are to eacV other as their numerators. A, B and C were to share in the proportion of 20, 15 and 12. But C dying, his 12 parts must be shared by A and B in the proportion of 20 and IS, or 4 and 3. 4 + 3 = 7. 7 : 4 : : 12 : ( ) = 6f , A's share of C's 12 parts ; 7 : 3 :: 12 : ( ) = 5i, B's " " " " " 20 + 6-|==26|, A's number of parts of the money ; 15 + 51 = 20 j, B's " " " " " 47 : 26 f :: $100000 : ( )= :$57l42.85f A's, ) .$42857.14f, B's, ) • 47 : 20 \' : $100000 : ( > Ex. 82. A' C c s + B's=:5 5-fB's=:7 s-B's=l Since A's + B's are t< B's + C's as 5 to 7, A and B together have 5 i.±J- =.4, C's proportion ; as often as B and C to- 7- -4r=3, B's u gether have 7. And 5- -3 = 2, A's u since C's -B's are to C's PROMISCUOUS EXAMPLES. 158 f B's as 1 to 7, 7 is the sum and 1 the difference of B and C's proportionate shares. Hence we find the proportionate share of each. Then 2 + 3 + 4 = 9, the sum of their proportions. And A has | of 135 = 30 sheep ; B " f " 135=45 « C " ^ " 135 = 60 " Ex. 83. 250x4 =1000 300x4^ = 1350 369x5 =1845 919 4195 4195-4-919 = 4f II, Ans. Ex. 84. The relative values of the work performed by one of each class, in the same number of days, are as follows : 1 boy 3x 8 = 24 ) (2 1 woman 4 x 9 = 36 >■ or •] 3 1 man 6 x 12 = 72 ) ( 6 The relative amounts of wages received by the whole num- ber of boys, women and men, in the same number of days, are as follows : Boys, $5 ; women, $10 ; men, $24. Hence the proportions of boys, women and caen ar*^ ex- pressed by the following quotients : 5-^2 = 21 boys; 10-^3=3i women ; 24- i-6=4 men. n :2i :: 59 : ( ) = 15 boys; H : 31 : 59 : ( ) = 20 women; n ; 4 : 59 : ( ) = 24 men. 7* 164 PROMISCUOUS EXAMPLEa Ex. 85. A, B and C fill ^ of it in 1 hour ; B, C " D " 1 " " " C, D " A " tV " " " D, A " B " j\ " " « A, B, C and D fill i of J/^=:-JJ^ of it in 1 hour. W, X and Y empty | of it in 1 hour ; X, Y " Z « } " " ** Y, Z '' W " \ " " " Z, W " X " 1 " " " W, X, Y and Z empty J of 111= Jt?^ of it in 1 hour. yVo"~TVo =tVo> ^^ emptying pipes gain of the filling pipes m 1 hour. Therefore, to exhaust the fountain will require 120-M9 = 63V hours, Ans, Ex. 86. 1 A. 1 R. 6 P. 181 gq. yd.=56250 sq. ft.; 56250 ^ , Ex. 87. 75x125 ' U^lbO, $ Ix = $ 2x 1 = 2 4x 2 = 8 8x 3 = 24 16 X 4= 64 32 X 5 = 160 64 X 6 = 384 128 X 7 = 896 256 X 8 = 2048 512X 9 = 4608 1024 X 10 = 10240 2048 X 11 = 22528 Cost, $4095. $40962, sum of products. 40962 -f-4095 = 10 mo., average term of credit. Jan. 1 4-10 mo. = Nov. 1, average time. PROMISCUOUS EXAMPLES. 155 Ex. 88. 44.32 x 36 = 1595.52 sq. ch. = 159 A. 2 R. 8.32 P.. Ans, Ex.89 l-|-i4-3-=lf. That is, if a number be increased by ^ and ^ of itself, the result will be If times the number. Hence, by the conditions. If times the number, plus 18, is equal to 2 times the number; consequently, 18 is 2 — l| = i of the number, and 18 x 6 = 108 is the number. Ex. 90. If that which is worth $.621 be rated at $.56, what ought that which is worth $.25 be rated at ? $.621 : $.25 :: $.56 : ( )=$.224 Therefore, m the barter for a pound of coffee at $.22, the mer- chant obtains that which is worth, ratably, $.224; hence he gains $.004 on $.22 ; and $.004-f-$.22 = .01/y, ^^s. ANOTHER SOLUTION. A pound of tea, in the barter, will buy 56-4-22 = 2/y pounds of coffee; and this is worth $.25 x 2y«y=:$.63yV But the pound of tea given is worth $.621; and $63/y~$.62i = $.01^2, the gain on $.621. Hence, lJL^62i=.01yV, the gain per cent. Ex. 91. First find how much ready money will cancel a debt of $1, due in 4 equal installments, for the times and at the rate mentioned, $ .25-^-1.011 =$.245901 -f, pres. worth of 1st installment; $ .25-^1.0375=$.240965+, *' " 2d " $ .25^1.05 =$.238095+, " « 3d " $ .25^1.081 =$.230769+, " « 4th " $1.00 $.955730 + , present worth of $1. Now ii' $1, payable as by the conditions named, be worth $.95573 in ready money, what sum will be worth, in readj money, $750, on the same conditions ? $1 : ( ) : : $750 : $.95573 ; $784.74 + , Ans. 156 PROMISCUOUS EXAMPLES. Ex. 92 |— 11=1 of the capital of either must be equal to $500. Therefore $500 x 8 = $4000, Ans. Ex.93. Due. Dec. 4 Feb. 9 Feb. 29 March 4 May 12 da. 00 67 87 91 160 Items. 240.75 137.25 65.64 230.36 36.00 710.00 Prod. 9195.75 5710.68 20962.76 5760.00 41629.19 41629.19-^710 = 59 da., average term of credit. Dec. 4, 1859 + 59 da.=Feb. 1, 1860, Arts. Ex. 94. When he had spent i of his fortune, he had f lelt I ofi = /^. He had spent, in all, i + 2^=4 of his fortune; consequently, the $2524 which he had left was ^ of his for- tune. $2524-^^ = $5889.33-f , whole fortune, Ans. Ex. 95 > By the conditions, the payments consist of five several parts or installments of the $3000 and interest on the first part or installment for 1 yr., on the second installment for 2 yr., on the third for 3 yr., on the fourth for 4 yr., and on the fifth for 5 yr. ; and the payments, each of which con- sists of the sum of one installment and its interest, are equal to each other. That is, the amount of first installment for 1 yr.= amount of second installment for 2 yr.= amount of third payment for 3 yr., and so on. But, as the payments are made annually, the interest must be added to the principal at the end of each year ; consequently the second year's interest is less than the first by the interest on the first installment, and the second installment must exceed the first by this in- terest, or by .07 times the first ; therefore, the second install* ment =1.07 times the first. For similar reasons the third installment =1.07 times the second, =1.07 x 1.07 times the first, and sa on to the last. Hence, the installments form a Geometrical Series, of which MENSURATION. 157 Ist installm.=: 1st term ; 5, (numb. ofpaym'ts)=No. of terms; 1.07=ratio; and $3000 = sum of the series. That is, we have the ratio, the number of terms, and the sura of the series, to find the first term. $3000 X (1 . 07-1'=).07 _^ (i.ovyi^i 1521.674 (1st installment) + $210 (int. on $3000 for 1 yr.) = 1^731.674, annual payment, Ans, :1st term=:$521.674-|- ; Ex.96. Due. mo. Itoms. Prod. Jan. 1, 1859 Sept. 1, " Apr. 1,1860 8 15 200 350 500 2800 7500 1050 10300 10300-7-1050 = 9 mo. 24 da. Jan. 1, 1859 4-9 mo. 24 da.=Oct. 24, 1859, Ans, Ex, 97. Make the notes given the Dr. side of an account, and the note received the Cr. side ; the balance will be the other note received, and the average maturity, its date. Br. Cr. Dae. da. Items. Prod. Due, da. Items. Prod. July 7, 1859 Oct. 4, " Feb. 20,1860 00 89 228 600 530 400 47170 91200 Novl5,1859 131 730 95630 1530 730 138370 95630 730956301 Balances 800 42740 42740-^800 = 53 da. July 7-^53 da.=Aug. 29, 1859, note due. 158 MENSURATION. MENSURATION OF LINES AJN'D SURFACES. (460, page 342.) Ex. 1. 3 X 12=36 inches long; 36 x 20=720, Ans. $32x198x150 Ex. 2. Ex. 3. 160 1000x100 =$5940, Ans. = 10 A., Ans, 10000 (469, page 343.) Ex.1. 20x12=240 sq. ch. ; 240x16 = 3840 sq. rd. m iie meadow, requiring 3840 rain, for mowing. 3840 min. = 6 da. 4 b., Ans. Ex. 2. 15 \/15^— 9^=12, the perpendicular; 15x12 = 180 sq.ft., Ans. 13 Ex. 1. Ex. 2. Ex.3. 16 + 9_ 2 16 + 8_ 2 40 + 22 (471.) = 12i in.=lJj ft, average width ; 12 xl2V = 12i sq.ft., Ans. = 12 in.=l ft., mean width ;' 1 x8 = 8 sq. ft., Ans, X 25=115 sq. ch.=77 A. 5 sq. ch., Ans. (47 5.) Ex. 1. ^^ X 45=3330 sq. fl.=370 sq. yd., Ans. Ex. 2. The two ends together are equal to a rectangle, 2H feet by 7 feet; hence 28 x 7 = 196 sq. ft.. Ans. MENSURATION. 159 (47 9^ page 344.) Ex. 1. 5x3.1416 = 15.708 ft. = 15 ft. 8.4 + in., Ans. Ex. 2. 721 X 3.1416 = 2265.0936 rd. = 7 mi. 25 rd. 1.54 + ft Ans, Ex. 3. 33 X .3183 = 10.5 + yards, Ans. (480, page 344.) Ex.1. 11^211^=10028.15, Ans. 4 Ex. 2. Reversing the rule (II,) 1 sq. mi.-^.7854= 1.2732 +, square ot aiamet«r, ^.2732 = 1.1284 +mi. = l mi. 41 rd. 1.4 +ft. Ex. 3. 84' X. 07958 = 561.5 +P. = 3 A. 81.5 + P., Ans. k: E y MISCELLANEOUS EXAMPLES PROGRESSIVE INTELLECTUAL ARITHMETIC. 97. 2. Since he sold -^ of his share,he had | left; and | of J is ^, Therefore, if a man owning f of a share in the Central Rail- road sold -f of it, he had | of a share left. 3. Since he gave J of it away, he had | of it left; and § of \ is i. Therefore, etc. 4. Since he gave | of it for a knife, he has | of it left, >»nd I of I is y^^. Therefore, etc. 5. Since $18 was | of what the watch cost, he lost J of the cost, which is i of $18 or 6. Therefore, etc. 6. Since $45 was f of the cost, he gained i, which is J ot $45, or $5. Therefore, etc. '7. Since -f of the cost was sacrificed, $120 is | of the cost; f of $120, which is $30, is |, and 3 times $30 is $90, the whole loss. Therefore, etc. 8. Since he lent i of the remainder, $22 is |, and i of ^22, which is $11, is i ; 3 times $11 is $33, or the whole of the remainder; and $33 is J^ of 4 times $33 which is $132. Therefore, etc. 9. Since $80 v^as ^ of | of 2 times, or | the cost, the losa was i which is J of $80, or $40 Therefore, etc. (US) 162 MISCELLANEOUS EXAMPLES IN THE 10. Since S^54 was ^ of 2 times, or -f the selling price, lie gained |, which is J- of $54, or $9. Therefore, etc. 11. Since 15 is f, ]• of 15, which is 3, is } ; 8 times 3 's 24, and i of 24 is 8. Therefore, etc. Or, i of that number of which \ of 15 is i. 12. Since 4 is |, i of 4, which is 2, is ^; 3 times 2 is 6, which is I of 2 times 6, or 12. 4 from 12 leaves 8, aL^d limes 8 is 16. Therefore, etc. 13. Since he sold J of his flock, 20 must be f ; | of 20, which is 10, is ^, and 5 times 10 is 50. Therefore, etc. 14. Since | of the remainder, or | of J which is |, was in the water, the 3 feet in the mud must be the remaining a ; and 3 feet is i of 5 times 3 or 15 feet. Therefore, etc. 15. J + 1 = 1- j, and the 4 years equals the remaining y^, and 4 years are /_ of 30 years. Therefore, etc. 16. Since $20 = 1 the cost of the coat plus $12, $20— $12 = $8, or I the cost; I of $8, or $4 = i, and 3 times $4 is $12. Therefore, etc. 17. If i the number + 80 was 5 more than 3 times the num- ber, 80 — 5 = 75, when added to I the number must have been 3 times the number. 3 — |, and J — i = f ; and 75 must have been f times the number. 75 is f of 2 times } of 75, or 30. Therefore, etc. 18. Since 16 is |, | of 16 or 8 is i, and 3 times 8 is 24, which is twice as many as James has ; then i of 24 or 12 is what James has after losing 16, and 12 + 16 = 28, or the number James had at first ; since J of John's equaled ^ of James's, ^ of ^ or I of James's=^ of John's ; | of 28 is 4, and 5 timea 4 is 20, or what John had at first. Therefore, etc. 19. Since J + 10 ycars=:l^ or J of his age, f — 3 — J or J must =10 years; and 2 times 10 years is 20 years. There- fore, etc. 20. Since | of $40 is /_, i of |, or i of $40 is j\ ; i of $40 is $8, and 11 times $8 is $88, which is 2| or | times i ; } of (145, 146) PROGRESSIVE INTELLECTUAL ARITHMETIC. 163 $88 or $11 is i of 3 times §11 or $33, and $33 is i of 4 times $33, or $132. Therefore, etc. 21. 4 times f is f ; and 25 is f of 4 times i of 25 which is 20. Therefore, etc. 22. Since he made away with f of | of it, the $10 left was the remaining | of | or I of the whole ; and 5 times $10 is $50. Therefore, etc. 23. Since | of $1500 is f, J of |, or j% of $1500, which is $200, is 1 ; 8 times $200 is $1600, which is 4 times the cost of the barn, and J- of $1600 is $400. Therefore, etc. 24. Since | of 500 or 300 men was | of yf ^, or -^^ of the force, 50 times 300 men or 15000 men, was the whole force. Therefore, etc. 25. Since f of 100 or 60 was 7i or J/» tj of 60 which is 4 is J ; 2 times 4 is 8, which is y^^ of 100 times 8 or 800. Therefore, etc. 26. Since f of 60 or 50 was 2^ or f times yi^, i of 50 or 10 men is |, and 2 times 10 is 20, which is j^q of 150 times 20 or 3000 men. Therefore, etc. 27. Since f of 100 or 80 was 1^ or y-, yV of 80 or 8 is |. 1 times 8 is 5Q, less 6 = 50 or ^\. And 20 times 50 is 1000. Therefore, etc. 28. Since 2i times 30 or 70 is 3i or V", rV of 70 or 1 is i, 3 times 7 is 21, which is j\ of 10 times 21 or 210. There- fore, etc. 29. Since i of 1200 or 1000 is 8^ or V', sV of 1000 or 40 is 1 ; 3 times 40 is 120, which is yf^ and i of 120 or 20 ia ^^0 ; 100 times 20 or 2000 is j of 10000, which lacking 1000 of being the whole army, 10000 + 1000 = 11000. There- fore, etc. (146,147) 104 MISCELLANEOUS EXAMPLES IN THE 98. 2. Since ^6 + $4 or $10, bouglit 40 bushels, $1 would bujf 1*0 of 40 bushels or 4 bushels, aud $6 would buy 6 times 4 bushels or 24 bushels, and |4, 4 times 4 bushels or 16 bushels. Therefore, etc. 3. Since they approach each other 4 miles + 3 miles or 7 miles an hour, they will meet in Y- or 7 hours, and the one who traveled 4 miles per hour would travel 7 times 4 or 28 miles, and he who traveled 3, Y times 3 or 21 miles. Therefore, etc 4 Since 3 weeks +2 weeks or 5 weeks' hire is $25, 1 week IS i of $25 or $5, 2 weeks 2 times $5 or $10, and 3 weeks 3 times $5 or $15. Therefore, etc. 5. Since 5 cows -f 3 cows or 8 cows' pasture cost $24, 1 cow's pasture cost | of $24, or $3, 5 cows' 5 times $3 or $15, and 3 cows' 3 times $3 or $9. Therefore, etc. 6. 2 horses for 2 weeks=l horse 4 weeks, and 2 horses 4 weeks = l horse 8 weeks; and since 12 weeks' pasture cost $12, 1 week's pasture costs y*2 of $12, or $1, 4 weeks' 4 times $1 or $4, and 8 weeks' 8 times $1 or $8. Therefore, etc. 7. Since 9 cents + 7 cents or 16 cents bought 32 figs, 1 cent would buy y*g of 32 or 2 figs, 9 cents would buy 9 times 2 or 18 figs, and 7 cents would buy 7 times 2 or 14 figs. There- fore, etc. 8. A's $10 for 5 months=:$5 for 1 month, B's $5 for 8 months=:$40 for 1 month; and since $50 + $40 or $90 gain $45, $1 will gain J^ of $45, or $i, $50 50 times $^, or $25, and $40, 40 times $i, or $20. Therefore, etc. 9. Since they paid in the proportion of $5, $4, and $3j. they own in the same proportion ; consequently the gain ii divided into 5 plus 4 plus 3, or 12 parts, and j\ of $24 is $2. A's portion is 5 times $2 or $10; B's, 4 times $2 or $8; and C's, 3 times $2 or §6. Therefore, etc. 10. Since A does 2 times 3 days, or 6 days' work, B 3 times 3 days, or 9 days' work, and C 3 times 2^ or 5 days' (147, 148) PROGRESSIVE INTELLECTUAL ARITHMETIC. 165 work, it takes 20 days to mow the field ; 1 day^s work c/jsts 5?o of |40, or $2 ; and A should receive 6 times $2, or $12; B 9 times $2, or 18 ; and C 5 times $2, or §10. Thei'efore, etc. 11. Since C took $10, or :^2 — -^V of the gain, he must have put in 2^^ of the stock, and A's $30 plus B's $50, or $80 = |f ; ^V of $80, or $5=^\, and 5 times $5 or $25=:iC's siock $42— $10=$32,A's + B'sgain; A's=ff or f of $32, or $12; end B'sr=f J or ^ of $32, or $20. Therefore, etc. 12. Since he put in f^J or f of the capital, he should also lake f of the gain; f of $240rr$150, and $150-$145=$5 loss. Therefore, etc. 13. Since 2 colts consume as much as 3 calves, 4 colts, or 2 times 2 colts=:2 times 3 calves, or 6 calves, and 5 calves plus 6 calves = ll calves. If 11 calves cost $11, 1 calf cost j\ of $11, or $1 ; 5 calves 5 times $1, or $5; and 6 calves 6 times $1, or $6. Therefore, etc. 14. Since C pays |- of the rent, he puts in i of the cows. Then A's 5 cows + B's 3 cows =8 cows--=| of the cows, and |- of 8, or 4 cowsr^C's number. And since C's 4 cows cost i of $42 or $14, 1 cow cost { of $14, or $3i ; 5 cows cost 6 times $31, or $17^; and 3 cows 3 times $31, or $10^. Therefore, etc. 15. Since 4 cows=3 oxen, 8 cows, being twice 4, = 2x3, or 6 oxen ; and since 5 calves = 4 cows, 10 calves, being twice 6, =2 X 4 or 8 cows. But 8 cows=:6 oxen; and 9 oxen + 6 oxen -f- 6 oxen==21 oxen, which cost $5B. 1 ox cost ^^ of $5G or $2| ; 9 oxen 9 times $2|, or $24 ; and 6 oxen 6 times $2 1, or 16 ; etc. 16. Since Mary wrote | as many lines as Melissa, Melissa's work IS divided into 8 parts, 7 of which =Mary's; then 8 + 7 = 15 ; and j\ of 60 is 4 ; j\ 8 times 4, or 32 ; and y''^ 7 times 4, or 28. Therefore, etc. 17. Since the boys received as many pears as the girls, tbey received i of 24, or 12. There were as many boys as 8.R.P. ' (148, 149) 166 MISCELLANEOUS EXAMPLES IN THE } is contained tioies in 12, whicli is 4 times; as many girls is 4 is contained times in 12, which is 3 times ; and 4 + 3 = 7, rherefore, etc. 18. Since each son received ^ as much as each daughter, the 2 sons received as much as 1 daughter ; then we have $96 divided into 3 + 1=4 parts; } of $96=$24=each daughter's portion ; and J of $24=$12=each son's portion. Therefore, etc. 99. 1 The 1st has 1 part, the 2d 1 part +2, and the 3d 1 part-T'2 + 6; then 3 parts + 2 + 2 + 6 = 76, or 76 = 3 parts + 10; and 76 — 10, or 66 = 3 parts; J of 66 or 22=what 1st boy had ; 22 + 2, or 24 = what 2d boy had ; and 22 + 2 + 6, or 30 = what 3d boy had. Therefore, etc. 2. Henry has 2 more than James, and Joseph having 2 more than Henry, has 4 more than James ; hence 72, the sum of all, is 2 + 4, or 6 more than if each had no more than James. l2 — Q = G6yAns, 3. If Henry had 2 more he would have as many as Joseph ; and James + 2=Henry, and +2 more= Joseph ; and 72 + 2 + 2 + 2 = 78, Ans. 4. If Joseph give James 2, Joseph's number will be dimin- ished and James's increased 2, when each will=Henry's. James will now have ^ of 72 — 6, or 22 ; Henry 22 + 2, or 24 ; and Joseph 24 + 2, or 26. Therefore, etc. 5. Since C paid as much as A and B, he paid ^ of $600, or $300; and B and A paid $300. And as B paid $100 more than A, $300 — $100==$200, or what each would have paid if they had paid no 'nore than A. J of $200 is $100, or what A paid ; and $100 + $100 = $200, what B paid. 6. The drum cost T part, the rifle twice as much, or 2 parts., and the watch twice as much as the rifle, or 4 parts ; hence $42 is divided into 1 part + 2 parts + 4 parts, or 7 parts. I of $42, or $6 = costof drum; 2 times $6, or $12 = cost of rifle; and 2 trmes $12, or $24 —cost of watch Therefore, etc. (149) PROGRESSIVE INTELLECTUAL ARITHMETIC. 167 Y. The harness cost 2 parts, the horses 4 times as much or 8 parts, and the wagon IJ times the harness, or 3 parts; and the harness 2 parts 4- the horses 8 parts, plus the wagon 5 parts=15 parts=$225. j\ of $225 is $15, and 2 times $]5, or $30==harness ; 8 times $15, or $120=:cost of horses ; and 5 times $15, or $75z=cost of wagon, etc. 8. Since he traveled ^ as far the 1st as the last 2 daya, the last 2 days' travel is divided into 2 parts ot which 1 = first day ; hence ^ of 114 miles, or 38 milesi^lst day; the same reasoning applied to the last day gives 38 miles, and leaves 38 miles for the 2d day. 10. The note of $20 was less than ^ of what remained due after the 1st payment, by the $20 that exceeded \ ; hence $20 + $20, or $40=r|. 2 times $40, or $80 = what remained after 1st payment, and $80 was less than ^ the debt, by the $10 the payment exceeded i ; $80 + $10, or $90= J ; and 2 times $90, or $180= the whole debt. 11. The 4 pennies left is less than \ of the remainder by the 1 penny more than ^ paid for the whip; then 4+1=5 pennies, or ^, and 2 times 5, or 10 pennies = remainder after purchasing top ; and since he paid 2 pennies more than \ of all for the top, 10 pennies + 2 pennies, or 12 peunies=| ; ^ of 12 or 6 pennies=i ; and 3 times 6 = 18 pennies. There- fore, etc. 12. Since he sold the whole, the \ gallon more than | the remainder sold was ^ of the remainder, and 2 times ^ or 1 gallon = remainder. The gallon left after 1st sale was less than 1 the keg by the | gallon more than the \ gallon sold ; then 1 gallon + i gallon or 1| gallons = ^ the keg, and 2 times 1 1^ or 3 gallons=the contents of the keg. 14. Since | of John's = J of Mary's, \ of John's=-J- of J or f of Mary's, and | or all of John's =3 times f or | of Mary's ; hence xMary's are divided into 8ths and John's =9 of them, and the whole = J/ of Mary's, y^y of 34 or 2 is | ; 9 times 2 or 18= John's, and 8 times 2 = 16 = Mary's. (1195 150) 168 MISCELLANEOUS EXAMPLES IN THK 15. Since | of A's plus 8=B's, B's— 8 = | of A's; and if 8 be taken from B's, the sum of both flocks will be 83 — 8 or 75. A has 3 parts, B 2, and both 5. ^ of 75 is 15. 3 times 15 or 45= A's; and 2 times 15 or 30, + 8 = 38, B's. 16. Since f of Mary's less 10 cents = Susan's, Susan's f 10 cents=:f of Mary's, and then both would have 39 + 10 or 49 cents. Mary having 4 parts, and Susan 3, they both have 7 parts. I of 49 or 7 = 1 part; 4 times 7 or 28=Mary's; and 3 times 7 or 21 — 10 = 11 =:Susan's. 17. Since J of Homer's=-f of Silas's, J of Homer's will=J- of f or f of Silas's, and f or the whole of Homer's, 5 times f or Jy" of Silas's; and since Homer's exceeds Silas's by ^ of Silas's, the 3 marbles must=^ of Silas's ; hence Silas has 1 marbles and Homer 10. 100. 1. Since the first drink a gallon in 8 days, he will drink i of a gallon in 1 day, and since the second drink a gallon in 4 days, he will drink J of a gallon in 1 day ; both will drink i+i or -{fj of a gallon in 1 day, and 1 gallon will last as many days as y'g, what they drink in 1 day, is contained times in j| or 1 gallon ; y\ is in y| 1-f times. Therefore, etc. 2. Since Julia can do it in 7 hours, in 4 hours she can do 4 of it, and Jane must do the remaining ^ ; and since Jane does if in 4 hours, she will do | in i- of 4 or li hours, and ^, or tht whole, in 7 times li, or 9^ hours. Therefore, etc. 3. Since the first can do it in 9 hours, he can do f in 5 hours, and the second must do the remaining |- ; and since the second pitches |^ in 5 hours, he can pitch J in | of 5, or 1 1 hours, and f, or the whole, in 9 times 1^, or 11; hours. 4. 3f=Y- and 7{=Y-. Since the second pipe can empty t in \^ hours, it can empty j\ of it in |, and |f in Y" l^ours, and the first must empty the remaining || ; and since the first can empty || in ^■^- hours, it can empty -^\ in gV of W or j hour, and |-|, or the whole, in 58 times |, or 7| hours. (151) PROGRESSIVE INTELLECTUAL ARITHMEIKJ. 169 6. Since A can make a vest in | of a day, he can make as many vests in a day as f is contained times in |, or 1^ vests ; B as many as f is contained times in J, or 1-^ vests ; and l| + li, or 3 vests=:what A and B can both do. C can make as many as f is contained times in |, or 1} vests, and 3— li = l|. Therefore, etc. 6. Susan can knit as many pairs as | is contained timei m I, or 1| pairs; Sarah can knit as many as -3 is contained times in ^, or 2i pairs; and l| + 2^ = 4 pairs. 7. Since Sarah can knit 2i or J pairs in a day, she can knit i of a pair in | of a day, which is the part she must knit for Susan. 8. Since Susan can knit 1| or f pairs in a day, she can knit ^ of a pair in i of a day, which is the part she must knit for Sarah. 9. Since Jason can hoe 10 rows in J of an hour, he can hoe 1 row in y^ of J, or j% of an hour, and as many rows in an hour as 3 is contained times in 40 or 131 rows. Since Jesse can hoe 10 rows in j of an hour, he can hoe 1 row in y'^ of f or j\ of an hour, and as many rows in an hour as 3 is contained times in 50, or 16| rows; and both can hoe 13i + 16|, or 30 rows, in an hour; 1 row in ^^ of an hour ; and 10 rows in i§ or ^ of an hour. 10. Smce Jesse can hoe 16| or \o rows in an hour, in I of an hour he can hoe | of */ or V=^3 rows; leaving 1| rows for Jason, who can hoe 13i or ^« rows in an hour, 1^ of a row in y^^ of an hour, and 1| or f rows in 5 times yV or I of an hour. 11. Since Jason can hoe 13^ or ^^^ rows in an hour, in J of an hour he can hoe ^ of */, or V=4f rows ; leaving 5f rowa for Jesse, who can hoe 16|, or Y=-^p rows in an hour, J of a row in yJ^ of an hour, and \% or 5f rows in 50 timea , jj of an hour. 12. See analysis of Example 9. 13. Since A and B can clear the field in 15 days, thev can (151. 152) 170 MISCELLANEOUS EXAMPLES IN THE clear f^ of it in 1 day, and -fj or J of it in 9 days ; and sinco A and B clear | of it in 9 days, C must clear the remaining } ; and if he clear f in 9 days, he will clear i in ^ of 9 or 4j days, and f or the whole field in 5 times 4^ or 22| days. 14. Since A and B can dig it in 6 days, they can dig } of it in 1 day ; since A and C can dig it in 8 days, they can dig J of it in 1 day ; and |— J or j\ of it, is what B does more in a day than C. As B and C dig it in 9 days, they can di^ i ^^ it in 1 day, and since B's day's work exceeds C's by ^■f of the well, J — aV ^^ 1^=^ ^^ ^'® ^^7^ ^.nd J of yV ^^ yf j =what C can do in 1 day ; hence C can do it in as many days as 5 is contained times in 144 or 28|^ days. Since B and G dig I of it in 1 day, and C digs yf y of it in 1 day, i — yf 4- or yU_=what B digs; hence B can dig it in as many days as 11 is contained times in 144 or 13tV davs. Since A and B dis ^ of it in 1 day, and B digs jW of it in 1 day, J — yVv or tV\ =:what A digs in 1 day ; hence A can dig it in as many days as 13 is contained times in 144 or llyj days. 15. Since A digs y^^^, B y^^, and C yf y of it in 1 day they will all dig yW + yW -f if?? or y^y of it in 1 day ; and it will take as many days as 29 is contained times in 144 01 m <^ays. 16. Since Patrick and Peter can dig it in 15 days, they can dig y'y of it in 1 day, and |{ or | in 10 days, and Philc must dig the remaining third ; and since Philo digs ^ in 10 days, he can dig f or the whole in 3 times 10 or 30 days. Since Philo can dig it in 30 or */■ days, he can dig ^V of it »n 1 of a day, and in 13^ or ^/ days he can dig 40 times ■^\ or |- of it, and Peter must dig the remaining f ; and since he digs f in 4Ji days, he will dig | in ]^ of V" or f days, and f or the whole in 9 times f or 24 days. Since Peter can dig it in 24 days, in 15 days he can dig ^f or f of it, and Patrick must dig the remaining | ; and since he digs | in 15 daysi be will dig | in i^ of 15 or 5 days, and f in 8 times 5 oz 40 days. As Patrick can dig 40 rods in 24 days, he can dig (152) PROGRESSIVE INTELLECTUAL ARITHMETIC. 171 iV of 40 or 1| rods in 1 day, and since Peter ,ian dig 40 rods in 40 days, lie can dig 1 rod a day, and it will take liirn as many days asl|-fl = 2|is contained times in 28, or 10|^ days. 17. Since 30 rods is JJ or J of 40 rods, it will take each man J as long to dig it. Since Patrick could dig it in 40 days, he can dig 30 rods in J of 40 or 30 days ; since Peter can dig it in 24 days, he can dig 30 rods in f of 24 or 18 days and since Philo can dig it in 30 days, he can dig 30 reds in I of 30 or 22i days. 18. Henry's work is divided into 4 equal parts, and since Harlan's exceeds Henry's by 1 of these parts, he must do 5 parts, and both of them 4 + 5 or 9 parts. Since Henry cuts f of it in 6| or ^^ days, he can cut ^ in ^ of -^- or f days, and f in 9 times | or 15 days. Since Harlan cuts f of it in 6| or 2-0 days, he can cut |^ in ]^ of ^/- or ^ days, and f in 9 times 1^ or 12 days. 19. Since the 3d does | as much as the 1st and 2d, the work of the 1st and 2d is divided into 5 parts; and since the 3d does | as much,. the whole is divided into 5-f 2 or 7 parts. Since the 3d does f of the whole in 10 days, he can do I in |- of 10 or 5 days, and -J in 7 times 5, or 35 da;'s. Since the 1st and 2d do ^ in 10 days, they can do 4 in } of 10 or 2 days, and ^ in 7 times 2 or 14 days. And since the 1st does J as much as the 2d, the whole is divided hto 7 parts, of which the 1st does 3, and the 2d 4 parts. Smce the 1st does ^ in 14 days, he can do | in |^ of 14 or 4 1 days, and ^ in 7 times 4| or 32| days. Since the 2d does \ in 14 days, he can do | in {^ of 14 or 3^ days, and ^ in 7 times 3i or 24^ days. 20. See Analysis of example 19. 21. Since the 1st can do it in 32| or *3^ days, he can do j'j of it in |- of a day or -^j in a day ; and since the 3d cau do it in 35 days, he can do j'j of it in 1 day ; and both can do ^8 + 3 J ^^ -Ns ^^ 1 ^^y» *^^ ^^^ whole in as many days ag 29 is contained times in 490 or l^-^j days. (152) 172 MISCELLANEOUS EXAMPLES IN THE 22. Since the 2d can do it in 24^ or Y" ^^ys, he can dc j^ of it in i of a day, or /^ in a day ; and since the 3d can io ^^-g of it in 1 day, the 2d and 3d can do /g +-3J, or g'^^ of it in 1 day, and they can do all of it in as many days as 17 is con- tained times in 245, or 14^'^^ days. 23. Since B and C can do it in 12 days, they can do y"^ cr I of it in 8 days, and A must do the other i ; and since A can do "I" in 8 days, he can do ^ in 3 times 8 or 24 days. Since A and B can do it in 10 days, they can do j% or J of it in 8 days, and C mast do the other \ ; and since C can do | in 8 days, he can do | in 5 times 8, or 40 days. Since A can do it in 24 days, he can do i| or /^ of it in 10 days, and B must do the remaining y^g ; and since B can do y\ i^ ^^ ^^J^i ^g can do j\ in I of 10 or 1^ days, and -ff in 12 times If or 17 j dayt. 24. Since the 1st and 2d will discharge it in 8 hours, they 'discharge f or i of it in 4 hours, and the 3d must discharge the other ^ ; and since it discharges -^ in 4 hours, it will dis- charge I in 2 times 4, or 8 hours. Since the 3d will discharge it in 8 hours, it discharges f or | of it in 6 hours, and tho 1st must discharge the other \ ; and since the 1st discharges } of it in 6 hours, it will discharge | in 4 times 6 or 24 hours. Since the 1st and 3d discharge it in 6 hours, they will dis- charge f or I of it in 4 hours, and the 2d must discharge the other i ; and since the 2d discharges J in 4 hours, it will dis^- charge f in 3 times 4 or 12 hours. 25. Since A and B can do it in 20 days, they do ij or f of it in 10 days, and C does the other i ; and since C does | in 10 days, he can do | in 2 times 10 or 20 days. Since B and C can do it in 15 days, they do j| or | of it in 10 days, and A does the other third ; and since A does ^ in 10 days, he can do f in 3 times 10 or 30 days. Since A can do it in 80 days and C in 20 days, they can both do 3V + 2V or y'^ of it in 1 day, and if in 12 times 1 or 12 days. 26. Since it would last them aH 30 days, they would eat 3'^ of it in 1 day, and 20 times gV or | of it in 20 days, leaving \ (152, 153) PROGRESSIVE INTELLECTUAL ARITHMETIC. 173 of it to be eaten by the sister. Since the brother and servant would eat it in 45 days, they would eat |f or | of it in 30 days, and the sister must eat the other J in 30 days. 101. 2. Since 2 plums was tbe increase given to 1 playmate, and 9—1 or 8 plums the increase given to all, there were as many !d\ ay mates as 2 is contained times in 8, which is 4 times. Therefore, etc. 3. Since the difference between 6 times and 3 times a num- ber is 3 times the number, 15 must be 3 times the number, and J- of 15, or 5 must be the number. Therefore, etc. 4. Since the difference per yard was 12 cents — 8 cents, oi 4 cents, she wanted as many yards as 4 is contained times in the whole difference, 11 cents + 17 cents, or 28 cents, which is 7 times. 5. Since the difference between 6^ times and 4 times a number is 21 times or f times the number, 15 must be | times the number; J of 15 or 3, i ; and 6, the number. 6. Since | — i = |, i of 4 or 2 must be | and 9 times 2 or ,18 = t. • • 7. Since the difference between 5^ times and 3| times a number is 2.1 times the number, -^j of 21, or 1, must be y^, and 10 times 1, or 10, |J, or the number. 9. If we let 1 or | represent the whole number of chickens, I times 5 + J- times 3=-'/, will represent the whole number of grains, that is, "3^=26. And since 26 is -y- times the number of chickens, | or the whole number of chickens, was 3 times | of 26, or 6. 10. Since 26 is 5 times | + 3 times ^, or ^-f of the number; }^ of 26 or 2 is ^, and 3 times 2 or 6 is f or the number. Therefore, etc. 11. Since the 1st condition gives 5 times ^ of a number, plus the 2d condition, which gives 3 times J of the same num- ber, plus the 3d, which gives 2 times | of the same n amber, (153, 151) 174 MISCELLANEOUS EXAMPLES IN THK plus the 4th, which gives once J the same number ; we have ^ of the class equal to 29, and ^^^ of 29 or 1 is ^ of 9 times 1 or 9. Therefore, etc. 13. Since 4 times ^ of a number, plus S^ times the num- ber, or -y- times the number, is equal to 28 plus 5, or 33, f\ of 33, or 3 is ^ of the number, and 2 times 3 or 6 is the num- ber. Therefore, etc. 14. Since the 1st condition gives ^ of his age plus 4, the 2d gives f, and the 3d gives ^ less 4, we have the sum -'J equal to 50 ; y^ ^^ ^^ ^^ 5 is ^ of his age, and 3 times 5 or 15 is J. Therefore, etc. 15. Since he paid $5 a head for ^ of the flock, the cost=:5 times ^ or 1^ of his flock; $4 a head for as many more = 4 times I or J of his flock; $3 a head for | of the remainder, or ^ = 3 times J or i of his flock ; and 82 a head for the rest, or J- = 2 times | or i- of his flock; and f + J + i + i = V' That is, the number of dollars the flock cost=:\3 of f}^Q num- ber of sheep, and 115 is ^ o^ ^ times gV of 115 which is 30 Therefore, etc. 16. Since he received 6 dimes each for I =1x6=1 for ^ of the remaining | and 3 more, 4 dimes each =(f + 3) x4=f + 12dime» for 1 of the rest (which is f less 1 on each eight), or 1 + 1 = (} + l)x3=:f+ 3 ** for the rest (which is f less 1 on each eight, also the 2 of last sale), or f less 4 =(|— 4) x 2=| — J ** the whole number of dimes is equal to -5/ of the baskets and 7 dimes besides, . . =-\^+ 7 ** hence $10 or 100 dimes less 7 dimes=93 dimes=3_i^ _i_ qj 93 or 3 = |-, and 8 eighths, 8 times 3 or 24. Therefore, etc. I 18. 6 times a number equals \^y Y times J of it plus 5 dmes 1 of it equals ^/, and Y less Y- = J or ^ of it, which, (154, 155) PROQKESSIVB INTELLECTUAL ARITHMETIC. 175 according to the condftion of the question, is 4 ; and 4 is | of 2 times 4 or 8. Therefore, etc. 19. 6 times the number, or Y» left 4 cents, but 5 times J oi it, or y , plus 7 times ^ oi it, or Y- was it all of it ; and by the condition of the question ^^ less y or \ equals 4 ; and f is 2 times 4 or 8. Therefore, etc. 20. 4 times a number equals Y, 5 times 4 of it equals y, and Y less Sy^- equals f of it, which-by the question is 6 ; an.l 6 is I of 7 times ^ of 6, which is 14. Therefore, etc. 21. 2 times a number equals | of it, 5 times i of it equals I, and this plus 2 times \ of it — which is | — equals | of it, and J less f , equals J of it, which by the conditions of the question is 60 ; and 60 is | of 2 times -J- of 60 which is 40. Therefore, etc. 22. 2 times } of a number equals f of it, which is 8 more than J' hence 8 is f or i of it, and 2 times 8 or 16 is the whole of it Therefore etc, 103. 2. Since 19 is the sum of two numbers whose difference is 8, 19 less 3, or 16, is twice the less number; ^ of 16 is 8, the less number, which, increased by 3, equals 11, the greater number. Therefore, etc. 3. Since 31 is the sum of two numbers whose difference is 9, 31 less 9 or 22, is twice the less number ; | of 22or 11 is* the less number, which, increased by 9, equals 20, the greater number. Therefore, etc. 4. Since 37|^ is the sum of two numbers whose difference it 5^, 3Y| less 5^ or 32, is twice the less number ; | of 32 is 16 the less number, which, increased by 5^, equals 21J-, the greater number. Therefore, etc. 5. Since 21 is the sum of two numbers whose difference is 5, 21 less 5 or 16, is twice the less number; |^ of 16 is 8; the number Homer had at first, plus 3, equals 11, or what (155, 156) 176 MISCELLANEOUS EXAMPLES IN THE he lias now ; and 21 less 11, or 10, equals what Horace h^ now. Therefore, etc. 6. Since Mary has twice as many as Martha, she has 2 parts, and Martha 1, they both have 3 parts; ^ of 12 quarts or 4 quarts, equals what Martha has, and twice 4 or 8 quarto equals what Mary has. Therefore, etc. 7. Since 47 is the sum of two numbers, one of which is S nore than twice the other, 47 less 5, or 42, equals 3 times he less number ; i of 42 or 14, equals the less, and twice 14 or 28 plus 5, which is 33, equals the greater. Therefore, etc. 8. If the small bin held 6 bushels more, it would contain J as much as the other, and both would hold 60 bushels, or 3 times as much as the small one ; ^ of 60 or 20, less 6, which is 14, equals the number of bushels in the smaller bin, and 2 times 20 or 40, equals the number in the larger bin. There- fore, etc. 9. Had the watch cost $4 more, both would have cost $100, or 4 times the cost of the chajn ; J- of $100, or $25, equals the cost of the chain, and $96 less $25, or $71, equals the cost of the watch. Therefore, etc. 10. Since Hiram received 11 times 2, or 22 dimes more than Harvey, 253 dimes, what both received, less 22 dimes, or 231 dimes, equals twice what Harvey received; | of 231, or 1151 dimes equals what Harvey received, and 115^ dimes, ■plus 22 dimes, or 137|^ dimes equals what Hiram received; y'y of 115 J dimes, which is $1.05, equals what Harvey received per day; and $1.05, increased by 2 dimes, equals $1.25, what at Hiram received. Therefore, etc. 11. Since B's age was 2 times A's 6 years since, 48 years^ the sum of their ages then, must have been 4 times A's age ; I of 48, which is 12, plus 6, or 18 years, equals A's age ; and 60 less 18; or 42 years, equals B's age. Therefore, etc. 12. Since the horse cost $4 more than 3 times the cost of the cow, $124 less $4, or $121, is 4 times the cost of the cow; \ of $121, or $30.25, equals the cost of the cow (156, 157) PROGRESSIVE INTELLECTUAL ARITHMETIC. 177 and $124 less $30.25, or $93.75, equals the cost of the lorso Therefore, etc. 13. Since the product is the same whichever factor be taken for the multiplicand, we will use } of the cost of the cow, which taken 4 times, equals J or the whole cost ; hence, i of the cost of the colt must be $4 ; twice $4, or $8, equals what he paid for the colt; and $24 less $8, or $16, equals what he paid for the cow. Therefore, etc. 14. Since the cost of the cover (which, by a condition of the question is ^ as much as the dish plus the difference), increased by the difference, equals the cost of the dish, the dish costs twice the difference plus I of itself, or the differ- ence equals J of the cost of the dish ; and f less |, or § of the cost of the dish equals the cost of the cover ; and 24 dimes equals | of the cost of the dish. | of 24 dimes=3 dimes, 5 times 3 = 15 dimes, the cost of the dish; and 24 dimes— 15 dimes =9 dimes, the cost of the cover. 15. Since the less number, — which by the question equals | of the greater plus the difference, — increased by the difference equals the greater, we have the greater equaling -i of itself plus twice the difference, or the difference equaling j\ of the gi-eater, and t6~"tV — tV ^^ ^^^ greater equals the less; |J 4-y«g=f-f ; 2 J of 25 pounds, or 1 p )und, is j\ of 16 pounds, the greater number, and 9 times 1 pound or 9 pounds is the less. Therefore, etc. 16. Since the sum of the difference and the less number equals the greater, the less must equal | of the greater, and both numbers f of the greater ; | of 10 = 2 is i of the greater number, 3 times 2 is 6, the greater ; and 10 less 6, or 4 is the less, 17. Since the cost of ironing, plus ^ of the difference, equals I*-, of the cost of the wood-work, the remaining ^j must equal i of the difference, and the difference equals j\ of the cost of the wood-work ; || less y\ equals j\ of the cost of irou- mg ; II plus Y®, or |^ times the cost of the wood-work equali (157) 178 MISCELLANEOUS EXAMPLES IN THK |38. i-'g of $38 or $2, is yV» ^ ^^^n^^s $2 or $22, is the cost o( the wood-work, and $38 less $22 or $16 is the cost of ironing, 18. Since the cost of the ribbon, — which by the question equals ^ of the cost of the lace, plus ^ the difference, increased by the difference between the cost of the lace and ribbon, — equals the cost of the lace ; we have -^ of the cost of the lace equal to f of the difference, or the lace costing a sum equal Ic } of the difference, and the ribbon ^ of the difference, and both 30 cents, or 5 times the difference. | of 30 cents, or cents, is the difference between the cost of the two ; 30 cents less 6 cents or 24 cents, is twice the cost of the ribbon, and ^ of 24 cents or 12 cents is the cost of the ribbon ; and 30 cents less 12 cents, or 18 cents is the cost of the lace. 19. Since the whole of the cost of the knife and once the difference equals the cost of the skates, and by the question ^ the cost of the knife plus twice the difference equals the same, once the difference must equal ^ the cost of the knife, twice the difference the whole cost, 3 times the difference the cost of the skates, and 5 times the difference equals 20 shillings, or the cost of both ; | of 20 shillings is 4 shillings, 2 times 4 shillings is 8 shillings, the cost of the knife ; and 3 time?* 4 shillings is 12 shillings, the cost of the skates. 20. Had the harness cost $1 more, both would have cost i J5, and the horse would cost ^ of $35 or $20, and the harnesi* ^ of $35 or $15, less $1 or $14. Therefore, etc. 103. 2. Had all been old sheep, he would have paid $84, or $8 more than he did ; each yearling made a difference of $1, hence there were as many yearlings as $1 (the difference on 1) is contained times in $8 (the difference on all), which i» 8 times ; and 28 less 8 equals 20, the number of old sheep. 3. Had all been first quality, he would have paid $9( ^ ox $8 more than he did ; and since the difference per barrel Ht» (157, 458) PROGRESSIVE INTELLECTUAL ARITHMETIC. 179 $•1, he bought as many barrels of poor quality as $1 is contained times in $8, or 8 barrels; and 20 less 8 equals 12 first quality. 4. Since he lost | of the cost, ^ of $18 or $9, must have been J of the cost, and 5 times $9 or $45, was the whole cost. 5. There were as many of each as 12 dimes (the number it took to pay one of each) is contained times in 72 dimes (the number paid to all). 12 is in 72 6 times, and 2 times 6 12, the whole number. 6. Since »he received 8 dimes for 1 of each, she sold as many of each as 8 dimes is contained times in 40 dimes, which is 5 times ; twice 5 is 10, the number of fowls she sold. 7. He bought as many bushels as $.50, the difference on 1 bushel of each is contained times in $7, the difference on all ; $.50 is in $7 14 times. Therefore, etc. 9. He was idle as many days as $3.50 (the difference made by 1 idle day) is contained times in $7 (the difference made by all the idle days), which is 2 times ; 20 days less 2 days is 18 days. Therefore, etc. 11. Since she gave | of the remainder to her teacher, the 2 left must be the other ^ ; 4 times 2 is 8, which was the J left after division among the playmates, and 4 times 8 or 32 equals the number she had at first. Therefore, etc. 12. J of 1 1 is y\ ; hence 12 is y\ of his flock, and 10 times 12, or 120 equals the number of sheep he had at first 13. Since he paid | of the remainder, $3 must be | of it. J of $3 is $1, and 5 times $1 is $5, the remainder, which by thefirst payment wants $5 of being i of the whole ; $5 plus $5 is $10, 1 of the whole, and 4 times $10 is $40, the whole Therefore, etc. li. Since he lent |, $3 plus $5 or $8, must have been i. 8 times $8 or $24 is what he had after paying for the watch ; $24 plus $12 or $36 equals what he had after paying for hia clothes, which lacks $10 of being i of his wages; $36 plua llO, or $46 is ^ ; and 2 times $46, or $92 equals his wages. (158,159) 180 MISCELLANEOUS EXAMPLE? IN THE 15. iSince in $1 there are 10 climes, he could be idle tm many days, for each day he worked, as 2 dimes, what he paid a day for board, is contained times in the amount his daily wages exceeded $1, which is once; hence he worked ^ of the time, 9vd was idle 10 days. 104. 2. The part standing was divided into 4 equal parts, 3 of ir!iich equaled the part broken off; the sum of both piec^ uras 7 equal parts, 1 of which was | of 56 feet or 8 feet, 3 parts were 3 times 8 or 24 feet, which was the part broken off; and 4 times 8 or 32 feet was the part standing. Therefore, etc. 3. Since Henry has 5 parts and Horace 4 parts, both have 9 parts ; ^ of 45 is 5 ; 4 times 5 or 20 equals the number Hor? ',e had, and 5 times 5 or 25 equals the number Henry had. Therefore, etc. 4. Since he left 5 parts and took out 3, he left f of 160, or 100 pounds. Therefore, etc. 5. Since he paid 5 parts for his lodging and 4 for his sup- per, his supper cost J of 63, or 28 cents. Therefore, etc. 6. Since 9 times | = | times the cost of wagon, equals the cost of the horse, both cost 8 plus 9, or y- times the wagon ; iV of $170 is $10 ; 8 times $10, or ^80 was the cost of the wagon ; and 9 times $10, or $90 the cost of the horse. Therefore, etc. 7. Since the second day's travel was 1^^=^ times the first, both equaled f times the second ; | of 140 miles is 20 miles; 3 times 20 equals 60 miles, the second day's travel; and 4 times 20 equals 80 miles, the first day's travel. 8. Since Bergen is 50 miles from Buffalo, 280 miles less 60 or 230 miles equals the distance from Bergen to Schenectady; and as the distance from Utica to Schenectady is l^zzz^j^ times the distance from Bergen to Utica, the whole distance from Bergen to Schenectady equals y plus |, or Y '^ _i_ of 230 miles is 10 miles, and 15 times 10, or 150 miles equals the (159, 160) PROGRESSIVE INTELLECTUAL ARITHMETIC. 181 distance from Schenectady to Bergen ; and 150 miles plus 60 miles, the distance from Bergeu to Buffalo, gives 200 miles from Buffalo to Utica. 9. Since the head was 3 inches long, 17 inches less 3 or 14 inches equals the length of the body and the tail; and as the body was divided into fifth's, 2 of which equaled the tail, we have body and tail divided into 7 parts ; | of 14 inches is 2 inches, and 2 times 2 or 4 inches equals the tail. 10. Since the less has 7 parts and the greater 11, both have 18 ; jV o^ '^6 is 2 ; 7 times 2 = 14, the less part; and 11 times 2 = 22, the greater. 12. If the distance from Victor to Rochester were 4 miles less, it would equal the ^j mentioned, and the whole distance would be 52 miles less 4 miles, or 48 miles ; from Geneva to Victor is 11 parts, from Victor to Rochester 5 parts, in all 16 parts; y^- of 48 is 4 ; 11 times 4 miles=44 miles, the distance from Geneva to Victor ; and 52 miles less 44 miles =18 miles, the distance from Rochester to Victor. 13. If the church were 6 feet lower, the whole distance would be 140 feet, of which the steeple would be 4 part's, the church 3 parts, and both 7 parts; | of 140 feet is 20 feet; and 4 times 20 is 80 feet, the height of the steeple. There- fore, etc. 14. Since the jar (which, by a condition of the question, weighs as much as i the cover plus 12 pounds) and the cover weighs 18 pounds, we have the cover, J the cover and 12 pounds equal to 18 pounds, or f of the cover weighing 6 pounds; ^ of the cover, ^ of 6 pounds, or 2 pounds; and |, times 2 pounds, or 4 pounds ; and 18 pounds less 4 pounds, i 14 pounds, the weight of the jar. Therefore, etc. 15 Had the vest cost ^3 less, both had cost but $16, of which the coat cost 3 parts, the vest 1, both 4 parts ; } of ^16 18 ^4 ; 3 times $4 is $12, the cost of the coat ; and $4 plus $3, or $7 is the cost of the vest. Therefore, etc. 17 Since | of George^s equaled J of Abel's, 2 halves would (160, 161) 182 MISCELLANEOUS EXAMPLES IN THE equal twice f , or J ; then Abel had 4 parts, George 6 parta^ and both 10 parts ; j\ of 50 cents is 5 cents ; 6 times 5 cents = 30 cents, George's money ; and 4 times 5 cents =20 cents, Abel's money. Therefore, etc, 18. Since f equaled 4, i would equal J of 4, orf, and f 5 times f or y ; then the black ones were 7 parts, the gray ones 10, and both 17 ; j\ of 34 is 2 ; 10 times 2 is 20, the num- ber of gray ones ; and 7 times 2 is 14, the number of black ones. Therefore, etc. 19. Since | equaled f, | would equal | of |, or i^g, and J, 3 times /^ or |f ; one number is divided into sixteenths, 15 of which equals the other, and ^ equal both ; Jy of 62 is 2 ; 16 times 2 is 32, the larger number; and 15 times 2 is 30, the smaller number. Therefore, etc. 20. Since J equals |, J would equal J of |, or -j^, and J, 4 times ^y or y"j ; the value of the contents is 15 parts, of thfl* purse 8 parts, and of both 23 paits ; ^3 of 46 shillings is 2 shillings ; 15 times 2 shillings is 30 shillings, the value of the contents ; and 8 times 2 shillings is 16 shillings, the value of the purse. 22. Since from midnight to 10 o'clock is 10 hours, and the past time is divided into 3 parts, the future into 2, and the whole iuto 5, we have 1 part equal to ^ of 10 hours, or 2 hours ; and 3 times 2 hours is 6 hours, the past time ; hence it was 6 o'clock. 23. Since | equals J, ^ must equal ^ of J, or f, and f, 3 times f , or f ; from midnight to 5 o'clock, p.m., is 17 hours, and as past time is 8 parts, future 9 parts, and the whole 17 parts, 1 part equals 1 hour, and 8 parts 8 hours ; hence it is 8 o'clock, A. M. 24. Since J equaled J, { would equal \ of J, or J, and }, 4 times I, or |. John's age was divided into fifths, 4 of which equaled Peter's, and both equaled f of John's ; J of 36 years = 4 years, J of John's; 5 times 4 years =20 years, John'i age; and 4 timcf 4 years =16 years, Peter's age, (161, 162) PROGRESSIVE INTELLECTUAL ARITHMETIC. 183 25. Since J equaled J^*, f would equal f ; and we have what :ras wanting divided into 5 parts, what was in the bin into 6 parts of the same size, and the whole capacity of the bin into 11 parts; ^j of 44 bushels=4 bushels, 1 part; and 5 times 4 bushels = 20 bushels, what was wanting to fill the bin. 26. Since f of what it exceeded equaled V of what it lacked. I would equal y- ; and we have what it lacked divided into 7 parts, what it exceeded into 15, or the whole, 83 miles— 39 miles=44 miles, divided into 22 parts ; ^V ^^ ^^ ™^^^^ ^^ ^ oiles; 7 times 2 miles is 14 miles, the distance it lacked of being S? miles ; and 83 miles less 14 miles is 69 miles, the dis- tance to Cincinnati. 27. Since f of what it lacks of being 150 miles equals what it exceeds 100 miles, we have, the excess, 3 parts plus the deficiency, 2 parts, or 5 parts in all, equal to 150 miles less 100 miles, or 50 miles ; } of 50 miles is 10 miles ; 3 times 10 miles is 30 miles ; and 100 miles plus 30 miles =130 miles, the distance from Charleston to Columbia. 105. 2. Since f equal 4 + 9, J will equal ^ of ^ 4- 9, which is f 4-8, I, and 4 times f + 3, which is ^ -h 12 ; hence, the mother's age is divided into 7 parts, and 8 of the same size + 12 years equals the father's age, or 15 parts + 12 years equals 72 years ; 72 years less 12 years is 60 years, j\ of 60 years is 4 years, and 7 times 4 years equals 28 years, the mother's age. 3. Since | equal f less 4 rods, -J- will equal ^ of J less 4 rods, which is | less 2 rods, and §, 3 times | less 2 rods, which is I less 6 rods ; hence what one built equals 6 rods less than ) of what the other built, and both built -y of the amount the eeccud did, less 6 rods ; 38 rods plus 6 rods, or 44 rods equals V ? fV ^^ ^4 r^^s ^^ ^ Todsy is A of 5 times 4 rods or 20 rode, what the second built; and 38 rods less 20 rods, or 18 rods fquftis what the first built. 4. Since \ was 4 more than J^ | would be 1 more than J, (162, 163) 184 MISCELLANEOUS EXAMPLES IN THE and -J, 7 more than J ; hence what Richard sheared are divided into 5 parts,- Hiram's into Y parts plus 7 sheep, and both into 12 parts plus 7 ; 67 less 7 is 60 ; /_ of 60 = 25, the number Richard sheared ; 67 less 25=42, the number Iliram sheared. 5. Since | of future time equaled | of the past 4- f f hours, j- would equal J of | + 1| hours, which is ^ -f j\, and |, 5 times ^4-tj ^ours, which is -3 +f hours; hence the future time equals f hours more than f of the past, and both past and future time equa. f of the past-hf hours, or 24 hours; 24 hours less |- hours is 21^ hours, and | of 21^ hours is 8 hours, or the past time; hence it was 8 o'clock a. m. 6. Since | of what his age lacked of being 100 years equaled f of what it exceeded 64 years, + 9 years, | of his age would equal i of J + 9 years, whicb is J--|- 1 year, and f , 8 times } + l year, which is | -f 8 years ; hence, what his age lacked of being 100 years equaled 8 years more than | of what it ex- ceeded 64 years, and \^ of what it exceeded 64, is 8 years less than the difference between 100 years and 64 years, or 36 years; 36 less 8 is 28 years, j\ of 28 years is 2 years, and 6 times 2 or 12 years, is what his age exceeded 64 years. 8. Since the body is as long as the head and tail, it must be I of the length of the fish ; the tail being as long as the head and ^ the body, must be ^ of the length of the fish plus 7 inches, and the 7 inches it exceeds the { with the 7 inches of the head, must equal the other ^ ; 14 is J of 4 times 14 or 66. Therefore, etc. 9. The first price plus the second, equal to J 4- 3 pounds, equals the third price; 2 times j+3, equal to | of it + 6, equals the whole of it, and 6 pounds must be f of it ; | of 6 pounds 01 2 pounds is } of it, and 5 times 2 = 10 pounds is the whole of it. Ji^^ Or it may be solved like the following, 10. Since the third dug as many as the other two, he dug ^ , and as the first two dug \ less 2 bushels + 5 bushels, or 3 bushels more than |^, those 3 bushels must equal the difference (163, 164) PROGRESSIVE INTELLECTUAL ARITHMETIC. 185 Detween |- and ^ of them, or } of tliem ; and 3 bushels is J oi 6 times 3 bushels which is 18 bushels. 11. Since the distance from Avon to Bath is 12 miles more than the sum of the other two distances mentioned, we have the whole distance equal to f of itself+ 60 miles ; hence | »f 60 miles or 20 miles is \ of the distance ; and 5 times 20, OT 100 miles is the whole distance from Batavia to Corning. 12. Since he took $24 more than i of the whole for shee]» and swine, and $7 less than f as much for cattle, he took for the cattle |2 more than \ of the whole ; and we have $18 4-3 of the whole, +$6, +1 of the whole + $2, or ^^ of the whole + $26, equal to the whole amount; hence -^^ of $26, 3r $2 is 2V, and 24 times $2, or $48 is what he took for all. 13. Of ^ that number t)f which |, of J and ^ of ^ of 12 is 1. 1 and i of ^ of 12 is 6, ^ of 6 is 2, and 2 is 1 of 8 times 2, or 16. Therefore, etc. 14. Since he earned | as much as he had spent, he only lacks I of |=J of the whole, of having as mucn as he had at first ; $16.50 is ^ of 6 times $16.50, or $99. Therefore, etc. 15. Since | equal f, | will cost i of | of an eagle, or $2. 16. Since C is | as old as A, he is 4 years more than ^ as old as B ; and as B's age equals the sum of A's and C's, we have i of it plus 6 years, plus ^ of it plus 4 years, or | of it + 10 years, equal to itself; hence 10 years must be } of B^s age, and 6 times 10 years is 60 years B's age ; ^ of 60 is 30, 30 + 6 is 36, A's age ; and | of 36, or 24 is C's age. Il, Since C owns ^ as much as A, he owns 6 acres more than 3 as much as B; and we have what A owns, 12 more than J as many acres as B, + what C owns, 6 more than f at many acres as B, equal to 18 more than f as many acres as B owns, or 24 acres more than his farm ; hence 6 acres equals J- of B's farm, 8 times 6 is 48 acres, B's ; f of 48 is 36 acres, and 36 + 12 equals 48 acres or A's ; and ^ of 48, or 24 acres equals C's. (164J 186 MISCELLANEOUS EXAMPLES IN THE 106. 2. Since the son's age is | of the father's, the 22 years tho father's age exceeds the son's must be ^ of the father's age ; 3 times 22 years is 66 years, the father's age, and 66 years less 22 years equals 44 years, the son's age ; at the son's birth the father was 22 years old, in 22 years from that time each would be 22 years older, and the son being 22, and the father 44 years of age, would answer the condition of the ques tion, and as the son is 44 now, 44 years less 22 years, or 22 years since, he was | as old as his father. 3. At Helen's birth her sister was 22 less 9, or 13 years of age, and in 13 years from that time Helen would be 13 and her sister 2 times 13, or 26 years of age; and as Helen has advanced through 9 of 13 years, she has 13 less 9 or 4 years more to advance. Or, for brevity, 2 times 9 is 18 ; 22 less 18 is 4. Therefore, etc, 5. Since he took as many from one field and put in the other as were there, and now both have twice as many as were there at first, the 60 sheep must have been three times ths number before removing ; ^ of 60 is 20, the number in the smaller ; and 20 plus 60, the number in the larger flock, equals 80, the whole number. 6. Since both bins now contain the same quantity, and each 2 bushels more than twice what was in the less at first, the larger must have had 4 bushels more than 3 times the less ; 62 less 4 is 48 bushels, ^ of 48 is 16 bushels, what was in the less; and 16 bushels + 52 bushels, what was in the larger, equals 68 bushels. Therefore, etc. 7. J less I is 2*;j-, which by the condition of the question, is 6 more than | of his age; ^*y less | is j'^, 6 is y'j of 72. Therefore, etc. 8. Since he received { of his wages for his summer's la- bor, \ as much, or } of them in fall -f S20, and $20 in springs $20 -f $20=^40 must be f or i of his wages, and 3 time* |40=$120, must be the whole amount. (165) PROGRESSIVE INTELLECTUAL ARITHMETIC. 187 10. Since A sold B J as much as B had, B now has J o) what he had i*t first, which is J of what A has left ; ^ of | =y''^ is I ; and 4 times j\, or | what A has now — f f, what A has left ; f | plus the J sold B, gives A f } of B's before the Bale; -^\ of 74 is 2, and 12 times 2 is 24, the number of acres B had before the sale ; and 24 acres plus f of 24 acres equals 42 acres, what he now has; and 74 less 18 acres leaves 5Q acres, what A has left. 11. Since | equal i, i will equal ^ pf J or J, and f, 3 times J or I ; hence as the turkeys equal J of the chickens, 10 must be the remaining \ ; 4 times 10 is 40. Therefore, etc. 12. Since | of the price of the coat equaled J of the price of the suit, ^ would equal ^ of j- or J-, and f , 3 times | or J of the suit ; and since the coat cost f of the price of the suit, $15 must be f of it ; ^ of $15, or $3 is |, and f are 8 times $3 or $24. Therefore, etc. 13. Since f times his brother's equaled J of his, ^ would equal | of J or y^^ of his ; f , which is 3 times jV, or j\ and the 14 more must be the remaining ^^^ | of 14 is 2 ; 10 times 2 or20 equals what Daniel caught, and 20 less 14 equals 6, what his brother caught. 14. Since by the conditions of the question we have ^ of 3^ times a number +15, equal to once the number +15, or, to avoid fractions, 2 times a number +30, equal to 3^ times he same number + 15, 2 times the number is equal to 2 times the same number, leaving 30 equal IJ times the number + 15, or 15 equal to J of the number; ^ of 15 or 5 is i of it; 2 times 5, or 10 is the less number; and 3^ times 10, or 35 is the larger. 15. Since f equal /j, | will equal i of j%, or /^ ; and }, 5 ♦.irnes /^ or | ; and as the buggy cost | as much as the horse, the difference, $40, must be i of the cost of the horse ; 3 times $40 or $120 is the value of the horse ; and | of it, or $80, is the value of the buggy. 16. 5 years since the mother's ago was 5 times Alice's, ana (166) 188 MISCELLANEOUS EXAMPLES IN THE by the first condition we have 5 times Alice's age +5 (the mother's age) equal to 3 times Alice's age +15 ; and since 3 times Alice's age equals 3 times her age, we have 2 times her age + 5 equal to 15, or 2 times her age equal to 10 ; ^ of 10 is 5, her age; 5 years since + 5 equals 10, her age now ; and 3 times 10 or 30 is her mother's age ; 2 times 10 is 20, and 30 less 20 is 10, the number of years in which she will be I as old as her mother. See Ex, 3, in this lesson, 17. Since Hobart has but | of his left, he lost \ of them to Dwight, which, by the condition of the question, was equal to ^ of D wight's ; % must have equaled all of D wight's, and the 20 Hobart's exceeded Dwight's must have been \ of Ho- bart's; 3 times 20 = 60 marbles Hobart had ; and 2 times 20 = 40 marbles Dwight had. 18. Since the difference between the numbers is 16, if 4 be taken from the larger difference will be 12, then added to the less it will be but 8 ; 2 J times this difference, or 19, is equal to 3J- times less 2f times = if times the less number; y^ of 19 or 1 is -^^ ; 24 times 1=24, the less number; and 24-fl6 = 40, the larger number. 19. Since he paid twice as much for the rifle as for the watch, and the watch cost $20, the rifle cost- 2 times $20, or $40. 20. Since C's age at A's birth was 5^ times B's, and is now equal to the sum of A's and B's ages ; and as the increase of C's age w^ould just equal A's age, and B's increase being the same, the increase must have been what B's age lacked of being equal to C's at first, or 4^ times B's age then ; hence we have A's age now equal to 41 times B's age at first ; J^'g « *4 a u gx ** " " " " Q)g u u u <' 10 ^^ ^^ ^^ ^^ ^^ ^ow if 4 years be added to B's age, | of the sum, or 4} times /i's age as first + 3 years, is equal to A's age, and 4^ times B's it first, which gives the 3 years, is equal to f of B's age at first; hence B was 8 years old then, and is now 5^ times 8, or 44 vears old ; A is 4| times 8, or 36 years old ; and C is 10 times 8, or 80 years old. (166, 167) PROaRESSIVB INTELLECTUAL ARITHMETIC. 189 107. 2. In as many hours as 2 miles, the number he gained in I hour, is contained times 2 times 5 miles, the distance to be gained ; 2 times 5 is 10 miles, and 2 is in 10 5 times. There- fore, etc. 3. As many times 9 rods as 2, the number of rods he gaina in running 9, is contained times in 28, the number to be gained; 2 is in 28 14 times, and 14 times 9 is 126 rods. Therefore, etc. 4. John will have as many times $7 as $2, what he gains on $7, is contained times in $30, the whole gain ; $2 is in $30 15 times, and 15 times $7 is $105, what John has saved ; and $105 less $30 is $75, what Henry has saved. 5. Since the distance B ran is divid'^d into eighths, 1 of which equaled the distance he was ahead of A, A must have run J as far as B ; | of 84 = 12 rods is \ of the distance B ran, and 8 times 12 = 96 rods, is B's distance 6. Since $25 is \ of what B and C paid, they paid 4 times $25 or $100, which, with the $25 A paid, makes $125, th© cost of the horse ; and since B paid | as much as A and C, they paid 3 parts and he 2 parts ; that is, 5 parts equal the whole cost ; f of $125 is $50, what B paid ; and $50 plus the $25 A paid equals $75 which, taken from $125, leaves $50, what C paid. 8. Since the minute hand passes over 12 spaces while the hour hand passes over 1, tbe minute hand gains 11 spaces on the hour hand for every 12 spaces it passes over, and it would pass as many times 12 spaces as 11, the number it gains in passing 12, is contained times in 45 spaces, the number to be gp^ned after 3 o'clock before they are opposite ; 11 is in 45 4 p"| timus, and 4Jy times 12 is 49 jV spaces. Therefore it would be 49y*y minutes past 3 o'clock. 9. Since 3 of the hound's leaps equal 6 of the fox's, 1 will equal I of 6, or 2 of the fox's, and 4, 4 times 2, or 8 of the fox's ; hence the fox will take as many times 7 leaps as 1, the 9.R.P. (167, 168) 190 MISCELLANEOUS EXAMPLES, ETC. number the hound gains on the fox in making 7, is contained times in 40, the number of leaps to be gained ; 1 is in 40, 40 times, and 40 times 1 is 280. Therefore, etc. 10. Since the distance the sheep ran was divided into 5 parts, 8 of which equaled the distance between them, the whole dis- tance equaled 8 parts ; | of 80 rods or 10 rods is 1 part, and 3 times 10 or 30 rods eq^ials 3 parts, or the distance betwee thorn. 11. Since the interest at 5 per cent., for 2 years 7 months and 6 days, is y'^^ of the principal, the amount will be m ; j{^ of $2260 is $20, and 100 times $20r=$2000, the sum at interest ; and since B's money equaled f of A's, the whole equaled | o^ A's ; } of $2000 is $400 or i of what A had in, which is I of all, and 5 times $400 or $2000 equals A's ; $400 equaled \ of what B had in, which is | of all, and 8 times $400 or $3200 equals B's. 12. Sinc(> B's fortune is 1^ times A's, \ of A's is equal to ^ of Ws, and Ih*^ mterest on it for 5 years at 6 per cent, would equal j\ of it ^ of $600, or $200, is yV of 10 times $200, or $2000; i ot $2000, or $1000*, equals what each had in ; 2 times $1000, or $2000, equals A's fortune ; and 3 times $1000, or $3000, equals B'c. 13. Since he lost 8 per cent, or -fj of the cost on the sale, he sold for || of the ca^ ; hence \ of his calves and | of his shf^ep cost $25, and 4 times $25, or $100 is the cost of all the calves and | of the sheep ; this exceeds the whole cost by $24, which must equal the cost of the | of the sheep ovei the whole number ; -J- of $24 is $C, and 5 times $8 is $40 which would buy 20 sheep at $2 ; ^76 lesj. $40 givep $36 for calves, which would buy 12 calves at $3. Therefore, etf (168) The Standard School Dictionaries OF THE ENGLISH LANGUAGE. By NOAH WEBSTER, LL.D. This popular Series is very justly regarded as the only National stand- ard authority in Orthography, Definition and Pronunciation, and as such, these works are respectfully commended to Teachers and others as the best Dictionaries in use. More than T£N times as many are sold of Webster's Dictionaries as of any other series in this country, and they are much more extensively used than all others combined. 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