LIBRARY OF Tin: UNIVERSITY OF CALIFORNIA. GIKT Mrs. SARAH P. WALS WORTH. Received October, 1894. ^Accessions No . 3Lr Class No. KEY TO THE PROGRESSIVE PRACTICAL ARITHMETIC, INCLUDING ANALYSES OF THE MISCELLANEOUS EXAMPLES IN THE PROGRESSIVE INTELLECTUAL ARITHMETIC, FOR TEACHERS ONLY. IVISOST, BLAftl^^rraYLOR & CO., NEW YORK AND CHICAGO. 1875. ROBINSON'S Mathematical Series. Graded to the wants of Primary, Intermediate, Grammar, Normal, and High Schools, Academies, and Colleges. sv$t!4 Progressive Table Book. Progressive Primary Arithmetic. Progressive Intellectual Arithmetic. Rudiments of Written Arithmetic. JUiYlOR-CLASS ARITHMETIC, Oral and Written. NEW. Progressive Practical Arithmetic. Key to Practical Arithmetic. Progressive Higher Arithmetic. Key to Higher Arithmetic. New 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 Conic Sections, Surveying and Navigation. Entered, according to Act of Congress, in the year 1860, by HORATIO N. ROBINSON, LL.D., Jn the Clerk's Office of the District Court of the United States for the Northern District of New York. PREFACE A KEY to any Mathematical work is not intended to su- persede labor and study, but to give direction to the latter and make it more effective and useful. In many examples and problems the 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 examples, and of s^ch questions as contain the application of a new principle. The arrangement * such as to be easily understood. NOTATION. ROMAN NOTATION. (17, page 9.) Ex. 1. Ans XL Ex. 3. Ans. XXV. Ex. 5. Ans. XLVHI. Ex. 7. Ans. CLIX. Ex. 9. Ans. MDXXXVIII. Ex. 2. Ans. XV. Ex. 4. Ans. XXXIX. Ex. 6. Ans. LXXVII. Ex. 8. Ans. DXCIV, Ex. 10. Ans. MDCCCCX. Ex. 1. Ans. 125* Ex. 4. Ans. 900. Ex. 7. Ans. 505. ARABIC NOTATION. (26, page 12.) Ex. 2. Ans. 483. Ex. 5. Ans. 290. Ex. 8. Ans. 557. (28, page 13.) Ex. 2. ^4ns. 5160. Ex. 5. Ans. 2090. Ex. 8. Ans. 9427. Ex.3. Ans. 716, Ex. 6. Ans. 809. Ex. 3. Ans. 3741. Ex. 6. Ans. 7009. Ex. 9. ^W5. 4035. Ex. 1. Ans. 1200. Ex. 4. ^tm\ 8056. Ex. 7. ^ws. 1001. Ex. 10. Ans. 1904. Ex. 11. ^4ws. 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 tbousand 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. Ans. 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, 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. 406420015vJ, 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 ten. Ex. 18. Ans. Fifty-seven million four hundred sixty-eight Jiousand one hundred thirty-nine. Ex. 19. Ans. Five thousand six hundred twenty-eight. Ex. 20. Ans. Eight hundred fifty million twenty-six thou- sand eight hundred. Ex. 21. Ans. Three hundred seventy thousand five. Ex. 22. Ans. 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. Ans. 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.5. Ans. 698. Ans. 898. SIMPLE NUMBERS. ADDITION. (4O, page 21.) Ex. 4. Ans. 967. (42, page 24.) Ex. 7. Ans. 1807. Ex. 8. Ans. 27246. Ex. 9. Ans. 4945. Ex. 10. Ans. 78313. Ex. 19. Ans. 145. Ex. 22. Ans. 69585. Ex. 23. Ans. 566. Ex. 24. Ans. 3746. Ex. 27. Ans. 4619. Ex. 28. Ans. 4915. Ex. 29. Ans. 4320. Ex. 30. Ans. 4623. Ex. 31. Ans. 3871. Ex. 35. Ans. 101500. Ex. 37. Ans. 50000000. Ex. 40. Ans. 1194. Ex. 44. Ans. 2773820. Ex. 45. Ans. 4403241 SUBTRACTION. (49, page 30.) Ex. 6. Ans. 353. Ex. 8. Ans. 205. Ex. 19. Ans. 123. Ex. 24. Ans. 4202. Ex. 26. Ans. 16348755. Ex. Ex. Ans. 210. Ans. 320. Ex. 22. Ans. 2113. Ex. 25. Ans. 11425. Ex. 27. Ans. 4014580, Ex. Ex. (51, page 32.) Ans. 721. Ans. 3769. Ex. Ex. Ans. 561. Ans. 269. SUBTRACTION. Ex. 7. Ex. 9 Ex. 11. Ex. 13. Ans. 4509. Ans, 1288. Ans. 21078. Ans. 762301. Ex 20. Ans. 220874. Ex. 8. Ans. 3449. Ex. 10. Ans. 30616. Ex. 12. Ans. 142. Ex. 19. Ans. 224130. Ex. 25. Ans. 181972. Ex. 31. Ans. 529509693. Ex. 34. Ans. 1902001 EXAMPLES COMBINING ADDITION AND SUBTRACT ION. (52, page 33.) Ex. 1. 2500+ 175 = 2675 5200 2675 = 2525 dollars, Ans. Ex. 2. 235 + 275 + 325 + 280 = 1300 1115 = 185 miles, Ans. Ex. 3. 4234 + 1700 + 962 + 49=6945; 87^6945 = 1807 dollars, Ans. Ex. 4. 4765 + 750 = 5515; f>515-34^5131 dollars, Ans. Ex. 5. 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, Ans. Ex. 8. 2324 + 1570 + 450 + 175=4519; 67844519=2265 miles, Ans. Ex. 9. 7375, first paid ; 7375+ 7375 = 14750, second paid; 7375 + 14750=22125; 35680-22125 = 13555, dollars, Ans. 10 SIMPLE NUMBERS. Ex.10. 750 + 379 + 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. Ex. 8. Ans. 416223. (64, page 41.) Ex. 5. Ans. 2508544. Ex. 6. Ans. 15731848. Ex. 7. Ans. 16173942. Ex. 9. Ans. 78798. Ex. 13. Ans. 937456. CONTRACTIONS. (67, page 43.) Ex, 2. 3472x6=20832; 20832x8 = 166656, Ans. Ex. 3. 14761x8=118088; 118088x8=944704,^5. Ex. 4. 87034x3=261102; 261102x3 = 783306; 783306x9 = 7049754, Ans. Ex. 5. 47326x6 = 283956; 283956 x 5 = 1419780* 1419780x4 = 5679120, Ans. MULTIPLICATION. 1 1 Ex. 6. 60315x8x3x4 = 5790240, Ex. 7. 291042x5x5x5 = 36380250, Ans. Ex. 8. 430x7x8 = 24416 miles, Ans. Ex. 9. 124 x 6 x 3 x 4 = 8928 dollars, Ans. Ex. 10. 5280 x 7 x 3 x 4 = 443520 feet, Ans. Ex. 11. 120 x 5 x 5 x 5 = 15000 dollars, Ans. (69, page 44.) Ex. 3. Ans. 13071000. Ex. 4. Ans. 890170000. (7O, page 45.) Ex. 12. 888000 dollars, Ans. EXAMPLES COMBINING ADDITION, SUBTRACTION, AND 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, Ans 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 ; tf4 x 16 = 1024 miles, Ans. b SIMPLE NUMBERS. Ex. 8. 14 x 26 x 43 = 15652 yards, Ans. Ex. 9. 4 x 365 = 1460, yearly expenses ; 3700 1460 = 2240 dollars, Ans. Ex.10. 2475, first; 2475 840=1635, second; 2475 + 1635=4110, third ; 8220 dollars, Ans. Ex.11. 336 (28x10) = 56 miles, Ans. Ex. 12. 23 x 14= 322, 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 3x850 = 2550 8454 dollars, Ans. Ex. 14. 14760 (1575 x 5)=6885 dollars, Ans. 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. Ex. 13. Ans. 10487951. Ex. 14. Ans. 635926f. Ex. 15. Ans. 2379839f. Ex. 16. Ans. 9355751*. Ex. 17. Ans. 245640}!. Ex. 18. Ans. 7014132f Ex. 19. 47645-^-5 = 9529 dollars, Ans. Ex. 20. 17675-4-7 = 2525 weeks, Ans. Ex. 21. 6756-=-6 = 1126 barrels, (Ans. Ex. 22. 46216464-^-12 = 3851372 dozen, Ans. Ex. 23. 347560-4-5 = 69512 barrels, Ans. Ex.24. 3240622-=-!! =294602 acres, Ans. Ex. 25. 38470-4-5 = 7694 dollars, Ans. (8O, page 54.) Ex. 5. Ans. 212 T 9 /4. Ex. 14. Ans. 1489|f. Ex. 15. Ans. 12152^. Ex. 16. Ans. 608301^. Ex. 17. Ans. 1210900-LfH. Ex. 18. Ans. 997f J$. Ex. 19. Ans. 1343iff|. Ex. 20. Ans. 5473|fff Ex. 21. Ans. 7500 T y T T V- Ex.27. 1850400-4-18504 = 100 dollars, Ans. Ex. 28. 723 20060-4- 10735 = 6736 T VWs dollars, Ans. Ex.29. 942321-4-213 = 4424^^ volumes, ^Lns. Ex. 30. 5937120^-22320 = 266 dollars, Ans. CONTRACTIONS. (81, page 56.) Ex. 2. (3690-7-3)-^5 = 246, Ans. Ex.3. (3528H-4)-J-6=147, Ans. Ex.4. (7280-f-5)-4-7=208, Ans. Ex. 5 (6228-^-6)^-6 = 173, Ans. Ex. 6. (33642-=-3)-J-9 = 1246, Ans. Ex. 7. (l53160-^-7)-^8 = 2735, ^4s. Ex.8. 15625^-5^-5-4-5 = 125, Ans. 11 SIMPLE NUMBERS. (8, page 57.) Ex. 2. 6)34712 7)5785 2 826---3x6 = 18_ 20, Arts. Ex. 3. 8)401376 8)50172 6271 ---4x8=32, Ans. Ex. 4. 3)139074 4)46358 6)11589 ...... 2x3= 6 1931- --3x4x3 = 36^ 42, Am. Ex, 5. 3)9078126 5)3026042 6)605208 ---2x3 = 6, .4ns. 100868 Ex. 6. 4)18730627 5)4682656 6)936531 ....... 1x4= 4 156088 ---3x5x4=60^ 67, Ant. Ex. 7 2)7360479 6)368C239 ............ 1 8)613373 ---- --1 x2= 2 76671 ---5x6x2 = 60 63, Ans. SIMPLE NUMBERS. 16 Ex. 8. 2)24726300 5)12363150 7)2472630 353232 - - - 6 x 5 X 2=6Q, An* Ex. 9. 7)5610207 2)801458 1 6)400729 66788--- 1x2x7 = 14 15, Ans. (83, page 58.) Ex. 2. Ans. 476. Ex. 3. Ans. 3620 7 W Ex. 4. Ans. 1306 T : VVo- Ex* 5 - Ans - 9>76 iiUo- Ex. 6. Ans. 2037l T 8 oWoV (85, page 59.) R*. 6. Ans. 14556 T \VoVV Ex. 7. ^ins. 10609 VoVo- Ex. 8. Ans. 114304^ffJ. Ex. 10. Ans. Ex. 11. 24898-^50=497|f hours, Ans. Ex. 12. 350000^-14000 = 25 dollars, Ans. EXAMPLES IN THE PRECEDING RULES. (Page 60.) Ex. 1. 1732+67 = 1799, Ans. Ex. 2. 1095^-365=3 dollars, Ans. Ex. 3, 141+47=188 dollars, Ans. Ex. 4. 500 + 17 + 98 + 121 = 736 acres owned? 736 325=411 acres, Ans. Ex. 5. 2300 625 = 1675 dollars, Ans. SIMPLE NUMBERS. Ex. 6. 60 45 = 15 dollars, saved in one month; 900 -M 5 60 months, Ans. Ex. 7. 87 x 9 = 783 days, Ans. Ex. 8. 4 first number ; 4x8= 32 second; 32x9 = 288 third; 324, Ans. Ex. 9. 2x2x7 = 28; 364-^-28 = 13, Ans. Ex. 10. 78 + 104 = 182; 182x2 = 364 acres, Ans. Ex. 11. 90 + 304-12 + 5 + 7 = 144; 144x27 = 3888 dollars, Am. Ex. 12. (2250 x 4)-f-3 = 3000 dollars, Ans. Ex. 13. 35 + 40 = 75 miles in one day; 75 x 6 = 450 miles, Ans. Ex. 14. 40 35=5 miles in one day; 5 x 6=30 miles, Ans. Ex. 15. 45 19=26 years, Ans. Ex. 16. 1000000000-^25000=40000 acres, Ans. Ex. 17. 384 + 1562 + 25 + 946=2917 ;2917 2723 104 194^97 = 2 ; and 2 x 142=284, Ans. Ex.18. 5280-f-3 = 1760 steps, Ans. Ex. 19. 2375 + 340 =2715 dollars, cost; 867 + (235 x 8)=2747 " sold foi 2747 2715 = 32 dollars gain, Ans. Ex. 20. 4500 1350 = 3150 to gain ; 800 450=350 yearly savings ; 3150^-350=9 years, Ans. Ex. 21. 1600 x 75 = 120000 ; 120000-^-40 = 3000 bushels, Ans. Ex. 22. 325 x 50 X 2 = 32500 dollars, Ans. SIMPLE NUMBERS. 17 Ex. 23. 225 75 = 150; 150 x 52 = 7800 cents, Ans. Ex. 24. 31383450-7-4050=7749, Ans. Ex. 25. 31647000-r-700=45210 dollars, Ans. Ex. 26 Reversing fourth operation, 10040= GO; Reversing third operation, 60x5 = 300; Reversing second operation, 300-^3 = 100 ; Reversing first operation, 100 x 7 = 700, Ans. Ex. 27. (54 x l7)-r-27=34 cows, Ans. Ex. 28. 56-(2 x 26) = 4 dollars, Ans. Ex. 29. 98 x 7 = 686 days, Ans. Ex. 30. 5301212-r-11137 = 476 dollars, Ex. 31. 60 39 = 21 gallons, gained hoi 840-^-21=40 hours, Ans} Ex. 32. 4500 x 24 = 108000, Ans. Ex. 33. 1900 1492=408 years, Ans. Ex.34. Maine, 31766; New Hampshire, 9280 ; Vermont, 10212 ; Massachusetts, 7800 ; Rhode Island, 1306 ; Connecticut, 4674 ; 651)38-47000 = 18038, Ans. Ex. 36. 25000^8 = 3125 pounds in the thread ; 3125 + 235 = 3360 pounds, Ans. Ex. 37. 8546+342=8888 ; 8888^-4=2222 dollars, Ans. Ex.38. 245x37=9065; 9065 + 230 = 9295, Ans. Ex. 39. 5190048^-72084 = 72, Ans. 18 PRIME NUMBERS. Ex. 4 . 109x73 = 7957, greater n umber ; 28x17=476, difference; 7957476 = 7481 less, Am. Ex.41. 360 114 = 246, greater; 246x114 = 28044, Ans. Ex. 42. 2568754 2473248 = 95506, Ans. Ex. 43. Wheat, 35x2 = 70 dollars ; Wood, 18x3 = 54 " 124 " Cloth, 9x4 = 36 " 88 dollars, Ans. Ex. 44. 684375 = 309 yearly savings ; 309 x 5 = 1545 dollars, Ans. Ex. 45. 58 + 10 + 5 + 28 + 3 = 104, cost of one barrel; 125 104 = 21 cents, Ans. Ex. 46. 286000-6000 = 280000 ; 280000-^-14 = 20000 dollars, Ans. Ex.47. 256x25 = 6400; 6400625=5775; 5775^-35 = 165, Ans. Ex. 48. 189 + (4 + 5) = 21 hours, Ans. PRIME NUMBERS. (91, page 68.) Ex 2. Ans. 2, 2, 3, 5, 19. Ex. 3. Ans. 3, 3, 5, 5, 7, 19 Ex. 4. Ans. 11, 13, 17. Ex. 5. Ans. 19, 23, 29. Ex. 6. Ans. 2, 3, 5, 7, 11 Ex. 7. Ans. 3, 3, 5, 7, 7. Ex. 8. Ans. 11, 31, 41. PRIME NUMBERS. 19 Ex, 2, 24= (92, page 69.) Ex. 3. Ex. 4. 40 r= i 2x12 3x8 4x6 2x3x4 2x2x6 2 x 2 x 2 >. 3 2x20 4x10 5x8 2x2x10 2x4x5 2x2x2x5 Ex. 5 72 = 125 5x25 5x5x6 2x36 3x24 4x18 6x 12 8x9 2x2x18 2x3x12 2x4x9 3x4x6 2x6x6 3x3x8 2x2x2x9 2x2x3x8 2x3x3x4 O x2x2x3x3 20 PROPERTIES OF NUMBERS. CANCELLA1 (95, p--e 72.) Ex. 3. n $ 3 Ex. 4. 4 n 3 It $ nt y$ y> dp 11 u t 14 20 14, Ans. 33, Ans. Ex. 5. 7 $4 4$ 16 Ex. 6. 30 ,*rt "# ' *4 K Kf' ft n 4 20 Pi? 13 t 7 64 3 13 9|, ^4^5. 4i, Ans. EX. 7. n 00 3 Ex. 8. ^0 200 9 40 $0 2 8 40 It 3 X$ $0 a:4 ^a: 8 81 2, As. 10}, A*. Ex. 10. 4 102 Ex. 9. 00 240 4 n at * 2 ~ 8, Ans. % 42 2 4, ^4n*. Ex. 11. X2 30 10 ta. 12. 00 4 4 * 10 tons, ^4n.- dred twenty-five four hundred twenty-eighths. Ex. 12. Ninety one hundredths ; three hundred twenty- five one thousandths; four hundred fifty one thousand tieo REDUCTION. 25 hundred fortieths twenty -five one thousand fiiv hundredth^ ; twelve two thousandths j seven hundred twenty -six three thousand four hundred seventy-fifths. Ex. 13. Seventeen one hundred fourths ; one ten thousand one hundred tenths ; nine hundred fifteen eighty-four thousand six hundred twenty-firsts; thirty-eight thousand sixty-five four million Jive hundred thirty-one thousand four hundred twenty-ninths. REDUCTION. (126, page 90.) Ex. 5. Iff =4, An*. Ex. 6. Ex. 7. TWV=f> Ans - Ex - 8 - Ex. 12. tH=fi ^*- Ex - 13 - Ex. 14. fHf =}, ^s. (127, page 91.) Ex.4. i.|J.=i53|, -4*. Ex.6. i-|p=:54if, Ex.7. J T V= 41 > Ans - ' Ex - 9- 7 H J -= 43 Tf (138, page 92.) Ex.4. 140=*^ Ans. Ex.6. 94 = t| Ex. 7. 180:=-4f-fi-S ^. ws . Ex. 9. 247=a.p, Ans. (129.) Ex. 8. 7lf =A$S ^ws. Ex. 5. Ex.9. 96 4 = ^Iws. Ex. 11. K P FRACTIONS. (13O, page 93.) Ex. 2. 15-^5 = 3; = T 4l?4 168 ~" 168 168' 3 FRACTIONS. Ex. 5. + 1 = 45 13 21 1274 + 2835 + 390 4499 404 T j ~~ 4095 4095 4095 ' ' 42 9 7 1 42 + 18 + 35 + 10 105 3 A =777;=7 Ans 140 140 4' 61 131 24 1 2 fcx ' 7 - 7^ + 150 +25 + 2 + 3 = 102 + 131 + 144 + 75 + 100 17 "" l3 ' ' 31245678 9 EX ' 8 - 4 + 2 + 3 + 5 + 6 + 7 + + 9 + IO = 1C90 + 1260 + 1680 + 2016 + 2100 + 2160 + 2205 + 2240 + 2268 2520 4 j^ 2 19_ 19 K "I f\ O OA AA O JL \J o *i\J DU 14 + 3+ 1 = 18 Ex. 11. 17819 _ 179 2520 ~" 2520' 1+10 + 5 = 16 Ex. 12. 17 + 18 + 26=61 , Ans. Ex. 13. _ V 1+3 = 4 SUBTRACTION. Ex. 14. 125 + 327 + 25=477 478^y , Ans. Ex. 15. iAo +| 7 +u + H+ i n=3 i f i, An8 . Ex. 16. T V + H-K + T'o = 4 3 + 2 + 40 + 10 = 55 554, ^4w5. Ex. 17. i + f + f = 2^ 125 + 96 + 48 = 269 27l 7 T yards, Ans. 5 + 3 = 8 9, 1 7 yards, ^4/i5. Ex. 19. A + tf + f + f + | = s 26 + 40 + 51+59 + 62 = 238 241 if acres, Ans. Ex. 20. ! + 4+i2 + 7 7. = 2 ^ 175 + 325 + 270 + 437 = 1207 1209/j bushels. 205 + 296 + 200 + 156= 857 $859||, dollara. SUBTRACTION. (135, page 99.) 85 3 1 . 8. -=-=-, Ans. 14 11 3 1 80 FRACTIONS. 20 6 14 49 36 13 -To-=75' 75-11 64 1 182 110 72 _ 6 "848" ~348~29' * (136, page 99.) 1 2 94 5 Ex. 2. --- = - = , Ans. 2 9 18 18' 15 2_75-48_ 27 _ 9 ~-~~- == A 3 4 51-32 19 EX ' 6 ' 84 4 49-8 41 T2b-3F = -7b- = 7-0' 1500 50 125 100 25 1728-72= 4T- = 144' 60 332 _720 83_ 637 ~~ 1068 ~1068' A Ex. 9. 8^ = 8-^ Ex. 10. 25f = 4}f , ^n. 163*0 = Ex.11. 4f=4|f Ex. 12. 6 J li 3}|, ^4ns. 4f, Ans. Ex. 13. 4501 =450$ Ex. 14. 3-^ = 330|f, I MULTIPLICATION. 31 Ex. 15. 75i Ex. 16. 227| 49 196| = 196| 261, Ans. Ex. 19. $7f $6 =$!-&, Ans. Ex. 20. 4 + 31 =4j\ Ex.21. 5? 4^ 12" Ex. 22. 7|-2f =4|f, ^5. Ex. 23. H~U= Ai Ans ' Ex. 24. 9121 + 5451 = $2000 $1457^=1542-^, Ans. Ex. 25. $136 T ^ + $350!=$487H cost. = $6011 receipts. Ans. MULTIPLICATION. (137, page 101.) Ex.4. J> T x>I=&=l Ex. 5. T 9 x 12 = W =H Ans ' Ex.6. 2 5 T x63 5 x3 = 15, Ans. Ex. 8. 7| x 12 = *J 6 = Ex. 9. T Vr x 8 = m^ Ex.10. T ^x51=f=2 Ex.11. 15fx 16 = 125x2 = 250, Ans. Ex.12. ifix22 = 1 | 1 = Ex.13. $8/oXl2=$ i n Ex. 14. $H x9 = $tt= Ex. 15. $f x27^$ L | 9 =$23f, Ans. 32 FRACTIONS. (138, page 103.) Ex. 2. Ex. 4. Ex. 7. 3 Ex. 3. 100 9 4 14 9 3 4 7 450 1J, Ans. 64f, ^*. 105 Ex. 5. 19 21 17 47 13 85, Ans. 47 247 - 5H, Ans. 42 Ex. 8. 80 4 39 16 233 2 819 1165, Ans. t Ex. 9. 39 40 9 , Ans. 156 Ex. 10. $8xf=6f dollars, Ans. 27 Ex. 11. 36 x lOf =384 miles, Ann, 108, Ans. Ex.12. $450 x T 7 2 Ex.13. $16x2*= Ex. 2. 4 (139, page 104.) 4 9 3 4 3 1 Ex. 3. 8 10 Ex. 4. 24 55 10 11 36 Ex. o. 6 7 F 7 o> <*n*. 21 6 18 3|, Ant. MULTIPLICATION. Ex. 6. 10 9 7 2 9 5 4 1 28 1 5 ! T , As. E^. 9 15 8 4 9 5 1 3 22 25 44 l^f, As. Ex. 11. 3 2 1 7 4 5 4 4 13 35 78 Ex. 13. 8 7 2 1 9 11 2 3 1 8 12 77 Ex. 7. 6 , Ans. Ex. 10. 7 1 10 3 2 16 7 80 3 256 851, Ant Ex. 12. 2 4 5 3 Ex. 14. 2 2 4 6 3 4 4 2, As. 25 11 27 16 7425 ^7ei 6 T 5 2 Ans. Ex. 15 $ x f =8|, As. Ex. 16. Ex. 17. f x li=$f, As. Ex. 18. x = x $f =$, A* 84 FRACTIONS. Ex. 20. 4 8 51 17 867 Ans., 108f. Ex. 21. 8 5 20 51 14 357 Ans., dollars. x f x y x$- i x V 3 x f x $Y-= Ex. 22. ? x 4 x I W =121 A, Ans. Ex. 23. Ex. 24. Ex. 25. Ex. 26. $*fa x =$25 A, Ex. 27. ^f- 3 - acres x f x |=49 T ^ acres, Ans. Ex. 28. H x f barrels =6 barrels, DIVISION. (14O, page 107.) EX. 6. Tj Ex. 9. $f- Ex. 11. -f ^- Ex. 14. *J Ex. 16. , Ans. Ex.10. Ex. 12. Ans. Ex. 15. (141, page 109.) Ex, 7. | x 9^f= Ex, 8. $21^-^ $49, Ans. Ex. 9. 16 x | = 10; 10-f-f =22^, Ex. 11. Ex. 12. Ex. 13. 15^-f =9, Ans. =1, Ana. x ^9-= iU VISION. Ev 14. f x 320 = 200; 200-S-V=S Ex. 15. $32x1 = 8; ^ Ex. 16. 183-^J-p=4, (149, page 110.) Ex. 2. 8 | 7 Ex. 4. 63 , Ans. 10 40 Ex. 3. 9 10 Ex. 5. 14 13 13 f?, Ans. Ex. 6. 3 2 Ex. 7. 6 5 27 28 4 5 81 56 24 25 f f , ^. 1ft. ^ Ex. 8. 3 5 Ex. 9. 19 17 3 7 7 19 9 35 7 17 01 /i Ex. 10. 20 13 Ex. 11. 7 7? * 2 5 16 2 5 25 52 A 4 2 g A 02 Ans 2fi7 21 40 FRACTIONS. Ex. 12. 10 13 325 432 HH, Fx 15 ?|_!! X 3_?! 8|~9 X 26~39' J?v 1A J y Of) !* V7. . A. a ^(V. T 5 T 5 5 25 = Ex. 18. |x|_2 3 2_. 25921 Ex.20. V x f= 14 Ex. 21. \ 3 xf=Y=6|, Ex. 22. 8 35 21 Ex.24. Ex. 23. 3 14 $17 $14f=$ 2$, An*. Ex. 25 10 20 98 35 11 1, Ant. PROMISCUOUS EXAMPLES. 37 Ex. 26. 10 10 2 bu., Ans. Ex. 27. 16 75 10 1905 127 Ans PROMISCUOUS EXAMPLES. (Page 112.) Ex.2. 91-^7 = 13; 4* j jj-ff, Ans. Ex. 3. 3, 40 | 3 :: 40 3x40 = 120, Ans. Ex. 4. 4+3 Ex.5. fxH==4A m>== 2 H l}ff, Ans. Ex. 6. 4756f + 128f =4885^, An*. Ex.7, f xfx|xV-=H=^A V xfxfx |^H= 4 . 2 ? Ex. 8. f x f =20, ^W5. ; f x J = V =1 8 Ex. 9. 1825f=-^f^; lAfJiixf^ 2 - 4 1^=3043} Ex.10. i+J=A? l 9 o-H; 77-^=140, Ex. 11. i x $V x V =*24|f Ex, 12. $V-xt=l23i, 38 FRACTIONS. Ex 13. 8 14701 14701 Ex. 14. 8 471 37803 628 $1, Ans. $12601, Ans. Ex. 15. Ex. 16. '_eji x i x = , As. 1 42 10 2 22 O_ _ _ v _ _ ^ __ _ _ . "I A _ _ - I A ~2 X 5 X 147~7' 7 7~ X f xf = 27, Ans. xyxf = W=24 1 V, Ans. xfx f=34i, Ans. Ex. 17. V Ex. 18. V Ex. 19. x y=- EX. 20. !-2 9 _3 = Ex. 34.*6-hl846= lr f 3, Ex 35. $ 2 T x x &=$ Ex 36. $ y x i x i x | x f-=$5, Ex.37. J + T V=H; 1-=A; A-i=A Ex. 38. if cents x y x = 100 cents, Ex. 39. i+f=i3. ; i_ 11 = ,, remainder. H* x 2T x $2-1^=^1 ^i=$4577 Ex. 40. If the horse cost li times as much as the wagon, the horse and wagon must cost 2i times the wagon. Hence, $270 -T- 21 =$120, Ans. Ex.41. JL. X =32; 32 20f = lli, ^1*. Ex. 42. $4-^1 x aV x $f=126, Ans. Ex. 43. If A can do as much as B, he can do the work in A of the time that B will require, and in l+|=i of the time be \h will require. Hence 14 days x i=32| days, A's time ; ) . 32| days x f = 241 days, B's time ; ) ^ Ex. '4. J^-xf xf=lli, Ans. Ex. 45. A, B, and C can do 1 of the work in a day ; B and C can do 1 of the work in a day; hence A alone can do i 1=^ of it in a day ; and be wW therefore require -*/-=13i days, Ans. Ex, 46. f + i+i=JL; 1--^=^, remainder; rV-rV^sV; $24-r-3-V=^20, Ana. Ex 4-1 V x A x V =4f DECIMALS. Ex. 48. i x i Ex. 49. A-i= T 3 o ; 30 feet-5- 1 ^ = 100 feet, Ex. 50. $i| x f x | =$6 J J, ^W5. Ex. 51. + =7 S 2 fraction of the post below water, 1& =7*3- " " " above " 21 -=--^=36 feet, Ans. Ex. 52. ^n= eldest son's fraction; f x 4 H= youngest son's fraction; 1 (? + i)= H= daughter's fraction; DECIMAL FRACTIONS. (145, page 11 8.) Ex. 1. Ans. .38. Ex. 2. ^te f. Ex. 3. .4ns. .325. Ex. 4. Ans 04. Ex. 5. Ans. .016. Ex. 6. Ans. .00074. Ex. 7. Jbts. .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; ore thouhdnd three hundred eighty-five millionths ; one million eighty-peret) ten- millionths. NOTATION AND NUMERATION. 41 (146, 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-fivt 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. .000425. 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 hundred 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 DECIMALS. thousand five hundred eighty-three ten-thousandths ; twenty seven, and forty-five hundred-thousandths. REDUCTION. (149, page 121.) Ex. 2, .1700000 Ex. 3. .700000 24.6000000 .024000 .0003000 .000187 84.0000000 .000500 721.8000271 108.450000 Ex. 4. 1000.001000 841.780000 2.600400 90.000009 6000.000000 (15O, page 122.) Ex.2. MJ v =i,Ans. Ex.3. T ^=^ Ans Ex. 4. TW T =H* Ans. Ex. 5. Ex. 6. = (151, page 123.) Ex. 4. .4/1*. .4. Ex. 6. Ans. .875. Ex. 9. Ans. .375. Ex. 10. Ans. .0375. ADDITION. (158, page 124.) Rx, 6. 26.26 Ex. 7. 36.015 300.0605 5.000003 60.0000087 87 04^7, Ans. 401.0755117, Ans. ADDITION, 43 Ex. 8. Ex. 10 Ex. 9. 71.399107, Ans. 61.843 143.75 218.4375 21.9 445.9305, Ans. Ex. 12. 21= 2.5 5 = 5.75 3 = 3.625 3.0642 8.925 Ex. 11. 23.8642 barrels, Ans. 18.375 41.625 35.5 95.500, Ans. 12f =12.75 18| =18.4 9=9 241 =24.125 4}f = 4.8125 8 T 9 o= 8.9 151^=15.65 93.6375, Ans. Broadcloth. Ex. 13. First suit, 2.125 Second " 2.25 Third " 5.0625 Sums Total Gassimere. 3.0625 2.875 Satin. .875 1.000 1.125 9.4375 5.9375 3.000 9.4375 + 5.9375+3 = 18.375, 44 DECIMALS. SUBTRACTION. (153, page 126.) Ex. 4. 714.000 .916 713.084, Ans. Ex. 6. 21.004 .75 20.254, Ans. Ex. 8. 900. .009 899.991, Ans. Ex. 10. 1. .000001 .999999, Ans. Ex. 12. .34 .034 .306, Ans. Ex. 5. 2.000 .298 1.702, Ans. Ex. 7. 10.0302 .0002 10.03, Ans Ex. 9. 2000. .002 1999.998, Ans. Ex. 11. .427 .000427 .426573, Ans. MULTIPLICATION. (154, page 127.) Ex. 4. 274.855, Ans. Ex. 8. 243.5, Ans. Ex. 12. .000030624, Ans. NOTATION AND NUMERATION. 45 DIVISION. page 129.) Ex. 5. .111. Ans. Ex. 6. 11.1, Ans. Ex. 8. 15.27 +, Ans. Ex. 9. 1; 10; 100; 1000, Ans. Ex. 10. 5.6814 + , Ans. Ex. 12. 3020, Ans. 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^-6.35-11.13, Ans. Ex. 12. T 6 oVo=i, Ans. Ex. 13. 26 T Vo s o=26i, Am. Ex. 17. 3.625 x 36.75 x $.85-$113.2359375, Ex. 18. 56.925-r-4.6 = 12.375 = 12f, Ans. DECIMAL CURRENCY. NOTATION AND NUMERATION. (16O, 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 ; one 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- lars 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. 7. Ans. 3008 mills. Ex. 8. Ans. 890 mills. (163, page 134.) Ex. 2. Ans. $15.04. Ex. 3. Ans. $138.75. Ex. 4. Ans. $16.525. Ex. 5. Ans. 52 .4 centa. Ex. 6. Ans. $6.524. Ex.2. ADDITION. (163, page 134.) $ 50.07 Ex. 3. $ 364.541 1000.75 486.06 60.003 93.009 .184 1742.80 1.01 3.276 25.458 $2689.686, Ans. $1137.475, Ans. ADDITION. Ex. 4. $ .92 .104 .357 .186 .444 .125 .99 126 Ans. Ex. 5. $89.74 13.03 6.375 19.625 $128.77, ]An*. Ex. 7. $2175.75 240.375 605.40 140.125 $3161.65, Ant. $10.3375, Ans. Ex. 8. $ 6.08 26.625 16.000 7.40 156.105, Ans. Ex. 9. $7425.50 253.96 170.09 $7849.55, Ans. Ex. 10. $3.625 1.75 1.375 .625 .875 $8.25, Ant. 48 DECIMAL CURRENCY. SUBTRACTION. (164, page 136.) Ex. 2. $365.005 Ex. 3. $50. 267.018 .50 $97.987, Ans. $49.50, Ans. Ex. 4. $100. Ex. 5. $1000. .001 .037 $ 99.999, Ans. $ 999.963, An*. 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 + $7015.875 = 10575.875 $1.215, Ans. $4149.325, Am, Ex. 10. $13.75 5.25 1.375 .875 $25 -$21.25 = $3.75, Ans. fcx. 11. $480 80.50 $560.50 $200 = $360.50, Ans. MULTIPLICATION. (165, page 137.) Ex. 2. $4.275 x 300 = $1282.50, Ans. Ex. 3. $2.45 x 175=$428.75, Ans. DIVISION. 49 Ex. 4. $1.28 x 800 =$1024, Ans. Ex.5. $.15 x 372 = $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, An*. Ex. 8. $32.50 x80 = $2600 34.25 x70= 2397.50 $4997.50 3975 $1022.50, Ans. DIVISION. (166, page 138.) Ex. 2. $41.25^-33=$!. 25, Ans. Ex. 3. $94.50 -=-27 =$3. 5 0, Ans. Ex., 4. $136-=-64 = $2.125, Ans. Ex. 5. $1.32^-$.12 = 11, Ans. Ex. 6. $405-r-$.54 = 760, Ans. Ex. 7. $180-^12=$15, Ans. Ex. 8. $2847.504-] 00=$28.475, Ans. Ex. 9. $80.46-=-894=$.09, Ans. Ex. 10. $1.125 x 120=$135 ; $185-^-27 $5, Ans. K. P. 3 50 DECIMAL CURRENCY. Ex.11.- $3.20 x 4=$12.80 .08x37= 2.96 $15.76 6.80 J-h$.16=56, Ans. Ex.12. $4.50 + $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 APPLICATION (168, page 139.) Ex. 2. 693 x $=$321, Ans. Ex. 3. 478 x =$239, Ans. Ex. 4. 4266 x $ T V=$355.50, Ans. Ex. 5. 1250 x $i=$156.25, Ans. Ex. 6. 3126 x $ T V=$195.375, Ans. Ex. 7. 1935 x $i=$322.50, Ans. Ex. 8. 56480 x$i=$7060, Ans. Ex. 9. 1275 x $}=$255, Ans. (169, page 140.) Ex. 2. $.09 x 864=$77.76, Ans. Ex. 3. $1.25 x 87=$108.75, Ans. Ex. 4. $1.45 x 400 = $580, Ans. Ex. 5. $.44 x 52 x 16 =$366.08, Ans. (17O, page 141.) Ex. 2. $l75^-25=$7, Ans. Ex. 3. $200^-48=$4.16|, Ans. ADDITIONAL APPLICATIONS. 51 Ex. 4. $1200^96 = $12.50, Ans. Ex. 5. $56.25-^-10 = $5.62, Ans. Ex. 6. $11.70-M8=$.65, Ans. Ex. 7. $10.07 -=-53 =$.19, Ans. Ex. 8. $1016-r-800=$1.27, Ans. Ex. 9. $874.65-^-343 =$2.55, Ans. Ex. 10. $684.375-:-365=$1.875, Ans. (171, page 142.) Ex. 2. $5.55-i-$.15=37, Ans. Ex. 3. $216-^$12 = 18, Ans. Ex. 4. $2178.75-J-$1.25 = 1743, Ans. Ex. 5. $643.50-^-$19.5 = 33, ^ln*. Ex. 6. $52.65-^$.45 = ll7, Ans. Ex. 7. $6336-=-$132=48, ^rcs. Ex. 8. $H77l5-^-$65 = 1811, Ans. page 143.) Ex. 2. $4.50 x 42.65 = $191.925, A. Ex. 3. $.85 x 24.89 = $21.156+, Ans. Ex. 4. $17.25 x 7.842=$135.274+, Anf. Ex. 5. $12.50 x 23.48 = 1293.50, Ans. Ex. 6. $3 x 1.728=$5.184, Ans. Ex. 7. $7 x 2.40 =$16.80 6.40 x .865= 4.671 .80X12.56 = 10.048 $31.519, Ans. Ex. 8. $4.375 x 14.76=$64.575, Ans. (173, page 144.) Ex.2. $7-j-2 = $3.50; $3.50 x 1.495 =$5.2825, Ant. 52 DECIMAL CFRRENCY. Ex, 3. $8.75-^-2=$4.375; $4.375 X .325=$1.421 -f, An*. Ex. 4. $3.84-:- 2 = $1.92; $1.92 x 3.142 =$6.032 + , Ans. Ex. 5. $5.60^-2=$2.80; $2.80 x 1.848=$5.1744, Ans. Ex. 6 $18-h2=$9; $9 x 125 x .148=$33.30, Ans. Ex. 7 $3.05-5-2=11.525; $1.525 x 31.640=$48.251, Ans< (174, page 145.) Ex. 1. $3.60 x 7=$25.20 1.125 x 9= 10.125 .90 Xl2= 10.80 1.375x24= 33.00 .65 x32= 20.80 $99.925, Ans. Ex. 2. $3.75 x 67=$251.25 2.62x108= 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, Ans. Ex. 8. $.07 x325=$22.75 .0625x148= 9.25 .05 x286= 14.30 .125 x 95= 11.875 2.75 x 50=137.50 3.625 x 75 = 271.875 2.85 x 12= 34.20 $501.76, Ant. PROMISCUOUS EXAMPLES. Ex. 4. Ex. 5. $15 x20 9.50 x 7.5 G.25 x 10.75 2.625 x 3.90 3.00 x 5.287 $.11 x25=$2.75 .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, = 8300. = 71.25 = 67.1875 = 10.395 = 15.861 $464.6935, Am Ans. PROMISCUOUS EXAMPLES. (Page 146.) Ex. 1. $124.35 x 62.75 =$7802.9625, Ans. Ex. 2. $.17 x 15=$2.55, Ans. Ex. 3. $1406.25-=-2250=$f, Ans. Ex. 4. $48.96-^12=$4.08, Ans. Ex. 5. 325 miles x. 45 = 146.25 miles, Ans. Ex. 6. 657-^36.5 = 18,^4^. Ex. 7 $105 + $125 + ($35x4)=$370 $400 $370 =$30, Ans. Ex. 8. $19 $15 = $4; $4x28=$112, Ans. Ex. 9. e 2 9 -x^3=V-=^ 5 Ans - Ex. 10. $9-f-$.3125 = 28.8, Ans. Ex.11. $3.50x365=$1277.50 $2000-1277.50=$722.50, Ans. 64 DECIMALS. Ex. 12. $687. $1630.89 $875.29 = $755.60, Ans. Ex.13 $!728-+2 = $864 1st half sold for ; 144x8 = 1152 2d " " " Ex. 14. $3.75 x .875 = $3.281 +, Ans. Ex. 15. $65.42-$46.56 = $18.86, gain per head; $3526.82^$18.86 = 187, Ans. Ex. 16. $54.72-^36.48=$1.50 ; $1.50 x 14.25 = $21.375, Ans. Ex. 17. $3548-r-4 = $887, Ans. Ex.18. 112.34-^$.82 = 137, Ans. Ex. 19. $3461.50^-46 = $75.25 ; $75.25 x 5 = $386.25, Ans. Ex. 20. $24000 x | x i =$3200, Ans. Ex.21. $1.25x160= $200 $5 x 80= 400 $600 $2.50x240= 600 Loss 000, Ans. Ex. 22. $1.70x48 = $81.60 72.90 $ 8.70, Ans. Ex.23. 1221 + 751 = 197^; 197|-60 = 137f ; $.9375 $.8125 =$.125, loss per bushel; $.125 x 137f = $17.218+ loss ; 12.50 gain ; $4.7 18 + , loss, Ans. Ex 24, $1.40x6 =$8.40 wages; $ .75x7= 5.25 expenses, $3.15 savings, Ans. PROMISCUOUS EXAMPLES. 55 $.08 x 39 Ex. 25. r-- = Ex. 26. $4.50 x 23.487 =$105^6915, Ans. Ex. 27 $1200-=-365=$3.287^f, Ans. Ex 28. $.f 7 X 56 X 28 = $266.56, Ans. Ex 29. $.07x26xl3xl6=$378.56, 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; 32 x 8 ^ = 21^ minutes, Ans. $32.3 4 15 " X T9 X T= $51 '^' $5.635 -r-.875 =$6.44; $6.44 x 9 =$59.57, Ans. $5000 $1200.25 x3 = $3600.75 1800.62x3= 5401.86 Ex. 33. Ex 34. Ex. 35. $14002.61 .87x2= 1901.74 $12100.87, Ans. Ex 36. $4.50 x 186.40 =$838.80, Ans. Ex. 58. $96.40-^2=$48.20 ; $48.20 x 1.375=$66.275, Ans. Ex. 39. r s o 6 oV7r=T 9 6> 5*5 COMPOUND NUMBERS. Ex.40. ^=.09375; .62i=.625; .37^ =-.370625; f =.375 ; .09375 + .625 + .370625 + .375 1.464375, Am. Dr. Cr. Ex. 41. $4.745 $2.765 2.625 1.245 1.27 .625 .45 3.45 5.285 1.875 $14.375 $9.96 = $4.415, Ans. Ex. 42. $.125 x 120 = 115.00 $1.50 .625 x 18= 11.25 1.27 .07 x 47= 3.29 1.87 .18 x 6= 1.08 2.30 $30.62 $6.94=$23.68, Ans. REDUCTION. (183, page 152.) Ex. 1. 14194 far. -7-4 = 3548 d. 2 far. ; 3548 d.-^- = 295 s. 8 d.; 295 s.-f-20 = 14 15 s. Ans. 14 15 s. 8 d. 2 far. Ex. 2. 14x20 + 15s.=295s.;295s.xl2 + 8d. 3548d.x4 + 2 far. = 14194 far., Ans. Ex.3. 15 x 20 + 19 s. = 319 s.; 319 s. x 12-flld. = 3839 d. ; 3839 d. x 4 + 3 far.=15359 far., Am Ex. 4. 15395 far. -1-4 = 3839 d. 3 far.; 3839 d.-i-12 = 319 s. 11 d. ; 319 s.-=-20 = 15 19 s. Ans. 15 19 s. 11 d. 3 far. Ex.5. 46 sov.x 20 + 12 s. = 932s.; 932s. x!2f2d. = 11186 d M Ans REDUCTION. 57 Ex. 6. 11186 d.-M2 = 932 s. 2 d. ; 932 s.-f-20=46 sov 12 s. Ans. 46 sov. 12 s. 2 d (185, page 153.) Ex.3. 5lb. x 12 + 7 oz.=67 oz. ; 67 oz. X20 + 12 pwt.= 1352 pwt.; 1352 pwt. x 24 + 9 gr. = 32457 gr., Ans Ex. 4 43457 gr. -7-24 = 1810 pwt. 17 gr. ; 1810 pwt.-f- 20=90 oz. 10 pwt.; 90 oz. -r-12 = 7 Ib. 6 oz. Ans. 7 Ib. 6 oz. 10 pwt. 17 gr. E* 5. 41760 gr. -f-24 = l740 pwt. ; 1740 pwt. -f- 20 = 87 oz.; 87 oz.-7-12 = 7 lb. + 3 oz., Ans. Ex. 6. 14 Ib. 10 oz. 18 pwt.=3578 pwt. ; 3578 x $.75=12683.50, Ans. Ex. 7. 5 Ib. 6 oz.=1320 pwt.; 2 oz. 15 pwt.=55 pwt. ; Ex. 8. 1 Ib. 1 pwt. 16 gr.=5800 gr. ; 4 pwt. 20 gr.= 1 16 gr. ; 5800-7-116 = 50; $1.25x50=$62.50, Ans. (1 86 5 page 155.) Ex.3. 16 Ib. X12 + 11 oz.=203 oz. ; 203 oz. x 8 + 7 dr. -* =1631 dr.; 1631 dr.x3 + 2 sc. =4895 sc.; 4895 sc. x 20+ 19 gr.=97919 gr., Ans'. Ex. 4. 47 Ib. x!2 + 6 =570 ; 570 xS + 43 = 45643 5 45643 x 3 = 13692 3, Ans. Ex.5. 20 gr. x5 x 365 =36500 gr. ; 36500 gr.-j-20=18253; 18253-^3=60831 3; 608 3 -^-8=76 ; 76 -1-12=6 Ib. 4 | . Ans. 6 lb.4| 13. (187, page 156.) Ex.8. 3 T.X20 + 14 cwt.=74cwt.; 74 cwt.xlOO + 74 lb.=7474 Ib.; 7474 Ib.xl6 + 12 oz.=119596 oz. ; 119596 oz. x!6 + 15 dr.= 1913551 dr., Ans. 58 COMPOUND NUMBERS. Ex. 4. 1913551 dr. -4-16 = 1 19596 oz. 15 dr.; 119596 oz. -f-16 = 7474 Ib. 12 oz. ; 7474 Ib. -=-100 = 74 cwt. 74 Ib.; 74 cwt.-r-20=3 T. 14 cwt. Ans. 3 T. 14 cwt. 74 Ib. 12 oz. 15 dr. Ex 5. 3 T. 15 cwt. 20 lb. = 7520 Ib. Ans. $.22 x 7520=$1654.40. Ex. 6. 115 lb.-j-2000=.0575 T. ; $10 x .0575 =$.575, Ans , Ex. 7. 217 Ib. x 10=2170 Ib. @ $.06 -$130.20 306lb.x 5 = 1530 Ib. @ $.07 =114.75 3700 Ib. @ $.08 =$296.00, which -$244.95 =$51.05, Ans. Ex. 8. 2 T. x 2000=4000 Ib. ; 4000 x $.12 = $500 ; $500 $360 = $140, Ans. Ex. 9. 10 T. X20 + 6 cwt. = 206 cwt. ; 206 cwt. x4 + 3 qr, = 827 qr. ; 827 qr. x 28 + 14 Ib.=23l70 Ib. $.06 buying price. $130-f-2000 = .065 selling price. $.005 gain per pound. $.005 x 23170 = $115.85, Ans. Ex. 10. 2352 Ib.-f- 56=42 bu.; $.90 x 42 x 2 =$75.60, Ans. Ex. 11. 300 bbl. x 196 = 58800 Ib., Ans. Ex. 12. $1.25 x 3 = $3.75 cost. $.0075 x 280x3= 6.30 $2.55, Ans. REDUCTION. 59 (191, page 157.) Ex. 1. 5 Ib. 10 oz. = 90 oz. ; $.50 x 90=$45.0C cost. 480 $33.75, Ans. Ex. 2. 424 dr.-=-8 = 53 oz. ; 53 oz.-f-12 = 4 Ib. 5 oz., Ans Ex.3. 20 Ib. 8 oz. 12 pwt. = 119328 gr. 119328 gr.-f- 7000 = 1 7^ lb -> Ex.4. $.40x16x20 =$128 cost $.50x320x437.5 ia .. fi o, ---------- --- r= 145.834- 480 __ L $ 17.831, (193, page 159.) Ex. 3. 7912 mi. x 63360 = 501304320 in., Ans. Ex.4. 168474 ft. +-3 = 56158 yd.; 56158 yd.+-5i = 10210 d. 3 yd.; 10210 rd. +-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. x 8 + 7 fur. = 255 fur.; 255 fur. x 40 + 10 rd.=10210 rd.; 10210 rd. x 5^ + 3 yd.-56158 yd.; 56158 yd. x 3 = 168474 ft,, Ans. Ex.6. 2500 fathoms x 6 = 15000 ft.; 15000 ft. -+16i = 909 rd. lift.; 909 rd.-=-40 = 22 fur. 29 rd. ; 22 fur. -+8 = 2 mi. 6 fur. ' , Ans. 2 mi. 6 fur. 29 rd. 11 ft. Ex. 7. 2200 mi. x 5280 = 11616000 ft. ; $.10 x 11616000 = $1161600, Ans. Ex. 8. 4 fathoms x 6 + 3 ft.=27 ft. ; 27 ft. x 12 +8 in.= 832 in Ans. Ex. 9 200 mi.=12672000 in. ; 18 ft. 4 in = 220 in. ; 12672000-^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. HI bands x 4 = 58 in., Ans. 60 COMPOUND NUMBERS. (194, page 160.) Ex.1. 3 ini.x 80 + 51 ch. = 291 ch. ; 291 ch. x 100 -I- 73 1. = 29173 1., Ans. Ex. 2. 29173 1. + 100- 291 ch. 73 1. ; 291 ch. + 80 = 3 mi 51 ch. Ans. 3 mi. 51 ch. 73 L Ex 3. 17 ch. 31 1. = 17;31 ch. 12 ch. 87 1 = 12.87 ch. 30.18 ch. half round the field. 30.18 ch. x 2 x 66 = 3983.76 ft., Ans. (196, page 163.) Ex.3. 87 A. x 4 + 2 R.= 350 R. ; 350 R. x 40 -f 38 sq. rd = 14038 sq. rd. ; 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. 28 sq.in.; 3S21910 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. 61 sq. yd. 6 sq. ft. 28 sq. in. But (| 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.; 111.1 sq. yd.-+30i = 367 sq. rd. 9| 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. 9} sq. yd. 1 sq. ft. ; or 2 A 1 R. 7 scj. rd. 9 sq. yd. 3J sq. ft. REDUCTION. 61 Ex. 9. 181x16 = 296 sq.ft.; 296 sq. ft.-:-9 = 32 sq. yd., Ans. Ex. 10. (18 + 16i) x 2 = 69 ft., distance round the room ; 69 x 9 = 69 sq. yd., in the walls; y 18x161 -=83 sq. yd., in ceiling; y 69 sq. yd. + 33 sq. yd. = 102 sq. yd. $.22 x 102 =$22.44, u&w. Ex. 11. 40x20x2 = 1600 sq. ft.=16 squares; $10xl6=$160, Ans. (197, page 164.) Ex. 2. 3686400 P. +102400 = 36 sq. mi., Am. Ex.3. 94 A. x 10 + 7 sq. ch.=947 sq. ch. ; 947 sq. ch. X16 + 12 P. = 15164 P.; 15164 P. x 625 + 118 sq. l. = 9477618 sq. 1., A *s. Ex. 4. 4550000 sq. 1. -=-10000=.455 A. $50 x 455=122750, Ans. (199, page 166.) Ex.1. 125 cu. ft. x 1728 + 840 cu. in. = 216840 ou. in., Ans. Ex. 2. 5224 cu. ft.-M28=40i Cd., Ans. Ex. 3. 3ft.2in.=38in.; 2ft. 2 in.=26 in.; 1 ft. 8 in. =2 Om.; 88 x 26 x 20 = 19760 cu. in., Ans. Ex.4. 6x6x6 = 216 cu. ft.; 216 en. ft. x 1728 = 373248 cu. in., Ans. .Ex.5. 60x20x15 = 18000 cu. ft. ; 18000 cu. ft.-f-128 = 140f- Cd., Ans. Ex. &. 10 x 3i x 3i=113f cu. ft., Ans. Ex. 7. 128-r-(3xl2) = 3f ft. high, Ans 62 COMPOUND NUMBERS. Ex. 8. 27 x 175 lb.=4725 lb.=2 T. 7 cwt. 25 lb., Ann. Ex. 9. 32 ft. + 24 ft.=56 ft. ; 56 ft. x 2 = 112 ft, girt; 112 x 1^x6 = 1008 cu. ft; 1008 cu. ft.^-24f--40 T 8 1 Feb.; $1.25 x 40 T 8 T =$50.909 +, Ans. Ex.10. $.15x 32X 2 2 7 4X g=$25.60, Ex 11. 10x9x8 = 720 cu. ft. ; 720-^-10=72 minutes, EJC. 12. 30 x 20 x 10 = 6000 cu. ft. ; 6000 -=12 minutes, Ans. 60x10 (2OO, page 168.) Ex. 3. 3 bbd. x 2016 = 6048 gi., Ans. Ex. 4. 6048 gi.-r 2016 = 3 hhd., Ans. Ex. 5. 13 hhd. x 63 + 15 gal. = 834 gal. ; 834 gal. x 4 x 1 qt. = 3337 qt. ; 3337 qt. X 2 = 6674 pt., Ans. Ex. 6. 6674 pt.-h2 = 3337 qt. ; 3337 qt.-h4 = 834 gal. 1 qt. ; 834 gal. -^-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. $3.84-h$.06 = 64 pt. = 8 gal, Ans. Ex. 10. 2 gal. 2 qt. 1 pt. = 21 pt.; 1 hhd. = 504 pt. ; 504-^21 = 24, Ans. page 169.) Ex.1. 49 bu. x4 + 3 pk.=199 pk. ; 199 pk. x 8 + 7 qt.= 1599 qt. ; 1599 qt. X 2 + 1 pt. = 3199 pt, Ans. REDUCTION. 63 Ex. 2. 3199 pt. -=-2 = 1599 qt. 1 pt. ; 1599 qt.-^-8 = 199 pk. 7 qt.; 199 pk.^-4 = 49 bu. 3 pk. Ans. 49 bu. 3 pk. 7 qt. 1 pt. Ex. 3. 1 bu. x4-f-l pk.=5 peck; 5 pk. x8 + l qt. 41 qt. ; 41 qt. X 2 + 1 pt. = 83 pt., Ans. Ex. 4. 83 pt.-^-2 = 41 qt, 1 pt.; 41 qt.-j-8 = 5 pk. 1 qt. f pk.-j-4 = l bu. 1 pk. Ans. 1 bu. 1 pk. 1 qt. 1 pt. Ex. 5. $.65 x 50=$32.50 cost ; $.25 x 4 x 50=450.00 sold for ; $17.50 Ans. page 170.) Ex. 1. 1 bu. (Dry Measure) =21 50f cu. in.; 2150| cu. in.-:-57 = 37if wine quarts; 3711 qts. 32 = 5i| qts., Ans. Ex. 2. 40 qt.^4 = 10 gal. ; 10 gal. x 282 2820 cu. in. , 2820 cu. in.^57f = 48f4. qts. Wine Measure: 48ff qts. 40 qts. = 8^4 qts., Ans. Ex. 3. 1 bn. Dry Measure =2150| cu. in. 32 qt. Wine Measure = 1848 cu. in. 302| cu. in., Ans. (2O6 5 page 171.) Ex. 1. 365 da.x24+-5 h.=8765 h. ; 8765 h.x604-48 min.=525948 min. ; 525948 min. x 60+46 se u=^ 31556926 sec., Ans. Ex. 2 31556926 sec. ^-60 = 525948 min. 46 sec.; 525948 min-j-60 = 8765 h. 48 min.; 8765 h.-f-24 = 365 da. 5 h. Ans. 365 da. 5 h. 48 rain. 46 sec, 64 COMPOUND NUMBERS. Ex.3. 5 wk.x7;fl da. = 36 da.; 36 da. x 24+1 h.- 865 h. ; 865 h. x60 + l rain. = 51901 niin. ; 51901 inin. x 60 + 1 sec. = 31 14061 sec., Ans. Ex. 4. 3114061 sec.-:- 60 = 5 1901 rain. 1 sec.; 51901 min. -7-60 = 865 h. 1 min.; 865 h.-7-24 = 36 da. 1 h. ; 36 da. -=-7 = 5 \vk. 1 da. Ans. 5 wk. 1 da. 1 h. 1 rain. 1 sec. Ex. 6. 10 mi. = 17600 yd.; 17600 sec.-f-60 = 293 min. 20 sec. ; 293 rain.-7-60 =4 h. 53 min. Ans. 4 h. 53 min. 20 sec. Ex.7. 29 da. X24+12 h.=708 h. ; 708 h. x 60 + 44 min. =42524 min. ; 42524 min. x 60 + 3 sec.= 2551443 sec., Ans. Ex.8. 40 yr.x 365^ = 14610 da.; 14610 da.x45 = 657450 min. gained. 657450 min.-^-60 = 10957 h. 30 min. ; 10957 h.-r- 24 = 456 da. 13 h. Ans. 456 da. 13 h. 30 min. (2O7, 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'; 310-7-30 = 10 S. 10. Ans. 10 S. 10 10' 10". Ex. 3. 11400'-7-60 = 190, Ans. Ex. 4. 190 x 69} = 13148 miles, Ans. ' Ex. 5. 360 x 60 = 21600', Ans. Ex. 6. 397'-f-60 = 6 37', Ans. (31O, page 174.) Ex. 1. 150000000-^12 = 12500000 doz.; 12500000 doz. -M2 = 1041666 gross 8 doz.; 1041666 gross-f- 12 = 86805 great gross+-6 gross. Ans. 86805 great gross 6 gross 8 doz. REDUCTION. 65 Ex. 2. 100000 sheets -=-24 = 4166 quires 16 sheets; 4166 quires -=-20 = 208 reams 6 quires; 208 reams -7- 2 = 104 bundles; 104 bundles^ 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 page*, Ans. Kx. 5. 32 pages x!0x2=640 pages, Ans. PROMISCUOUS EXAMPLES IN REDUCTION. Ex. 1. 6 yd. 3f qr. = 27 qr.; 333 yd. = 1332 qr. ; 1332-=-27f =48 suits, Ans. Ex. 2. 1 oz. 15 pwt.=35 pwt. ; $.70x35 =$24.50, Ans. Ex. 3. 2 Ib. 3 5 5 3 13 10 gr.=13290 gr.; 13290-^15 = 886, Ans. Ex. 4. 1 T. 11 cwt. 12 lb.=3112 lb.; 3112x$.01i=$38.90, Ans. Ex. 5. 1456 lb. -=-32 = 45.5 bu. ; $.375 x 45.5=$17.0G25, Ans. Ex. 6. 45 lb. x 1000=45000 lb. 45000 lb. -7-196 = 229 bbl. 116 lb., Ans. Ex. 7. 2430 lb.H-60=40.5 bu. ; $1.20x40.5=$48.60, Ans. Ex. 8. $12.50-f-200 = $.06}, 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 ch.(4 rd.)=808 ck ; 808 nh. x 100 + 20 L= 80820 L, Ans. i)6 COMPOUND NUMBERS. Ex. 11. 25x100x144 = 360000 sq. in.; $.01 X 360000 = ^3600, Ans. Ex.12. 50x25 x 10 = 12500 cu. ft. ; 12500 cu. ft. -=-16 = 781 ccl. ft. 4 cu. ft.; 781 cd. ft. -=-8=97 Cd. 5 cd ft. ; Ans. 97 Cd. 5 cd. ft. 4 cu. ft. Ex 13. 10x10x10x1728 = 1728000 cu. in. ; 1728000 cu. in.-=-231 = 74804f gal.; 7480ff gal.-+63 = 118 bhd. 46^ gal., Ans. Ex. 14. 8 x 5 x 4^ = 180 OIL ft. = 3 11040 cu. in. ; 311040 cu. ^.+2150.4 = 144/4 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., Ans. Ex. 16. 1296000 sec.+-86400 = 15 da., Ans. 9.0 v 1 9 Ex.17. ~=40 yd., Ans. Ex 18. 4 reams x 20 + 10 quires =00 quires; 90 quires x 24 + 10 sheets=2l70 sheets, Ans. Ex.19. 16 ft. 6 in.=l rd. ; 1 mi. = 320 rd. ; 320-=- 1 = 320 times in 1 mi. 320 x 42 = 13440 times, Ans. Ex. 20. 1000000 sec. + 60 = 16666 min. 40 sec. ; 16666 min.-r-60 = 277 h. 46 min. ; 277 h. -7-10 = 27 da. 7 h. ; Ans. 27 da. 7 h. 46 min. 40 sec. Ex. 21. 6 x 4i =27 sq. mi ; -= 216 farms, Ans. 80 Ex. 22. 10 mi. 176 rd. = 3376 rd. $21.75 x 3376 ^$73428, Ans. REDUCTION. 67 11, page 176.) Ex. 2. n * 5f x V x -V 2 = & d., Ans Ex. 3. TT { yy wk. x T x V- x V-=T Ex. 4. T o^ h lld - x V- x t x \ x f =1 Ex 5. ff i 7 oz. x V x V =\ g r -> Ans - Ex 6. TO o o o c mi. x f x V x V x Jf Ex. 7. | x | x f- Ib. x Y=l oz - ^ W5 * Ex. 8.' F ^- hlid. x 6 T 3 x T x f- = f | pt., Ex. 9. TT 7 7 A. x | x V=-2- rd - Ans - page 177.) Ex. 2. J- ft. x ^3 = T h rc l- Ans - Ex. 3. f dr. x T V x T V~ T/O" Ex. 4. | ct. x T J--=r^^^ E., Ex. 5. % ft. x 3 T V o = TJ ^ T o mi., Ex. 6. ^ x f- pwt. x T V x j2 = 2K Ex.7. Jpt.x'ix|x^= T i hlid. T J bhd.=fU hhd., Ans. Ex. 8. f in. x ^_i ? _=: TJ Vo o mi -> ^*- Ex. 9. | oz. x T V=jft lb. = 2-V <> f 2 lb - ; and 2T of 2 lb - is | of | of J T of 2 lb., or | of of 2 lb., ^ra. Ex. 10. f oz. x T V= 2 T lb.- aV o f 2 lb - ; and T of 2 lb - ia i of f of J T of 2 lb., or J- of f of 2 lb., ^w*. (SI 3, page 178.) Ex. 2. 4 mo.x30 = l7j da.; } da. x 24 = 3^ h. ; ^ h. x 60 = 254 rain. ; f min. x 60 = 42-f sec. Ans. 17 da. 3 h. 25 min. 42-fi sec. Ex. 3 f x 20=84 s - ? 4 s - x 12 = 6^ d. ; 4 d. x 4=34 8 s. 6 d. 34 far. 68 COMPOUND NUMBERS. Ex. 4. bu.x4 = lf pk. ; f pk. x8 = 4f qt.; qtx2~ 12 pt. Ans. I pk. 4 qt. If pt. Ex.5. ^ of 15 cwt. = 12$ cwt. ; 4 cwt. x 100 = 854 lb.; | Ib. x 16 = 11$ oz. ; 5 oz. x 16 = 6$ dr. .d/w. 12 cwt. 85 Ib. 11 oz. 6f dr. Ex. 6. | x | x 1 a=4|$ oz. ; f$ oz. x 16 = llf$ di. Ans. 4 oz. 11 |f dr, Ex. 7. | A. x 4=3i R. ; $ R. x 40 = 131 p. ^tws. 3 R. 13i ?. Ex. 8. if da. x 24 = 16 T 8 3 h - 5 A b - x 60 = 36if min. ; }a min. x 60 = 55^ sec. Ans. 16 h. 36 min. 55 T 5 3 sec. Ex. 9. | Ib. x 12 = 7i oz. ; 1 oz. x 20 = 4 pwt. Ans. 7 oz. 4 pwt. Ex. 10. % f-y-T.-4 T 5 T. ; & T.x20 = 5f cwt.; cwt. x 100 = 55 Ib. Ans. 4 T. 5 cwt. 55f Ib. Ex. 11. f ofY A.= lf A.; f A.x4 = li R. ; 1 Rx40= 20 P. Ans. 1 A. 1 R. 20 P. (214, page 179.) Ex. 2. 6 fur. 26 rd. 3 yd. 2 ft. =4400 ft. ; 1 mi. =5280 ft.; |f mi.=f mi., Ans. Ex. 3. 13 s. 7 d. 3 far. = 655 far. ; l = 960 far. ; Ex. 4. 10 oz. 10 pwt. 10 gr.=5050 gr.; 1 lb.=5760 gr.; .= lb - Ans - Ex. 5. 2 cd. ft. 8 cu. ft.=40 cu. ft. ; 1 Cd. = 128 en. ft. ; T W Cd. = t V Cd., Ans. Ex.6. lbbl.lgal.lqt.lpt.lgi.=1053gi.; lhhd.=2016gi.; i^ff hhd.=H} hhd., Ans. Ex. 7. 4 yd. li ft.=27 half-feet; 2 rd. = 66 half-feet; REDUCTION. 69 Hi 82, Lx. 8. - bu.= =- bu., Ans. Ex. 9. ^V = , Ans. Ex. 10. 2 yd. 2 qr.=10 qr ; 8 yd. 3 qr. = 35 qr. ; i| yd.=f yd., Ans. (215, page 180.) Ex.2. .217 x 60 = 13.02'; .02' x 60 = 1.2*. Ans. 13' 1.2*. Ex.3. .659 wk. x7 = 4.613 da. ; .613 da. x 24 = 14.712 h.; '712 h. x 60=42.72 min.; .72 min. x 60 = 43.2 sec. . 4 da. 14 h. 42 min. 43.2 sec. Ex. 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. x4 = 2gi. 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." Ans. 12 s. 6 d. 3 far. Ex. 7. .22 hhd. 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. Es 8. .67 lea. x 3 = 2.01 mi. ; .01 mi. x 8 =.08 fur. ; .08 fur. x 40 = 3.2 rd.; .2 rd. x 5.5 = 1.1 yd.; .1 yd. x 3 = .3 ft. ; .3 ft, x 12 = 3.6 in.=3f in. Ans. 2 mi. 3 rd. 1 yd. 3 j 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. 70 COMPOUND NUMBERS. Ex. 10. .78875 T.x 20=15.775 ewt.;. 775 cwt.x 4=8.1 qr.; .1 qr. x 28 = 2.8 Ib. ; .8 Ib. x 16 = 12.8 oz. Ans. 15 cwt. 3 qr. 2 Ib. 12.8 oz. Ex. 11. .88125 A. x 4=3.525 R. ; .525 R. x 40=21 P.; Ans. 5 A. 3 R, 21 P. Ex 12. .0055 T.x 2000 = 11 Ib., Ans. Ex. 13. .034375 bundles x 40 =1.3 75 quires; .-<>75 quires x 24 = 9 sheets ; Ans. I quue 9 sheets. , page 181.) Ex. 2. Ex.4. Ex.6. . 4 1.00 gi. 2 1.250 pt. 4 3.625 qt. Ans. .90625 gal. . 2 1.2 pt. 4 .6 qt. 63 .150 gal. Ans. .00238 + hhd . 40 12.56 P. 4 3.314 R. Ex. Ex. 3. 24 20 12 9.000 gr. 13.375 pwt. 10.66875 oz. An 5. 8 4 *. .8890625 Ib 1.12 qt. 3.14 pk. Ans. .785 bu. 40 12.56 P. Ex. 7. 12 3 5.5 40 8 6 in. 4 3.314 R. 1.5 ft. Ans. .8285 A. 17.5 yd. 3.1818 81+ rd. .0795 545+ far. Ans. .0095 &] Ex. 8. .32 pt.-=-64=.005 bu., Ans. Ex. 9. 4.875 ft.-5-6 = .8125 fathoms, Ans. Ex. 10. 150 sheets-f-480=.3125 Rm., Ans. Ex. 11. 47.04 lb.-M96=.24 bbl flour., An* Ex. 12. .33 ft.-=-r>280 = .0000625 mi., AM. ADDITION. Ex. 13. 60 60 24 57.6 sec. 36.96 min. 5.616 h. Ans. .234 da. ADDITION. (217, page 183.) Ex 3. 43 10. 1 3 2 3 16 gr., Ans. Ex. 5. 68 bu. 3 pk. 1 qt. 1 pt., Ans. Ex. 6. 21 mi. 5 for. 23 rd. 1 yd. 4 in., Ans. Ex. 10. 627 hhd. 7 gal. 1 qt. 1 pt., Ans. Ex. 11. 187 bu. 3 pk. 1 pt., Ans. Ex. 16. 152 en. yd. 9 cu. ft. = 152i cu. yd. $.16x1521 =$24.37f Ex. 17. 2564 Ibs. 2713 " 3000 " 3109 " 11386 Ibs, -=-56 = 203.3214+ bu. $.80x203.3214=:$162.657 + , ATM. bbls. gal. qt. pt. gi. Ex. 18. 1 4 1 30 2 1 2 15 Ana. 3 49 2 1 1=4613 gi. $.09 x 4613=1415.17, Ans. (318, page 185.) Ex. 2 $ rd. = 12 ft. 41 in. | ft. = _9_ " 13 ft. li in., Ans. 72 COMPOUND NUMBERS. Ex. 3. i mi. = 7 fur. I fur.= 26 rd. U ft. rd. = 13 " U in. 7 fur. 27 rd. 7 ft. in. ; or 7 fur. 27 rd. 8 ft. 3 in. Ant. Ex.4, | =13 s. 4d. s. = 6 2| far. 13s. 10 d. 2| far., Ans. Ex. 5. f T. =12 cwt. | cwt.= 42 Ib. 134 oz. 12 cwt. 42 Ib. 13f oz., Ex. 6. da. =9 h. i h. = 30 min. 9 li. 30 min., Ex. 7. wk. = l da. 4 h. da. = 18 " | h. = 15 min. 1 da. 22 h. 15 min., Ans. Ex. 8. $ hhd.=54 gal. gal. = 3 qt. 54 gal. 3 qt., ^ws. Ex. 9. 4 cwt.=64 Ib. 8f Ib. = 8 " 13 oz. 5 dr. . = 3 " 14 " 73 Ib. 1 oz. 3} dr., Ant. Ex. 10. ^ mi. =3 fur. |yd.= 2 ft. ft. = 9 in. 3 fur. 2 ft. 9 in., Ans. SUBTRACTION. 78 Ex. 11. i of A. = i A.- 26 P. 181 | sq. ft. 60f rd. = l R. 20 " 204^ " f A. =1 " 5 " 194 if " A. = 13 " 90 3 R. 26 P. 126 T VV sq. ft., Ex. 12. li T. 1 T. 3 cwt. 33 Ib. 5 oz. ly^ T. =1 " 3 " 75 " 18f cwt.= .18 " 83 " 51 '- 3 T. 6 cwt. 91 Ib. lOf oz., Am. SUBTRACTION". (919, page 187.) Ex 4. 3 T. 18 cwt. 70| Ib., Am. Ex. 6. 2953 2 s. 7f d., Ans. Ex. 11. 365 da. X5 + 2 da. = 1827 da. 1 hhd. = 63 gal. 1827 gi. =57 " qt. pt. 3 gi. 5 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., Ans. Ex. 13. 16 Cd. 5 cd. ft. 75 Cd. 6 cd. ft. 24 " 6 " 12cu.ft. 69 " 2 " 12 cu. ft. 27 " 7 " 6Cd.3cd.fT4cn.ft. 69 Cd. 2 cd.ft. 12cu.ft. Ans. K. p. 74 COMPOUND NUMBERS. Ex. 14. 10 gal. 1 qt. 1 pt. 63 gal. 15 " 1 pt. 40 " 1 qt. 14 " 3 " 40 gal. 1 qt, Ans. I, page 189.) 22 gal. 3 qt., Ans. yr. mo. da. yr. mo. da. Ex. 2. 1799 12 14 Ex. 5. 1815 18 1732 2 22 1775 6 17 67 9 22, Ans. 40 1, Am yr. mo. da. h. inin. 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. From Nov. 6 to April 6, 151 da. From Apr. 6 to Apr. 15, 9 " 160 da., Ans. Ex. 9. From Aug. 20 to June 20, 304 da. Subtract 5 " 299 da., AIM (891, page 190.) Ex. 2. \ rd. = 8 ft. 3 in. ft. = 9 " 7 ft. 6 in., Ans. Ex. 3. f = 11 s. 1 d. li far. s. = 6 " 10 s. 7 d. li far. An*. MULTIPLICATION. Ex. 4. lea.=2 ini. 75 TO mi.= 5 fur. 24'rd. 1 mi. 2 fur. 16 rd., Ans. Ex. 5. 8 T 9 7 cwt. =8 cwt. 3 qr. 16 Ib. 12 oz. 12f dr 1 qr. 2f Ib. = 1 qr. 2 " 6 " 13 4 " Ans. 8 cwt. 2 qr. 14 Ib. 5 oz. 15/j d>- Ex. 6. i wk.^rl da. 9 h. 36 min. da. = 4 " 48 " 1 da. 4 h. 48 min., Ans. f| of 120 mi. = 41 mi. 7 fur. 9 rd. 8 ft. 74 in., An* Ex.8. 1-i-f; Joff = A; T 4 3 of 96 gal.=25 gal. 2 qt. 3} gi., Ans MULTIPLICATION. (222, page 192.) Ex. 4. Ans. 23 13 s. 4 d. Ex. 5. Ans. 23 Ib. 4 oz. 6 pwt. 10 gr. Ex. 6. Ans. 163 T. 1 cwt. 36 Ib. 4 oz. Ex. 7. Ans. 128 35 r 15". Ex. 9. Ans. 20ft> 1 3 3 1 3 16 gr. Ex. 10. Ans. 235 mi. 6 fur. 7 rd. 4 ft. bu. pk. qt pt. Ex. 13 45 3 6 1 Ex. 14. 367 2 4 8 2941, Ans 178 12 17 16 6 10 5, Ans. 76 COMPOUND NUMBERS. Ex. 15. $4800-f-$80 = 60 = 6 x 10. A. R. P. Bq. yd. sq.fl 4 3 26 20 3 6 29 2 1 10 295 10, Ans. Ex. 17. Ans. 359 45' 40.45*. Ex. 18. Ans. 6 hhd. DIVISION. , page 194.) Ex. 7. Ans. 1 oz. 17 pwt. 4 gr. Ex.11. 5 10s. 10d.=1330d.;. 537 10s. 10d.=129010 d. 129010-^1330 = 97, Ans. Ex. 12. - -- =11 cu. yd. 3 cii. ft.. Ans. 27 x5 x 6 K, 13 . ^- 106 A. 2 R. 26 P. 20 sq. yd. 1 sq. ft. 72 sq. in., Ans. Ex. 14. 4 bu. 3 pk. 2 qt. = 154 qt. ; 336 bu. 3 pk. 4 qt. = 10780 qt. 10780-^-154 = 70, Ans. Ex 15. 60 yd. 2.25 qr. = 242.25 qr.; 242.25 qr.x4 = 969 qr. ; f of 969 qr.=646 qr. ; 9 yd. 2 qr. = 38 qr. ; 646-^-38 = 17, Ana. LONGITUDE AND TIME. 77 LONGITUDE AND TIME. (229, page 196.) Ex. 2. 84 24' Ex. 3. 155 74 1 18 28' 10 23' 173 28' 4 4 41 min. 32 sec., Ans. 11 h. 33 min. 52 sec., Ans. Ex.4. 77 1' Ex.5. 118 + 122=240 ; 30 19 360 240 = 120; " 7 120-M5 = 8 h 7 h. 9 min. 20 sec., Ans. Ex.6. 12 h. 77 1'= 5 " 8 min. 4 sec. 6 h. 51 min. 56 sec., A. M., Ans. Ex. 7. 90 15' 63 36 4 h. 26 39'= 1 h. 46 min. 36 sec. 2 h. 13 min. 24 sec., P. M., Ans. Ex. 8. 124 67^57 = 3 h. 48 min., Ans. Ex. 9. 99 5 68 47 30 18'=:2 h. 1 min. 12 sec. difference of time. Time at Bangor, 1859 yr. 1 mo. 1 da. 1 h. min. sec. A. M Subtract 2 h. 1 min. 12 sec. Time at Mexico, 1858 yr. 12 mo. 31 da. 10 h. 58 min. 48 sec. P.M. Ans. NOTE. In the above subtraction, borrow 31 days, the month being December. 78 COMPOUND NUMBERS. (26, page 197.) Ex. 2. 11 h. 33 min. 52 sec. = 693 inin. 52 sec. ; (693 min. 52 sec.)-;- 4 = 173 28', Ans. Ex. 3. 7 h. 9 rain. 20 sec.=i429 min. 20 sec. ; (429 min. 20 sec.)-h4 = 107 20', Ans. Ex.4. 16 h. 30 min. at St. Petersburgh ; 8 h. 32 min. 36 sec. at New Orleans ; 7 h. 57 min. 24 sec.=477 min. 24 sec. (477 min. 24 sec.) -4-4 = 119 21', Ans. Ex.5. 74 1' West; 8 h. 40 min.=130 4 h.=60 74 l f West, 1st Ans. 14 1' West. 2d Ans. 55 59' East 13 h. 25 min.=r201 15' 74 1' 3d Ans. 127 14' East. DUODECIMALS. MULTIPLICATION. (939, page 200.) Ex. 2 13 ft 9' Ex:. 3. 11 ft. 9' 11' 1 .3' 12ft. 7' 3*, Ans. 2 11' 11 9 14ft. 8' 3' 4 58 ft. 9', Ans. DUODECIMALS. 79 Ex. 4. 12 ft. 11' 6 ft. 2' 4 2 ft. 4' 51 ft. 8' length of walls; 2 0' 9 ft. 3' 12 4' 12 11 14 ft. 4' 8" 465 3 477 ft. 11' area of walls; 43 ft. 2', windows and door; 43 ft. 2' 434 ft. 9' = 48 Ex. 5 30 ft. 4' 25 ft. 6' sq. yd. 2 sq. ft. 9', Ans. Ex. 6. 18 ft. 6' 12 ft. -7-128 = 32| sq. yd .551, Ans 15 2 758 4 222 ft: 5 ft. 6' 773 ft. 6' 12 ft. 5' 111 1110 322 3 6" 9282 1221 ft. 1221 cu. ft.- Ans. 9 Cd ' 69 cu ' Ex. 8. 32 ft, 8 9 ft. 9604 ft. 3' 6", Ex. 7. 36 ft. 10' 22ft. 3' 926 810 " 4 294 sq. ft. 294 sq. ft. ^9 = &.l7x32f = $5 819 ft. 6' 6" 5 ft. 2' 136 7 1 4097 8 6 4234 ft. 3' 7" ; 156 cu. yd. 22 cu. ft. 3' 7', Ans. 80 COMPOUND NUMBERS. Ex.9. 33ft. 9' 27' __3 48ft. 12x3~4 36 1584 180-^-f = 240 yd., An*. 1620 ft. = 180sq. yd. DIVISION. (23O, page 201.) Ex. 2. 16 ft. 8')44 ft. 5' 4"(2 ft. 8', Am. 33 4 11 1 4 11 1 4 Ex. 3. 40 ft. 11' 4*)184 ft. 3' 0*(4 ft. 6', Ans. 163 9 4 20 5 8 20 5 8 Ex.4. 2 ft. 7')14 ft. 6' (5 ft. 7' 4* + , Ans. 12 11 170' 1 6 1 11 0' 10 4 8'", rem. PROMISCUOUS EXAMPLES. 81 Ex. 5. 3 ft. 7' 8 ft. 11' 6")64 ft. 2' 5"(7 ft. 2', 2ft. 6' 62 8 6 72 1 5 11 196 1 5 11 8 ft. 11' 6" PROMISCUOUS EXAMPLES. (Page 202.) Ex 1. 115200 gr. 4-5760=20 lb., Ans. Ex. 3. 1560 bu. x 4-6240 pk. ; 3 bu. 1 pk.=13 pk. , 62404-13=480, Ans. Ex.4. 295218 in.4-12 = 24601i ft.; 24601| ft.4-16=- 1491 rd. ; 1491 rd. 4-40=37 fur. 11 rd. ; 37 fur. 4-8=4 mi. 5 fur. Ans. 4 mi. 5 fur. 11 rd. Ex. 6. 3 x 20 x 24=1440, Ans. $3.25 x4x20x6 x 10 Ex. 7. =$121.871, Ans. Ex. 8. 1 bbl.=1008 gi. ; 1 qt. 1 gi.=9 gi. ; 10084-9 = 112, Ans. Ex. 9. $26.40 x \ 3 x f =$980.10, Ans. Ex. 11. 336 bu. 3 pk. 4 qt.= 10780 qt. ; 4 bu. 3 pk. 2 qt.: 154 qt.; 107804-154=70, Ans. Ex. 12. 3 qt. 1 pt, Ans. Ex. 13. 1 mi.= l760 yd.; 2 fur. 36 rd. 2 yd.=640 yd. ; T y 6 x f x | x |=$226.56f , .471*. page 208.) Ex. 2. 904-450 = .20 = 20 per cent., Ans. Ex 3 175-7-1400=.125 = 12| per cent,, Ans. Ex. 4. 165-7-750 = .22 = 22 per cent., Ans. Ex. 5. 13.20^240 = . 055 = 51 per cent, Ans. Ex. 6. .154-2":=.075 = 7i per cent., Ans. Ex. 7. 6 bu. 1 pk.=200 qt.; 4 bu. 2 pk. 6 qt.= 130 qt. 150-f-200 = .75 = 75 per cent., Ans. Ex. 8. 15 lb. = 240 oz. ; 5 Ib. 10 oz. = 90 oz. ; 90-^-240 = . 375 = 37i per cent., Ans. Ex. 9. 40^-250^.16 = 16 per cent., Ans. Ex. 10. 100 + 90 = 190; 190-^-760n:.25 = 25 per cent., Ans, Ex. 11. | of =i = .50 = 50 per cent., Ans. (337, page 209 ) Ex. 2. 164- .08 = 200, Ans. Ex. 3. 42-f- .07=600, Ans. Ex. 4. 75 -=-.125 = 600, Ans. Ex. 5. 33-=- .0275 = 1200, ^Iras. Ex. 6. $281.25-^.375 = 8750, Ans. Ex. 7. 50-=-.20 = 250, Jws. Ex.8. |59.75-=-.125=$478, Ans. Ex. 9. $975 -Kl 5 =$6500, ^4ws. Ex. 1C .40x.25 = .l $1246.50-=-.! =$12465, Ans. Ex. 11. 2000^- .40 = 5000; 5000-2000 = 3000, Am. COMMISSION AND BllOKERAGE. 87 (238, page 211.) Ex.2. 1.00 + .18 = 1.18; 1475-M. 18 = 1250, Ans. Ex.3. 1.00 + .25 = 1. 25 $4.00-=- 1.25 = $3. 20, Ans. Ex. 4. 1.00 + .15 = 1.15; $6900-^-1. 15=$6000, Ans. Ex. 5. 1.00 + .08 T 1 g = 1.08 T 1 ff ; $432250-M.080625 = $400000, Ans. Ex. 6. 1.00 + .04J = 1.0425; $8757-=- 1.0425 =$8400, Ans. Ex. 7. Since he increased his capital the first year by 20 % of itself he must have had 100 % + 20 %, or 120 == 144 yd., Ans. Ex. 6. 3 Ib. 12 oz. = 60 oz. 10.50 x 60 oz. 3.50 = 180 oz. = ll Ib. 4 oz., Ex. 7. 8 bu. 2 pk. = 34 pk. ; 76 bu. 2 pk. = 306 pk. $38.25x34 Ex. 8. V 1 x H a x rf 3 = Ex. 9. x x = Ex. 10. T V x I x f f SIMPLE PROPORTION. (399, page 287.) Ex. 1. 48 Cd. : 20 Cd. :: $120 : ( ) $120x20 ( ) = =$50, Ans. Or. $120 x = $50, Ans. r ^ ^>J &> <* * *4 %ri 116 PROPORTION. Ex. 2. 6 bu. : 75 bn. : : $4.75 : ( ) ( )= !i!l>iL 5 = $ 5 9.37i, An,. \ Or, $4.75 x Y L =$59.37, Ans. Ex. 3. $3i : $50 : :. 8 yd. : ( ) ( ) = 0^ =sll4f yd, Ans. Ex. 4. 12 : 20 : : 42 bu. : ( ) 42 bu. x 20 ( ) ---- =70 bu., Ans, Or, 42 bu. x H ' 70 bu " Ans. Ex. 5. $.75 : $9.00 : : 7 Ib. : ( ) 900 x 7 Ib. ( )=- - =84lb.,^w. Ex. 6. 3 Ib. 12 oz. : 11 Ib. 4 oz. : : $3.50 : ( ) 60 oz. : 180 oz. :: $3.50 : ( ) Ex. 7. 1 ft. 6 in. : 75 ft. : : 3 ft. 8 in. : ( ) 1| ft. : 75 ft. : : 3| : ( ) ft. x f-1831 ft.-183 ft. 4 in., Ant. Ex. 8. $2.75 x VV=$ 19 - 64 7> Ans. Ex. 9. $13.32 : $51.06 : : 12 bu. : ( ) 51.06 x 12 bu. Ex. 10. 15hhd. = 945 gal. 945 gal : 28.5 gal. : : $236.25 : ( ) SIMPLE PROPORTION. 117 Ex. 11. 6 mo. : 11 mo. :: 7 bbl. : ( ) 11x7 bbl. ( )= --- = 12 bbl., Ans. Ex. 12. 5 12 s. : 44 16 s. : : 9 yd. : ( ) x 9 yd. x 896 - -- = Ex. 13. $3100x^W=$ 310 Ans - Ex. 14. 100 Ibs. coffee=:100x } = 160 Ibs. sugar; 2 : 160 : : $.25 : ( ) $.25 x 160 ( )= - - - =$20, Ex. 15. 13 10' 35" : 360 : : 1 da. : ( ) 47435" : 1296000" : : 1 da. : ( ) ( ) J-|^IAIJL da. = 27 da. 7 h. 43 mm. 6.06 + sec., Ans. Ex. 16. 8f : 131 : : 84.20 : ( ) x y- X ,& = $6.48, Ans. Ex. 17. 6| d. : 10 6 s. 8 d. : : If yd. : ( ) ( ) =2.0.9. x yd. x 2 4 j = 6 94| yds., Ans. Ex. 18. 121 cw t. : 48f cwt. : : $42i : ( ) = $-L x -3-i x 2 2 $163.50 + , Ans. Ex.19. $lf : $317.23 ::8|lb.: ( ) ( ) = 3l7.23x8.4 Ib.x4-1522.7+ lb. 15 cwt. 22.7+ lb., Ans. Ex. 20. $1561 : $95.75 : : 15f bu. : ( ) ( ) fljyii x ifi bu. | y ^9.575 bu.= 9 bu. 2 pk. 2 1 qt., Ans. Ex.21. | bar. : $ bar. :: fy$ : ( ) K. P. 118 PROPORTION Ex. 22. "4 rd. : llf rd. : : A. : ( ) ( ) = a A. x V- x i=2^ A.=2 A. 28 rd., Ex. 23. 13 cwt. : 12 cwt. : : $421 : ( ) ( )=*4*. x -V 2 - x T V =$39, Ans. Ex. 24. 16 oz. : 12 oz. : : $28 : ( ) Ex. 25. 16 oz. 14}i oz.^lfV oz., cheat in 16 oa. 16 oz. : 1 T 5 F oz. : : $30 : ( ) Ex. 26. 1 yr. 6 mo. : 3 yr. 9 mo. : : $750 : ( ) , An, Ex. 27. 10 mo. x Yjo 30 mo -> Ex. 28. $25 : $30 : : $28 : ( ) ( )=$5 T 8- x V- X V $34.10, Ans. Ex. 29. 1 yr. 4 mo. = l = f yr. ; I yr. x f| = 2 yr.r=2 yr. 9 mo. 10 da. Am COMPOUND PROPORTION. (4O1, page 292.) Ex 1. 16 50 16 50 128 90 ) = 72 bu., Am. COMPOUND PROPORTION. 119 Ex. 2. 3 i < 12 J ' : i W Ex. 12. 54 } 75 ^ X ? / X -^5 X -j = *. Ex. 13. 24 ^ 217 j 33 189 [: 5i V : : 5f } : 3| 14) ( ) ) 31 PARTNERSHIP. 121 PARTNERSHIP. (4O7 5 page 295.) Ex. 2, $ 8000 T 8__ 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 " Ex. 4. $20000 : $13654 :: $3060 : ( )=$2089.062 $20000 : $13654 :: $1530 : ( )=$1044.531 Ex. 5. 16 + 24 + 28 + 36 = 104 $13x T Y T =$2, A pays; $l3xftt=$3, B pays; $13 x y 2JL-$3.50, C pays ; $13 x ^=$4.50, D pays. Ex. 6. 14 + 6 + 12 = 32 shares. $2240 x if = $980. Captain's share ; $2240 x ^=$420, Mate's share ; $2240 x if =$840, divided among the sailors, $840-7- 12 =$70, each sailor's share. Ex. 7 $3475.60 $2512=$963.60, lost to the owners ; $963.60 x|=$120.45, A's $963.60 x i = $240.90, B's [ Ans. $963.60 xf = $602.25, C's Proof 1 =$963.60 1 22 PARTNERSHIP. Ex 8. 6, C's proportional share. 4, D's 6 f 4=10; 10x1= 2, E's " " $2571.24 x W=$1285.62, C's ; $2571.24 x yV =$ 857.08, D's ; Ans. $2571.24 XfV=$ 428 .62, s ; \ .08, D's ; ( .54, E's;) Ex. 9 $7500-($2000 + $2800.75 + $1685.25) = $1014, D's gain ; gain. cap. gain. cap. $1014 : $3042 :: $2000 : ( )=$6000 A;) $1014 : $3042 :: $2800.75 , ( )=$8402.25, B; ( Ans t $1014 : $3042 :: $1685.25 : ( )=$5055.75, C ; ) (4O8, page 297.) Ex. 2. $250 x 6 =$1500, B's product ; 275x8= 2200, C's " 450x4= 1800, D's " $5500 $825 x || =$225, B's share of gain ; 825 x f|= 330, C's " " " 825x-]-|= 270, D's u " " Ex.3. $1000 x 8=$ 8000 $1500 x 4=$ 600C 1600x10= 16000 1200x14= 16800 A's product, $24000 B's product, $22800 $2 4000 + $22800 = $46800, sura 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 bushe1s-M=93 busliels to be divided. 62 : 20 : : 93 : ( )=30 bu., A's; ^ 62 : 18 :: 93 : ( )=27 " B's ; ) Am. 62 : 24 :: 93 : ( )=36 " C's; ) Ex. 5. From Jan. 1, 1856, to Apr. 1, 1858, is 27 mo., Gal lap's time ; " Mar. 1, 1856, " Apr. 1, 1858, " 25 " Decker's " " July 1, 1856, u Apr. 1, 1858, " 21 " Newman's" $3000 x 27=$81000, Gallup's product ; 2000 x 25= 50000, Decker's 1800x21= 37800, Newman's " $168800, sum of products. $168800 : $81000 : : $4388.80 : ( )=$2106, Gallup's gain; $168800 : $50000 : : $4388.80 : ( )=$1300, Decker's " $168800 : $37800 : : $4388.80 : ( ) =$982.80, Newman's u Ex. 6. -T- 8 =$70, A's monthly profit ; -M0=$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 : ( )=$2986.66|, 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 \- of B's, and to i of C's, and the proportion of the shares will corres- pond to the conditions. 124 ANALYSIS. 2+3+4=9 $117 xf=$26, A's gain $117 x =$39, B's " $mx$ = |52, C's " If we now divide the proportional shares of tht 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 same proportion as the respective shares of the. capital ; 2-+3=|, A's proportional share of capital ; 3-f-5=, B's " " " " 4-5-7=4, C ' s " " " " Iff* sum of proportional shares. : a' :: $1930 :( )= $700, A's capital; : f : : $1930 : ( )=$630, B's " : A : : $1930 :( )= $6 00, C's " ANALYSTS. (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. Ex. 6. We multiply 19 (pence) by 28 for the cost of the butter (in pence), and iivide by 7 times 12 (pence) the price of the tea. 56 7 3 126 4 $27, Ans> 28 19 T9 ANALYSIS. 125 Ex 7. 10s. 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 $648=$108, Ana. Ex.8. 2s. 6d.-30d. Then2x3x 7 x 3 Oncost in pence. Divide by 60 to reduce to Decimal currency. 60 2 3 7 30 $21, Ans. Ex. 9. 20 x 3 x 12= 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. 75 20 3 12 5 | 48 9|, Ans. Ex. 10. 96 160 18 90 160 24 Cost, $30 sold for $42.66f. $42.66f-$30=*$12.66f, Ans. Ex. 11. 431 = V ; 10 s. 6 d.=126 d. ; 8 s. 3 d. = 99 d. Ex. 12. 9s. 4 d. = 112 d. ; $1=96 d., Mich, currency 11 96 87 126 609 Ans. 300 112 $350, Am ."56 ANALYSIS. Ex. 16. Dividing 128 by 16, we ob- tain what 1 horse will consume in 50 IQ days ; dividing this result by 50, we ob- 50 128 5 90 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. 17. Divide 4f (=/) b 7 10 * 3 | u i'=V) to find what amount of wood $1 21 j 2 4 i QQ will buy ; then multiply by 24ff=y) to ' find how much $24 will buy. U * Ex. 18. 52 x 3 x J-f *= the money given for the cloth. Divide this result by 65, the number of yards, to find the price 3 65 52 3 100 Ans. 80 cents. Ex. 19. A shadow of 1 foot will require an object of 3 feet in length ; and a shadow of 46| feet will require an object 46 1 times of 3 feet in length ; hence x } x J-f A 28 feet > -A ns - Ex. 20. 8 sheep x 7|= 60 sheep for 1 mo., A's use of the pasture ; 12 " x4| = 50 " " " ,B's " " " 15 " x6f=:100 " " " , C's " " * . 210 " " " , total " " " 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 8 T $18> A. must pay ; $63 x |f = 130, C " u ANALYSIS. 127 Ex. 21. 1 bu. oats = dollars ; 1 bu. rye =-'/ bu. oats = J / x | dollars ; 1 bu. wheat= y bu. rye=Y x J / x e dollars. If we divide $30 by the price of 1 bushel of wheat, we shall have the number of bushels which $30 will buy ; hence 3 T n - x f x fV x T 7 o =15 bu. Ans. Ex. 22. If $480 gain $84 in any time, tc gain $21 in the same time will require fJ of $480 ; and if fi of $480 gain $21 in 30 mo., to gain the same amount in 15 mo. will require ff of f 1 of $480. | 480 $ 240 ' AnS ' Ex. 23. 28 x = 21 sq. yd., contents of the 28 yd. ; 21-f-f =31 yd. of that which is f yd. wide. Ex. 24. If 130 miles require 3 days, 390 miles will require ff of 3 days ; and if 14 hours a day require 3 days, 7 hours a day will require J T 4 - of 3 days. Ex. 25. If 6 men cut 45 cords in any time, 8 men can cut of 45 cords in the same time ; and if in 3 days any number of cords be cut, in 9 days there will be cut | times as many cords. 3 390 14 Ex. 26. A's age + B's age= C's age=2 T 1 times this sum And the sum of all their ages, or 93 Hence, 93-j-7f = 12 yr., A's age ; 12x11=18 " B's " 12x51 = 63 " C's " =2| times A's; =5| " " Ans. J -8 ANALYSIS. Ex. 27. 1 day of = T k - da. of D. ; 1 day of B= '/ da. of C = V 2 x T ^ da. of D. ; 1 day of A= da of B =f x V 2 x A da. of D ; hence 5 days of A=f x x y x T 8 T da. of D=8 da. of D, Ans. Ex. 33. If the cost of 12 oranges and - L - cta - iO lemons is 54 cents, the cost of one half ^ jo 54 the lot, or 6 oranges and 5 lemons, will be -g r ^7 27 cents. 2 6 But the cost of 6 oranges and 7 lemons 1 lemon =3 cts. is 33 cents. And, by subtracting, we 1 orange=2 cts. find the cost of 2 lemons to be 6 cents, which gives the cost of 1 lemon 3 cents. From the first ex- pression, 6 oranges and 21 cents (equal to 7 lemons) is equal to 33 cents; hence 6 oranges are worth 12 cents, and 1 orange is worth 2 cents. 18 20 1000 Ex. 34. 18x20x1000 = the whole number of ounces of provisions ; and since this quantity is to supply 1600 men 30 16QQ days, we divide by 30 to find the daily al- ' lowance for the army, and this result by ^ ns ' H oz< 1600 to find the daily allowance to each man. Ex. 35. If we add 6 bushels to the smaller bin, there will be 60 bushels in both ; but as the larger will then contain 2 times as many bushels as the smaller, the two together will contain three times the number in the smaller ; hence 3 times the smaller =60 The smaller =20 The larger =20 x 2=40, Ans. ANALYSIS. 12f Ex. 36, We take the difference of two numbers from the greater to find the less. The greater diminished by 1 of the greater equals the less, which must be f of the greater. And if the less be | of the greater, their sum, 20, is 1| times ths greater. Hence we have 20 -^- If = 12, the greater, Ans. tix. 37. 1 day of C = da. of B ; 1 day of A=f da. of C= f x f da. of B ; hence 6 days of A =6 x f x f da. of B ; and, 6 weeks of A =6 xf x wk. of B. 6 x f x f wk. = 1 1 1 wk., Ans. Ex. 38. 36 x 1 j=45 sq. yd. to be lined. 45 yd.-j- = 60 yds., Ans. Ex. 39. 80 x 3i x 96 = value of the broadcloth, in pence ; 104 x 10 = value of one sack of coffee, in pence ; and to obtain the number of sacks we divide the former product by the latter. 4 104 10 80 13 96 24, Ans. Ex. 40. If the time past since noon is equal to 1 to midnight, both intervals, or 12 hours, must be time to midnight ; hence 12 h.-^-li =10 h. to midnight. 12 h. 10 h.= 2 h. P. M., Ans. Ex. 41 She bought one half for \ cent apiece ; And the other half for \ cent apiece. (^+) -7-2 = ^, average buying price ; 3-^-5 =|, selling price. f ^ =J-i, gain on one peach. 55 -f 11 = 300, Ans. of the time times the 1 30 ANALYSIS. Ex. 42. A can build the boat in 18x10 = 1 80 hours ; B " " " " " 9 x 8= 72 " A " " T { of the boat in an hour ; Bu i a a a a 72 A and B can build yio' + TV s~lo f the boat m an hour ; A and B can build -g^- Q x 6 = g- 7 - of the boat in a zlay of 6 hours. It will, therefore, require as many days as 7 is con- tained times in 60 ; hence 60 -f- 7 =8| days, Ans. Ex. 43. He spent at first , and he had left. He then spent i of this , and he had f of this \ left ; hence | of \ = of his money, which is $10, and the whole is $30, Ans. Ex. 44. 4 times the work will require 4 times as many men, and of the time will require 5 times as many men ; hence 30x4x5 = 600, Ans. Ex. 45. If $3.25 buy 16.25 lb., $1.25 will buy if} of 16.25 Ib. ; hence 16.25 Ib.x iff = 6.25 = 61 lb., Ans. Ex. 46. On every idle day he lost the forfeit, $1, and his wages, $2.50, which together amount to $3.50. Had he la- bred every day, he would have received $2.50 x 20 =$50. $50 $43 =$7, what he lost by being idle ; and $7-^-$3.50 =2, the number of idle days. Hence 20 2 = 18, the number of days he labored, Ans. Ex. 47. A, B, and C perform yL- in an hour A and B " T V " " Hence, C performs r V rV^T? " " Again, A, B, and C perform ^ " A and C < T V " " Hunce, B performs ^j TV=sV " *' Therefore, B and C perform T ^ + -U y i T And in 9^ hours they will perform T 7 x y =, ALLIGATION. ALLIGATION. (414, page 305.) Ex. 2. $1.00xl2 1.50 x 5= 7.50 20 )$19.50 .975, Ans. Ex. 3. $1.25x52 = 13 65 65 Mixture, $1 per gal. 65 x 32 x $.061=^130, receipts ; $130 $65=$65, gain. Ex. 4. 8x10= 80 Ex. 5. 12 x 7i= 90 9x12 = 108 . lOix 8 = 84 11x16 = 176 11 X 9 = 99 38 364 10 x 101 = 105 88x10 = 380 35 378 Ans. 16 cents. 378x11=567 Ans. 567-^-35 = 161 cts. Ex.6. 50 x 4=200 Ibs. ; $.13 x 200 = $26.00 4CX 10 = 400 Ibs.; .10x400= 40.00 25x24 = 600 Ibs.; .07x600 42.00 1200)$108.00 $.09 average cost per Ih f.095-$.09=$.005 ; 1200 x $.005 =$6, Ans. ALLIGATION. (416, page 310.) Ex. 3. 12 10 11 14 1 1 2 Ans. 1 Ib. at 10, and 2 Ibs. at 11 and 14 cents. 1 I Ex. 4. 90 120 TfV Ans. I gal. of water to 3 gal. of wine. Ex.5. 275^ f200 250 300 [400 1 Ans. 5 of the 1st kind, 1 of each of the 2d and 3d, and 3 of the 4th. (80 To" 6 6 84 ) 6 6 87 * 4 4 94 * 3 3 96 * } 10 6 16 Ans. 6 of the first 2 kinds, 4 of the third, 3 of the fourth, and 16 of the fifth. (417, page 311.) Ex. 2. 80 Ans. 20 Ibs. of each of the first three kinds, at / > DO Ibs. the fourth. 40 T'O 2 2 20 60 . A 2 2 20 75 i 2 2 20 90 rV T'O TV 8 4 1 13 130 Ex. 3. 4 24 24 72 Ans. 24 at $3, and 72 at $5. ALLIGATION. Ex. 4. 90 1 60 130 40 4 4 12 60 60 180 Ans. 60 gallons each of alcohol and water. Ex. 5. 40 35 50 150 75 Ans. 150. Ex. 6. 7i > 81 ! 10 ' II 6 80 60 20 Ans. 60 Ibs. at 81 cts. and 20 Ibs. at 10 cts. (418, page 312.) Ex. 2. 14 Ans. 60 at 9 s., 40 at 12 s. ; 20 at 18 s., and 50 at 20 Ex. 3. 22 Ans. 6 ounces each of the first three, and 33 ounces of the la,t 9 i 6 6 60 12 i 2 4 4 40 18 * 2 2 20 20 i 5 5 50 17 170 16 1 2 2 6 18 2 2 6 21 1 2 2 6 24 i i 6 4 1 11 33 IT 51 Ex. 4. $178.50-^210 $.85, average price ; ( 50 i 3J 1113 13 78 85-^ 75 TV 13 13 78 ( 150 61 eVll 2 9 54 35 210 Ans. 78 bu each of oats and corn, and 54 bu. of wheat 184 EVOLUTION. Ex. 5. 48 (45 51 ( 54 21 3600 1200 1200 6000 Ans. A 3600 bu. ; B and C each 1200 bu. Ex. 6. $84-:- 56 =$1.50, average daily wages. (50 lio 6 6 24 i8 7 1 3 1 3 4 12 300 TJo 4 4 16 14 56 Ans. The boys 24, 4, and 12 days, respectively, and the man 16. EVOLUTION. ! SQUARE ROOT. (434, page 319.) Ex. 9. Ans. 234135. (441, page 321.) Ex. 3. 200 x lj = 225 sq. yd. ; 1/225 = 15 yd. = 45 ft., Ans. Ex.4. 10 A.=1600 rd.; 1/1600=40 rd., length of one side ; 40 x 4 = 160 rd., Ans. Ex. 5. 45 3 =2025, square of the base ; 60 a =3600, square of the perpendicu.ar ; 5625, square of the hypotenuse. = 75, Ans. SQUARE ROOT. 135 Ex. 6. 39 2 =1521, square of the hypotenuse 15 a = 225, square of the base ; 1296, square of the perpendicular. 4/1296=36, height of the stump. 36 ft. + 39 ft. =75 ft., Ans. Ex. 7. 4/40 2 33 2 = 22.60 + , from foot of ladder to one side ; 4/40' 21 2 =34.04+ " " " " " other" 56.64 + , Ans. Kx. 8 52 a =2704, square of hypotenuse ; 48 a =2304, square of perpendicular ; 400, square of the base ; 4/400 = 20, Ans. Ex. 9 1 mi. =3 20 rd., length of 1 side of the park. S20 2 =102400 102400 204800; 4/204800=452.5+, diagonal. 320 x 2 = 640, distance abound the park to the opposite corner. 640452.5 = 187.5, distance between A and B, when A ar- rives at the opposite corner. 187.5 + -^-2=937+, Ans. Ex. 10. 20 a + 16 a = square of the diagonal of the floor. The diagonal of the floor and the height of the room will form the base and perpendicular of a right-angled triangle, of which the diagonal from the lower corner to the opposite upper corner !s the hypotenuse. Hence Ifi2 _ I square of the diagonal of the floor ; 12* =144 square of the perpendicular; 800 square of the required diagonal ; 4''80b = 28.28427l + feet, Ans. 136 EVOLUTION. Ex. 11. 2:3:: (63.39) 3 : ( )=6027.43815 V / 6027.43815=77.63 + rods, Ans. Ex. 12 1 : 2 :: 5 3 : ( ) = 50 ; 4/50 = 7.07106+ feet, Ans. CUBE ROOT. (444, page 327.) APPLICATIONS IN CUBE BOOT. Ex. 1. 1/1331 = 11 ft., Ans. Ex. 2. 1/373248 = 72 in.=6 ft., Ans. Ex. 3. V474552 = 78 in.=6i ft., length of 1 side ; 6 x 6=42i sq. ft., Ans. Ex. 4. Vf|=f ft-, length of 1 side ; x f sq. ft. = area of 1 side ; xx6x= sq. yd., area of 6 sides, Ans. Ex. 5. If the bin be divided by a vertical section equi- distant from the ends, the two parts will be cubes, each of a capacity of one half of the bin. 125 x 2150.4=268800 cu. in. in the bin ; 268800-^2 = 134400 cu. in., contents of one half. Vl34400 = 51.223+in., width and depth. 51.223 x 2 = 102.446 in., length. Ex. 7. 1 : 2=(14.9) 8 : ( )=6615.898 V6615.898 = 1 8.7 + inches, Ans. Ex. 8. 16 : 25 : 8 : ( ) = 12.5 cube of the diameter. = 2.32+ ft. ARITHMETICAL PROGRESSION. 137 ARITHMETICAL PROGRESSION. (451, page 329.) Ex.1. (19 l)x 3 = 54; 54 + 4=58, Ans. Ex.2. (13 I)x5 = 60; 75 60=15, Ans. Ex 3. 2=firstterm; 3=com. diff. ; 18=No. terms. (18 I)x3 = 51; 51+2=53 cents, Ans. Ex. 4. (40 1) x i=9f ; 9f +i = 10i, Ans. Ex. 5. 20= first term ; 5= com. diff. ; 9= No. terms. (9 I)x5=40; 40 + 20=60, Ans. Ex. 6. 100=first term ; 7=com. diff. ; 46=No. terras. (46 1) x 7 = 315; 315 + 100=$415, Ans. (452, page 330.) Ex.1. 172 = 15; 15+5 =3, Ans. Ex. 2. 14 2 = 12 ; 12-^6=2 years, Ans. Ex. 3. 501 1=491 ; 49i-:-33 = li, Ans. Ex. 4. 3=first term ; 91= last term ; 14=No. terms. 9i3 = 6i; 61-7-13 = 1, com. dif. (453.) Ex.1. 43 7 36; 36-i-4=9; 9 + 1 = 10, Ans. Ex.2. 40-21 = 371; 37i~7i=5; 5 + 1 = 6, Ans. Ex. 3. 6=first term ; 226 = last term ; 4=com. diff. 2266 = 220; 220-^-4=55; 55 + 1=56, Ant. (454, page 331.) Ex. 1. (5-h32)x-^= 222 Ans. Ex. 2. (l+12)xY=^ 8 . Ans > 1&5 GEOMETRICAL rivuGRESSION. Ex. 3. 24=first term; 1224=last term; 52==No. terms. ($1224 + $24) x */=$32448, Ans. Ex. 4. 4 =first term, or twice the distance to the first apple , 400= last " " " u " " last " 100 = No. terms. (400 yd. + 4 yd.) x ^=20200 yd., Ans. GEOMETRICAL PROGRESSION. (458, page 333.) Ex. 1. 4 x 3 8 = 26244, Ans. Ex. 2. 1024 x U) T =TVWr =A. Ans - Ex.3. 1 first term; 2=ratio; 9 = No. terras. 1 mill + 2 8 = 256 mills=$.256, Ans. Ex.4. 7x'== Ex. 5. 1= first term ; 1.07= ratio ; 5 = No. terras. 1 x (1.07) 4 =$1.40255 + , Ans. Ex.6. 3= first term; 3= ratio; 7= No. terms. $3 x 3 9 =$2187, Ans. (459, page 334.) Ex. 1. (512x3)-2 = 1534; 1534^-2=767, Ant. Ex 2 (262144 x4)-4 = 1048572; 1048572^-3 = 349524, Ans. Ex. 3. (162 x 3)-2=484 ; 484-^2 = 242, Ans. Ex. 4. _u^-^i, ratio; (iL X 5) = 1; 1-^-4 = 1, Ans. PROMISCUOUS EXAMPLES. 139 Ex. 5. -2= first term; 6-r-2 = 3, ratio ; 12= No. terms. $2 x 3 n = $354294, last term ; ($354294 x 3) 2 = $1062880 ; $1062880-^2=1531440, Ans. Ex, 6. 7= first term, or yield of the first year; 7 = ratio ; 12 = No. terms, or the number of years to yield. 7x7 n =7 13 =13841287201, last term, or 12th year's produce. (13841287201x7) 7 ^ -'- =16148168400, sum of all terms. 6 16148168400-i-1000 = 16148168.4 pt. 16148168.4 pt.=252315 bu. 4-t qt., Ans. Ex. 7. 200^20 = 10, the number of times the family doubled its number. 10+1 = 11, No. terms; 2 ratio. 6x2 10 =6144, Ans. PROMISCUOUS EXAMPLES. (Page 334.) Ex. 1. 800 x 2 = 1600, the sum ; and 200 x 2= 400, the difference. The greater of any two numbers is equal to the less 4- the difference ; and the greater and the less, or the sum of the numbers, must be composed of twice the less and the differ- ence. Hence 1600-400=1200, twice the less; 1200-7- 2= 600, the less ; 600 + 400=1000, the greater. 140 PROMISCUOUS EXAMPLES. Ex. 2. f f T 3 T sV If A f a number be added to itseh, the result must be 1/j times the number. Hence, 61-hl/ T 55, Ans. Ex. 3. 3 h. 21 min. 15 sec. = 12075 sec. ; 1 da.=86400 sec. ; ., Ans. Ex. 4. 3 bu. 3 pk. 3 qt. 10 38 bu. 1 pk. 6 qt. 7 269 bu. pk. 2 qt., Ans. Ex. 5. A and B together have 3 times A's ; . C and D together have $300 + $500 =$800 ; And they all have 3 times A's +$800. Therefore, $1100 $800=$300=3 times A's. $300-^-3=$100, Ans. Ex. 6. B has A's votes H- 200 C has A's votes + 1000 A B and C have 3 times A's votes +1200 Therefore, 30001200=1800, 3 times A's votes. 1 800 -r- 3 =600, Ans. Ex. 8. i -1=-^-. Hence 10 is ^ of the number; and the number must be 560, Ans. Ex. 9. $28.35^$.35 = 81 gal. mixture. 8163 =18 gal. water added. Ex. 10. When A had gained J, he had | of the original stock. B, after his loss, had as much, or f of the original stock ; hence he had lost f ; the $200 which he lost was | of his stock; and his whole stock must have been 200-7-f = $500. PROMISCUOUS EXAMPLES. 141 Ex. 11. $.35 x 13=$4.55 ; $31.55 $4.55=$27, cost of the whole, if the wheat had been at the same price as the barley. 17 + 13 = 30, whole number of bushels. $27-^-30=$ .90, price of barley, $.90 + $.35 =$1.25, price of wheat, Ex. 12. 4 mo. 11 da. 7 h. 5 min. 3 20 15 21 v Arts. 21 da. 15 h. 44 rain., Ans. NOTE. Borrow 31 days for March. Ex. 13. The point of time divides the whole 12 hours into two intervals, which are in the ratio of 9 to 11. Hence, by Partnership, 9 11 20 : 9 : : 12 h. : ( ) = 5 h.24 min., Ans. Ex. 14. The least common multiple of 63, 42 and 31 ; or, since 63 is 2 times 31, the least common multiple of 63 and 42, which is 126, Ans. Ex. 15. The least common multiple of 8, 9, 15 and 16, which is 720, Ans. Ex. 16. Since B gets in debt $10 yearly, his income would enable Imr to spend $30 $10 = $20 a year more than A spends. Hence $20 is } of the income ; and $20 x 8 = $160, income, Ans. Ex. 17. $2.19xf^o_$ 2 .40, Ans. Ex. 18. * : 3 * - : : . j : - : : $3.37f : ( )=$52.779, Ans. Ex. 19. $1000 : $200 : : 6 mo. : ( ) = ! mo., Ans. K. r. 7 142 PROMISCUOUS EXAMPLES. Ex.20. $2356.80-=- .40 = 15892, left; $5892 + $2356.80=|8248.80, Am. Ex. 21. = ^. x a x f =!=12i per cent., Ans. Ex. 22. 1 private has 1 share ; 60 privates have 60 shares 1 subaltern" 2 " 6 subalterns " 12 " 1 lieut. " 6 " 3 lieut's "18 " 1 commander has 12 " All have 102 shares. $10200-M02=$ 100, share of a private. $100 x 12 =$1200, share of the commander. Ex. 23. 19 16 = 3 ; 51-^3 = 17 hours, Ans. Ex. 24. 40 30 50 54 14 20 10 133 95 95 190J [Ans. Ex. 25. $33.75-^22.5 =$1.50, selling price per bu. $22.50-M8 =$1.25, buying " $ .25, profit on 1 " $.25x240=$60, Ans. Ex. 26. The wagon is worth 4 times the harness ; the horse is worth 8 times the harness ; hence the horse, wagon and harness together are worth 8 + 4 + 1 = 13 times the harness Therefore, $1 69 -^ 1 3 = $1 3, harness, Ans. Ex. 27. 18 in : 42 ft. : : 40 in. : ( )=93i ft., Ans. Ex. 28. 25 rd. : 40 rd. : : 4 rd. : ( )=6| rd., Ans. PROMISCUOUS EXAMPLES. 148 Ex. 29. }, ft, and if =if, if, and if. And since fractions having a common denominator are pro- portional to their numerators, we have 15 shares for A and B ; 18 " " A " C; 13 " " B " C; 46, twice the number of shares for A, t> and C. 462 =23 shares for A, B, and C. no TO 1 A U U A Xi. j 23 18= 5 " " B; 23-15= 8 " " C. $26.45 x if =$11.50, A's portion ; $26.45 x /3=$ 5.75, B's " } Ans. $26.45 x V=$ 9.20, C's " Ex. 30. 2^ 6' j : 15 j- : : 12 : ( )=480 oz.=30 Ib. Ans. Ex. 31. $6300x|=$ 900, A's, $6300 x =$1260, B's; $6300 x =$1400, C's; $900+41400 =$2300, D's; $5860 $6300 $5860=$440 $440 xf =$165, E's, $440 xf= 275, F's. Ex. 32. S200xl.593848=$318.769 + , Ex. 33. At the time of the dismissal, the provisions on hand would supply 360 men 1 month ; they would aupplv \ us many men 5 months. Hence 360-^-5 = 72, the number that remained; 360-72 = 288, dismissed, Ans. 144 PROMISCUOUS EXAMPLES. Ex. 34 $1.338226 =amt. of $1 at compound int. for 5 yrs, at 6 per cent. $669.113-^1.338226 = $500, principal. $669.113 $500 = $169.113, interest. $500x.06 = $30,-simple int. of $500, for 1 year at 6 per cent. 169.113-r-30=5.637l yr. = 5 yr. 7 mo. 19.356 + da., Ans. Ex. 35. $148.352^-9.728 -$15.25, Ans. Ex. 36. It is evident that the product of two numbers must ontain each common factor to the two numbers twice, and sach factor not common once. 483-7-23 21, product of the factors not common. 23 x23 x 21 =11 109, Ans. Ex. 37. 8 7 15 12 9 :: 1 : 2 12 9 2 8 7 15 9 140 154, Ex. 38. 36 ) 60 ) 36 9 j ' 27 j ' 1 Ans. 144 yards. Ex. 39. Ex. 40. 1 x 4 = 4, A's product ; 3 x 2 = 6, B's in I7i : 4 : . $52.50 : ( )=$12, A's share ; I7i : 6 : : $52.50 : ( )=$18, B's share; 17 : 71 : : $52.50 : ( )=$22.50, C's share, | and |=| and f. 4 + 5 = 9. 9 : 4 :: $45 : ( ) = $20, A's; 9 : 5 :: $45 : ( )=$25, B's. PROMISCUOUS EXAMPLES. 146 Ex. 41. $.35, interest of $1 for 5 years at 7 per cent,; $33.25-h. 35 =|95, Ans. Ex. 42. 6 Ex.43. 2 + 3 + 4 9. 9:2:: $360 : ( )=$ 80 } 9:3:: $360 : ( )=$120 ( Ans. 9:4:: $360 : ( )=$160; Ex. 44. 8i : 6| : : 12| : ( )=9 T and 5 times $5 or $25=C's stock. $42-$10=$32,A's + B'sgain; A's=f or f of $32, or $12; and B's= $ or of $32, or $20. Therefore, etc. 12. Since he put in or of the capital, he should also take f of the gain; f of $240=$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 =11 calves. If 11 calves cost $11, 1 calf cost T V 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 i of the rent, he puts in 1 of the cows. Then A's 5 cows + B's 3 cows=8 cows | of the cows, and % of 8, or 4 cows=C's number. And since C's 4 cows cost -i of $42 or $14, 1 cow cost of $14, or $3 ; 5 cows cost 5 times $3, or $l7i; and 3 cows 3 times $3, 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 5, =2 x4 or 8 cows. But 8 cows =6 oxen; and 9 oxen + 6 oxen + 6 oxen = 21 oxen, which cost $56. 1 ox cost ^ T of $56 or $2f ; 9 oxen 9 times $2f, or $24 ; and 6 oxen 6 times $2|, or 16 ; etc. 16. Since Mary wrote J as many lines as Melissa, Melissa's work is divided into 8 parts, 7 of which = Mary's ; then 8 + 7 = 15 ; and y 1 ^ of 60 is 4 ; T 8 T 8 times 4, or ? ' md T 7 ^ 7 times 4, or 28. Therefore, etc. 17. Since the boys received as many pears as the girla, they received of 24, or 12. There were as many boys aa (148, 149) 174 MISCELLANEOUS EXAMPLES IN THE ) is contained times in 12, which is 4 times ; as many girla is 4 is contained times in 12, which is 3 times; and 4 + 3 = 7. Fherefore, 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 $9 6 =$2 4 Breach daughter's portion ; and of $24=$12 Breach son's portion. Therefore, etc. 99. 1 The 1st has 1 part, the 2d 1 part +2, and the 3d 1 part-r2 + 6; then 3 parts + 2+2 + 6 = 76, or 763 parts + 10; and 76 10, or 66=3 parts; 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. 72 6 = 66, Ans. 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 1 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 more than A. of $200 is $100, or what A paid ; and $100 + $100 =$200, whatB paid. 6. The drum cost 1 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. 4 of $42, or $6 = costof drum; 2 times $6, or $12 = cost of rifle; and 2 tr'mes $12, or $24 -cost of watch Therefore, etc. (149) PROGRESSIVE INTELLECTUAL ARITHMETIC. 175 7. The harness cost 2 parts, the horses 4 times as much or 8 parts, and the wagon 1 times the harness, or 3 parts; and the harness 2 parts + the horses 8 parts, plus the wagon 5 parts=15 parts=$225. -Jj of $225 is $15, and 2 timea $15, or $30=harness ; 8 times $15, or $120=cost of horses; and 5 times $15, or $75 = cost of wagon, etc. 8. Since he traveled ^ as far the 1st as the last 2 days, the last 2 days' travel is divided into 2 parts of which 1 = first day; hence of 114 miles, or 38 miles = 1st 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 1 of what remained due after the 1st j^yment, by the $20 that exceeded \ ; hence $20 + $20, or $40=. 2 times $40, or $80=what remained after 1st payment, and $80 was less than the debt, by the $10 the payment exceeded \ ; $80 + $10, or $90=| ; 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 1 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 pennies=|; \ 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 \ the keg by the \ gallon more than the \ gallon sold ; then 1 gallon + 1 gallon or \\ gallons = the keg, and 2 times 11- or 3 gallons=the contents of the keg. 14. Since | of John's = f of Mary's, of John's=i of f or f of Mary's, and or all of John's=3 times f or f of Mary's ; hence Mary's are divided into 8ths and John's =9 of them, and the whole = J J L ^ Mary's. T ' T of 34 or 2 is | ; 9 times 2 or 18 John's, and 8 times 2 = 16=Mary's. (149, 150) 176 MISCELLANEOUS EXAMPLES IN THK 15. Since f 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 of Mary's less 10 cents= Susan's, Susan's + 10 cents = 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. | 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 of Homer's =-f of Silas's, of Homer's will=i of -f or f of Silas's, and f or the whole of Homer's, 5 times $ or J T of Silas's ; and since Homer's exceeds Silas's by ^ of Silas's, the 3 marbles must=^ of Silas's ; Jience Silas has 7 marbles and Homer 10. too. 1. Since the first drink a gallon in 3 days, he will drink i of a gallon in 1 day, and since the second drink a gallon in 4 days, he will drink of a gallon in 1 day ; both will drink -L+i or T 7 j of a gallon in 1 day, and 1 gallon will last as many days as T \, what they drink in 1 day, is contained times in j| or 1 gallon ; T 7 is in }| If times. Therefore, etc. 2. Since Julia can do it in 7 hours, in 4 hours she can do -| of it, and Jane must do the remaining | ; and since Jane does ^ in 4 hours, she will do | in ^ of 4 or 1^ hours, and ^, or thi whole, in 7 times 1, or 9 hours. Therefore, etc. 3. Since the first can do it in 9 hours, he can do in 5 hours, and the second must do the remaining ; and since the second pitches f in 5 hours, he can pitch | in | of 5, or 1 1 hours, and , or the whole, in 9 times 1|, or 11{ hours. 4. 3f =\ 9 - and 7| = - 5 8 8 -. Since the second pipe can empty it in ^8- hours, it can empty jV of it in |, and f f in *Ji hours, and the first must empty the remaining |f ; and since the first can empty |f in \ 9 - hours, it can empty j\ in -j of 2 T *> or J hour, and f f , or the whole, in 58 times |, or 7| hours. (151) PROGRESSIVE INTELLECTUAL ARITHMETIC. 177 5. Since A can make a vest in of a day, he can make as many vests in a day as | is contained times in f, or 1 vests ; B as many as is contained times in , or 1^ vests ; and l^ + lj, or 3 vests what A and B can both do. C can make as many as is contained times in $, or Ij- vests, and 3 1=1. Therefore, etc. 6. Susan can knit as many pairs as f is contained times m |, or 1| pairs; Sarah can knit as many as f is contained times in ^, or 2 pairs ; and If + 2^=4 pairs. 7. Since Sarah can knit 2i or 1 pairs in a day, she can knit ^ of a pair in \ of a day, which is the part she must knit for Susan. 8. Since Susan c|n knit If or f pairs in a day, she can knit ~ of a pair in 1 of a day, which is the part she must knit for Sarah. 9. Since Jason can hoe 10 rows in of an hour, he can hoe 1 row in T ! 7 of , or -fa 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 | of an hour, he can hoe 1 row in T ' 7 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 + 16f, or 30 rows, in an hour; 1 row in j\ of an hour ; and 10 rows in if or i of an hour. 10. Since Jesse can hoe 16| or ^ rows in an hour, in i of an hour he can hoe | of 5 j or ^5 gi rows; leaving If rows for Jason, who can hoe 131 or 4 ^ rows in an hour, i of a row in ^ of an hour, and 1| or f rows in 5 times ?\ or | of an hour. 11. Since Jason can hoe 13^ or ^ rows in an hour, in of an hour he can hoe 1 of 4 /, or 4 / 4 t rows 5 leaving 5f rows for Jesse, who can hoe 16|, or *^-=% rows in an hour, of a row in T ]^ of an hour, and ^-, or 5f rows in 50 times , J 3 of an hour. 12. See analysis of Example 9. 13. Since A and B can clear the field in 15 days, they can (151. 152) 178 MISCELLANEOUS EXAMPLES IN THE clear T ' f of it in 1 day, and y 9 ^ or of it in 9 days ; ana since A and B clear f of it in 9 days, C must clear the remaining | ; and if he clear f in 9 days, he will clear in of 9 or 4| days, and 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 f of it in 1 day ; and | or ^ of it, is what B does more in a day than C. As B and C dig it in 9 days, they can dig of it in 1 day, and since B's day's work exceeds C's by J I T of the well, 5*4 or j 5 2=^ of C's days, and of -f^ or T f T =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 C dig -i of it in 1 day, and C digs T T of it in 1 day, T f f or T y4=whatB digs; hence B can dig it in as many days as 11 is contained times in 144 or ISyL days. Since A and B dig of it in 1 day, and B digs T y T of it in 1 day, T y 4 - or y 1 ^ =what A digs in 1 day ; hence A can dig it in as many days as 13 is contained times in 144 or lly 1 ^ days. 15. Since A digs yW, B T W, and C T T of it in 1 day they will all dig T V 3 4 +TT4 ^T4T? or rVf f ^ * n * day ; and it will take as many days as 29 is contained times in 144 01 4|| days. 16. Since Patrick and Peter can dig it in 15 days, they can dig y'j of it in 1 day, and }| or f 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 in of a day, and in 13^ or \ days he can dig 40 times ^ or f of it, and Peter must dig the remaining ; and since ha digs in 4J>- days, he will dig in % of *g- 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 f ; and since he digs | in 15 dayS| he will dig | in A- of 15 or 5 days, and | in 8 times 5 or 40 days. As Patrick can dig 40 rods in 24 days, he can dig (152) PROGRESSIVE INTELLECTUAL ARITHMETIC. 179 gij- of 40 or If rods in 1 day, and since Peter jan dig 40 rods in 40 days, he can dig 1 rod a day, and it will take him as many days as 1| + 1 = 2f is contained times in 28, or 10 days. 17. Since 30 rods is f or of 40 rods, it will take each man ^ as long to dig it. Since Patrick could 'dig it in 40 days, he can dig 30 rods in 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 rods in f of 30 or 2 2i 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 of it in 6| or % days, he can cut in j of 2 / or f days, and f in 9 times f or 15 days. Since Harlan cuts of it in 6 1 or Y days, he can cut in } of \- or f days, and in 9 times | or 12 days. 19. Since the 3d does f as much as the 1st and 2d, the work of the 1st and 2d is divided into 5 parts; and since the 3d does f as much, the whole is divided into 5 + 2 or 7 parts. Since the 3d does & of the whole in 10 days, he can do ^ in % of 10 or 5 days, and ^ 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 as much as the 2d, the whole is divided nto 7 parts, of which the 1st does 3, and the 2d 4 parts. Since the 1st does -f in 14 days, he can do | in ^ of 14 or 4| days, and % in 7 times 4| or 32| days. Since the 2d does | in 14 days, he can do | ii: | of 14 or 3^ days, and % in 7 times 3| or 241 days. 20. See Analysis of example 19. 21. Since the 1st can do it in 32 1 or \ 8 - days, he can do jL of it in i of a day or -/j in a day ; and since the 3d can do it in 35 days, he can do ^ T of it in 1 day ; and both can do tfV + ST or TVo i n 1 d av an d ^ ne whole in as many days as 29 is contained times in 490 or 16/ days. (152) 180 MISCELLANEOUS EXAMPLES IN THE 22. Since the 2d can do it in 24 or ^ days, he can dc T *j of it in i of a day, or fa in a day ; and since the 3d can io Jj of it in 1 day, the 2d and 3d can do fa + Jj, or J^. 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 T 7 T days. 23. Since B and C can do it in 12 days, they can do j% or | of it in 8 days, and A must do the other 1 ; and, since A can do i in 8 days, he can do f in 3 times 8 or 24 days. Since A and B can do it in 10 days, they can do ^ or | of it in 8 days, and C must do the other ; and since C can do in 8 days, he can do f in 5 times 8, or 40 days. Since A can do it in 24 days, he can do if or f^ of it in 10 days, and B must do the remaining T 7 ; and since B can do ^ m 10 days, ^ e can do j\ in I of 10 or If days, and jf in 12 times If or 17 j days. 24. Since the 1st and 2d will discharge it in 8 hours, they discharge f- or \ of it in 4 hours, and the 3d must discharge the other 1 ; arid since it discharges ^ in 4 hours, it will dis- charge | 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 the 1st must discharge the other \ ; and since the 1st discharges \ of it in 6 hours, it will discharge f in 4 times 6 or 24 hours. Since the 1st and 3d discharge it in 6 hours, they will dis- charge f or | of it in 4 hours, and the 2d must discharge the other i ; and since the 2d discharges \ 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 \l or \ of it in 10 days, and C does the other \ ; and since C does \ in 10 days, he can do f in 2 times 10 or 20 days. Since B and C can do it in 15 days, they do j| or f 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 30 days and C in 20 days, they can both do ^V + aV or TI ^ it in 1 day, and |f in 12 times 1 or 12 days. 26. Since it would last them all 30 days, they would eat ^ of it in 1 day, and 20 times ^ or f of it in 20 days, leaving \ (152, 153) PROGRESSIVE INTELLECTUAL ARITHMETIC. 181 of it to be eaten by the sister. Since the brother and servant would eat it in 45 days, they would eat f f or f of it in 30 iays, and the sister must eat the other in 30 days. 101, 2. Since 2 plums was the increase given to 1 playmate, and 9 1 or 8 plums the increase given to all, there were as many playmates as 2 is contained times in 8, whicL 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 of 15, or 5 must be the number. Therefore, etc. 4. Since the difference per yard was 12 cents 8 cents, 01 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 2^ times or f times the number, 15 must be | times the number; of 15 or 3, ; and 6, the number. 6. Since f f, ^ of 4 or 2 must be % and 9 times 2 or 18=|. 7. Since the difference between 5| times and 3| times a number is 2.1 times the number, -^\ of 21, or 1, must be T V, and 10 times 1, or 10, }, or the number. 9. If we let 1 or | represent the whole number of chickens, | times 5 + i- times 3=-^, will represent the whole number of grains, that is, 0^=26. And since 26 is -L- 3 - times the number of chickens, | or the whole number of chickens, was 3 times of 26, or 6. 10. Since 26 is 5 times f + 3 times J, or ~y of the number, V L - 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 -i- of the same num- ber, plus the 3d, which gives 2 times f of the same number, (153, 154) 182 MISCELLANEOUS EXAMPLES IN THE plus tho 4th, which gives once the same number ; we have V of the class equal to 29, and ^ f 29 or 1 is of 9 times 1 or 9. Therefore, etc. 13. Since 4 times 1 of a number, plus 3 times the num- ber, or -y- times the number, is equal to 28 plus 5, or 33, T \ 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 i of his age plus 4, the 2d gives , and the 3d gives less 4, we have the sum -L equal to 50 ; r \ of 50 or 5 is i of his age, and 3 times 5 or 15 is f. Therefore, etc. 15. Since he paid $5 a head for ^ of the flock, the cost =5 times i or of his flock; $4 a head for as many more =4 times i or | of his flock; 83 a head for | of the remainder, or J-r=3 times or \ of his flock ; and $2 a head for the rest, or | = 2 times or J of his flock; and f + f + | + i = V- That is, the number of dollars the flock cost = \ 3 of the num- ber of sheep, and 115 is ^- 3 of 6 times ^ of 115 which is 30 Therefore, etc. 16. Since he received 6 dimes each for i ={ x 6=f for i of the remaining | and 3 more, 4 dimes each r=(| + 3) x4=f + 12 dimes for i of the rest (which is f less 1 on each eight), or i + 1 = (i + l)x3 = f+ * " for the rest (which is f less 1 on each eight, also the 2 of last sale), or f less 4 =(%*) X 2 ^1~1_? " the whole number of dimes is equal to 2J- of the baskets and 7 dimes besides, . . =- 3 g L + 7 " hence $10 or 100 dimes less 7 dimes = 93 dimes^Y* ^ T ol 93 or 3rr|, and 8 eighths, 8 times 3 or 24. Therefore, etc. 18. 6 times a number equals y, 7 times of it plus 5 times | of it equals ^ and ^ less V i or i f ^ which, (154, 155) PROGRESS 13 INTELLECTUAL ARITHMETIC. 183 according to the condition of the question, is 4 ; and 4 is \ of 2 times 4 or 8. Therefore, etc. 19. 5 times the number, or \, left 4 cents, but 5 times 01 it, or y , plus 7 times oi it, or *- was it all of it ; and by the condition of the question *- less \ or equals 4 ; and f is 2 times 4 or 8. Therefore, etc. 20. 4 times a number equals 2 T 8 > 5 times 4 of it equals Y-, and zf less A equals ^ of it, which by the question is 6 ; and 6 is ^ of 7 times 1 of 6, which is 14. Therefore, etc. 21. 2 times a number equals f of it, 5 times of it equals |-, and this plus 2 times i of it which is f equals of it, and f less f, equals f of it, which by the conditions of the question is 60 ; and 60 is f of 2 times of 60 which is 40. Therefore, etc. 22. 2 times f of a number equals f of it, which is 8 more than f ; hence 8 is f or ^ of it, and 2 times 8 or 16 is the whole of it. Therefore etc. 1O2. 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 greatei number. Therefore, etc. 4, Since 37^ is the sum of two numbers whose difference is 5, 37 less 5 or 32, is twice the less number ; of 32 is 16 the less number, which, increased by 5, equals 21 , 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) 184 MISCELLANEOUS EXAMPLES IN THE he Las now ; and 21 less 11, or 10, equals what Horace has 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 quarts equals what Mary has. Therefore, etc. 7. Since 47 is the sum of two numbers, one of which is 6 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 | as much as the other, and both would hold 60 bushels, or 3 times as much as the small one ; i 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 84 more, both would have cost $100, or 4 times the cost of the chain ; 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 115^ dimes equals what Harvey received, and 115^ dimes, plus 22 dimes, or 137| dimes equals what Hiram received; T ' T of 115i 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 ; 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. 185 and $124 less $30.25, or $93. 75, equals the cost of the horse 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 or the whole cost ; hence, 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. J.4. 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 } of itself, or the differ- ence equals f of the cost of the dish ; and f less f , or f of the cost of the dish equals the cost of the cover ; and 24 dimes equals f 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 | of itself plus twice the difference, or the difference equaling T 7 g- of the greater, and |f T 7 g-= T 9 F of the greater equals the less; | +yV ff i IT of 25 pounds, or 1 pound, is T 'g- 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 f of the greater, and both numbers f of the greater ; | of 10 = 2 is 1 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 1 of the difference, equals ft of the cost of the wood-work, the remaining T 2 T must equal - of the difference, and the difference equals T 3 T of the cost of the wood-work ; || less T 3 T equals T 8 T of the cost of iron- ing ; || plus T 8 i or |f times the cost of the wood-work equali (157) 186 MISCELLANEOUS EXAMPLES IN THB $38. r V of $38 or $2, is T ' T , 11 times $2 or $22, is the cost of 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 to | of the difference, and the ribbon f 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 times 4 shillings is 12 shillings, the cost of the skates. 20. Had the harness cost $1 more, both would have cost & 15, and the horse would cost 4 of $35 or $20, and the harness $ 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 wero as many yearlings as $1 (the difference OH 1) is contained times in $8 (the difference on all), which is 8 times ; and 28 less 8 equals 20, the number of old sheep. 3. Had all been first quality, he would have paid $9( -, or $8 more than he did ; and since the difference per barrel *aa (157, 158) PROGRESSIVE ^NTELLECTUAL ARITHMETIC. 187 $0., 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 f of the cost, ^ of $18 or $9, must have been 1 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 is 12, the whole number. 6. Since she 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. | of 1 1 is T \ ; hence 12 is ^ of his flock, and 10 times 12, or 120 equals the number of sheep he had at first. 1 3. Since he paid f of the remainder, $3 must be f of it. i of $3 is $1, and 5 times $1 is $5, the remainder, which by thefirst payment wants $5 of being | of the whole ; $5 plus $5 is $10, i of the whole, and 4 times $10 is $40, the whole Therefore, etc. 14. Since he lent f , $3 plus $5 or $8, must have been A. 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 his clothes, which lacks $10 of being 1 of his wages ; $36 plus 110, or $46 is 1 ; and 2 times $46, or $92 equals his wages. (158, 159) 188 MISCELLANEOUS EXAMPLES IN THE 15. Since in $1 there are 10 dimes, he could be idle as 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, *d was idle 10 days. 1O4. 2. The part standing was divided into 4 equal parts, 3 of which equaled the part broken off ; the sum of both pieces was 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 ; i of 45 is 5 ; 4 times 5 or 20 equals the number Horp ',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 of 63, or 28 cents. Therefore, etc. 6. Since 9 times |=f times the cost of wagon, equals the cost of the horse, both cost 8 plus 9, or y- times the wagon ; fo 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 l=f times the first, both equaled |- 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 50 or 230 miles equals the distance from Bergen to Schenectady; yzd as the distance from Utica to Schenectady is 1| J T S times the distance from Bergen to Utica, the whole distance from Bergen to Schenectady equals Y plus f, or 2 3 ; gV f ^30 tniles is 10 miles, and 15 times 10, or 150 miles equals the (159, 160) PROGRESSIVE INTELLECTUAL ARITHMETIC. 189 distance from Schenectady to Bergen ; and 150 miles plua 60 miles, the distance from Bergen 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 arid the tail ; and as the body was divided into fifths, 2 of which equaled the tail, we have body and tail divided into 7 parts ; | of 14 inches is 2 mches v and 2 times 2 or 4 inches equals the tail. 10. Since the less has 7 parts and the greater 11, both have 18 ; T V of 36 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 milea less, it would equal the / T 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 ; T L of 48 is 4 ; 11 times 4 iniles=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 parts, 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 the cover plus 12 pounds) and the cover weighs 18 pounds, we have the cover, i 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 f , times 2 pounds, or 4 pounds ; and 18 pounds less 4 pounds, i \\ 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 is $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 \ of Abel's, 2 halves would (160, 101) K. P. Q 190 MISCELLANEOUS EXAMPLES IN THE equal twice , or ; then Abel had 4 parts, George 6 partis,, and both 10 parts ; T ' 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 j, } would equal of 4, orf, and f 5 times f or y ; then the black ones were 7 parts, the gray ones 10, and both 17 ; T ' T 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 f equaled f, | would equal of , or -^-, and f, 3 times -j^ or {f ; one number is divided into sixteenths, 15 of which equals the other, and f equal both ; ^ T 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 equals f , would equal i- of f , or /j, and -, 4 times ^_, or T \ ; the value of the contents is 15 parts, of the purse 8 parts, and of both 23 parts ; -fa 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 into 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 f equals f, must equal of f, 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 equaled , 1 would equal -f of |, or , and J-, 4 times i, or |. John's age was divided into fifths, 4 of which equaled Peter's, and both equaled f of John's ; of 36 years 4 years, } of John's ; 5 times 4 years =20 years, John's age ; and 4 timeF 4 years' =16 years, Peter's age. (161, 162) PROGRESSIVE INTELLECTUAL ARITHMETIC. 191 ^5. Since f equaled J T 4 ) f would equal f ; and we have what Tf'ds 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; T \ of 44 bushels=4 bushels, 1 part; and 5 times 4 bushels 20 bushels, what was wanting to fill the bin. 26. Since | of what it exceeded equaled f of what it lacked, | would equal J T 5 - ; 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 ; ^ of 4-4 miles is 2 miles; 7 times 2 miles is 14 miles, the distance it lacked of being 8? miles ; and 83 miles less 14 miles is 69 miles, the dis- tance to Cincinnati. 27. Since f of Avhat 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. 2. Since equal -f + 9, { will equal of -f 4-9, which is $ -f8, , and 4 times f + 3, which is -| + 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 f equal less 4 rods, J- will equal ^ of f less 4 rods, which is f less 2 rods, and f, 3 times f less 2 rods, which is f less 6 rods ; hence what one built equals 6 rods less than $ of what the other built, arid both built -y of the amount the second did, less 6 rods ; 38 rods plus 6 rods, or 44 rods equals u ; J T of 44 rods or 4 rods, is } of 5 times 4 rods or -20 rods, what the second built; and 38 rods less 20 rods, or 18 rods what the first built. 4. Since $ was 4 more than A, | would be 1 more than , (162, 163) 192 MISCELLANEOUS EXAMPLES IN THE and |, 7 more than f ; hence what Richard sheared are divided into 5 parts, Hiram's into 7 parts plus 7 sheep, and both into 12 parts plus Y ; 67 less 7 is 60 ; ^ of 60 = 25, the number Richard sheared ; 67 less 25=42, the number Hiram sheared. 5. Since f of future time equaled f of the past + { hours, | would equal ^ of 1 4- }| hours, which is ^ 4 T 8 5 , and |, 5 times ^+ T 8 j hours, which is | -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-f-f hours, or 24 hours; 24 hours less f hours is 211 hours, and f of 21 i hours is 8 hours, or the past time ; hence it was 8 o'clock A. M. 6. Since f of what his age lacked of being 100 years equaled | of what it exceeded 64 years, + 9 years, } of his age would equal of f + 9 years, which is + 1 year, and f , 8 times J- + 1 year, which is f + 8 years ; hence, what his age lacked of being 100 years equaled 8 years more than f of what it ex- ceeded 64 years, and ty 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, 7 ! T 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 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 | of 4 times 14 or 56. Therefore, etc. 9. The first price plus the second, equal to -f- 3 pounds, equals the third price; 2 times j+3, equal to f of it + 6, equals the whole of it, and 6 pounds must be f of it ; of 6 pounds or 2 pounds is } of it, and 5 times 2 = 10 pounds is the whole of it. Jp~ 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 bush els + 5 bushels, or 3 bushels more than , those 3 bushels must equal the difference (163, 164) PROGRESSIVE INTELLECTUAL ARITHMETIC. 193 between and | of them, or | of them ; and 3 bushels is of 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 i of 60 miles or 20 miles is | of the distance ; and 5 times 20, or 100 miles is the whole distance from Batavia to Corning. 12. Since he took $24 more than ^ of the whole for sheep 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 + of the whole, +$6, + } of the whole + $2, or |{ of the whole + $26, equal to the whole amount; hence T ' T of $26, or $2 is ^ and 24 times $ 2 or $ 48 is wnat ne to k for all. 13. Of ^ that number of which i, of i and \ of 1 of 12 is . i and i of i of 12 is 6, of 6 is 2, and 2 is j of 8 times 2, or 16. Therefore, etc. 14. Since he earned f as much as he had spent, he only lacks | of f =| 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 -, will cost \ of | of an eagle, or $2. 16. Since C is f 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 \ of it plus 6 years, plus \ of it plus 4 years, or f- 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 f of 36, or 24 is C's age. 17. Since C owns \ as much as A, he owns 6 acres more than f as much as B; and we have what A owns, 12 more than \ 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 i 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. (164) 194 MISCELLANEOUS EXAMPLES IN THE 106. 2. Since the son's age is f of the father's, the 22 years tiio 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 18 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 the number before removing ; i 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 ; 52 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. Y. | less | is 2^, which by the condition of the question, is 8 more than | of his age; ? less | is y 1 ^, 6 is y'j of 72. Therefore, etc. 8. Since he received { of his wages for his summer's la- bor, i- as much, or of them in fall +$20, and $20 in spring, $20 + $20 =$40 must be f or 1 of his wages, and 3 time* f>4Q=$120, must be the whole amount. (165) PROGRESSIVE INTELLECTUAL ARITHMETIC. 195 10. Since A sold B f as much as B had, B now has J of what he had ut first, which is of what A has left ; of \ i 7 j i g an d 4 times T 7 , or | what A has now f-f, what A has left ; f f plus the f sold B, gives A ff of B's before the sale; ^ 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 Y4 less 18 acres leaves 56 acres, what A has left. 11. Since f equal , 1 will equal \ of or , and f, 3 times |- or | ; hence as the turkeys equal f of the chickens, 10 must be the remaining { ; 4 times 10 is 40. Therefore, etc. 12. Since f of the price of the coat equaled \ of the price of the suit, i would equal |- of \ or |, and f , 3 times | or f 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 j, and f are 8 times $3 or $24. Therefore, etc. 13. Since f times his brother's equaled f of his, would equal | of or y 1 ^ of his ; , which is 3 times T V, or T 3 7 and the 14 more must be the remaining T 7 ; | 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 1^ times the number + 15, or 15 equal to f of the number ; 1 of 15 or 5 is \ of it ; 2 times 5, or 10 is the less number; and 3 times 10, or 35 is the larger. 15. Since f equal T 8 ^, } will equal \ of T 8 ^, or T 2 j ; and f, 5 times y* 5 or | ; and as the buggy cost | as much as the horse, the difference, $40, must be ^ of the cost of the horse ; 3 times $40 or $120 is the value of the horse ; and | of it, or $60, is the value of the buggy. 16. 5 years since the mother's age was^S times Alice's, ana (166) 1.96 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 as old as her mother. See Ex. 3, in this lesson. 17. Since Hobart has but f of his left, he lost of them to Dwight, which, by the condition of the question, was equal to i of Dwight's ; | must have equaled all of Dwight'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 ; 2f times this difference, or 19, is equal to 3J- times less 2f times=if times the less number; T ^ of 19 or 1 is ^ T ; 24 times 1=24, the less number ; and 24 + 16 = 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 would 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 4i times B's age then ; hence we have A's age now equal to 4i times B's age at first ; J^'g 44 it gJL H It It II Q'g it it tt " ] " " " " " Now if 4 years be added to B's age, f of the sum, or 4} times B's age as first + 3 years, is equal to A's age, and 4 times B's *t 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 ] times 8, or 80 years old. (166, 167) PROGRESSIVE INTELLECTUAL AlilTHMETIC. 197 107. 2. In as many hours as 2 miles, the number he gained in 1 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 gains in running 9, is contained times in 28, the number to be gained; 2 is in 28 14 times, and 14 time? 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 divided into eighths, 1 of which equaled the distance he was ahead of A, A must have run I 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, tho cost of the horse ; and since B paid f 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, the 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 gained after 3 o'clock before they are opposite ; 11 is in 45 4 T *| times, and 4^ times 12 is 49 T 1 T spaces. Therefore it would be 49 T 1 T minutes past 3 o'clock. 9. Since 3 of the hound's leaps equal 6 of the fox's, 1 will equal \ 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 (107, 168) 11)8 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 7 is 280. Therefore, etc. 10. Since the distance the sheep ran was divided into 5 parts, 3 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 equals 3 parts, or the distance betwee thorn. 11. Since the interest at 5 per cent., for 2 years 7 months and 6 days, is -ffa of the principal, the amount will be ||f ; _i_ of $2260 is $20. and 100 times $20 = $2000, the sum at interest ; and since B's money equaled -| of A's, the whole equaled f c^ A's ; i of $2000 is $400 or | of what A had in, which is ] of all, and 5 times $400 or $2000 equals A's ; $400 equaled i- of what B had in, which is | of all, and 8 times $400 or $3200 equals B's. 12. Sincp B's fortune is 1 times A's, i of A's is equal to | of B's, and th^ hiterest on it for 5 years at 6 per cent, would equal ft of it | of $600, or $200, is T V of 10 times $200, or $2000; \ 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's. 13. Since he lost 8 per cent, or -/j of the cost on the sale, he sold for f f of the co.'t ; hence { of his calves arid f O f his sheep cost $25, and 4 times $25, or $100 is the cost of all the calves and f of the sheep ; this exceeds the whole cost by $24, which must equal the cost of the f of the sheep ovei the whole number ; -J- of $24 is $, and 5 times $8 is $40 which would buy 20 sheep at $2 ; $76 less, $40 givep $36 for calves, which would buy 12 calves at f 3 Therefore, etc (168) &(p ^mcriran (Kfruratiarml ROBINSON'S PROGRESSIVE COURSE OF MATHEMATICS. This Series, being the freshest, most complete and scientific course of Mathematical Text-Books published, is more extensively used in the Schools and Educational Institutions of the United States, than any competing series. 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