• ElKiiEY A "^ 1^ LIBRARY unVemity Of iDOaTxoK Limu ^ A 0^ ,V «• * WENTWORTH'S SERIES OF MATHEMATICS. First Steps In Number. Primary Arithmetic. Grammar School Arithmetic. High School Arithmetic. Exercises in Arithmetic. Shorter Course in Algebra. Elements of Algebra. Complete Algebra. College Algebra. Exercises in Algebra. Plane Geometry. Plane and Solid Geometry. Exercises in Geometry. PI. and Sol. Geometry and PI. Trigonometry. Plane Trigonometry and Tables. Plane and Spherical Trigonometry. Surveying. PI. and Sph. Trigonometry, Surveying, and Tables. Trigonometry, Surveying, and Navigation. Trigonometry Formulas. Logarithmic and Trigonometric Tables (Seven). Log. and Trig. Tables (Complete Edition). Analytic Geometry. Special Terms and Oircnlar on Applioation. HIGH SCHOOL ARITHMETIC (Wentwortu & Hill's Practical Arithmetic). BY G. A. WENTWORTH, A.M., POPE880R OF MATHEMATICS IN PHILLIPS EXETER ACADEMT. FOR HIGH SCHOOLS AND ACADEMIES. JOHFS^mLL Qdl & Mechanical Eng^ieer. SAU FRANCISCO, CAL. BOSTON: PUBLISHED BY GINN & COMPANY. 1888. Entered accortiing to Act of CongresB, in the year 1888, by CI. A. Wkntwoutii, iu the oflice of the Librarian of CougresB, at Washingion. Typoqiuphy by J. 8. CusHiNO k Co., Bop-ism. PBKH8WOUK UY UiNN & Cu., Bu8TUN. GIFT \AJ48 eouc LIBRARY PREFACE. THIS edition is intended for teachers, and for them only. The publishers will make every effort to keep the book from pupils ; and teachers are urged to exercise the utmost care not to lose their copies, or to leave them where pupils can have access to them. It is hoped that young teachers will derive great advantage from studying the systematic arrangement of the arithmetical work, for such attention has been paid to this as the limitation of the page would allow. It is also expected that many teachers, who are pressed for time, will find great relief by not being obliged to work out every problem in the Arithmetic. G. A. WENTWORTH. Phillips Exetee Academy, August, 1888. 916 AEITHMETIO, 45. Find the following sums : 231 + 764 ; 341 + 57.8 ; 430.31 + 58.61 ; 512.87 + 36.84 + 12.78 + 711.56 + 415.86. 512.87 36.84 12.78 231 341. 430.31 711.56 764 57.8 58.61 415.86 995 398.8 488.92 1689.91 46. Add 1543.1 to 164.7; to 1728 ; to 402.56 ; to 1897.3; to 475.34 ; to 6897.65. 164.7 1728. 402.56 1897.3 475.34 6897.65 1543.1 1543.1 1543.1 1543.1 1543.1 1543.1 1707.8 3271.1 1945.66 3440.4 2018.44 8440.75 47. Add 1897.3 to 475.34 ; to 6897.65 ; 1 bo 1728 ; to 402.56 ; to 164.7 ; to .5236 ; to 2.71828. 475.34 6897.65 1728. 402.56 164.7 1897.3 1897.3 1897.3 1897.3 2299.86 1897.3 2372.64 8794.95 3625.3 2062. 0.5236 2.71828 1897.3 1897.3 1897.8236 1900.01828 48. Find the following sums : .7854 + 3.1416 + 2.71828 ; .7854 + 3.1416 + 30,103 ; 2.71828 + 402.56+1897.3; 2.7113 + 27.53 + 341.586. 0.7854 0.7854 2.71828 2.7113 3.1416 3.1416 402.56 27.53 2.71828 30,103. 1897.3 2284.57828 341.586 6.64528 30.106.937 371,8273 ARITHMETIC. 49. Add 737.87 to each of the following numbers : 111 2304 ; 222 ; 263 ; 373 ; 262.13 ; 561.2 ; 32.35 ; 604.3. 111. 737.87 848.87 1011. 737.87 1748.87 2304. 737.87 3041.87 222. 737.87 959.87 1011; 263. 737.87 1000.87 373. 737.87 1110.87 262.13 737.87 1000. 561.2 737.87 1299.07 32.35 737.87 770.22 604.3 737.87 1342.17 50. Find the five sums : 230.8 + 223 + 2.63 + 373.8 + 56.123 ; 32.358 + 821.9 + 23.04 + 73.7 ; 202.3031 + 71.575 + 65.813 + .0078 + 7.377; 653.03 + 65.303 + 6.5033; 939.303 + 65.746 + 8.2794 + 681.28. 230.8 223. 2.63 373.8 56.123 886.353 32.358 821.9 23.04 73.7 950.998 202.3031 71.575 65.813 0.0078 7.377 347.0759 653.03 65.303 6.5033 724.8363 939.303 65.746 8.2794 681.28 1694.6084 52. 2.7182818 3.1415927 0.7853982 6.6452727 0.4342945 4.8104774 2.5399772 7.7847491 3.2808693 2.5399772 4.8104774 10.6313239 1.6093295 15.4323487 3.785 20.8266782 0.3047973 0.3010300 0.6213768 1.2272041 0.3937043 0.3819660 0.4342945 1.2099648 53. 0.3010300 0.6180340 0.3819660 1.30103 0.6180340 2.2360680 1.7320508 4.5861528 0.3819660 1.7320508 1.4142136 3.5282304 2.2360680 1.4142136 15.4323487 19.0826303 15.4323487 0.8450980 0^ 16.5414467 3.785 0.6213768 1.6093295 6.0157003 TEACHERS EDITION. 0.6213768 1.6093295 0.3047973 3.2808693 3.2808693 0.3937043 0.3047973 0.3937043 2.5399772 4.2070434 5.2839031 3.2384788 0.3937043 0.3047973 2.7182818 3.2808693 0.4342945 3.1415927 4.8104774 0.5235988 0.5235988 8.485051 1.2626906 6.3834733 54. 2.7182818 3.1415927 0.7853982 3.1415927 0.5235988 0.4342945 0.7853982 4.8104774 4.8104774 0.5235988 2.5399772 0.3937043 0.4342945 0.3937043 11.4093504 0.3047973 7.603166 6.7286717 0.3937043 1.6093295 1.6093295 3.2808693 0.3047973 0.6213768 1.6093295 0.3937043 3.785 0.4342945 0.5235988 0.264 4.8104774 0.4342945 15.4323487 10.528675 3.2657244 21.712055 0.6213768 3.785 15.4323487 0.264 0.264 1.4142136 15.4323487 15.4323487 2.2360680 0.3937043 1.4142136 0.3819660 3.2808693 1.7320508 22.6276131 0.6180340 19.9922991 20.0826303 65. 0.4771213 2.7182818 0.3010300 0.2908882 3.1415927 0.6180340 1.6093295 0.7853982 0.3819660 0.8450980 0.4342945 2.2360680 0.3819660 4.8104774 1.7320508 0.6180340 2.5399772 1.4142136 0.3010300 0.3937043 14.8237261 15.4323478 4.523467 22.1157111 1 AiMinmriiiu. 1.6093295 0.3937043 3.785 0.6213768 0.3047973 15.4323487 3.785 3.2808693 0.6213768 0.264 1.6093295 1.4142136 15.4323478 0.6213768 3.2808693 1.4142136 3.785 0.3047973 1.7320508 0.264 4.8104774 24.8583194 10.2590772 29.6490831 57. Add by doul ble columns -. 45.68 154.31 73.86 73.91 296.85 453.71 78.54 736.48 137.64 534.69 345.19 98.87 134.70 782.34 643.48 581.43 78.43 462.71 1448.95 2393.60 1870.27 498.50 65.42 621.65 17.37 638.34 167.32 684.29 763.43 856.96 231.56 809.31 718.83 210.10 798.83 501.49 671.54 835.78 315.72 643.53 356.47 768.44 2956.89 4267.58 3950.41 791.52 32.54 763.80 504.83 254.63 78.23 879.26 63.27 345.61 243.97 131.56 26.73 732.86 506.72 489.56 47.95 283.54 812.36 856.43 345.83 607.28 497.65 643.46 219.07 541.26 708.91 68.72 616.72 463.73 216.78 857.94 67.74 436.74 6570.39 3501.93 4064.96 TEACHERS EDITION. 69. 8-3-2=3; (8 -3) -2 = 5-2 = 3; 8 -(3 - 2) = 8-l = 7. 8-2-3 = 3; (8-2)-3 = 6-3 = 3; 8-(2-3) = 8-(-l) = 9. 18-(7-3)=18-4 = 14; 18 - (3 - 7) = 18 -(-4) = 22 ; 18 - 3 + 7 = 22. 70. The following questions will illustrate the meaning of minus numbers : Starting 90 miles south of Chicago, I go 50 miles due north ; and the next day 80 miles, still north. How far from Chicago am I now? - 90 + 50 + 80 = - 90 + 130 = 40, the number of miles north. Ans. With only $67 I undertake to pay three bills, of |47, of $13, and of $11. Can I pay the bills ? How much shall I lack ? 67 - (47 + 13 + 11) = 67 - 71 = - 4. I shall lack $ 4. Ans. 73. Subtract 123 from each of the numbers: 234, 343, 424, 555, 234 343 424 555 676 725 123 123 123 123 123 123 111 220 301 432 553 602 839 999 1000 10101 5120 123 123 123 123 123 716 876 877 9978 4997 74. Subtract 456 from each of the numbers : 789, 879, 978, 6378, 6855, 6853, 7797, 7006, 3542, 4334, 9790, 3455. 789 879 978 6378 6855 6853 456 456 456 456 456 456 333 423 522 5922 6399 6397 7797 7006 3542 4334 9790 3455 456 456 456 456 456 456 7341 6550 3086 3878 9334 2999 75. What is the difference between 779 and 974 ? 368 and 249 ? 479 and 2301 ? 2731 and 929 ? 708 and 394 ? 1 123 and 1072 ? 891 and 773 ? 8103 and 5621 ? 19,001 and 3456 ? 792 and 2180? (5 ARITHMETIC. 974 368 2301 2731 708 779 249 479 929 394 195 119 1822 1802 314 1123 891 8103 19001 2180 1072 773 5621 3456 792 51 118 2482 15545 1388 76. Subtract: $183.45 $716.43 $647.51 $270.04 76.47 628.74 549.64 128.31 $106.98 $87.69 $97.87 $141.73 $125. $247.93 $641.87 $56.27 101.50 129.47 333.95 29.89 $23.50 $118.46 $307.92 $26.38 77. Subtract from 7854 each of the numbers : 788, 879, 567, 5006, 6107, 578, 867, 894, 463, 4603. 78. 7854 788 7854 879 7854 567 7854 5006 7854 6107 7066 6975 7287 2848 1747 7854 578 7276 7854 867 6987 7854 894 6960 7854 463 7391 7854 4603 3251 3.1415927 2.7182818 0.7853982 0.5235988 4.8104774 0.4342045 0.4233109 0.2617994 4.3761829 2.5399772 0.3937043 0.3937043 0.3047973 3.2808693 0.3047973 2.1462729 0.088907 2.976072 3.2808693 1.6093295 3.785 0.6213768 15.4323487 0.264 1.6715398 3.1636232 15.1683487 TEACHERS EDITION. 79. 80. 1.7320508 1.4142136 0.3178372 2.2360680' 1.7320508 0.5040172 3.1415927 0.7853982 2.3561945 0.7853982 1.5707963 0.7853982 0.7853981 0.3819660 0.6180340 2.2360680 0.3819660 1.854102 2.2360680 0.6180340 1.618034 3.1415927 0.5235988 2.6179939 0.5235988 2.0943951 0.5235988 1.5707963 1.4142136 0.6180340 0.7961796 O.JioUolO 0.3010300 0.317004 0.3819660 0.3010300 0.080936 1.5707963 0.5235988 1.0471975 0.5235988 0.5235987 0.6180340 0.3819660 1. 81. In a school of 83 pupils, 37 are girls ; the rest, boys. How many boys are there ? 83 — 37 = 46. Ans. 82. Take 1787 from 21,205, and what is the remainder ? 21,205-1787=19,418. Ans. 83. Into a bowl containing 338 fine shot I poured a handful more, and the bowl then contained 720. How many did I pour in ? 720-338 = 382. Ans. 84. From a box containing 209 oranges I took a basketful, and left 163 oranges. How many did I take in the basket ? 209-163 = 46. Ans. 85. The minuend being 1718.754, and the subtrahend 1389.328, what is the remainder ? 1718.754 - 1389.328 = 329.426. Ans. 86. If the minuend was 6532.18, and the remainder 1916.47, what was the subtrahend ? 6532.18 - 1916.47 = 4615.71. Ans. 87. How many must be taken from 729,434 in order to leave 613,488 ? 729,434 - 613,488 = 115,946. Ans, ARITHMETIC. 88. How many must be taken from 1,000,000 to leave 817,259 ? 1,000,000-817,259 = 182,741. Ans. 89. Subtract 4187.94 from 8010.101. 8010.101-4187.94 = 3822.161. Ans. 90. Find the difference between 8,765,420 and 9,873,210. 9,873,210 - 8,765,420 = 1,107,790. Ans. 91. In a till are if 391 in bills, $67.50 in gold, $39.75 in silver, and $2.77 in copper and nickel. How much money is in the till ? $391 + $67.50 + $39.75 + $2.77 = $501.02. Ans. 92. Starting out with $315.75 in one wallet and $54.37 in another, I pay the grocer $127.38; the butcher, $64.17; the shoemaker, $21.40 ; the landlord, $50; the tailor, $35. What ought I to have left? $127.38 $315.75 64.17 54.37 ^^■"^^ $370.12 ^^- 297.95 35. — — — ; $72.17 Ans. $297.95 93. On a bill of $753.43, I pay $517.87. How much do I still owe ? If I owe $817.87, and have but $637.50, how much do I lack of being able to pay ? $753.43 $817.87 517.87 637.50 $235.56 $180.37 94. If a man was born January 1, 1812, how old was he January 1, 1878 ? How old December 31, 1857? 1878 1 1 1857 12 31 1812 1 1 1812 1 1 66 45 11 30 95. America wa.s discovered in 1492. How many years after its discovery was each of the following events? Settlement of Florida, 1565 ; of Virginia, 1607 ; of Massachusetts, 1620; of Quebec, 1608; French and Indian War, 1756; Declaration TEACHERS EDITION. of Independence, 1776 ; inauguration of Washington, 1789 ; war with England, 1812 ; Mexican War, 1846 ; Civil War, 1861. 1565 1607 1620 1608 1756 1492 1492 1492 1492 1492 73 115 128 116 264 1789 1812 1846 1861 1492 1492 1492 1492 284 297 320 354 369 96. How many days in common years, and in leap-years, between January 1 and March 1 ? January 4 and April 4 ? February 5 and May 5 ? February 7 and October 7 ? January 4 and July 4 ? March 4 and July 4 ? Between January 1 and March 1, 58 days Between January 4 and April 4, 89 days Between February 5 and May 5, 88 days Between February 7 and October 7, 241 days Between January 4 and July 4, 180 days Between March 4 and July 4, 121 days 59 in a leap-year, 90 in a leap-year. 89 in a leap-year. 242 in a leap-year. 181 in a leap-year. 121 in a leap-year. 97. The sum of two numbers is 3 ; their difference, 1. What are the numbers ? The sum of two numbers is 5 ; their difference, 1. Eequired the numbers. What two numbers added together make 8, if the difference of the numbers is 2 ? If the difference is ? if 4 ? if6? (3-fl)^2 = 2) (5-hl)-^2 = 3| (8 + 2) ^2 = 5 (3 - 1) -- 2 = 1 r (5 - 1) - 2 = 2 ) (8 - 2) -- 2 = 3 (8-f0)--2 = 4| (8 +4) ^2 = 6) (8 + 6) -2 = 7 (8 - 0) -f- 2 = 4 i (8 - 4) -^ 2 = 2 i (8 - 6) ^ 2 = 1 98. If the minuend is 9874, and remainder 3185, what is the sub- trahend ? The subtrahend being 7659, and remainder 675.68, what is the minuend ? 9874 minuend. 7659. subtrahend. 3185 remainder. 675.68 remainder. • 6689 subtrahend. 8334.68 minuend. 10 ARITHMETIC. 99. The smaller of two numbers is 7.957.64328 ; their diflference is .00087692. What is the larger number ? 7.95764328 + 0.00087692 = 7.9585202. Ans. 100. The larger of two numbers is 7.95764328, and their difference is 7.153485. What is the smaller number ? 7.95764328 - 7.153485 = 0.80415828. Am. 101. A hired man pumps out of my cistern in one hour 243.75 gallons ; in the next hour, 227.5 gallons ; in 45 minutes more, an additional 137.75 gallons ; and the cistern is empty. How much was in it ? 243.75 + 227.5 + 137.75 = 609. ^ g^,^ ^^ 102. From what number must I subtract 5 to leave 7 ? 8 to leave 9? From what number must I subtract 5.1736 to leave 8.1964? 6.231 to leave 9.6648 ? 74.213 to leave 25.787 ? 5 8 5.1736 6.231 74.213 7 9 8.1964 9.6648 25.787 12 17 13.37 15.8958 100. 103. What must be subtracted from 1 to leave .5 ? to leave .53 ? to leave .532 ? to leave .5236 ? to leave .5235988 ? 1. 1. 1. 1. 1. 05 053 0532 05236 0.5235988 05 047 0.468 04764 04764012 104. I start on a journey of 3433 miles. The first day I make 428 miles ; the second day, 511 miles ; the third, 497 miles ; the fourth, 513. How many miles of my journey remained for me at the close of each day ? How many miles had I gone at the close of each day ? 3433 428 3005 after first day. 428 end of first day. 511 511 2494 after second day. 497 1997 after third day. 513 939 end of second day. 497 1436 end of third day. 513 1484 after fourth day. 1949 end of fourth day. teachers' edition. 11 105. Subtract 76,343 from the sum of 61,932, 51,387, 5193, 4674, and 8199 ; then subtract 23,657 from the remainder. 61,932 + 51,387 + 5193 + 4674 + 8199 = 131,385. 131,385 - 76,343 = 55,042 ; 55,042 - 23,657 = 31,385. Ans. 106. J. bought a farm and stock for 1 7633.90 ; sold off the stock for 1305.75 ; then sold the farm for $ 7325. What did he lose ? 1 305.75 + 1 7325 = 1 7630.75 ; $7633.90 -1 7630.75 = $3.15. Ans. 107. If I gave |4375 for my land, and paid for house, barn, sheds, and fences, 1 2789. 50; also $973.75 for horses, cattle, tools, etc.; what did my farm and stock cost ? If I sold part of the land for $675, and some cattle, etc., for $ 217.50, what may I estimate as the cost of what I have left ? $4375 + $ 2789.50 + $973.75 = $8138.25. $675 + $217.50 = $892.50. $ 8138.25 - $ 892.50 = $ 7245.75. Ans. 108. Alfred the Great died at the age of 52, a.d. 901. In what year was he born ? William the Conqueror began to reign a.d. 1066, and reigned 21 years. In what year did he die ? Socrates was born B.C. 469, and died at the age of 70. In what year did he die ? Plato was born B.C. 429, and died at the age of 82. In what year did he die ? Demosthenes died at the age of 60, B.C. 322. In what year was he born ? The battle of Marathon was fought B.C. 490 ; 560 years later Jerusalem was destroyed by Titus. In what year was Jeru- salem destroyed ? 901 1066 469 429 322 560 52 21 70 82 60 490 A.D. 849 A.D. 1087 B.C. 399 b.c. 347 b.c. 382 a.d. 70 109. John has 158 cents, James has 271 cents ; James gives John 56 cents. Which has more than the other, and how many more ? 158 271 215 _56 56 214 214 John's cents. 215 James's cents. 1 James's excess. 12 ARITHMETIC. 116. Multiply 111 by 5 ; 123 by 3 ; 231 by 2 ; 114 by 3 ; 421 by 4 ; 512 by 5 ; 4328 by 4 ; 1187 by 6 ; 1782 by 8 ; 8.287 by 7 ; 9.6198 by 3 ; 62.818 by 7 ; 9.2758 by 8 ; 52.134 by 9. Ill 123 231 114 421 5 3 2 3 4 555 369 462 342 1684 512 5 2560 4328 4 17312 1187 6 7122 1782 8 14256 8.287 7 58.009 9.6198 3 62.818 7 9.2758 8 52.134 9 28.8594 439.726 74.2064 469.206 117. Multiply 0.5235988 by 6; 0.7853982 by 4 ; 3.14159265 by 5, and the product by 5. 3.14159265 5 0.5235988 3.1415928 0.7853982 4 3.1415928 15.70796325 5 78.53981625 118. Multiply 3.1416 by 11 ; by 12 ; by 10 and by 3, and add the two results ; by 10 and by 4, and add the results ; by 9 and by 6, and add the results. Multiply 2.236068 by 11 ; by 6 and by 7, and add the results ; by 8 and by 9, and add the results ; by 10 and by 7, and add the results (compare the sum of these two products with the sum of the last two product*) ; by 10 and by 8, and add the results ; by 12 and by 7, and add the results. 3.1416 11 34.5576 3.1416 12 37.6992 3.1416 10 31.4160 3.1416 3 9.4248 31.4160 9.4218 40.8408 3.1416 10 31.4160 3.1416 4 12.5664 31.4160 12.5664 43.9824 3.1416 9 28.2744 3.1416 6 18.8496 TEACHERS EDITION. 13 28.2744 18.8496 2.236068 11 2.236068 6 2.236068 7 15.652476 13.416408 15.652476 47.1240 24.596748 13.416408 29.068884 2.236068 8 2.236068 9 17.888544 20.124612 2.236068 10 2.236068 7 17.888544 20.124612 38.013156 22.360680 15.652476 22.360680 15.652476 2.236068 10 2.236068 8 22.360680 17.888544 40.249224 2.236068 12 38.013156 22.360680 17.888544 26.832816 2.236068 7 26.832816 15.652476 15.652476 42.485292 120. How much is 10 times 3.14159265? 100 times? a million times? What will 10 barrels of apples cost, at |3.75 a barrel? at $ 2.17 ? at $ 5.875 ? How much will 100 barrels cost at each of these prices, and at $3,375 ? at ? 5.125? 10 X 3.14159265 = 31.4159265 ; 100 X 3.14159265 = 314.159265 ; 1,000,000 X 3.14159265 = 3,141,592.65 ; 10 x $3.75 = $37.50 ; 10 X $2.17 - $21.70 ; 10 X $5,875 = $58.75 ; 100 X $3.75 = $375 ; 100 X $2.17 = $217 ; 100 X $5,875 = $587.50 ; 100 x$ 3.375 = $337.50; 100 x $5,125 = $512.50. 122. What is a tenth of 2.36 ? a hundredth of 2.36 ? a thousandth of 0.63. Write the second members of the following equations, and then read them : 0.01 X 7.8 = 0.1 X 0.065 = 0.001 X 4.31 0.01 X 0.012 0.0001 X 23.31 = 0.1 X 2.36 = 0.236 ; 0.01 x 2.36 = 0.0236 ; 0.001 x 0.63 = 0.00063 ; 0.01 X 7.8 = 0.078 ; 0.001 x 4.31 = 0.00431 ; 0.0001 x 23.31 = 0.002331 0.1 x 0.065 = 0.0065 ; 0.01 x 0.012 = 0.00012. 14 ARITHMETIC. 123. Find the cost of 30 barrels of flour, at |3.27 a barrel ; of 70 barrels, at $4.58; of 90 barrels, at $6.76; of 100 barrels, at |7.84; of 120 barrels, at 1 8.57. BOX $3.27 = $98.10; 70 X $4.58 = $320.60 ; 90 X $6.76 = $608.40; 100 x$ 7.84 = $784; 120 x $ 8.57 = $ 1028.40. 124. Find the cost of 0.03 of a barrel of oil, at $27,875 a barrel ; of 0.7; of 0.009; of 0.17 ; of 0.019; of 0.13; of 0.8 ; of 0.83 ; of 0.014 of a barrel ? 0.03 X $27,875 = $0.83625 ; 0.7 X $27,875 = $19.5125 ; 0.009 X $27,875 = $0.250875 ; 0.17 X $27,875 = $4.73875; 0.019 X $27,875 = $0.529625 ; 0.13 x $27,875 = $3.62375 ; 0.8 X $27,875 = $22.30 ; 0.83 X $27,875 = $23.13625; 0.014 X $27,875 = $0.39025. 126. What is the numerical value of the expressions : 30x8.75? 700x7.81? 300x7.85? 0.07 X 6.975 ? 8000 x 65.432 ? 0.0009 x 10356.78 ? 30 X 8.75 = 262.5 ; 700 x 7.81 - 5467 ; 300 x 7.85 = 2355 ; 0.07 X 6.975 = 0.48825 ; 8000 x 65.432 = 523,456 ; 0.0009 X 10356.78 = 9.321102. 129. Multiply 0.785398 by each of the following numbers: 2; 20; 3 ; 300 ; 5 ; 0.5 ; 0.005 ; 737 ; 7.37 ; 856 ; 85.6 ; 0.0856 ; 10 ; 1001 ; 1.001 ; 954 ; 0.00954. 0.785398 2 0.785398 20 0.785398 3 0.785398 300 1.570796 0.785398 5 15.707960 0.785398 0.5 0.3926990 -5.78838326. 2.356194 0.785398 0.005 235.619400 0.785398 737 3.926990 0.785398 X 7.37 = 0.003926990 5497786 2356194 5497786 578.838326 TEACHERS EDITION. 15 0.785398 856 4712388 3926990 0.785398 X 85.6 = 67.2300688. 0.785398 X 0.0856 = 0.0672300688. 6283184 0.785398 1001 0.785398 672.300688 954 785398 3141592 785398 3926990 786.183398 7068582 749.269692 0.785398 X 1.001 = 0.786183398. 0.785398 X 0.00954 = 0.00749269692. 0.785398 10 7.853980 130. Multiply 2150.42 by 0.1 ; by 0.001 ; by 0.75 ; by 0.075 ; by 0.083. 50.42 0.1 2150.42 0.001 2150.42 0.75 1075210 1505294 1612.8150 2150.42 0.075 1075210 1505294 161.28150 • 2150.42 0.083 5.042 2.15042 645126 1720336 178.48486 131. Multiply 1 .4142136 by 0.7 ; by 0.707 ; by 0.7071 ; by 0.707107. Multiply 1.41421 by 1.4; by 1.4142; by 1.41422. Multiply 1.732 by 1.732 ; 2.23607 by 2.236 ; 0.618 by 618 ; 0.618034 by 0.618035. Subtract this last product from 1. 1.4142136 1.4142136 1.4142136 1.4142136 0.7 0.707 0.7071 0.707107 0.98994952 98994952 14142136 98994952 98994952 98994952 98994952 14142136 0.9998490152 98994952 0.99999043656 98994952 1.0000003360552 16 AEITHMETIC. 1.41421 1.41421 1.41421 1.732 1.4 1.4142 1.41422 1.732 565684 282842 282842 3464 141421 565684 282842 5196 1.979894 141421 565684 565684 141421 12124 1732 141421 1.999975782 565684 141421 2.999824 2.0000040662 2.23607 0.618 0.618034 1. 2.236 618 4944 0.618035 0.38196664319 1341642 3090170 0.61803335681 670821 618 1854102 447214 3708 4944272 447214 4.99985252 381.924 618034 3708204 0.381966643190 133. Find the value of the expressions : 88 X 718.54 ; 96 x 6.8193 ; 6.3 X 71.569 ; 1.32 x 234.769. 718.54 6.8193 71.569 234.769 11 12 0.9 0.12 7903.94 8 63231.52 81.8316 654.6528 64.4121 7 450.8847 28.17228 11 309.89508 134. Multiply 291.47 by 16, and the product by 625. In like manner, find the continued products • 8 X 125 X 278.56 ; 8 X 3.75 X 3.33333 ; 8 X 625 X 1.5708. 8 X 125 X 278.56 = 1000 X 278.56 «= 278,560. 8 X 3.75 X 3.33333 = 30 X 3.33333 - 99.9999. 8 X 625 X 1.5708 - 5000 x 1.5708 -7864. 291.47 16. 174882 29147 4663.52 625 2331760 932704 2798112 2914700.00 TEACHERS EDITION. 17 135. One mile measures 5280 feet. How many feet in 3 tenths of a mile ? in 0.7 ? in 0.17 ? in 0.573 ? in 0.846 of a mile ? 0.3 X 5280 = 1584 ; 0.7 X 5280 = 3696 ; 0.17 X 5280 = 897.6 ; 0.573 X 5280 = 3025.44 ; 0.846 x 5280 = 4466.88. 138. Multiply (using complements) 0.7854 by 9.9 ; by 0.99 ; by 0.099. Multiply 0.5236 by 99.7 ; by 9.989 ; by 9.87. Multiply 8537 by 0.0097 ; by 0.9995. 0.7854 X 10 = 7.854 0.7854 X 1 = 0.7854 0.7854 X 0.1 = 0.07854 0.7854 X 0.01 0.7854 X 0.99 = 0.007854 0.7854 X 9.9 = 7.77546 = 0.777546 0.7854 X 0.1 = 0.07854 0.5236 X 100 = 52.36 0.7854 X 0.001 = 0.0007854 0.5236 X 0.3 0.5236 X 99.7 = 0.15708 0.7854 X 0.099 = 0.0777546 = 52.20292 0.5236 X 10 = 5.236 0.5236 X 10 = 5.236 0.5236 X 0.11 = 0.0057596 0.5236x0.13 0.5236 X 9.87 = 0.068068 0.5236 X 9.989 = 5.2302404 = 5.167932 8537 XO.OI = 85.37 8537 Xl = 8537 8537 X 0.0003 = 2.5611 = 82.8089 8537 X 0.0005 8537 X 0.9995 = 4.2685 8537 X 0.0097 .-= 8532.7315 139. Multiply 0.61803 by 147 ; by 373 ; by 7.56 ; by 8.93 ; by 9.93. Multiply 0.5236 by 5.99 ; by 7.99 ; by 8.997 ; by 699.98. 0.61803 0.61803 0.61803 0.61803 0.61803 147 373 185409 7.56 8.93 9.93 432621 370818 185409 185409 247212 432621 309015 556227 556227 61803 185409 432621 494424 556227 90.85041 230.52519 4.6723068 5.5190079 6.1370379 18 ARITHMETia 0.5236 5.99 47124 47124 26180 3.136364 0.5236 7.99 47124 47124 36652 4.183564 0.5236 8.997 699.98 0.5236 140. Multiply 0.7854 by 0.618 ; Multiply 2.718 by 0.618 ; by 0.382 0.7854 0.7854 0.618 0.382 36652 47124 47124 41888 419988 209994 139996 349990 4.7108292 366.509528 by 0.382 ; by 0.7854 ; by 0.302. by 0.7854 ; by 0.607. 0.7854 0.7854 0.7854 0.302 62832 7854 47124 0.4853772 2.718 0.618 21744 2718 16308 1.679724 15708 62832 23562 0.3000228 2.718 0.382 5436 21744 8154 1.038276 31416 39270 62832 54978 0.61685316 2.718 0.7854 10872 13590 21744 19026 2.1347172 15708 23562 0.2371908 2.718 0.607 19026 16308 1.649826 141. Find the continued products : 0.477 X 101 X 0.708 ; 15.43 X 0.4343 X 3 ; 4 X 0.175 X 3.28 ; 0.615 x 0.771 X 10 ; 3.2809 x 5 x 0.71 ; 0.785 X 0.7 X 0.202 ; 0.471 X 0.807 X 22 ; 3.28 x 25 x 0.909. 15.43 0.175 0.615 0.477 101 0.4343 0.771 477 4629 0.7^ 615 477 6172 3.28 4305 48.177 4629 2.296 4305 0.708 6172 0.474165 385416 6.701249 3 10 337239 4.74165 34.109316 20.103747 TEACHERS EDITION. .362134 19 3.2809 5 0.785 0.7 0.5495 0.202 0.471 0.807 3297 3768 0.380097 22 3.28 25 16.4045 0.71 1640 656 164045 1148315 10990 10990 82.PP 0.909 11.647195 0.1109990 760194 760194 738 738 74.538 144. Find the product, to the fifth fractional place, of 3.14159265 by 2.236. Find 1414.2136 x M142.136, to the second place ; 0.618034 by 0.618034, to the sixth place ; 2.236068 by 2236.068, to the third place; 1.73205 by 1732.0508, to the second glace. 3.14159265 1414.2136000 0.6180340 6322 63124141 14142136000 4308160 6283185 3708204 628318 5656854400 61803 94248 141421360 49442 18849 56568544 2828427 141421 42426 8485 185 7.024600 24 0.3819658 0.381966. Ans. 20000001.063 2.2360680 1.732050 8006322 80502371 44721360 1732050 4472136 1212435 670820 51962 134163 3464 1341 87 178 1 4999.9998 2999.999 5000. Ans. 3000. Ans. 20 ARITHMETIC. 147. 1. What will a man earn in a year if he has $2 a day, omit- ting Sundays ? Suppose that the year begins on Sunday ? Suppose the year to be leap-year, and not begin on Sunday ? Suppose it leap- year, and to begin on Saturday ? (365 - 52) X $2 = $626 ; (365 - 53) x $2 = $624 ; (366 - 52) X $2 = $628 ; (366 - 53) x $2 = $626. 2. If a field of corn averages 2 ears to a stalk, how many ears on 673 stalks ? 673 2 real multiplicand. 1346 1346 ears. Ana. 3. At 27 bushels an acre, how much wheat to the square mile of 640 acres, deducting 47 acres for roads and waste land ? 640 _47 593 27 real multiplicand. 4151 1186 16011 16,011 bn. Ana. 4. How much money would be required to give $ 7000 to each of 7568 men? 7568 7000 real multiplicand. 5297G000 $52,976,000. Am. 5. In a certain book of 378 pages, the words average 7 letters to a word, and 10 words to a line. There are, on an average, 29 lines to a page. How many letters in the book ? 378 29 real multiplicand. 3402 756 10962 10 real multiplicand. 109620 7 real multiplicand. 767340 767.340 letters. Ans. teachers' edition. 21 6. How many bushels of wheat in a township of 37 square miles, if we deduct 47 acres to the square mile for roads and waste, and suppose that half the remainder is in wheat averaging 23 bushels to an acre ? 47 640 37 37 329 4480 141 1920 1739 23080 1739 21941 0.5 10970.5 23 real multiplicand. 329115 219410 252321.5 252,321.5 bu. Ans. 7. If 5700 persons, each paying 1 cent toll, and 324 carriages, each paying 5 cents toll, pass over a bridge in a day, how much money will be received ? 5700 324 0.01 real multiplicand. 0.05 real multiplicand. 57.00 16.20 57 73.20 $73.20. Ans. 8. A merchant bought 960 pounds of cheese at 7 cents a pound, and 147 pounds of butter at 20 cents. He gave in payment 12.5 yards of cloth at 1 dollar a yard, 2 barrels of sugar, each weighing 226 pounds, at 9 cents a pound, and the remainder in cash. How much money had he to pay ? 22 ARITHMETIC. 960 226 0.07 real multiplicand. 2 real multiplicand. 67.20 452 0.09 real multiplicand. 40.68 12.50 147 53.18 0.20 real multiplicand. 29.40 96.60 67.20 53.18 96.60 43.42 $43.42. Ans. 148. 1. Express the product of 7^ X 7^ 8^ x 8 ; 2^ x 2 ; 5* x 5^ 75x73^78. 82x8 = 83; 2«x2 = 29; 5*x52 = 5«. 2. Express the product of : 3.0Px 3.01 ; 0.672x 0.678 ; 0.208x0.208'. 3.0P X 3.01 = 3.013 ; 0.67^ x 0.678 = 0.67i«> ; 0.208 X 0.208' = 0.208*. 3. Express the product of : 2.0032x2.003*; 20.033x20.03; 20.03x20.032. 2.0032 X 2.003* = 2.003« ; 20.03' x 20.03 = 20.03* ; 20.03 X 20.032 = 20.03'. 153. Divide 963 by 3 ; 846 by 2 ; 846 by 3 ; 846 by 6 ; 848 by 4 ; 52.05 by 5 ; 84.028 by 7 ; 13.31 by 11 ; 1.728 by 12. 3)963 2)846 * 3 )846 321 423 282 6 )846 4 )848 5)52.05 141 212 10.41 7 )84.028 11 )13.31 12 )1.728 12.004 1.21 0.144 158. 1. Divide 0.003 by 0.07 ; 0.003 by 110 ; 110 by 0.003. 7 )0.30000 11 )0.000300 SJUOOOOOOOOO 0.04286 0.000027 36666.66667 2. Divide 0.07 by 0.003 ; 110 by 0.07 ; 1.3 by 0.07. 3) 70.00000 7 )11000.()00(X ) 7 ) 130.00000 23733333 1571.42857 18.57143 teachers' edition. 23 3. Divide 1.7 by 0.07 ; 0.07 by 110 ; 1.3 by 110. 7 )170.0000 11 )0.00700 11 )0.13000 24.28571 0.00064 0.01182 4. Divide 1.7 by 110 ; 0.07 by 1.2 ; 0.003 by 1.2. 11 )0.17000 12 )0.70000 12 )0.0300 0.01545 0.05833 0.0025 5. Divide 110 by 1.2 ; 1.7 by 1.2; 17 by 1.2. 12 )1100.'00000 1 2)17.00000 12 )170.00000 91.66667 1.41667 14.16667 " 6. Divide 136 by 0.06 ; 136 by 0.12 ; 136 by 1100. 6 )13600.00000 12 )13600.00000 11 )1.36000 2266.66667 1133.33333 0.12364 7. Divide 256 by 0.8 ; 2.56 by 0.08 ; 0.0256 by 0.008. 8 ) 2560 8 )256 8 )25.6 320 32 3.2 8. Divide 256 by 8000 ; 1.06 by 0.9 ; 1.06 by 9000. 8 )0.256 9 )10.60000 9 )0.00106 0.032 1.17778 0.00012 160. 1. Divide 1.6093295 by 0.479 ; by 0.917 ; by 0.017 ; by 0.0087. 479) 3.35977 1609.3295 1437 917) 1.75499 1609.3295 917 6923 6419 94.66644 17)1609.3295 153 79 68 113 102 112 102 109 102 75 68 87) 184.98040 16093.2950 87 1723 1437 2862 2395 739 696 5042 4585 433 348 4679 4311 4579 3668 852 783 3685 3353 332 9115 8253 862 699 696 350 34« 24 ARITHMETIC. 2. Divide 3 by 1.7; by 1.73; by 1.732; by 1.7321. 1.76471 1.73410 17)30.0000 173) 300.0000 17 173 130 1270 119 1211 110 590 102 519. 80 710 68 692 120 180 119 173 1 7 1.73210 173200 1732)3000.0000 17321)30000.0000 1732 17321 12680 12124 5560 5196 3640 3464 1760 1732 126790 121247 55430 51963 34670 34642 280 28 3. Divide 1.60932nr) l,y r.i>so, and the quotient by 12. 0.00o;U)l.S 0.0O()0254 528)0.16093295 12)0.0003048" 1584 2532 2112 4209 TEACHERS EDITION. 25 4. Divide 2 by 1.4142 ; 5 by 1.41423 2.236. 2236) 2.23614 14142) 20000.00000 14142 5000.0000 4472 58580 56568 5280 4472 8080 6708 13720 13416 20120 14142 59780 56568 32120 28284 3040 2236 3836 804 165. Perform the work in the following questions by the use of reciprocals : 1. 8x0.25 =8h-4 = 2. 2. 171 -H 0.25 =171x4 = 684. 9. 567 ^ 625 = (567^ 5)x 0.008 = 113.4 X 0.008 = 0.9072. 3. 876x1.25 =876^0.8 = 8760 - 8 = 1095. 4. 132x2.5 =132-^-0.4 = 1320 ^ 4 = 330. 5. 591 H- 2.5 = 591 x 0.4 = 236.4. 6. 756^0.125=756x8 = 6048. 7. 268x25 =268-0.04 = 26800 - 4 = 6700. 8. 753-5-25 =753x0.04 = 30.12. 10. 1764x0.025 = 1764-40 = 44.1. 11. 5381^0.025 = 5381x40 = 215,240. 12. 7452 ^ 0.875 = 7452 x 8 h- 7 = 59,616-^7 = 8516.6. 13. 651 X 0.33333 = 651 -5- 3 = 217. 14. 456 X 6.66667 = 456 ^ 0.15 = 45,600 H-15 = 3040. 15. 1554x0.16667= 1554 -^ 6 = 259. 26 ARITHMETIC. 16. 432 + 1.33333 = 432x0.75 17. 375 + 16.667 = 375x0.06. = 324. » 22.50. 18. 225 + 6.6667 = 225 x 0.15 = 33.75. 167. 1. Taking 7 as unity, what would be the value of 14? of 28? of 35? of 3.5? of 2.8105? of 6.31415? 1]U 7)28 7)35 7}3^ 7)2.8105 7 )6.31415 2 4 5 0.5 0.4015 0.90202 2. If the side of a square is 10 inches, and its diagonal 14.14214, express the side in terms of the diagonal as unity. 0.70710 1414214) 1000000.00000 9899498 10050200 9899498 1507020 1414214 928060 3, If the diagonal of a square is one foot, what decimal of a foot must its side be ? i ^ 1.414214 = 0.70710. (See 167. 2.) 4. If the diameter of a circle is 11.3 inches, and its circumference 35.5 inches, what is the circumference in terms of the diameter i* "What is the diameter in terms of the circumference ? 3.14159 0.31831 113)355.00000 355)113.00000 339 1065 160 650 113 355 470 2950 452 2840 180 1100 113 1065 670 350 S65 1050 1017 teachers' edition. 27 5. What decimal fraction of 87 is 47? 53? 43.5? 29? 0.54023 0.60919 0.5 87)43.5 0.333 87)47.00000 87)53.00000 87) 29.000 435 622 43.5 261 350 800 290 348 783 261 200 170 290 174 87 261 260 830 783 29 6, How many times 393 is 587 ? 7857? 131? 196.5? 1.49 19.99 0.33 0.5 393)587.00 393)7857.00 393)131.00 393) 196.5 393 393 1179 196.5 1940 3927 1310 1572 3537 117? 3680 3900 131 3537 3537 363 7. How many 684'8 are there in 1368 ? in 1760 ? in 342 ? in 77 ? in 6.84? in 0.0785? 2 0.5 0.01 684)1368 684)342.0 684)^84 1368 342 6 84 2.57 0.1126 684) 0.00011 684) 1760.00 1368 684) 77.000 684 860 684 1760 1368 0.0785 684 3920 3420 101 5000 4788 392 28 ARITHMETIC. 168. 1. Divide 11.4285285 by 3.1415927 to six decimal ])laces. 3.637813 '6HXm^) 114285285 94247781 20037504 18849556 4. Divide 0.0053 by 72.654 to eight decimal places. 0.00007294 72^^^)5.30000 508578 21422 14531 1187948 942478 6891 6539 245470 219911 by ices. 5. Divide 6 decimal places 352 300 25559 25132 427 by 0.1573 to three 38.144 314 113 94 2. Divide 0.004239239 3.2783278 to five decimal pk 0.00129 160000.0 4719 12810 12584 2260 1573 32783^7^)42392.39 3278328 629 960911 655675 6. eight Divide 0.11 by 1937.^ decimal places. 305236 295049 \nm) 0.00005677 11.0000 3. Divide 437 by 215.253 to three decimal places. 2.030 21^3)437000 43051 96872 13128 11624 1504 1356 649 646 148 133 teachers" edition. 29 7. Divide 46 by 0.00751515151 to three decimal places. 6120.968 75;^;^;:^^) 4600000000000 450909091 9090909 7515152 1575757 1503030 72727 67636 5091 4509 582 169. 1. Find the value of 100 ; iqi . 102. iqs . iq* ; iQS . iqs. 10» = 1; 101 = 10; 102 = 10x10=100; 10^ = 10x10x10=1000; 10* = 10 X 10 X 10 X 10 = 10,000 ; 10^ = 10 X 10 xlO xlO xlO = 100,000 ; 10« = 10 X 10 X 10x10x10x10 =1,000,000. 2. Find the value of lO^ ; 10-^ ; IO-2 ; 10-^ ; 10"* ; IQ-^ ; 10-«. 10» = 1 ; 10-1 _ 0.1 ; 10-2 = 0.12 _ 0.01 ; 10-3 _ o.l^ = 0.001 ; 10-* =0.1* = 0.0001; 10-5 = 0.15 = 0.00001; 10-6 = 0.16 = 0.000001. 3. Find the value of 100 X 100 ; lOixlQ-i; 102x10-2; 10^x10 '; 10* X 10-5. 100 X 100 = 100 = 1 . 101 X 10-1 _ 100 = 1 ; 102 X 10-2 _ 100 = 1 ; 103x10-3=100 = 1; 10* X 10-5 = 10-1 = 0.1. 4. Find the value of 10^ - lO-i ; IO-2 - IO2 ; lO^i -5- lO"-^ ; 20-2 ^ 10-*. 103 -^ 10-1 _ 10* = 10,000 ; 10-2 -V- 102 = 10-* = 0.1* = 0.0001 ; 10-1 ^. 10-3 = 102 = 100 ; 10-2 -=- 10-* = 102 _ 100. 5. Findthe value of 10-3x102; 103-^102; 10-3-^102; 10-2-^-10-3. 10-3 y 102 = 10-1 = 0.1 ; 103 ^ 102 = 101 =-10 ; 10 3 + 102 = 10-5 _ 0.15 _ 0.00001 ; 10-2-10-3 = lOi = 10. 30 ARITHMETIC. 6. Find the value of 102 + 10 ; 102 -h 10^ ; 10»^10 ^ lO'^ + lO-^ 10' + 10 = 10; 102 -^10' = 10-1 = 0.1; 100^10-1 = 10; 10-1 + 10-1 = 10» = 1. 7. Find the value of l.OP-r- 1.01-1; l.OPxl.Ol-i; l.Ol-'-hl.Oii. l.OP + 1.01-1 _ 1.013 _ 1.030301 ; 1.01« X 1.01-1 = 1 01 ; 1.01-2 ^ 1.01-1 ^ 1.01-1 = 1 -i- 1.01 = 0.99009. Exercise I. 1. Express in words, 327.244. Three hundred twenty-seven and two hundred forty-four thou- sandths. 2. Express in words, 80.9056. Eighty and nine thousand fifty-six ten-thousandths. 3. Express in words, 0.390012. Three hundred ninety thousand twelve millionths. 4. Express in words, 20000.002. Twenty thousand and two thousandths. 6. Express in words, 0.0000008. Eight ten-millionths. 6. Express in words, 41.27105. Forty-one and twenty-seven thousand one hundred five hundred- thousandths. 7. Write in figures, two hundred thirty-five and eight hundred thirty -five thousandths. 235.835. 8. Write in figures, seventy-four and two hundred three thou.-Jaiid six millionths. 74.203006. 9. Write in figures, twelve hundred and eight thousand three tjn- millionths. 1200.0008003. teachers' edition. 31 10. Write in figures, five thousand sixty-four millionths. 0.005064. 11. Write in figures, one million and four tenths. 1000000.4. 12. Write in figures, six hundred-millionths. 0.00000006. 13. Multiply and divide 789.365 by 10 ; by 100 ; by 100,000. 7893.65; 78.9365; 78936.5; 7.89365; 78,936,500; 0.00789365. 14. Multiply and divide 0.004 by 100 ; by 10,000 ; by 1000. 0.4 ; 0.00004 ; 40 ; 0.0000004 ; 4 ; 0.000004. 15. Multiply and divide 436 by 1,000,000 ; by 1000 ; by 10. 436,000,000; 0.000436; 436,000; 0.436; 4360; 43.6. 16. Multiply and divide 0.1 by 10 ; by ten millions. 1; 0.01; 1,000,000; 0.00000001. 17. Find the value of 21.3706 + 15.243 + 1.8954 + 0.26891 + 5.328 H- 29.74. 21.3706 15.243 1.8954 0.026891 5.328 29.74 73.603891 18. Find the value of 57 + 0.0057 + 6.8 + 1200 + 0.847 + 159.2 + 3. 57. 0.0057 6.8 1200. 0.847 159.2 3. 1426.8527 32 ARITHMETIC. 19. Find the value of 0.0012 + 10 + 5.8281 + 5 + 39.43 + 0.6827 + 1. 0.0012 10. 5.8281 5. 39.43 0.6827 1. 61.942 20. Find the value of 23.9875 - 12.4764 ; 35.14732 - 27.62815. 23.9875 35.14732 12.4764 27.62815 11.5111 7.51917 21. Find the value of 102.1274 - -83.072; 39.801- 102.1274 39.801 83.072 17.9645 17.9645. 19.0554 21.8365 22. Find the value of 30 - 5.2817 ; 1.7 - 0.8469. 30.0000 1.7000 5.2817 0.8469 24.7183 0.8531 23. Find the value of 1 - 0.54237 ; 100-0.00176. 1.00000 100.00000 0.54237 0.00176 0.45763 99.99824 ind the value of 24.271 - 3.5485 + 15.271 - -13.256- 3.6485 24.271 13.256 39.542 15.271 14.125 31.0295 14.125. 39.542 31.0295 8.5125 TEACHERS EDITION. 33 )5. Fmd the value of 52 + 0.52 - 17.8946 - 30.254 - 0.5 + 21.12. 52. 17.8946 0.52 30.254 73.64 21.12 0.5 48.6486 73.64 48.6486 26. Find the value of 41.289 x 0.5 ; 0.268 x 0. 41.289 0.268 0.5 0.9 24.9914 0.112 X 0.2. 0.112 0.2 20.6445 0.2412 0.0224 27. Find the value of 2.435 x 4.23 ; 71.651 x 3.37 ; 0.251 x 0.04. 28 0.0768. 2.435 71.651 0.251 4.23 3.37 501557 0.04 7305 0.01004 4870 214953 9740 214953 10.30005 241.46387 Find the value of 0.0012 x 0.005 ; 2.26823 x 200 ; 0.0012 2.26823 5.6125 0.005 200 0.0768 0.000006 453.646 449000 336750 392875 29. 0.43104 Find the value of 0.7 X 7 X 0.07 ; 0.15625 x 23.7 X 0.00192 x 5. 0.7 7 0.15625 23.7 109375 46875 31250 3.703125 0.00192 4.9 0.07 0.343 7406250 33328125 3703125 3.703125 0.00711 5 0.03555 30. Find the value of (2.465 + 1.121) x (3.2- - 2.89). (2.465 + 1.21) X (3.2- -2.89; ) - 3.675 X 0.31 = 1.13925. 31. Find the value of (3.01)» ; (0.045)' ; (0.0081)» ; (5.1004)» ; (0.76)». 3.01 0.045 0.0081 3.01 0.045 0.0081 301 225 81 903 180 648 9.0601 0.002025 0.00006561 S.1004 0.76 ft. 1004 0.76 456 204016 51004 532 255020 26.01408016 5.1004 10405632064 2601408016 13007040080 132.682214448064 0.5776 0.76 34656 40432 0.438976 32. Find the value of (0.125)« x (0.32)». 0.125 0.125 625 250 125 0.015625 0.32 0.32 64 96 0.1024 0.32 2048 3072 0.032768 0.032768 0.015625 163840 a5536 196608 163840 32708 0.000512 teachers' edition. 35 33. Divide 291.84 by 6 ; 0.12936 by 12 ; 7.92801 by 0.9. 6)291.84 1 2)0.12936 9)7.92801 48.64 0.01078 0.88089 34. Divide 58.383 by 0.39 ; 0.28744 by 0.08 ; 491.205 by 0.065. 149.7 7557 )) 5838.3 8)28.744 65)491205 39 3.593 455 193 362 156 325 378 370 351 325 273 455 273 455 35. Divide 68.325 by 6.25 ; 0.732 by 1.6 ; 1208.88 by 0.438. 10.932 16j 0.4575 438) 2760 625)6832.500 7.3200 1208880 625 64 92 / 876 5825 3328 5625 80 120 3066 2000 2628 1875 112 2628 1250 80 1250 80 36. Divide 498 by 0.0125; 7 by 0.007 ; 1000 by 0.0001. The reciprocal of 0.0125 is 80. 498 7 )7000 1 )10000000 80 1000 10000000 39840 36 ARITHMETIC. 37. Divide 0.235 by 10.24 ; 27 by 12 ; 0.00507702 by 0.0283. 0.02295 0.1794 1024) 23.;-;0000 12 )27.00 283)50.7702 2048 2.25 283 3020 2247 2048 1981 9720 2660 9216 2547 5040 1132 1132 38. Divide 89.3 by 0.00752 ; 74.1 by 0.0256 ; 1 by 0.128. 11875 2894.53125 7.8125 752) 8930000 256) 741000.00000 128) 1000.0000 752 512 896 1410 2290 1040 752 2048 1024 6580 2420 160 6016 2304 128 5640 1160 » 320 5264 1024 256 3760 1360 640 3760 1280 640 800 768 320 256 640 512 1280 1280 39. Divide 0.39842 by 3.7164 ; 281.5 by 13.789 ; 0.0005 by 0.0028. TEACHERS EDITION. 37 0.10720 20.41482 0.r78«7 37164) 3984.20000 37164 13789} 281500.00000 27578 28) 1 5.00000 28 267800 260148 57200 55156 220 196 76520 74328 20440 13789 240 224 21920 66510 55156 113540 110312 160 140 200 196 32280 27578 40. Divide 63.04 by 0.0059. 0.06905 128 by 493) 912.85; 287.209 58257.40365 28720900.00000 2465 by 59; 0.00493; 2000 338983.05084 91285)6304.12800 547710 ) 20000000.00000 177 827028 821565 546300 456425 4070 3944 1269 986 230 177 530 472 2830 2465 580 531 3650 3451 490 472 1990 1972 180 177 1800 1479 300 295 3210 2958 500 472 2520 2465 280 236 38 ARITHMETIC. Exercise II. 1. 1.4 + 2.08 + 3.895 - 3. 1.667 + 0.4 + 0.286 + j^ +0.636+0.931 = 2*08 1-G67 3.895 OA 7.375 0.286 6.08 0.636 0.931 2. 2.8 + 2.08 + 0.28 + 0.028 + 0.812 = 2.8 10. 2.08 28 4. 6.125-0.57 = 0.028 6.125 0.812 0.57 6. 5.555 5. (4.625 + 1.146) -(1.2 + 3.57) 6. 6.913 - (2.85 - 0.937) = 5.771-4.771 =6.913-1.913 = 1. =5. 7. 24 - 2.4 + (5 - 3.508) - 3.092 = 24 - 2.4 + 1.492 - 3.092 = 25.492 - 5.49^ = 20. 8. 10 - (4.25 - 2.5 + 2 - 0.625 - 0.4 - 2.02) - 0.295 = 10 - (6.25 - 5.545) - 0.295 = 10 - 0.705 - 0.295 = 10-1 = 9. 9. 1.5x0.08x0.5= 10 0.1204 xO:0168x 100 1.5 0.08 0.12 0.5 0.06 0.1204 0.0168 9632 7224 1204 0.00202272 100 0.202272 teachers' edition. 39 11. 0.04x3.25x0.06 = 12. 36x0.00 2x2.05x0.00765 = 3.25 36 0.1476 0.04 0.002 0.00765 0.13 0.072 7380 0.06 2.05 360 144 8856 0.0078 10332 0.00112904 0.1476 13. 0.139x28+42x0.002 + 6 X 0.004 - 0.05 X 20 = 3.892 + 0.084 + 0.024 -1 = 4-1 = 3. 14. (10 - 1.25) X 0.2 + 0.02 X 2.8 + (80.3 X 0.1- 5.3) X 10 -805.3x0.02 = 8.75 X 0.2 + 0.02 X 2.8 + (8.03 - 5.3) X 10 - 805.3 X 0.02 = 1.75 + 0.056 + 27.3 - 16.106 = 29.106 - 16.106 = 13. 15. 28.3696-5-1.49= . 17. 8.8779-^175.8 = 19.04 , 0.0505 149)2836.96 1758)88.7790 149 8790 1346 8790 1341 8790 596 596 16. 0.27-5-0.00225= 18. 0.0427 -^ 92.3 = 120 0.00046 225)27000 923)0.42700 225 3692 450 5780 450 5538 40 ARITHMETIC. 19. 0.28744 ^ 800 = 8)0.0028744 0.0003593 20. 491.205^650 = 22. 0.732-^16,000 = 0.7557 0.00001575 65)49.1205 16)0.00073200 455 04 362 02 325 80 370 120 325 112 455 80 455 80 21. 68.325^6250 = 23. 1208.88-0.438 = 0.010932 2760 625) 6.832500 438)1208880 625 876 5825 3328 5625 3066 2000 2628 1875 2028 1250 1250 24. 2 ^ 0.01-(0.2 -^ 0.02 + 0.8 -^-10) + 36.48 ^ 8 - (4 ^ 0.05-2+0.0^1 .25 = 200 - (10 + 0.08) + 4.56 - (80 - 2 + 0.48) = 200 - 10.08 + 4.56 - 78.48 « 204.56 - 88.56 - 116. 25. 72.2 -H 10 - 2 -h (0.5 ^ 1.60) + 2.125 -!- (1.75 - 0.5) = 72.2^10-2-^0.3125 + 2.125^1.25 . = 7.22-6.4 + 1.7 = 8.92 - 6.4 teachers' edition. 41 Exercise III. 1. What number subtracted 88 times from 80,005 will leave 13 as a remainder ? 909 80,005 88) 79992 13 792 79,992 792 792 2. If 7 men can build a wall in 16 days, how many men will it take to build a wall three times as long in half the time ? 7 _3 21 42 3. How many minutes are there between 25 minutes past 8 in the morning and midnight ? 35 180 720 935 4. The velocity of sound being 1090 feet per second, at what dis- tance is a gun fired, the report of which I hear 11 seconds after seeing the flash? (5280 feet make a mile.) 2.27083 1090 11 1090 1090 5280) 11990.00000 10560 14300 10560 11990 37400 36960 44000 42240 17600 15840 42 ARITHMETIC. 5. How long would it take to travel 30.2375 miles at the rate of 8.85 miles per hour ? 3.4167 885) 3023.7500 2655 3687 3540 1475 885 5900 5310 5900 6195 6. The circumference of a circle being 3.1416 times the diameter, find the circumference of a circle whose diameter is 6.8 feet ; also, find the diameter of a circle whose circumference is 20 inches. 6.366 3.1416 31416)200000.000 6.8 188496 251328 115040 188496 94248 21.36288 207920 - 21.363 ft. Ans. 188496 194240 188496 7. How much wire will be required to make a hoop 30 inches in diameter, allowing two inches for the joining? 3.1416 30 84.248 _2 96.248 TEACHERS EDITION. 43 8. How many times would such a hoop turn in going half a mile? 336. 2 )5280 2640 2640 12 5280 2640 31680 94248)31680000. 282744 340560 282744 578160 565488 9. Cork, whose weight is 0.24 of that of water, weighs 15 pounds per cubic foot. What is the weight of 6 cubic feet of oak, the weight of oak being 0.934 of that of water ? 62.5 24) 1500.0 62.5 144 6 60 48 120 120 375. 0.934 1500 1125 3375 350.25 10. From what number can 847 be subtracted 307 times, and leave a remainder of 49 ? 847 307 5929 2541 260029 49 260078 11. What is the 235th part of 141,235? 601 235)141235 1410 235 235 44 ARITHMETIC. 12. What will 343 barrels of flour eost, at $6.37 a barrel ? $6.37 343 1911 2548 1911 $2184.91 13. 12 make a dozen, and 12 dozen make a gross. How many steel pens in 28 gross ? What will a gross of eggs cost, at 27 cents a dozen ? 144 $0.27 28 12 1152 288 4032 54 27 $3.24 14. How much must be added to $4429 in order to make the sum 43 X $241? $241 43 723 964 10363 4429 $5934 15. What number deducted from the 26th part of 2262 wi 1 leave the 87th part of the same number ? 87 26 26) 2262 87) 2262 87 208 174 26 182 522 61 182 522 16. At an ordinary rate, 123 words a minute, how long will it take a man to deliver a speech of 15 pages, each of 28 lines, and each line containing 11 words? How long would it have taken Daniel Webster to deliver the same speech, at the rate of 93 words a minute? 37.6 49.7 123)4620.0 15 28 120 30 420 11 420 420 369 930 861 690 93)4620.0 372 900 837 630 4620 TEACHERS EDITION. 45 17. How long would it take a railway train to go from New York to San Francisco, 3310 miles, at the rate of 1973 feet a minute ? 8858 3310 1973) 17476800 5280 15784 L'64S00 16928 6620 15784 inr)50 11440 17476800 9865 July 4, how much money will there be in the box ? 15750 18. How long will it take to count a million, at the rate of 67 a minute ? 14925.4 67) 1000000.0 67 330 268 620 603 170 134 360 335 250 19. If you put into a box 17 cents a day including Sundays, beginning January 1 and ending 31 185 28 0.17 31 30 31 30 1295 185 31.45 4 . =$31.45 185 20. If a man's income is 1 3000 a year, and his daily expenses average 1 7.68, what does he save in a year ? $7.68 365 3840 4608 2304 12803.20 13000. 2803.20 $196.80 21. In a question of division the quotient was 87.83, the divi- sor 759. What was the divi- dend? 87.83 759 79047 43915 61481 66662.9- 46 ARITHMETIC. 22. It is 3.1416 times as far round a wheel as across it. How many times will a wheel 4.5 feet across turn round in going 23 miles of 5280 feet each ? 8590 5280 23 3.1416 4.5 141372^ 1214400000 1130976 15840 10560 121440 157080 125664 14.1372 834240 706860 1273800 1272348 14520 23. How many gallons of 231 cubic inches are contained in a cubic foot (1728 cubic inches)? in a bushel of 2150.42 cubic inches? ITow many cubic feet in a bushel? How many bushels in 31.5 gallons ? (i.) (ii.) 7.48 9.309 231)1728.00 1617 231, ) 2150.420 2079 1110 924 714 693 1860 1848 2120 2079 (iii.) (iv.) 1.244 31.5 231 21504; 3.38 .728)2150.420 1728 2) 727650.00 645126 4224 3456 315 945 630 7276.5 825240 645126 7682 6912 1801140 1720336 TEACHERS EDITION. 47 24. Seven children had left to them $7186 apiece; one died, and his share was divided among the surviving six. How much had each then ? 6 )17186.00 $1197.67 7186 $8383.67 26. What is the nearest num- ber to 7196 that will contain 372 without a remainder ? 19 372)7196 372 3476 3348 7196 128 7068 128 26. How long will it take 2 men to do what 1 man can do in 6 days ? what 4 men can do in 3 days ? what 3 men can do in 4 days ? 6 days -7-2 = 3 days. 2x3 days = 6 days. (3x4 days) -i- 2 = 6 days. 27. Divide $1.80 am.ag Thom- as, Richard, and Henry in such a way that Henry shall receive 3 cents for every 5 cents that Thomas gets, and Richard shall receive 2 cents for every 3 cents that Henry gets. 2 3 5 10 $0.18 3 10 ) $1.80 $0.18 2 $0.36 = R.'8. $0.18 5 $0.54 = H.'s. $0.90 = T.'s. 28. Divide $87.84 between B and C so that C shall get $19 as often as B gets $17. 2.44 36) 87^84 19 17 72 36 158 144 144 144 $2.44 $2.44 19 17 2196 1708 244 244 $46.36 = B's. $41.48 = 29. Three partners received for goods : one, $371.63 ; the second, $285.40; the third, $411.91. They paid for the goods $879.34, and divided the balance equally among them. How much did each receive ? $371.63 285.40 411.91 $1068.94 $1068.94 879.34 3) $189.60 $ 63.20 48 ARiTHMETIC. 30. At 12 inches in a foot, how many inches long is a wall 35 feet in length ? A brick and its share of mortar being 8.4 inches long, how many bricks in length is the wall '' 35 ftO 12 84)4200 70 420 35 420 31, A brick and mortar being 2.4 inches in height, how many bricks are required to build the wall 12 feet high, if the wall be two bricks wide? 12 60 60 12 24)U40 _^ 144 ]^ 3000 2 6000 32. What is the total weight of the wall, if a brick and its share of the mortar weigh 4.13 pounds ? What is the weight after a long rain, when the weight is increased to 1.27 pounds for r>acli brick'* 4.13 HOOO 24780 6000 2r)(^)20 33. llow many pounds does carli foot in Vnt^th of tlio wall weigh ? 708 35)24780 245 280 280 35)25620 245 ~112 105 'Id 70 34. If 60.98 cubic inches of brick weigh 4 pounds, how many cubic inches of brick weigh 1 pound ? How many pounds would a cubic foot (1728 cubic inches) weigh ? 4^60.980 15.246 113.35 15245) 1 72801 lOoo io24r. 20350 15215 5ia50 46735 ~53l50 45735 74160 35. If a cubic foot of wat«M- weigh 62.5 pounds, how many times as heavy as water is brick' 1.8 625)1133.5 625 5085 5000 TEACHERS EDITION. 49 36. Light moves through the air at 186,500 miles in a second. How many times can it go around the earth in a second, the distance round the earth being 24,897.714 miles ? 7^ 24897714)186500000.0 174283998 122160020 37. Light moves through the air at 300,190 kilometers in a second. How many times can it go around the earth in a second, the distance round the earth be- ing 40,007.5 kilometers ^ 7.5 400075)3001900.0 2800525 2013750 2000305 38. A minute is 60 seconds. How many miles and how many kilometers can light travel through air in a minute? 300190 km. 18,011,400 km. 186,500 mi. 60 11,190,000 mi. 39. An hour is 60 minutes. How many miles and how many kilometers can light travel in an hour? 18,011,400 km. 60 1,080,684,000 km. 11,190,000 mi. 60 671,400 000 mi. 40. The distance round the earth, given in Ex. 37, is meas- ured on a north and south line. Around the equator the distance is 40,075.45 kilometers. How many times could light move round the equator in one min- ute ? . 7.49 400754 5):5001 9000.00 28052815 19661850 16030180 36316700 36067905 7.49 60 449. 41. Find the reciprocal of tlie between 31.24 and difference 31.23768. 31.24 31.23768 0.00232 50 ARITHMETIC. 431.034 232)100000.000 928 720- 696 240 232 800 690 1040 928 42. The Hanoverian mile is 25,400 Hanoverian feet long, each foot being 0.9542 of an Eng- lish foot. Find to four places of decimals the fraction that an English mile of 5280 English feet is of a Hanoverian mile. 0.9542 25400 3816800 47710 19084 24236.6800 0.2178 2423668)528000.0000 4847336 4326640 2423668 19025)720 16965(>76 21610440 19389*44 43. Express in inches the length of a meter, given that a meter is one ten- millionth of a quarter of the earth's circumfer- ence, that the circumference is 3.14159 times the diameter, that the diameter is 7911.7 miles, and that a mile is 5280 x 12 inches. 5280 12 10560 5280 63360 7911.7 443520 63360 63360 570240 443520 501285312. 3.14159 4511567808 2506426560 501285312 2005141248 501285312 1503855936 4 )1574832923.32608 393708230.83152 0.0000001 39.370823083152 - 39.3708 in. Aiu. 44. How must a number be altered to double its reciprocal ? Divided by 2. teachers' edition. 51 . 45. What effect is produced on the sum of two numbers, if each number is increased by the same number ? What effect on the differ- ence It is increased by two times the number ; not any. 46. What effect is produced on the product of two numbers, if both numbers are multiplied by the same number ? What effect on the quotient*^ It IS multiplied by the square of the number ; not any. 47. What effect is produced on the remainder, if both divisor and dividend are multiplied by the same number? If both are divided by the same number ? It is multiplied by the number ; it is divided by the number. 48. In going from one planet to another light probably moves faster than in air. Suppose it moves at 309,800 kilometers a second, how long would it take light to perform each of the following journeys : Moon to Earth 375,500 kilometers. Sun to Karth 147.250,000 Sun to Mercury 56,900,000 Sun to Venus 106,400,000 Sun to Mars 224.100,000 Sun to the Asteroids 400,000,000 Sun to Jupiter 765,400,000 Sun to Saturn 1,403,000,000 Sun to Uranus 2,817,000,000 Sun to Neptune 4,421,000,000 Sun to the nearest star . . 24,000,000,000,000 1.21 475.3 183.7 3098)3755.00 3098)1472500.0 3098) 569000.0 3098 12392 3098 6570 23330 25920 6196 21686 24764 3740 16440 11560 3098 15490 9294 9500 22660 9294 21686 52 ARITHMFTIC. 343.4 723.4 1291.1 3098)1064000.0 3090) 2241000.0 3098)4000000jO 9294 21686 3098 13460 7240 9020 12392 6196 10440 6196 10680 28240 9294 9294 27802 13860 12460 3580 12392 12392 3098 4820 3098 2470.6 3098)7654000.0 6196 14580 12392 4528.7 3098)14030000.0 12392 16380 15490 9092.9 3098)28170000.0 27882 28800 27882 21880 21686 8900 6196 9180 6196 19400 18588 27040 24784 29840 27882 22560 ^1686 14270.5 77469335 3098)44210000.0 3098 3098) 2 2 40000000000 1686 13230 12392 23140 21686 8380 6196 14540 12392 10380 9294 21840 21686 21480 18588 10860 Q294 15400 15490 28920 27882 15660 15490 teachers' edition. 53 49. A kilometer is about 0.6214 of a mile. How many miles is each of the planets from the sun ? 14725 5690 10640 6214 6214 6214 58900 22760 42560 14725 5690 10640 29450 11380 21280 88850 34140 63840 b, 91, 50], 150 Mercury, 35,357,660 Venus, 66,116,960 22410 76540 6214 6214 89640 306160 22410 76540 44820 6214 153080 134460 40000 Asteroids, 248,560,000 459240 , 139,255,740 Jupiter, 475,619,560 140300 281700 442100 6214 6214 6214 561200 1126800 176S400 140300 281700 442100 280600 563400 884200 841800 1690200 2652600 Saturn, 871,824,200 Uranus, 1,750,483,800 Neptune, 2,747,209,400 Exercises on Page 73. 1. ('onvc'i-t 5427"" into kilometers; into millimotei's ; into centi- iiieLei s. 5427"' = 5.427''"' = 5427000°'"' = 542700°°'. 2. 6853""'" contain how many meters ? how many centimeters ? what part of a kilometer ? 6853«"'» = 6.853«« - 685. 3«"' = 0.006853'"»\ 54 ARITHMETIC. 3. Write 49.7"' as centimeters ; as millimeters ; as part of a kilo- meter. 49.7'° = 4970"" = 49700™" = 0.0497"™. 4. How many centimeters in 12.4*""? how many millimeters? 12.4km _ 1,240.000"° = 12,400,000™"'. 5. Change 1280 meters into kilometers ; into centimeters. 1 230™ = 1 23''™ = 1 23,000™. 6. Write 1230«™ as meters ; as millimeters. 1230c™ = 12.3™ = 12,300™™. 7. 0.435"' + 852c™ ^ 4263™™ + 0.1595"™. 0.435™ 8.52 4.26 159.5 172.718^ 8. 0.927''™ - 6495c™ . 4.37cm _ 42.87™™. 927.™ 0.0437™ 64.95 0.04287 862.05™ 0.00083™ 9. 8x0.0457"™; 3.04x60.93c™; 5.43 X 67.2™™. 0.0672™ 0.6093™ 5.43 3.04 15T6 45.7™ 24372 2688 8 18279 3360 365.6" 1.852272™ 0.364 H96™ 10. 38,OI9n>"'-- 0.097: 0.41"™ + 25.625. 16" 25625)410000 25625 1437r>0 143750 391.948« 97)38019.000 291 891 873 189 97 920 873 470 388 820 776 11. At $1.87 the met^r what is tlie cost of 6.20™ of clotli ".' 11.87 6.2 "374 1122 111.594 111.59. Ans. TEACHERS EDITION. 65 12. At 10.75 the meter what is the cost of GO"" of cloth ? $0.75 $45.00 13. From a piece of cloth con- taining 47.60™ a tailor cuts off three pieces : the first of 3.80™, the second of 1.30™, and the third of 45«™. How much of the cloth is left ? 3.8™ 1.3 47.6™ 0.45 5.55 5.55™ 42.05™ 14. What is the value of 60«™ of cloth, worth $5.20 a meter ? $5.20 0.6 $3.12 . 15. If $6.00 are paid for a railroad ticket to travel 440"^™, what is the fare per kilometer ? 0.0138 = $0,014 440) 6.000 440 1600 1320 3800 3520 280 16. If a train run 288^^™ in 9 hours, how many meters does it run in a minute? 60 533.33' 9 540 54) 28800.00' 270 180 162 180 162 180 162 180 162 17. If a man walk at the rate of 6^™ an hour, what part of an hour will it take to walk 420 meters ? 6km = 6000™ 0.02 6000)420.00 42000 18. A railroad carried 412 passengers 18 kilometers, and received $88,992; at the same rate, what will it receive for carrying 350 passengers 35 kilo- meters ? 412 0.012 18 7416)88.992 3296 7416 412 14832 7416 14832 350- 12250 35 $0,012 1750 24500 1050 12250 12250 $147,000 56 ARITHMETIC. Exercises on Pages 75 and 76. 1. Convert 1,854,2761°' into hektars ; into square kilometers. l,854,276i» = 185.4276h»; = 1.8542761'"". 2. How many hektars in 2.7856 square kilometer,-? 2.78561'"" = 278.56''a. 3. Write 1.7431» = 174.6751''™. 5. How many square meters in 1.36141'"° ? 1.36141'^'" = 1,361,4001"'. 6. How many square meters in 2.25 hektars ? 2.25h» = 22,5001"'. 7. How many square centi- meters in 0.0137 of a square meter ? 0.01371" = 1371"'". 8. Write 3.571i'='" as souare millimeters. 3 571qom ^ 357,lqmin Exercises on Page 77. 1. How many cubic centi- meters in 2.25<'»>'»? 2.25°'>™ = 2,250,000"'". 2. How many cubic meters in 2,162,875''«»? 2,162,875«'"« = 2.162875*'"". Exercises on Page 78. 1. How many liters in 1.7«'""? in 157,854««'» ? 1.7"^'" = 1700'; 157,854«<='» = 157.854>. 2. How many cubic centi- meters in 9.5»? in 0.015»? 9.5> = 9500««'» ; 0.015^= 15«5». 3. Change 1.25''' to cubic cen- timeters ; to the fraction of a cubic meter. 1.25"=125,000««»; — 0.125«'»"'. 4. Convert 431.88» into hekto- liters ; into the fraction of a cubic meter. TEACHERS EDITION. 57 431.881 = 4.3188'^i; = 0.43188<=i"". 5. Write 0.375"^™ as liters ; cubic centimeters. 0.375«bm _ 3751 . = 375,000'''"». 6. Write 734,159.651'=«'« as lit- ers ; as hektoliters ; as cubic meters. 734,159.651<'<"» = 734.1596511 ; = 7.34159G51hi; = 0.734159651 7. How many meters in 8,573,412.867°°™? 8,573,412.867°<=°» = 8.573412867°"". 8. Change the expression 0.734578912°''«» into cubic centi- meters ; into liters. 0. 73457891 2°i"« = 734,578.912°°'^ ; = 734.5789121. 9. Change 1731.5 liters into cubic meters ; into cubic centi- meters. 1731.51 =1.7315°bn>; = 1,731,500°°°\ Exercises on Page 79. 1. How many kilos in 1.73*? in 0.341 of a ton ? 1.73*= 17301^8 ; 0.341*= 341''g. 2. How many" kilos will a hektoliter of water wei^ ;h? 100^8. 3. Convert 13,756'»k into grams ; into the fraction of a kilo. 13,756'°8= 13.7566; = 0.0137561^8. 4. What is the weight in grams of 346.1°°™ of water? 346.18. 5. Give the weight in kilo- grams of 0.37615°i>™ of water. 376.15kg. 6. Change 0.67781^8 into milli- grams. 677,800™K.. 7. How many milligrams in the third part of 17.4 grams? 5,800™8. 58 ARITHMETIC. Exercise IV. 1. Add 17.3™, 87.41' SSO""™, and 1.79™. 17.3» 87.41 2.71 0.38 1.79 271* 109.59^ 2. What is the sum of $15.87, $39.46, $47.52, $75.38, $75.89? $15.87 39.46 47.52 75.38 75.89 $254.12 3. Add 187°", 49.3™, 317" and 6.138™. 1.87™ 49.3 0.317 6.138 57.625™ 4. The door-sill being 3«™ high ; the door, 2.34™ ; the finish over it, 13.7"™ ; and the distance from finish to coiling, 930™ . how far from floor to ceiling ? 0.03« 2.34 0.137 0.93 3.437« 5. The distance to the post- office is 3.31^^™ ; thence to the mill, 1.71P™ ; thence to the store, 3.718^™; thence home, 2.543''™. How long is the circuit? 3.31"™ 1.711 3.718 2.543 11.282''" 6. From Portland, Me., to Boston is about 132"™ ; Boston to Albany, 320"™; Albany to Bufi'alo, 480"™ ; Buffalo to Chi- cago, 800"™ ; Chicago to Omaha, 800"™; Omaha to Cheyenne, 780"™ ; how far from Cheyenne to Portland ? to Albany ? from Boston to Chicago ? from Boston to Cheyenne ? (1) (2) 132"™ 132km 320 320 480 452"™ 800 800 3312"™ 780 452 3312"™ 2860"™ (3) (4) 320"« 3312"™ 480 132 800 3180"" 1600"' TEACHERS EDITION. 59 7. If I travel 789.7"'^ a day, how far shall I go in 7 days ? in 8.5? in 19.6 ? in 27.8 ? in 365 days ? 789. 7'^'^ 7 789.7>^- 789.^ 8.5 19.( 39485 4738^ 63176 71073 ^km 789.7'^°» 5 27.8 } 63176 55279 15794 789. 7'^'" 365 5527.9'^'^ 39485 47382 6712.45'^'" "^^^^ 15478.1 23691 2^-^ 21,953.66'^- 288,240.5'''»' 8. How m cost, at $1.^ uch will 3"' of cloth J7 a meter? How at $2.63 a meter? $4,875 153 much 5.38'", 14625 $1.37 3 $2.63 5.38 24375 4875 $4.11 2104 789 $745,875 = $745.88. Ans. 1315 $14.1494 = $14.15. Ans. 9. How much will 13.4i'g of opium be worth, at $ 8.48 a kilo ? 28.79i^«, at$7.96akilo? $8.48 13.4 3392 2544 848 $113.63 28.79 $7.96 17274 25911 20153 $229.1684 = $229.17. Ans. 10. A man bought 153 barrels of flour, at $4,875 a bbl. What did the whole cost him ? 11. He gave for it 6 shares of stock, at $113.50 a share, and the rest in cash. How much money did he pay? $113.50 6 $681.00 $745.88 681. $ 64.88 12. He paid $13.75 for stor- age ; also, 75 cents a barrel for freight. How much do these expenses amount to ? 60 ARITHMETIC. 153 $075 765 1071 $114.75 13.75 $ 128.5a 13. It was sold, 49 bbls. at $6.50 a bbl.; the rest at |6.25. "What were the gross receipts? $6.50 $6.25 49 104 5850 2600 $318.50 153 _49 104 2500 625 $650.00 318.50 $968.50 14. He paid for commissions, etc., $17.50; and counts his los-s of interest at $29.30. What then is his net profit ? $745.88 128.50 17.50 29.30 $921.18 $968.50 921.18 $ 47.32 15. Find the circumference of a circle having a diameter of 1"». 3.1416 3.1416- 16. Find the circumferences of circles of which respectively 83'"; 3.71'°; 32.8°'; 10.4<"° ; 11.8<='° Give each to the nearest tenth of a millimeter. 3.1416 3.1416 83000'°'° 3710°'™ the diameters are ; 167.1°'°'; 39.3°'°'. 3.1416 32800°'°' 94248000 251328 260752.8°'°' 314160 219912 94248 25132800 62832 94248 11655.3°' m 103044.4800°'°' = 103,044.5°'°'. Ans. 3.1416 3.1416 3.1416 3.1416 104mm 118°'°' 167.1°'°' 31416 39.3°"' 125664 251328 94248 31416 31416 219912 282744 326.7264™°' 31416 188496 94248 326.7°"°. Ans. 370.7088°'°' = 370.7°'°'. Ans. 31416 123.46488°'°' 524.96136°'°' = 123.5-°'. Ans. = = 525°«'. Ans TEACHERS EDITION. Gl 17. What is the length of the (2) earth's orbit, to the nearest = 4.115"^ Ans. meter, if the diameter of the orbit is 294,481, 21 7'^'^? (3) 294481 21 7'^'^ Ills'" 3.1416 17 1766887302 28805 294481217 4115 1177924868 69.955'n. j{ns. 294481217 883443651 20. How often must that wheel 925,142,191.3272'^'» turn in going 69.429'"? 73.5 13"^ / -925,142,191,327'". Ans. 17.2? on? 18. How far round this world. (1) 17 nearly if its diameter is 12,734'™? 12734''°^ 4115)69429 3.1416 4115 28279 12734 50936 12734 38202 Ans. 40,005.1344'^"'. 19. If a carri age- wheel is 1.31'" in diameter, what is its circumference ? How far will it go, if it roll without slipping. in turning once? 17 times? (1) 3.1416 1.31- 31416 94248 31416 4.115496™ 4.115™. Ans. (2) 18 nearly 4115)73513 4115 32363 (3) 4197 nearly 4115)17270000 16460 8100 4115 39850 37035 28150 21. Find the reciprocal of I 3.14159 to the 5th place. 62 ARITHMETIC. 0.31831 24. How thick through is a 314159)100000.00000 tree which has a girth of 2.97»'? 942477 0.31831 575230 2.97» 314159 222817 2610710 286479 2513272 63662 974380 0.9453807» 942477 = 0.915'". Am. 319030 314159 25. What is the diameter of a circular field two kilometers iu circumference ? 22. What is the diameter of the circle whose circumference is 0.31831 314.159'""? 2000 lOQcm 636.62" 314159)31415900 314159 26. What is the diameter of a rope of which the circumference 23. What is the diameter of is 20"'"? the wheel which revolves 1£ .5 0.31831 times in going 107.25">? 20cm 5.5"» 6.3662«»» 195)1072.5'^ 975 27. In a park is a fountain 975 whose basin is 75"" in circuni er- 975 ence. What is the diameter of the basin ? 0.31831 0.31831 5.5« 75'" 159155 159155 159155 222817 1.750705"' 23.87325'" = 1.75'". A718. - 23.87"'. Am. TEACHERS EDITION. 63 Exercise V. 1. Find the area of a rectangle 17"™ by 19«". 19cm 133 19 323i« 2. In a rectangular township 16^ by 7^™, how many hektars ? If there are in it 47.3^™ of highway, averaging 11.7™ wide, how much land is left for other uses ? 47300™ 11.7 331100 473 473 5534101™ 55.34 P* 11144.659»'» 3. In a rectangular field, 751.3™ long and 189.3™ wide, is a straw- berry bed 31.4™ by 17.8™. How many hektars in the field? How many, exclusive of the strawberry bed ? 751.3™ 31.4™ 189.3™ 17.8™ 22539 2512 67617 2198 60104 314 7513 558.921™ 142221.091™ - 0.056'>» : 14.222'»^ Ans. 14.2221'* 0.056»"' Ans. 14.1661'*. 64 ARITHMETIC. 4. If my garden contain 941.65«", and my neighbor's 748.37*i"', what is the area of both in hektars ? 941.65 614656'i°» 27««' in diameter? of one which 784m 0.7854 is 1°^ in diameter ? 3136 2458624 (1) 6272 3073280 27cm 0.7854 729qcm 70686 5488 4917248 27cm 614656*1'° 4302592 189 482750.8224^" 54 15708 54978 ="48.275h^ Arts. 729qcm 572.5566™ 1375. 6064". 4. If a liter of grain weigh 0.81 of the weight of a liter of water, how much will the grain in that bin weigh ? TEACHERS EDITION. 75 1375.6064W = 137560.64'^8 0.81 13756064 110048512 111,424.1184'^g 5. A bin measuring 16™ by 9.7'", and 2.8"> deep, is full of oats, worth $0.98 a hektoliter. What is the whole worth ? 16°> 4345.6^1 9.7"» $0.98 347648 391104 $4258.688 $4258.69. Ans. 112 144 155.2°^ = 4345.6J>i. 6. A vat 197"°^ long, 87<='» wide, and 63«'» deep, holds how many- liters ? What would be the weight of water required to fill it ? 197cm 17139i''m 87cm g3cm 1379 1576 17139qcm 51417 102834 1079757°«"» 1079.75? 1079.757'^8. 7. Add 1341<=«™, 2311, and 2.13*'\ and give the sum in terms of each of the three units. 134lccm 2311 = 231000«'™ 2.13''i = 213000O'"" 445341°««' = 445.3411 = 4.45341^1. 8. If a spring pours out 467.8^ each minute, how many hekto- liters will it deliver in 60 min- utes ? in 37 minutes? in 78 minutes ? 467.81 4.678" = 4.678" 37 60 280.68" 32746 14034 173.086" 4.678" 78 37424 32746 364.884" 9. If 67.31 of oil in a vat with perpendicular sides fill it to a depth of 173'°'", how deep will 13.7 times that quantity fill it? and how many hektoliters will there be ? 173" 13.7 1211 519 173 2370.1^ 2.3701« 67.31 = 0.673" 13.7 4711 2019 673 9.2201" 76 ARITHMETIC 10. Into a round cup 10»" tin cu p 95'""» across and 11.08<"» acroK }, with perpendicular sides, deep? pour 3il until it is 1*^ deep ; then 95mm = 9.5«»» 0.7854 there are 78.54«»» of oil in the How many cubic centi- 9.5<"» 90.25 7088235 ' 785.3764380««"' 11. What is the capacity of a = 0.785'. Am. 12. What are the capacities of two cylindrical vessels, one being 1G.24<"» across and 19.95<»° deep, th e other 75 4ram across and 87.9"™ deep? 1G.24<''° 4132.433«°» 75.4'"«» 1().24<'™ = 4.132» 75.4mm 6496 3016 3248 3770 9744 5278 1624 5685.16*1"^ 263.7376'K" 0.7854 0.7854 2274064 10549504 2842580 13186880 4548128 21099008 3979612 18461632 4465.124664'«"«' 207.1395 -4465.125«»»» 19.95<« 87.9°^ 10356975 40196126 18642555 31255875 18642555 35721000 2071395 392485.4875«- 4132.433025«« - 0.392'. TEACHERS EDITION. 77 13. How many cubic centi- meters in a ball 10°™ in diam- eter ? How much less if you take the more exact multiplier ? 10°™ 10°™ 0.5236 1000°°™ 1001°™ 10°™ 523.6°°™ 1000°°™ '1 3.1415927 0.5235988 1000°°™ 523.5988°°™ 523.6°°™ 523.5988°°™ 0.0012°°™ 14. Into a cubical box 20°™ on a side, and full of water, an iron ball 20°™ in diameter is gently lowered until it touches bottom. How much water is left in the box? Answer in liters and in cubic centimeters. 20°™ 0.5236 20°™ 8000°°™ 4188.8° 400i°™ 20 8000°°™ 4188.8°°™ 3811.2°°™ 3.81121. 15. One cask contains 171.41 of oil ; another, 209.3i ; a third, 73.81; while a square vat, 137°™ each way, is filled to a depth of 69°™. How much oil in all the ? in liters and in hekto- 1 iters. 137°™ 187691°™ 137°™ 69°™ 959 411 168921 112614 137 1295061°°™ 18769i<^"' =1295.0611. 171.41 209.31 73.81 1295.061J 1749.5611 17.49561i'i. 16. How many liters of air in a room 7.8™ long, 6.23™ wide, and 3™ high ? 6.23™ 7.8™ 4984 4361 48.5941™ 3™ 145.782°»>™ == 1457821. 17. If a person's breathing spoils the air at the rate of 0.2175°!'™ a minute, how long will it take 3 persons sitting 78 ARITHMETIC. in the room, closed, to spoil the air? 0.2175 3 0.6525 223.42 6525)1457820.00 13050 15282 13050 22320 19575 27450 26100 13500 18. How long, at the same rate, would the air in a hall 22'" long, 16" wide, and 7°* high, last an audience of 280 persons ? 22'» 2175 16" 280 132 22 352i« •7in 2464«^™ 174000 4350 60.9000 40.5 609) 24640.0 2436 2800 19. How many cubic meters of wood in a round stick of equal size throughout, 37*™ in diameter and 8.4™ long ? 37« 0.7854 0.10752126'» 7854 = 0.9032«»>'°. 1075.2 126 - 73.8*« l«b»- lOOO'^K 1—-0.001M of TEACHERS EDITION. 79 2. If a man buys half a ton of potatoes for $20, and retails them all, without waste, at 5 cents a kilogram, what profit does he make on the whole ? 10.05 500 $25.00 20. $5. 3. What is the weight of water required to fill a vat 98°™ long, 71cm ^i(je, and 38°°» deep ? 98««» 6958i°°> 71cm 38 686 6958i<"" 55664 20874 264.404«'°' 264.4041'g. 4. If the vat of the last ex- ample were filled with brine weighing 1. 04^^8 10 the liter, what would be the weight of the brine ? 264.404^^8 1.04 1057616 264401 274.981^8 5. If the vat of Example 3 were filled with wine weighing 0.981''8 to the liter, what would be its weight? 264.404^^8 0.981 264404 2115232 2379636 259.38^^8 6, What is the total weight of 13 men averaging 73.48''8 each ? 73.48^8 13 22044 7348 955.24''8 7. How many kilograms, and how many tons, would 3.61 75'=^™ of brick weigh, at 2 tons to a cubic meter ? at 2.34 tons ? 3.6175 3.6175 2* 2.34* 7.235* = 72351^8 144700 108525 72350 8.46495* = 8464.951^8 8. From a barrel containing 67''8 of granulated sugar were taken three parcels of 2.75''8 each, and four parcels of 7.50^^8 each. How much is left in the barrel ? 2.75^8 7.5^8 3 4 8.25^8 30.0^8 80 ARITHMETIC. 30.0kK 24 8.25 325)7800 38.25''« 650 67'"5 38.25 1300 1300 2'^ 7~/'^^ 10. A mass of 21.8« is divided into 60 pills. What is the weight 9. Into how many pills of of each pill ? 325'nK each can a mass of 7.88 6)2180.000»8 be divided ? 363.333°»8 Exercise VIII. 1. If a stone weighs 1.3^8 in air and 0.68''8 in water, and the stone and a block of wood together weigh 1.55''8 in air and 0.63"^ in water, what is the specific gravity of the block of wood ? 1. 55118 _ 1.3kg ^ o.25''«, the weight of the wood in the air. 1.55kg _ o.es^s = 0.92''8, the weight of water displaced by the stone and the wood. 1.3kg _o.68''8 = 0.62''8, the weight of water displaced by the stone alone. Therefore 0.92''8 - 0.62''8 = 0.3''8, the weight of water displaced by the wood. And 0.25 -4- 0.3 = 0.833, the specific gravity of the wood. 2. What is the weight of 8.17" of alcohol, specific gravity 0.83 ? 8.17»>i = 817''8 0.83 2451 6 536 678.11*8 8. What will 97' alcohol weigh, of specific gravity 0.817? of speci- fic gravity 0.819? of specific gravity 0.823? 0.838? 0.847? 0.817 0.819 0.823 0.838 0.847 97*8 97*8 97*8 97*8 97''« 5719 7353 5733 7371 7'.»'143*8 r.7<;i 7107 79.831*8 rtXi\i\ 5920 7r.2:^ 7',t L'l '■""-' 81.286*8 s TEACHERS EDITION, 81 4, A bar of aluminum 113"™ long, 17™™ wide, and 13™™ thick, is said to bo of specific gravity 2.57. What does it weigh ? If it really is of specific gravity 2.67, what does it weigh ? 113™™ 19211™™ 24.9738 24.9738 17mm 13mm 2.57 2.67 791 113 5763 1921 174811 124865 49946 174811 149838 19211™™ 249 73«™™ = 24.973«c™ = 24.9738 49946 64.188 66.677918 = 66.688 5. What would be the specific gravity of the bar of the last example if it weighed 65.1378? 9fi08 24973)65137.000 49946 151910 149838 207200 199784 6. What is the weight of a bar of aluminum 371™™ by 63™™ by 84™™, specific gravity being 2.63? 371™™ 1963332°™™ 63™™ weigh 1113 1.963332'^e 2226 2.63 233731™™ 5889996 84 11779992 93492 3926664 186984 5.163563161^8 IQCQQQOfimm = 5.1636^^8 7. An irregular mass of cop- per, gently lowered into a pail brimful of water, caused 1.374^ to run over. What did it weigh if of specific gravity 8.91 ? if 8.89 ? 1.374>'8 1.3741^8 8.91 8.89 1374 12366 10992 12.242^^8 12366 10992 10992 12.21486''8 12.215'^8 8. What was the specific grav- ity of that copper if the mass weighed 12.30161^8? 8.953 1374)12301.600 10992 13096 12366 7300 6870 4300 4122 82 ARITHMETIC. 9. A plate of iron 137*™ long, 64.3<"" wide, and 4.31«» thick, weighs 277.54''8. What is its spe- cific gravity ? What would the same mass weigh at specific grav- ity 7.47 ? at 7.79 ? 137*'"" 8809.1i«" 64.3C"" 4.31 411 548 822 8809.1 88091 264273 352364 37967.221«'«' 37.9? . 7.309 3797) 27754.000 26579 11750 11391 35900 34173 37.96722^8 7.47 265770547 151868884 265770547 283.615'' a minute fills a tub in 16 minutes, how long should a tap delivering 5' a minute be in filling the same tub ? 16 3.5 80 48 5 )56.0 m in. 11.2 min. TEACHERS EDITION. 87 11. If both taps of the last ex- ample be opened at once, how soon will they fill the tub ? ^ 4.2 2.5 3.5 6.6 18 336 42 18)45.6 5. 8.5 85)560.0 510 36 96 500 75.6 30 90 12. If 3 men can dig 378"° of ditch in 2 days, how long will it take 5 men, at the same rate, to dig 787^ 13. 48^ one tap is delivering water at the rate of 3.7' a minute ; while out of it, by another tap, the water is running at 2.5' a minute. How long will it take to fill the tub, beginning with it empty ? 3.71 2.51 1.21 40 min. 12)480 14. A tap discharges into a tub 4.2' a minute ; from the tub water is also running, by a sec- ond tap; the water in the tub gains 30' in 18 minutes. How fast is the second tap discharging ? 45.6 ^m? 2.5 15. If a wheel how often will it one kilometer ? 3.1416 1.2™ 62832 31416 is 1.2'" across, turn in going 378°» 189'» 315)787.0 630 1570 tub which will hold 265 63'" 5 315™ 377)100000 754 2460 3.76992°* = 3.77'" = 0.00377k'" 2262 Into a 1980 1885 16. How many times in a minute does the wheel of the last example turn, when the carriage is driven M""" an hour ? 14 ^ 60 = 0.23 265 0.23 795 530 60.95 = 61 times. AEITHMETIC. 17. What is the weight of the water in a tank if it would take a flow of 8.7^ a minute 1 hour and 38 minutes to empty it? 60 min. 38 rain. 98 min. 8.? 686 784 ' 852.6» - 852.6»« 18. Replace the bulk of water with oil worth $18.75 a hekto- liter, and what will the contents of the tank be worth ? 852.6^8 = 8.526" 118.75 42630 59682 68208 8526 1159.86 Exercise X. 1. A train leaves Paris at 11 o'clock a.m., and reaches Lyons at 10 o'clock P.M. How many meters does it travel in a hour, the dis- tance from Paris to Lyons being 512.7'"°? There are 11 hours between 11 a.m. and 10 p.m. Therefore the train runs 512.7"™ -5- 11 = 46.609'"" = 46,609°'. 2. A railroad has a single track 11.450*™ long. How many rails 4.569™ in length did it require to lay the track ? There are two lines of rails. Therefore length of rails is 2 X 11.450kn» = 22.900^ = 22.900™. Therefore the number of rails is 22,900 + 4.569. 5012 4569)22900000 22845 5500 4569 9310 9138 number of rails required was 5013. Ans. teachers' edition. 89 3. A book is 2.1<"^ in thickness ; each leaf is 0.05™"" thick. Find the number of pages in the book. The number of leaves is 21 ^ 0.05 - 420. The number of pages is 2 X 420 = 840. 4. The cost of opening a canal amounts to $25,400 a kilometer. How much would a canal cost which was 113. 253'"'" long ? If it cost ? 25,400 to open V^, to open 113.253 it will cost 113.253 X $25,400. 113.253 $25400 45301200 566265 226506 .2,876,626.200 5. The expense of laying out a paved road is $12,500 a kilometer. How much would a road cost which was 72,053^™ long? If it cost $12,500 to lay out 1^"^, to lay out 72.0531''" it will cost 72.053 X $12,500. 72.053 $12500 36026500 144106 72053 $900,662.50 6. The cost of building a railroad is about $78,000 a kilometer in France, and only $25,000 in the United States. How much would it cost in each country to make a road 295.67P™ long? If it cost $78,000 to build 1"^, to build 295.671''"" it will cost 295.671 X $78,000 = $23,062,338. If it cost $25,000 to build P'", to build 295.67P'" it will cost 295.671 X $ 25,000 = $7,391,775. 295.671 4 )29567100 $78000 $7,391,775 2365368000 2069697 $23,062,338,000 90 AEITHMETIC. 7. If you must go up 211 steps to reach the top of a tower, and each step is 195"^ high, what is the height of the tower ? 195inin = 0.195™. If one step is 0.195" high, 211 steps are 211 X 0.195" high. 0.195" 211 195 196 390 41.145" 8. A house has 5 stories, each story has 19 stairs, each stair is 16"" in height. Calculate how high the floor of the fifth story is from the ground. 16«" = 0.16". If one step is 0.16" high, 19 steps are 19 X 0.16" = 3.04", and 4 flights of 19 steps are 4 x 3.04" = 12.16". 0.16« 19 144 16 3.04 4 12.16 9. A ream of paper contains 20 quires, each quire has 24 sheets, the ream is 13.5"" in thickness. Find the thickness of each sheet. In one ream there are 20 x 24 = 480 sheets. If 480 sheets are 13.5«» thick, 13.5«" + 480 = 0.028«", thickness of one sheet. 0.028 48)1.350 96 390 384 teachers' edition. 91 10. The equator on a terrestrial globe measures 0.80™ in circumfer- ence. By the aid of a tape-measure we find that the distance between two cities on this globe is 0.046™. What is really the distance in kilo- meters between the two cities? (The earth's equator is 40,075.45'^™.) The ratio of distance on globe between the two cities to the equator is 0.046™ -V- 0.80™ = 0.0575. Therefore the actual distance between the two cities is 0.0575 x 40,075.45^™ = 2304.338''™. 8 )0.4600 40075.451^™ 0.0575 0.0575 20037725 28052815 20037725 2304.338'^™ 11. Upon a military map we find that the distance from Paris to St. Denis is 78™™. What is the distance in kilometers from Paris to St. Denis? The map is made on the scale of 1 to 80,000™ ; that is, 1™ on the map represents 80,000™ of actual measurement upon the ground. The actual distance is 80,000 times the distance on the map ; that is, 80,000 X 78™™ = 6,240,000™™ = 6.241^™. 12. Give the number of revolutions made by the wheels of a car- riage in travelling 82^^™. The wheels are 1354™™ in diameter. 82km ^ 82,000,000™™. The circumference of the wheels is 3.1416 X 1354™™ = 4253.7264™™. The number of revolutions is the total distance divided by the cir- cumference of the wheel, or 82,000,000™™ ^4253. 7264™™ = 19,277 times. 3.1416 19277 1354 ™™ 42537264)820000000000 125664 42537264 157080 394627360 94248 382835376 31416 117919840 85074528 4253.7264" 328453120 29776084<^ 306922720 297760848 92 ARITHMETIC. 13. IIow many hektars in a square kilometer ? how many ars ? how many square meters ? Iqkm = 100''*, = 10000* = 1000000*. 15. A piece of land 1224.5™ square is sold at $140 a hektar. How much does the land bring ? 1224.5 149.94 1224.5 1140 61225 599760 48980 14994 ^^^^^ $20,991.60 24490 12245 1499400.251™ = 149.94"*. 16. The total surface measurement of the glass in the windows of a house is 182*i"». How many panes of 53«™ by 48"^ will it take to supply the windows ? 182i'» = l,820,000<»« 53«"» 715.4 48°» 2544)1820000.0 m ^7808 212 3920 2544 2544qom^ area of one pane 13760 12720 10400 10176 .'. it will take 716 panes. teachers' edition. 93 17. How many square slabs of marble ISO^o"^ on the surface will it require to pave a court whose area is 25.35^™ ? 25.351"' = 253,500» 21. The railroad from Paris to Orleans has a double track ; each rail is 4™ long, and the distance from Paris to Orleans is 121'"". What is the number of rails used in laying the track ? The width of the road is 15™ ; how many hektars of land does the road include ? There are four lines of rails. 4 X 121''™ = 484"™ = 484,000™ of rails. If one rail is 4™ long, in 484,000™ there are 484,000^-4 = 121,000 rails. 15™ = O.OIS''™. The area of road is 121"™ X 0.015"™ = 1.815^"™ = 181.5''^ Ans. 121"™ 4 )484000 121^» _J 121000 rails. ^j^ 484km 605 121 1.8151^ 22. Calculate the number of ars in a surface which a ream of paper (180 sheets) will cover. The sheets are 30.3"™ long and 195™™ wide. 1{)5""" = 19.5«™. The area of one sheet is 30.3°™ X 19.5°™ =. 590.85<»°™. The area of 480 sheets is 480 X 590.85'«°» = 283,608«»««» = 0.283608*. 19.5 590.85 30.3 480 ~585 4726800 585 236340 590.85<>«» 283608.00V" teachers' edition. 95 23. A pile of wood is 4^.25^ long, 1.33"^ thick, and 2.60«» high. How many sters are there in it? 1^ = 1<'^'". In the pile of wood there are 4.25 X 1.33 X 2.60 = 14.6965«l""=14.696^ Ans. 4.25 5.6525 1.33 2^ 1275 339150 1275 113050 425 14.69650c^°» 5.6525 24. A beam is 7.070™ long ; its two other dimensions are 0.258"^ and 87*"™. Find its volume. 87mm = 0.087°'. Its volume is 0.258"» x 0.087™ X 7.070™ = 0.15869«^™. 0.258 0.022446 0.087 7.07 1806 157122 2064 157122 0.022446 0.15869322'=^'™ 25. A bar of iron 3™ long measures 45™™ square on the end where it has been evenly cut. The bar is heated and drawn out to a greater length by being passed through an orifice 24™™ square. What is the length of the bar after the operation ? 45™™ = 0.045™. 24™™ = 0.024™. The volume of the bar is 0.045™ x 0.045™ X 3™ = 0.006075«bm. The area of the end, after the bar has been boated is 0.024™ X 0.024™ = 0.0005 7()i™. Therefore the length of the bar is 0.006075 -h 0.000576 = 10.547™. 0.024™ 10.547™ 0.024 576)6075.000 96 576 48 0.002025*1™ 0.0005761^ 3150 2880 3 2700 0.006075°^™ 2304 3960 3892 96 ARITHMETIC. 26. A reservoir is 1.50™ wide, 2.80™ long, and 1.25™ deep. Find how many liters it contains when full, and to what height it would be necessary to raise it that it might contain lO**^™, The volume of the reservoir is 1.5 x 2.8 X 1.25 = 5.25«^™ = 5250». The area of the bottom is 1.5 X 2.8 = 4.2°^ will make 1.375 X 1.18«^°» = 1.6225<=bra. In twenty-four hours its volume will be 1.01 X 1.6225«^"i = 1.6387«b"\ 025^^™ 1.375cbm 1.6225 55 _L1^ 101 — 11000 16225 lor 1^'^^ 16225 ±:5_ 1375 1.375«^'" 1.62250«b'« 1.6387"''™ 30. A reservoir is 2.80™ long, 1.50™ wide, and 1.25™ deep. How many liters will be required to fill 0.80 of it? 1.5™ 4.201™ 52501 2.8 JL25 0.8 120 ^^^^ 4200.0 30 840 420 4.201™ 5.2500<=b™ = 5250\ volume. 98 ARITHMETIC. 31. A man buys 1415" of wheat for $ 3.50 a hektoliter ; but the measure used proves too small, the mistake amounting to 3* in every hektoliter. What wa.s the quantity of wheat delivered to the pur- chaser, the cost, and the reduction which ought to be made to him on account of the error? The mistake was 3^ in 100', or he received only 0.97 of 1415" = 1372.55". If 1" of wheat cost $3.50, 1415" cost 1415x|3.50 = $4952.50. A reduction of 0.03 of $4952.50 = $ 148.58 ought to be made. 1415" 1415 $4952.50 0.97 $3.50 0.03 9905 70750 $148.5750 12735 4245 1372.55" $4952.50 32. The dimensions of a tile are as follows : length 22*"", width 11""", thickness 55"'"'. Find the volume of the tile, and the number of tiles in a pile of 25<='"". 55min ^ 5 5cm. The volume of a tile is 22 x 11 X 5.5 = 1331«^". 25obm = 25,000,000«'"». In the pile there will be 25,000,000 h- 1331 = 18,732 tiles. 22«'" 18732 y 1331)25000000 22 1331 22 11690 242 10648 55 10420 1210 9317 1210 11030 1331.0ocm 10648 3820 2662 33. The measurement of a pile of wood shows that a ster could be filled from it 25.08 times. Give the volume of the pile in cubic meters, reckoning the length of the logs to be 1.15". teachers' edition. 99 The volume of a pile is 1 X 1 X 1.15 x 25.68 - 29.532<=bm. Ans. 25.68 l_15cbin 12840 2568 , 2568 29.5320«to°» 34. A liter of air weighs 1.273s. How much does a cubic meter of air weigh ? icbm = looQi. Therefore l^^™ of air weighs 1000 x 1.273s = 1273^ = 1.2731^8. Ans. 35. A package of candles which weighs 465^ is sold at 28 cents. What is the price of a kilogram of candles ? IK of candles costs i^O.28 ^ 465 = $0.000602. Therefore l^s costs 1000x^0.000602 = $0,602. 36. How many times would 3.243' of water 611 a liter? As 1' of water will fill a cubic meter, 3.243' will fill 3.243«^'n = 32431. 3243 times. Ans. 37. Give the weight in kilograms of 43.4578«°™ of pure water. 43.4578<=°°» of water weigh 43.4578s = 0.0434578''«. 38. The volume of an engine's axletree is 0.245'='^'". Find its weight, the specific gravity of the iron being 7.8. 0.245c*"n of water weigh 0.245*, and 0.245°^'" of iron weigh 7.8x0.245^=1.911*. 0.245 7.8 1960 1715 1.9110* 39. Calculate the volume of a gram of the following substances : proof spirit, specific gravity 0.865; tin, specific gravity 7.291 ; lead, specific gravity 11.35; copper, specific gravity 8.85; silver, specific gravity 10.47; cork, specific gravity 0.240. 100 ARITHMETIC. The volume equal vided by the specific 8 1«"», which, filled wi gravity. th water, weighs 1«, di- (i) (lii.) 0.088«'» (V.) 0.095«« 865) 1000.00 865 1135) 1( )0.000 9080 1047) 100.000 9423 1350 865 9200 9080 5770 5235 4850 (vi.) (iv.) 0.113<«» 885) 100.000 4 lOyccm (ii.) 0.14ccm 24) 100.000 96 7291)1000.00 - 885 40 7291 27090 1150 885 24 160 2650 144 160 40. Olive oil costs 60 cents a kilogram. Wiiat is the price of a liter? The specific gravity of olive oil is 0.914. As 1^1 costs $1.87, r costs 0.792 X$ 1.87 = $1.48. $1.87 0.792 V of olive oil weighs 0.914*8. As !"«? costs .f 0.60, V costs 0.914 X 10.60 = $0,548. 374 1683 1309 0.914 $0.60 $0.54840 41. Pure alcohol copts $ 1 .87 a kilogram. What is the price of a liter? The specific gravity of alcohol is 0.792. 1» of alcohol weighs 0.792>«. $1.48 42. A man wants to build a shed large enough to hold 135»* of wood ; if the shed is to be 3™ high and 5" wide, how long must it be ? 135-» - 135«t»'>». The ground area is 3x5 = 15*»"». Therefore the height must be 135 +15 = 7™. TEACHERS EDITION. 101 43. In a country where fire- wood is cut LIS'" long what height must the sides of the ster be to hold a cubic meter ? Tlie height must be Icbrn^ll^qm^ 0.86207"". 0.86207'" 116)100 00000 928 720 696 240 232 800 44. If a ster of cork cost 120.00, how much would lOOi^s cost, the cork weighing one quarter as much as water? l«t of cork weighs 250^s, and costs $20.00. lOO'^g will cost 100 2:50' = $8.00. 45. A liter of powder weighs 825«. What would be the vol- ume of a charge for a gun if the charge weighed 5*?? Calculate the volume in cubic centimeters. The specific gravity of powder is 0.825. It takes (1 ^ 0.825)««'" of powder to weigh 1^ ; therefore to weigh 58 it takes 5'^'^ ^ 0.825 6.06 825)5000.00 4950 5000 4950 46. Out of gold which weighs 19.362 times as much as water, sheets of gold foil are made which are O.OIO"*"* in thickness. What surface would 3^ of gold cover? 0.010»^°» = O.OOlc"^. The vol- ume of the gold is 3°«™ -^ 19.362 = 0.154943"°'". Therefore, the surface is 0.154943<'«'" -- 0.001°'" = 154.943i«'>\ 0.154943 19362)3000.000000 19362 106380 96810 95700 77448 182520 174258 82620 77448 51720 47. Find the weight of an oak board 3.25"! long, 0.31 "Mvide, and 0.04™ thick ; the specific gravity of the oak being 0.808. 102 ARITHMETIC. The volume of the board is 3.25 X 0.31 X 0.04 = 0.0403«»'«n. P*"" of oak weighs 0.808* ; there- fore 0.0403«''"» weigh 00.403 X 0.808*= 0.0325624« = 32.5624^8. 3.25 0.31 325 976 1.0075 0.04 0.040; lOO^"™, volume. 0.0403 0.808 3224 3224 0.0325624 48. Find the weight of a bar of iron having the following dimensions: length 3.0", width 6''"», thickness 2*^; the specific gravity of the iron being 7.8. 3.6'" = 360^™. 360 6 2160 2 4320ccm^ volume. 4320 7.8 34560 3024 33696.0* «= 33.696k« 49. How many lead balls each weighing 27« could be obtained by melting a mass of lead, cubic in form, the edge measuring 0.356", the specific gravity of the lead being 11.35? 0.356™ = 35.6«°». 35.6<«»» 45118.016 18966 35.6 11.35 27)5120S9 2136 225590080 27 1780 135354048 242 1068 45118016 216 1267.36 35.6 45118016 512089.481601 260 OA9 760416 633680 380208 45118.016<«», volume. 178 162 169 162 teachers' edition. 103 50. Marble costs 1 30.95 a cubic meter, and the specific gravity of marble is 2.73. If a block of marble weighs 1260'^^^ what is its vol- ume, and what is it worth ? I'^to'" of marble weighs 2.73'. 1260'^^ = 1.26'. 0.45i5cbm 0.4615 273) 126.0000 ^ 30.95 1092 23075 1680 41535 1638 13845 420 114.28 273 1470 1365 51. Sea-water contains 28 parts, by weight, of salt in 1000. A liter of sea-water weighs 1.025^s. How many kilograms of salt could be obtained from 126.276842'=»"» of sea-water ? V^s of sea-water contains 0.028^8 of salt. 126.276842 129433.753 1.025''8 0.028''8 631384210 1035470024 252553684 258867506 ^^^^^^^^^ 3624.145084'^g 129.433753050''g 52. An empty cask weighs 17.06'^K; when filled with water it weighs 275.8''8. How many liters does it hold? How many casks of this size would it require to receive the wine from a vat containing 3.008°^'^ ? The cask will hold 275.8'^s - 17.06''« = 258.741^8 of water. It takes 258.741 of water to weigh 258.74i^g. Therefore the cask will hold 258.741. 3.008<'i>°' = 30081. If one barrel holds 258.741, to hold 30081, it will take 3008 - 258.74 = 12 barrels. 275. 801^8 ^2 17.06 25874)300800 25874 258.74i'8 42060 104 ARITHMETIC. 53. It takes about 204.8' of wheat to sow a hektar. How many cubic meters would it take to sow a square kilometer ? iqkm = looha. P* will require 100 x 204.8» = 20,480' = 20.48«»>'». Aru. 54. A piece of road l""" long and 7"* wide is to be macadamized ; the macadamizing is to be 33«" deep ; it costs 43 cents a cubic meter to prepare the stones. What will enough for the road cost? Itm = IQQQm . 330m = O.SSn*. 0.33 2310 7 $0.43 2.31 6930 1000 9240 2310.00 $993.30 55. A gasometer holds 28,000*5*° of gas. How many jets would this gasometer feed, when each jet burns 125' an hour, and is used 4 hours every evening? Each jet will burn 4 X 125' = 500' each evening. 28,000«»>°» = 28,000,000'. The gasometer will feed 28,000,000 ^ 500 = 56,000 jets. 56. The city of Venice is situated in the midst of a great lake of salt water, communicating with the sea, and all the rain- water is caught for the cisterns. Ordinary years the fall of rain in Venice is 82*=™ ; the surface of the city, after the canals have been deducted, is 520''*; reckoning the population at 115,330, how many liters a day of rain-water could each inhabitant have? 520'"' = 5,200,000<»°» ; 82<=™ = 0.82'". The average amount of rain-water is 5,200.000 X 0.82 = 4,264,000«'»«» = 4,264,000,000'. Each person can use per year 4,261,000,000 + 115,530, or, per day, 4,264,000,000 + (115,530 X 365) = 101.118'. 0.82 115,530 365 1U1.11»' 5200000 4216845)426400000.000 16400000 577650 4216845 410 693180 4715r)00 4264000.00 346590 4216845 42168450 4986550 4216845 7697050 42I6.S45 34802050 33734760 teachers' edition. 105 57. Find the weight of a bar of iron 5.35" long, 4.56°'" thick, and 3.51"^™ wide. Find, also, the width of an oak beam 4.30°* long, 9.12*=°^ thick, which has the same weight. The specific gravity of the oak to be reckoned at 1.026, that of the iron 7.788. 5.35m _ 535cm^ 4 3om _ 430cm_ The volume of the oak beam is 67,258.5969928 ^ 1.0262 = 65,554.1 88«=°\ The area of one side of the oak beam is 430 X 9.12 = 3921.6i<"" ; therefore thickness is 65,554.188<=«'" -j-3921.6^°i" = 16.72°'". 4.56<='^ 16.1424 8636.184 3.54 535 7.7888 1824 807120 69089472 2280 484272 69089472 1368 807120 8636.1840<"''», , volume. 60453288 16 1424q°°» 60453288 65554.188 67258.596992s = 67.259'^g. 1026)67258596.992 6156 16.72<'«^ 6698 392: 16)655541.88 5130 5685 5130 5559 39216 263381 235296 51.30 280858 4296 274512 4104 1929 63468 1026 9039 8208 8212 58. Give the specific gravity and volume of a body weighing 35'^8 in air and 30^^ in water. The weight of water displaced by the body is 5^^. The weight of body in air is 35^8. Therefore specific gravity is 35 -5- 5 = 7. 35^ of water weigh 35^^^ ; 35 -T- 7 = 5\ volume. 106 ARITHMETIC. 69. A Bter of piled oak wood weighs 42o*« ; the specific gravity of the wood is 0,74. What is the volume occupied by the spaces between the logs ? For how much must lOO^s of separate sticks be sold in order to bring the same amount as when sold by the ster ; aster selling for $2.20? If there were no spaces between the logs, the ster of wood would weigh 740''K. Therefore the spaces, if filled with wood, would weigh 740^8 _ 425»'8 = SIS"*. Therefore volume of spaces is 315 ^ 740 = 0.42568<""». lOQi^K ought to be sold for |f f of $ 2.20 = $ 220 h- 425 = $0,518. 0.425680"°' $0,518 74 Jsi. 50000 425) $220,000 296 2125 190 ^ 148 425 420 370 3250 500 444 560 60. Wrought iron sells for $7.00 per lOO'^R. A bar of iron 4.5o°» wide, 3.3°™ thick costs $5.08; what is its length, reckoning the spe- cific gravity of the iron at 7.4 ? $7.00 per lOO''* is the same as $0.07 per kilogram. An iron bar that costs $5.08 must weigh 5.08 ^ 0.07 = 72.57143*8, and its volume is 72.57143 -i- 7.4 = 9.8066» = 9806.5''cm. The area of an end of the bar is 4.5«™ X 3.3<"" = 14.85vm Therefore the length is 9806.6 + 14.85 = 660.4''™ = 6.604°*. 9.8066> 660.4°» 74)725.7143 1485)980650.0 666 8910 f7 • -^ ^4 ^ 444 5500 l03 666 TEACHERS EDITION. 107 61. Experiment shows that water weighs 770 times as much as air ; and the specific gravity of mercury, in comparison with water, is 13.6. How many liters of air will it take to weigh as much as a liter of mercury ? "Water is 770 times as heavy as air, and mercury is 13.6 times as heavy as water. Therefore mercury is 13.6 X 770 times as heavy as air. 13. 770 ters, the surface which can be covered by the sheets thus ob- tained. The specific gravity of the lead is 11.3. The volume of the lead is 753 H- 11.3 = 66.637' = 0.066637<"'°». imm _ 0.0001™. The surface of the lead is 0.066637'='"" H- 0.0001 °» = 666.37i"*. 66.63? ^'^s- 113) 7530.000 678 750 678 720 678 420 339 810 797 9520 9520 10472.0 62. A mass of lead weighing 7531^8 is made into sheets O.l''^"* thick. Calculate, in square me- 63. A rectangular sheet of tin of uniform thickness is 85*'™ wide, 1.35'" long; it weighs 268«. What is its thickness, reckoning the specific gravity of tin at 7.3 ? The volume of the tin is 268 -=- 7.3 = 36.7109<=<'«' ; 1.35°» = 135<'°'. The area of the tin is 135<=™ X 85""= 11,475<*<'™ ; therefore its thick- ness is 36.7109«"» -^ ll,475«i<"" = 0.0032«°'. 36.7109««°' 135 0.0032^ 73) 2680.0000 85 11475)36.7109 219 675 1080 34325 490 23859 438 520 11475qcin 22950 511 90 73 700 657 108 ARITHMETIC. 64. The fine coal which collects about the shafts of the mines and in the coal-yards, was for a long time wasted, because it could not be burned in stoves and grates. Now, this dust is mixed with tar in proportion of 92''k of dust and 8^^ of tar ; the mixture is heated, and afterwards pressed in rectangular moulds of 14. 75*™, 18.5<=°», and 29<'™ ; each one of these blocks weighs lO''^ ; they are sold at $3.00 a ton, and make excellent fuel for heating steam boilers. Give the specific gravity of this fuel ; also, the sum which would be realized in thus utilizing 800,000* of coal dust, the cost of tar, mixing, etc., being f 0.50 a ton. Volume of a block is 14.75 x 18.5 x 29 = 7913.305<^ = 7.9133051. Specific gravity is 10 -r- 7.913305 = 1.264. 800,000* of coal dust will make 800,000' ^ 0.92 = 869,565.217* of the mixture. 869,565.217* at $2.50 per ton = 869,565.217 X $2.50 = $2,173,913.04. 14.75«° 18.5 7375 11800 1475 272.875 29 2455805 545750 7913.305~» 1.264 79133) 100000.000 79133 208670 158266 504040 474798 292420 869565.217 92) 80000000.00 736 640 552 880 828 520 460 600 552 480 460 200 184 160 92 680 644 869565.217 $2.50 43478260850 1739130434 $2,173,913.04 TEACHERS EDITION. 109 65. A. bar of iron a millimeter square on the end will break un- der a tension of SO'^s, Find the length at which a suspended bar of iron will break from its own weight, the specific gravity of the iron being 7.8. 301^8 = 0.03*. The volume of the iron bar is 0.03 ^ 7.8 = 0.00384615'^t.in The area of an end of the bar is 1mm y^ 1mm ^ ^qram ^ O-OOOOOl^l™. Therefore the length of the bar is 0.00384615«^«^ -^ O.OOOOOli'" = 3846.15">. 0.00384615°^'^ 78)0.30000000 234 660 624 360 312 480 468 120 78 420 390 66. Fifty-three kilograms of starch are obtained from lOO'^s of wheat. A hektar of land pro- duces 1363 of wheat; a hekto- liter of wheat weighs 78^8. If the wheat harvested from a field measuring 2*'* and 33i»» is taken to a starch factory, how much starch will be made from it? 0.53'^s of starch are obtained from Ps of wheat. 1^ of wheat weighs O.VS^K. P^produces 1363 X 0.78^^ of wheat = 1063.14i^«, 2ha 33qm _ 2.0033iia. 2.0033^* produce 2.0033 x 1063.1 4"^^ = 2l29.7883f>2kg of wheat. The amount of otarch is 1128.7878'^k. 1363 0.781's 10904 9041 10G.;.14'^8 2.0033 318912 318942 212628 2129.788362»'e 2129.788 0.53 6389364 10648940 1128.78764''K 67. A gardener wishes to pro- vide glass for his hot-beds. The beds cover 2.65* ; the panes will cover 0.75 of the whole surface, the rest being taken up by the frames and alleys. First, find how many panes measuring 45<=™ by 37°™ it will take to cover the beds ; then find the price of the glass, at a cost of 95 cents a square meter. 110 ARITHMETIC. 45«« = 0.45°' ; 37"° = 0.37™ ; 2.65» = 265«i'°. Total area of the glass is 0.75 of 265'«'° = 198.75«»"». The area of one pane is 0.45 X 0.37 = 0.1665*i™. Therefore the number of panes needed is 198.75 + 0.1665 = 1194. At |0.95 per square meter, 198.75'»'° will cost 198.75 X $0.95 = $ 188.81. 0.45"' 1194 198.75 $0.95 99375 178875 0.37" 315 135 1665)1987500 1665 8225 1665 15600 14985 0.16651" $188.81 6150 68. A jar full of water weighs 1.325*«; filled with mercury it weighs 12.540''8, What is the capacity of the jar, and its weight? The specific gravity of the mercury is 13.59. The weight of the jar and the jar full of mercury is 12.540''-. The weight of the jar and the jar full of water is 1.325''8. Therefore the difference in weight between the mercury and the water is 12.540>'8 - 1.325''8 = 11.215''8. 13.59 - 1 = 12.59, the specific gravity of a liquid of which the jar full without the jar weighs 11.215''«. Hence the capacity of the jar is 11.215 -f- 12.59 = 0.890791. 0.89079> of water weigh 0.890791^8. Therefore weight of jar is 1.325 - 0.89079 - 0.43421''K = 434.21*. 12.540k« 1.325 1259) 0.890791 1121.50000 10072 1.325001^3 0.89079 11.215k« 0.4312i*« 11430 11331 9900 8813 10870 10072 TEACTTERS' EDITION. HI 69. A hektoliter of rape-seed weighs GoI^k, and 32^ of oil can be extracted from it. How many kilograms of the seed will it take to make a hektoliter of oil ? Ihi =. lOQi. If 32^ of oil can be extracted from eS^« of seed, V of oil can be extracted from 63 -4- 32 = 1.96875'^« of seed, and 100' of oil can be extracted from 100 X 1.96875'^g = 196.875''s of seed. 1.96875 32)63.00000 32 310 288 220 192 280 256 240 224 160 160 70. Common burning gas is 0.97 of the weight of air, and a liter of air weighs 1.293s. In a shop there are 65 jets, each one of which burns 123' an hour, and is used 5 hours in the winter evenings. Calculate the weight of the gas used in a month, and the expense of lighting the shop, when gas costs 6 cents a cubic meter. 1' of gas weighs 0.97 X 1.2938 = 1.25421«. 65 jets, each burning 1 23' an hour, and used 5 hours an evening for 30 days, will use 65 X 5 x 30 X 123' = 1,199,250', the weight of which is 1,199,250 x 1.25421k = 1,504,111.348 =1504.11134'^«. 1,199,250' = 1199.25«'»n. The ex- pense at 1 0.06 per cubic meter is 1199.25 x |0.06 = $71.96. 1.2938 123' 1199250 1199.25 0.97 65 1.254218 $0.06 9051 615 738 1199250 239850 $71.9550 = $71.96 79951 479700 L.254218 5 599625 39975' 239850 30 119925 1504111.342508 1199250 112 ARITHMETIC. 71. A merchant buys one kind of wine at 30 cents a liter, another kind at 21 cents a lit«r ; he mixes the two kinds by putting 5' of the first with 8^ of the second. For how much a liter must he sell the mixture in order to gain $3.75 a hektoliter? 5> at $0.30 per liter cost $1.50. 81 at $0.21 per liter cost $1.68. Therefore 13» of the mixture cost $1.50 + $1.68 = $3.18, and I'costa $3.18 ^ 13 = $0.2446. Again, if $3.75 per hektoliter is equivalent to a gain of $0.0375 per liter, to make $3.75 per hektoliter the mer- chant must sell the wine for $0.0375 + $0.2446 = $0.2821 per liter. $0.30 $0.21 0.2446 5 ? 13)3.1800 $1.50 11-68 26 1.50 58 $3.18 52_ GO 52 80 72. If it requires 360 tiles to drain an ar of land, what will it cost to drain 17.784''*, when the tiles cost $20 a thousand, and the ex- pense of laying is the same as the cost of the tiles? The expense of laying the tiles and their cost is $40 per thou- sand. 17 784»"' = 1778.4*. To drain 1778.4* of land 1778.4x360 tiles = 640,224 tiles = 640.224 thousand are needed. 640.224 thou- sand at $40 per thousand cost 640.224 X $40= $25,608.96. Ans. 1778.4 640.224 360 $40 1067040 $25608.960 53352 640224.0 73. It is found in building that hewn stone of medium durability ought not to support, as a permanent weight, more than 0.07 of the teachers' edition. 113 weight that it would require to crush it. A certain kind of stone used for building will be crushed under a weight of 250^^ a square centimeter. What is the greatest height to which a wall constructed of this material can be safely carried, the specific gravity of the stone being 2.1 ? 250^^^ per square centimeter is equivalent to 250,000^ per square centimeter. 0.07 of 250,000s = 17,5008 ought to be the pressure on a square centimeter. Therefore volume of imaginary prism ought to be 17,500 ^ 2.1 = 8333.33««'", or the height ought to be 8333.33'='^ = 83.333°^. 8333.33«<=°» 21)175000.00 168 "to 63 "To 74. Several kinds of wines are mixed as follows : 245^ at 20 cents a liter, 547' at 15 cents a liter, 344' at 25 cents a liter. How much does the mixture cost a liter ? 2451 at $0.20 per liter cost $49.00 547' at $0.15 per liter cost $82.05 344' at $0.25 per liter cost $86.00 1136' of the mixture cost $217.05 'herefore 245 $0.20 1» costs $217.05- 547 $0.15 -1136 = $0,191. 344 $0.25 Ans. 1136) $0,191 217.050 $49.00 2735 547 $82.05 1720 688 1136 10345 $86.00 10224 1210 1136 75. A farmer wishes to drain a field of 8.75'*'^. Each hektar re- quires 750"" of ditches. The opening of these ditches costs 10 cents a running meter ; the tiles are 30*=™ long, and cost $15 a thousand. He pays 2 cents a meter for laying the tiles, and 4 cents a meter for fill- ing the ditches. What is the cost of draining the field? 114 AKITIIMETIC. There are required 8.75 x 750"* = 6562.5°» of ditches. The expense of opening the ditches, laying the tiles, and filling the ditches is $0.10 + .$0.02 + $0.04 = $0.16 per meter. 6562.5°> will cost 6562.5 X $0.16 = $ 1050.00. SO^-" = O.S". For 6562.5'°, 6562.5 -f- 0.3 = 21,875 tiles are necessary. The tiles cost $15 per thousand. Therefore 21.875 thousand cost 21.875 x $15 == $328.13. Hence cost of draining the field is $1050.00 +$328.13 =$1378.13. 8.75 6562.5 21.875 $1050.00 750» $0.16 $15 328.13 43750 393750 109375 $1378.13 6125 (K5fi25 21875 6562.50°» $1050.000 $328,125 = ' $328.13. Exercise XI. 1. Find the prime ftictors of 148 ; 264; 178; 183; 173; 187; 346 343. 2^ 1148 2»[264 2[178 3| 183 37 3 [^ 89 61 22 X 37. Ans. 11 2 x 89. Ans. 3 X 61. Ans. 23x3x11. Ans. 1 |173 173 1x173. Ans. 11 1 187 17 11 X 17. Ans. 2 1 346 173 2x173. Ans. 7' 1 343 1 7'. Ans. 2. Find the prime factors of 210 ; 353; 5280; 231; 31,416; 1369; 1368. 210 105 35 7 2x3x5x7. Am. 2» 353 1x353. Ans. 5280 165 _55 11 2'»X 3x5x11. Ans 31231 7 1 77 n 3x7x11. Ans. 31416 3927 1309 3711369 37 37x37. Ans. 1368 171 187 19 2»x3»xl9. Ans. 17 2'x3x7xn xl7. Ans. TEACHERS EDITION. 115 3. Find the prime factors of 247 ; 327 ; 179 ; 83 ; 2125 ; 2353 ; 2333. 13 1 247 3 1 327 1 [179 1^83 19 109 13 X 19. A71S. 3 X 109. Ans. 179 1x179. Ans. 83 1 X 83. Ans. 5^ 12125 17 5^ X 17. Ans. 13 1 2353 181 13x181. Ans. 1 1 2333 2333 1x2333. Ans. 4. Find the prime factors of 165 ; 168 ; 2148 ; 16,662 ; 321 ; 1551 ; 38. 31165 5155 11 168 21 7 2212148 3 I 537 179 21 16662 31 8331 3x5x11. Ans. 2^ X 3 X 7. Ans. 2^ x 3 X 179. Ans. 2 x 3 x 2777. Ans. 3 1 321 107 3 X 107. Alls. 31 1551 11 1 517 47 3 X 11 X 47. Ans. 2[33 19 2 X 19. Ans. 5. Find the prime factors of 82 ; 129 ; 72 ; 66 ; 68 ; 65 ; 76 ; 86 ; 3; 142. 2j66 22 [68 3133 17 11 22 X 17. Ans. X32. Ans. 2x3x11. Ans. 2[82 3|129 23 72 41 43 32 9 2 X 41. Ans. 3 X 43. Ans. 1 2^x32. A 5[65 22[76 2186 13 19 43 23 1 11 2 1 142 7L 5x13. ^ns. 22x19. ^ns. 2x43. ^Ins. 23x11. ^ns. 2 X 71. ^ns. 6. Find the prime factors of 326 ; 368 ; 464 ; 292 ; 362 ; 365 ; 730 ; 42. 21326 2*1 368 2* 1464 22 1 292 163 23 29 73 2 X 163. Ans. 2*x23. Ans. 2*x29. Ans. 22 X 73. Ans 2|362 5|365 2 730 2 142 181 73 5 365 3 21 2 X 181. Ans. 5 X 73. Ans. 73 7 2 X 5 X 73. Ans. 2x3x7. Ans. 116 ARITHMETIC. 7. Find the prime factors of 868 ; 999 ; 822 ; 1346 ; 7641 ; 6234 234. 2^1868 3«|999 2 1822 2 |1346 7 [217 37 SgU 673 31 3'x37. ^715. 137 2x673. Ans. 2'' X 7 X 31. Ans. 2 x 3 x 137. Ana. 3^ 17641 283 33x283. Ans. 6234 3117 1039 2x3x1039. Ans. 2 234 32 [117 13 2x3^x13. Am. 8. Find the prime factors of 579; 577; 212; 126; 128; 8192; 8190. 3 1 579 1 |577 2^1212 2 1126 193 577 53 32 [^ 3x193. Am. 1x577. Am. 2^x53. Am. 7 2x32x7. Am. 271128 21' 18192 2 8190 1 1 32 4095 21 Ans. 2". Am. 5 455 7 91 13 2x32x5x7x13. Am. 9. Find the prime factors of 8197; 3125; 2401; 1331; 78,309; 25.179. 7|8197 1171 7x1171. Am. 32 S'^jSl 5^ A 78309 25 1 m. Am. 7* 7*. 2401 1 Ans. 3x7 3 7 11 1P|1331 1 IP. Am. 25179 7 8701 8393 11 1243 1199 32x7x 113 11x113. X^ 109 11X109. Am. TEACHEES EDITION. 117 Exercise XII. 1. Find the prime factors of 8.4 ; 7.6; 1.08; 0.144; 0.036; 0.037; 21.45. 8.4 = 84x0.1. 7.6 = 76x0.1. 1.08 = 108x0.01. 0.144 = 144x0.001. 22 184 22 [76 22M08 2^ 144 3 m 19 32 1^ 32 9 7 1 1 22x3x7x0.1.^ns. 22xl9x0.1.^ns. 22x32x0.01.^ws. 2*x 32x0.001. ^?7s. 0.036 = 36 X 0.001. 0.037 = 37 X 0.001. 22 1 36 1[37 3^\_9 37 1 1x37x0.001. Ans. 22x32x0.001. Ans. 21.45 = 2145x0.01. 2145 715 148 13 3x5x11x13x0.01. Ans. 2. Find the prime factors of 14.6; 2.61; 21.2; 78.54; 0.5236; 0.00052. 14.6 = 146x0.1 2.61 = 261x0.01. 21.2 = 212x0.1. 2|146 32 1 261 22 1 21 2 73 29 53 2 X 73 X 0.1. Ans. 32 x 29 x 0.01. Ans. 22x53x0.1. An 78.54 = 7854 x 0.01. 0.5236 = 5236x0.0001. 2 7854 22 7 11 5236 3 3927 1309 187 1309 7 187 11 17 17 22x7xllx 17x0.0001. Ans. 2x3x7x11x17x0.01. Ans. 0.00052 = 52x0.00001 22 [52 13 2^X13X0.00001. Ans. 118 ARITHMETIC. 3. Find the prime factors of 86.7 ; 48.3 ; 99.99 ; 5.04 ; 1.485 ; 0.216. 86.7 = 867x0.1. 48.3 = 483x0.1. 99.99 = 9999x0.01. 3 IT' 867 289 1 3 1483 71161 23 32 11 9999 UU 101 3x172x0.1. ^ns. 3 X 7 X 23 X 0.1. ^ns. 3^x11 X 101 x 0.01. ^rw. 5.04 = 504x0.01. 1.485 = 1485x0.001. 0.216 = 216x0.001. 231504 3^ 1 63 7 1485 55 11 216 27 1 2^x32x7x0.01. Ans. 33x5x11x0.001. Ans. 23x3^x0.001. Am. 4. Find the prime factors of 34.87 ; 32.4; 5.115; 71.2; 2.993. 34.87 = 3487x0.01. 32.4 = 324x0.1. 5.115 = 5115x0.001. 11 [3487 317 11x317x0.01. Ans. 22 1 324 3*1 81 1 22 X 3* X 0.1. Alls. 5115 . 1 705 341 31 71.2 = 712x0.1. 23 [712 89 23x89x0.1. Am. 3x5x11x31x0.001. Am. 2.993 = 2993x0.001. 41 1 2993 73 41x73x0.001. Am. Exercise XIII. 1. Find the G. C. M. of 27 and 33. 3[27 33 9 11 3. Am. 2. Find the G. CM. of 13 and 39. 13[13 39 1 3 13. Am. 3. Find the G.C.M. of 8 and 28. 2* 1 8 2^ 2 7 2^ = 4. Am. 4. Find the G. C. M. of 27 and 45. 3«[27 45 3 6 3« = 9. Am, TEACHERS EDITION. 119 5. Find the G. CM. of 81 and 108. 3^ 181 108 3 4 33=27. Ans. 12. 6. Find the G.C.M. of 4, 10, 2 [4 10 12 2 5 6 2. Ans. 7. Find the G. C. M. of 4. 6. 10. 2| 4 6 10 2. 3 5 2. Ans. 8. Find the G. C. M. of 9, 12, 21. 3|9 12 21 3 4 7 3. Ans. 9. Find the G. C. M. of 10, 15, 25. 5|10 15 25 2 3 5 5. Ans. 10. Find the G. C. M. of 14, 98, 42. 2 14 98 42 7 7 49 21 1 7 3 2x . 7 = 14. Ans. 11. Find the G. C. M. of 30, 18, 54. 2 130 18 54 3 115 9 27 5 3 9 2x3 = 6. Ans. 12. Find the G. C. M. of 14, 56. 42. 2 14 56 42 7 7 28 21 1 4 2x7 = 14. 3 A77S. 13. Find the G. C. M. of 96, 36, 48. 22 96 36 48 3 24 9 12 ^X 8 3 = 3 4 12. Ans 14. Find the G. C. M. of 84, 105, 63. 84 105 63 28 35 21 3 7^ ^4 5 3 3x7 = 21. Ans. 15. Find the G. C. M. of 24, 60, 84, 128. 2^ 124 60 84 128 6 15 21 32 22 = 4. Ans. 16. Find the G. C. M. of 45, 81, 27, 90. 32 1 45 81 27 90 5 9 3 10 32 = 9. Ans. 17. Find the G. C. M. of 78, 18, 54. 42. 2 78 18 54 42 3 39 9 27 21 13 3 9 7 2x3 = 6. Ans. 120 ARITHMETIC. 18. Find the G. C. M. of 98, 28. 70. 42. 98 28 70 42 49 14 35 21 7 2 5 3 2x7=14. Ans. 19. Find the G. C. M. of 96, 112, 80, 32. 2^1 96 112 80 32 6 7 5 2 2* =. 16. Ans. 20. Find the G. C. M. of 24, 96, 48, 120. 2^ 1 24 96 48 120 3 I 3 12 6 "~l5 14 2 5 23x3 = 24. Ana. 21. Find the G. C. M. of 84, 252, 108, 210. 2 SI 252 168 210 3 42 126 84 105 7 U 42 28 35 2 6 4 5 2 X 3 X 7 = 42. Ans. 22. Find the G. C. M. of 33, 88, 77, 55. 11 1 33 88 77 55 3 8 7 5 11. Ans. 23. Find the G. C. M. of 252, 315. 420. 504. 252 315 420 504 84 105 140 168 12 15 20 24 3x7 = 21. Ans. 24. Find the G. C. M. of 128, 192, 320, 368, 432. 2*1128 192 320 368 432 8 12 20 23 27 2* = 16. Ans. 25. Find the G. C. M. of loO, 204, 357. 459. 17 1 136 204 357 459 8 12 21 27 17. Ans. 26. Find the G. C. M. of 909, 1414, 2323, 4242. 101 1909 1414 2323 4242 9 14 23 42 101. Ans. Exercise XIV. 1. Find the G.C.M. of 2479 and 3589. 2479)3589(1 2479 lOl lllO 3 1 HI 37)2479(67 222 37. Ans. 259 259 2. Find. the G.C.M. of 3015 and 6195. 3045 6195 609 1239 203 ) 413(2 406 7)203(29 14 5x3x7 = 105. Ans. (J3 63 TEACHERS EDITION. 121 3. Find the G.C.M. of 568 and 712. 8|568 712 71 ) 2 32 8. Ans. 89(1 71 18 9 1 4. Find the G.C.M. of 11,023 and C)493. 6493) 11023 (1 6493 10 4530 3 453 151)6493(43 604 453 151. A 453 ns. 5. Find the G. C. M. of 1485 and 2160. 1485 2160 297 432 11 16 5x33 = 135. Ans. 6. Find the G. C. M. of 7040 and 7392. 32 7040 7392 10 220 231 22 11)231(21 22 Ti n 32 X U = 352. Ans. 7. Find the G. C. M. of 2760 and 4485. 3 5 2760 4485 920 1495 184 299 23)299(13 23 ~69 ()9 3 X 5 X 23 = 345. Ans. 8. Find the G.C.M. of 1177 and 2675. 1111177 107)2675(25 214 ~535 535 107. Ans. 9. Find the G.C.M. of 78,473 and 94,653. 78473)94653(1 78473 10 16180 1618 809) 78473 (97 7281 ~5663 5663 809. Ans. 122 ARITHMETIC. 10. Find the G. CM. of 36,143 and 10,283. 10283) 35143 (3 30849 2 19 4294 2147 113)10283(91 1017 113. Ans. 11. Find the G.C. and 61,087. 44323)61087(1 44323 113 113 M. of 44,323 4 16764 3 4191 11 1397 127. Ans. 127)44323(349 381 622 608 1143 1143 12. Find the G. CM. of 232,353 and 39,699. 11139699 232353 9 3609 401 ) 21123 2347(5 2005 342 11x9 = 99. Ans. _57 19)401(21 38 21 19 2)19(9 18 1 13. Find the G. C. M. of 33,853 and 35,017. 33853)35017(1 33853 4 1164 291 97)33853(349 291 475 388 873 873 97. Am. 14. Find the G.C.M. of 5115 and 7254. 3 5115 7254 5 11 1705 2418 341 31)2418(78 217 248 248 3x 31 = 93. Ana. 15. Find the G.C.M. of 2268 and 3348. 4 9 3_ 21 31 4x9x3=. 108. Am. 2268 3348 567 837 63 93 TEACHERS EDITION. 123 16. Find the G. C. M. of 1003 and 2419. 18. Find the G. CM. of 30,072 and 133,784. 1003)2419(2 8 7 3 30072 133784 2006 3759 l()723 7| 413 59)1003(17 59 413 59. Ans. 413 537 2389 179)2389(13 179 599 537 2 3 42 21 17. Find the G. C. M. of 419 and 52,.301. 419) 52301 (124 419 1040 7 1|179 179 838 3 5 2021 1676 345 115 21 7 1|419 419 19. Fi and 10,8: 9 11 9x4 Qd the G.C.M. of 4527 56. 4527 10836 3 7 1. Ans. 473 43 3 = 387 7 1 1204 ) 172(4 172 Ans. 20. Find the G. C. M. of 17,104 and 27,794. 17104 27794 8552 13897 1069)13897(13 1069 3207 3207 2 X 1069 --2t6^. Ans, 124 ARITHMETIC. Exercise XV. 1. Find the G. C. M. of 855, 1197, 1596. 855 1197 1596 285 399 95 7 |13^ 19 19 3X19 = 57. Ans. 41532 7 |T33 19 2. Find the G. C. M. of 3864, 3404, 3657. 3864 340 ; 9(i6 851 161 23}_ 851 3 )3657 1219 1219 37 23. Ans. 53 3. Find the G. C. M. of 15,561, 11,115, 13,585. 13 111115 13585 15561 855 5 95 11 1045 7 209 9 1197 171 19 19 13x19 = 247. Am. 19 4. Find the G.C.M. of 2943, 2616, 4578. 3 2943 2616 4578 9 1 981 8 1 872 21 1526 109 109 7 1 763 109 3x109-327. Aiu. 5. Find the G. C. M. of 1177. 1391, 1819. 11|1177 107)1819(17 107 749 749 107)1391(13 107 321 321 107. Ans. 6. Find the G. C. M. of 4939, 1347, 3143. 11 14939 449)1347(3 1347 449)3143(7 3143 449. Am. 7. Find the G. C. M. of 740, 333, 296. 21 740 9 1 333 8 |296 10 [370 37 37 37 37. Am. 8. Find the G. C. M. of 833, 1785, 1309. 7 1 833 3 11785 11 1 1309 119 5 I 595 119 119 119. Am. TEACHERS EDITION. 125 9. 6 Find the G. C. M 7326 7 1221 11 . of 735 8547 6, 8547, 9768, 22,755. . 8 9768 11 1221 5 41 22755 11 1221 4551 10 214^ 111 111. Ans Find the )94 3 G.C.J 7491 111 .1. of4£ 4 11 94, 7491, 9988 2497 111 9988, 12,485, 5 12485 11 2497 227 111 16,571. 73 1112497 11 2497 227) 16571 227 227 227. Ans. 227 1589 681 681 Exercise XVI. 21. 1. Find the L.C. M. of 6, 14, 2 [6 14 21 ? J 21 2 X 3 X 7 = 42. Ans. 2. Find the L. C. M. of 8, 12, 3,24. ^ ;^ 3 24. 24. Ans. 3. Find the L.C. M. of 6, 10, 15. 2 |6 10 15 3 ^ 15 2 X 3 X 5 = 30. Ans. 4. Find the L. C. M. of 9, 12, 18,4. 2 1^ 12 18 i 3 1 6 9 2 3 22 X 32 = 36. Ans. 5. Find the L. C. M. of 15, 21, 35. 3115 21 35 J 35 105. Ans. 24 3x5x 6. Find the L. C. M. of 12, 20, ;? 20 24 10 12 6 5 3 23 X 3 x 5 = 120. Ans. 7. Find the L.C.M. of 14, 24, 28. 22] ;^ 24 28 6 7 23 X 3 X 7 = 168. Ans. 20. 8. Find the L. C. M. of 12, 15, 3 1 12 15 20 ^ ^ 20 22x3x5 = 60. Ans. 126 ARITHMETIC. 32. 77. \)\). 13. 9. Find the L. C. M. of 16, 24, 2» |;^ 24 32 3 4 2* X 3 = 96. Am. 10. FindtheL.C.M.of21,33, 3 |21 33 77 T W 77 3x7x11 = 231. Ans. 11. Find the L. C. M. of 27, 33, 3^ 127 3'^ 99 3 11 33x11 = 207. Ans. 12. FindtheL.C.M. of7, 11, 17 11 13 7x11x13 = 1001. Ans. 13. Find the L.C. M. of 77, 55, 35. 6 |77 55 35 77 ;; /f 5x7x 11 = 385. A71S. 14. Find the L.C. M. of 16, 18, 27, 72. 2»| 16 X^ 27 72 2 27 JJi 2* X 3» - 432. Ans. 15. 22,33 2 FindtheL.C.M. of 10, 12 60. ;0 X;Z 22 33 60 3 ;; 33 30 22 X 11 10 3x5x11 = 660. Ans. 16. Find the L. CM. of 15, 16, 18, 20, 22, 24. 2 15 16 18 20 22 24 2 15 8 9 10 11 12 2 15 4 5) ^ 11 6 3 15 2 9 11 3 5 2 3 11 2^x3^x5x11 = 7920. Ans. 17. Find the L. C. M. of 56, 64, 70, 84, 112. 2 ^iS 64 70 84 112 2 32 35 42 56 22 16 35 21 28 7 4 35 21 7 4 5 3 2«x 3x5x7 = 6720. .4ns. 18. Find the L. C. M. of 48, 54, 81, 144, 162. 2 ^^ H H 144 162 32 72 81 8 9 2* X 3* =1296. Ans. 19. Find the L. C. M. of 75. 100, 120, 150, 180. n 100 120 150 180 2 10 12 15 18 3 ^ 6 15 9 2 5 3 23 X 32 X 5» » 1800. Ans. TEACHERS EDITION. 127 20. Find the L. C. M. of 112, 168, 196, 224. 2^ m 168 196 224 2 42 49 56 7 21 49 28 3 7 4 25x3x72 = 4704. An8. 21. Find the L.C.M. of 7, 14, 15, 21, 45. 3 |J 14 l^ 21 45 14 7 15 2x3^x5x7 = 630. Am. 22. Find the L.C.M. of 16, 25, 81. [16 25 81 16x25x81 = 32,400. Ans. 23. Find the L. C. M. of 26, 39, 52, 65. 13 |$Z^ 39 52 65 3 4 5 22x3x5x13 = 780. Am. 24. Fi 72, 225, ^ 2=^ nd the L. C. M. of 80 18. 80 72 225 48 2 10 9 225 6 ^ ^255 3 2* X 32 X 52 = 3600. Ans. 25. Find the L. C. M. of 10, 20, 30, 40, 50, 60. 2 2 5^ 2 5 3 23x3x52 = 600. Am. X0 %^ 30 40 50 60 20 25 30 10 25 15 26. Find the L. C. M. of 30, 42, 105, 70. 2 1 30 42 105 70 r^ n 105 0^ 2x3x5x7 = 210. Ans. 27. Find the L. C. M. of 36 t, 35, 20. 22 36 24 35 20 3 9 6 35 ^ 3 2 35 23 X 32 X 5 X 7 = 2520. Ans. 28. Find the L. C. M. of 7, 11, 14, 15. 1 / 11 14 15 2x7x11x3x5 = 2310. Am. 29. Find the L.C.M. of 12, 18, 27, 63, 28. 2 11 2 18 27 63 28 2 6 ^ 27 63 14 32 3 27 63 3 7 22x33x7 = 756. Am. 30. Find the L. C. M. of 34, 26, Qb, 85, 51, 39. 2 34 26 65 85 51 39 l^ JL^ 65 85 51 39 n i^ 51 39 17 13 2x3x5x13x17 = 6630. Am. 31. Find the L.C.M. of 12, 18. 96. 144. 23 n %?> 96 144 2 12 18 3 6 9 2 3 2^x32 = 288. Am. 128 ARITHMETIC. 32. Find the L. C. M. of 84, 156. 63, 99. 22 84 156 63 99 3 n :}'J 63 99 3 l.i 21 33 i;^ 7 11 2»x3'X 7x11 X 13=36,036. Am. 33. Find the L. C. M. of 17. 51, 119, 210. 17 |;T 51 119 210 3 T 210 2x3x7x5x17 = 3570. Am. 34. Find the L. C. M. of 16, 30, 48, 56, 72. 2 n 30 48 56 72 2» 15 24 28 36 3 15 6 7 9 5 2 7 3 2*x 32x5x7 = 5040. Am. 36. Find the L. C. M. of 27, 33, 64, 69. 132. 2 m n 54 69 132 3 1 27 69 66 9 23 22 2«X3>X 11x33 -27,324. Am. 36. Find the L. C. M. of li 26. 39, (55, 180. 2 l^ 26 39 65 180 3 n 39 65 90 5 n 65 30 i:i 6 2«x3«x 5x13-2340. Am. 44 126 198 280 330 22 63 99 140 165 n 63 99 70 165 21 33 70 55 3 38 10 55 37. Find the L. C. M. of 44, 126, 198, 280, 330. 2 2 3 7 5^ 33 2 ;; 2» X 3« X 5 X 7 X 11 = 27,720. Am. 38. Find the L. C. M. of 50, 338, 675, 975. 5 5 3^ 3.38 9 n 2 X 3» X 5"'' X 132 ^ 228,150. Am. 39. Find the L.C.M. of 552, 575, 920. 50 338 675 975 10 338 135 195 ;2 338 27 39 2» 552 575 920 5 69 575 115 2» 2»x3x, 69 115 ?3 3 5 5" X 23 -13,800. Am. 40. Find the L. C. M. of 228. 304, 342. 21 228 304 342 2 1114 152 17 1 19 1 g7 76 171 4 9 2« X 3» X 19 - 2736. Am. 41. Find the L. C. M. of 1080 and 1260. 1080 1260 10 2 3« 108 126 54 63 2»x3»x5x7-7560. ilrw. TEACHERS EDITION. 129 42. Find the L. C. M. of 600 and 480. 23 600 480 3 75 60 5 25 20 25 X 5 3x52 = 4 = 2400. Ans. 43. Find the L. C. M. of 1564 and 1032. 22 1564 1932 23 391 483 17 21 22 X 17 X 3 X 7 X 23 = 32,844. Ans. 44. Find the L. C. M. of 2530 and 1760. 212530 1760 5 1 265 880 11 253 176 23 16 25 X 5 X 11 X 23 = 40,480. Ans. 45. Find the L. C. M. of 936 and 2925. 32 1936 2925 13 rioi 325 8 25 23 X 32 X 52 X 13 = 23,400. Ans. 50. Find the L. C. 71 539 11 77 46. Find the L. C. M. of 3432 and 4032. 3432 4032 429 504 143 168 2* X 35 X 11 X 13 = 576,576. Ans. 47. Find the L.C.M. of 1875 and 2425. 52 1 1875 2425 75 97 3x5^x97 = 181,875. Ans. 48. Find the L. C. M. of 1632 and 2976. £3 1632 2976 £2 204 ;^,72 3 51 93 25x3x1 49. Fin and 2233. 11 7 17 31 7x31 = 50,592. Ans. 1 the L. C. M. of 1001 1001 2233 91 20:5 13 29 7 X 11 X 13 X 29 = 29,029. Ans. M. of 539 and 1463. 1463 209 7 19 7^x11x19 = 10,241. Ans. 130 ARITHMETIC. Exercise XVII. 1. Find the L. C. M. of 424 5. Find the L.C.M. of 1003 and 583. and 1357. 8|424 53)583(11 583 G. C. M. = 53. L. CM. = 11x424 = 4664. Ans. 1003)1357(1 1G03 6\ 354 59)1003(17 50 2. Find the L. C. M. of 319 413 and 407. 413 111319 407 29 37 L. C. M. = 17x 1357= 23.069. Am. G.C.M. = 11. L.C.M. = 29x407 = ll,803. ^ns. 6. Find the L. C. M. of 899 3. Find the L.C.M. of 1679 and 1932. and 961. 899)961(1 4 3 7 1932 483 161 23)1679(73 161 69 899 2| 62 31)899(29 62 279 279 G.C.M. = 23. «? L.C.M.=73xl932=141,036. Am. L. C. M. = 29 X 961 = 27.869. Am. 4. Find the L. C. MT of 1003 and 2419. 7. Find the L.C.M. of 407, 1003)2419(2 2006 7|413 59)1003(17 59 703, 444. 11|407 37)703(19 37 333 413 333 L.C.M. 413 -17x2419-41.123. i4n«. L.C.M.-llxl9x444-92.796. Ans. TEACHERS EDITION. Idi 8. Find the L.C.M. of 411, 2 322 959, 2055. 7 161 ill ^59 2055 23)851(37 959)2055(2 69 1918 161 137)959(7 161 959 L. C. M. = 7x 2055 = 14,385. Ans. 23 X 3 X 7 X 23 X 37 X 53 = 7,577,304. Ans. 9. Find the L.C.M. of 22 1 and 351. 12. Find the L. C. M. of 539 221)351(1 and 253. 221 Ill 253 539 10 130 23 49 13)221(17 L. C. M. = 23 X 539 = 12,397. Ans. 221 L.C.M. = 17x351 = 5967. Ans. 13. Find the L. C. M. of 2943, 2616, 4578. 10. Find the L.C.M. of 1426 and 989. 8 1 2616 327)2943(9 211426 2943 4 3 276 69 23)713(31 69 23 23 L.C.M.=2x31x989=61,318. Ans. 11. Find the L. C. M. of 3864 3404, 3657. 22 3864 3404 3657 3 966 851 3657 23 322 851 1219 14 37 53 2 327 2616 2943 45^ 1308 2943 2289 4 9 7 2x4x7x9x327 = 164,808. Ans. 14. Find the L. C. M. of 2863 and 1151. L.C.M. = 1151x2863 = 3,295,313. Ans. 15. Find the L. C. M. of 1177, 1391, 1819. 107 11177 1391 1819 11 13 17 132 ARITHMETIC. 1111177 18. Findthe L.C.M. of 23,309 107)1301(13 and 10,753. 107 L. C. M. = 10,753 x 23,309 321 = 240,631,677. Ans. 321 L. C. M. =13x17x1177=260,117. 19. Find the L.C.M. of 4939 Ans. and 3143. 7 1 3143 16. Find the L. C. M. of 5317 and 2863. 7 1 2863 409)5317(13 409 449)4939(11 449 449 449 L. C. M. = 11x31 43 = 34,573. ^rw. 1227 1227 20. Find the L.C.M. of 4199 L.C.M.=13x 2863-37,219. ^ns. and 6137. 13 [4199 323)6137(19 17. Find the L.C.M. of 12,703 323 an4 12,879. 2907 L.C.M. = 12,703x12,879 2907 = 163,601,937. Ans. L.C.M. = 19x4199=79,891..1n«. Exercise XVIII. 1. Reduce to whole or mixed numbers ^. J^=lf. Am. 2. Reduce to whole or mixed numbers V- V-=-2|. Ans. 3. Reduce to whole or mixed numbers ^^. V-6J. Ans. 4. Reduce to whole or mixed numbers -^^. i^^9^. Ans. 5. Reduce to whole or mixed numbers -^j'j*. W=13i Ans. 6. Reduce to whole or mixed numbers ^^. TEACHERS EDITION. 133 7. Reduce to whole or mixed 11. Reduce to whole or mixed numbers -y/-. numbers ^^V/- W = I'iff -i^s- -Wi- = mi ^ns. 12. Reduce to whole or mixed 8. Reduce to whole or mixed numbers -2//-. numbers -\y-. % = 13. Ans. 4M- = 37. Ans. 13. Reduce to whole or numbers -\^^. mixed 9. Reduce to whole or mixed -¥r = 182V ^^s- numbers -^f f-^. 14. Reduce to whole or mixed ^fP = 50ff. Ans. numbers ^^^f-. 3^i- = 18|0-. Ans. 10. Reduce to whole or mixed 15. Reduce to whole or mixed numbers -yVy*- numbers -^||-^. -V.¥ = 2G^v^^«. 8 9|5=359. ^ns. Exercise XIX 1. Reduce to improper frac 3ns 3|. 3| = V- -^ns. 2. Reduce to improper frac- tions 5j^^. 3. Reduce to improper frac- tions 12y\. 12t*x = Vt-- ^«"'- 4. Reduce to impro];)er frac- tions lOy'^^. 5. Reduce to improper frac- tions 8f. ^^-^^. Ans. 6. Reduce to improper frac- tions 12^|. 1211 = -W-. ^^^«- 7. Reduce to improper frac- tions 84 J-^. 8. Reduce to improper frac- tions 8tU-^f. 864^f = -s-^g^fi. Ans. 134 ARITHMETIC. 9. Reduce to improper frac- tions 41y§8^. 10. Reduce to improper frac- tions 41t^^7^. 11. Reduce to improper frac- tions 400H^. 400H* = l^'^-. Am. 12. E-educe to improper frac- tions 50005§3.^. 5000^^5 = i^fger^. Arts. 13. Reduce to improper frac- tions lOOOOjf 10000}! = J^f^. ■^^^«- 14. Reduce to improper frac- tions 300l7%3^. 15. Reduce to improper fractions 73||. 16. Express 8, 7, 3, 5, 12, 13, 18, 29, 25 in the form of fractions, each having 5 for a denominator. 8 7 3 5 12 13 18 20 25 = ^. =¥• =-^^- =¥• =¥• =¥• =-?-• =^^- =^P. 17. Express 21 in the form of fractions, having for denominators 3, 5, 7, 8, 12, 13, 20. 25, 30, 37. 21 21 21 21 21 -¥. = i§i. = ^fl. = ifA. =w- 21 21 21 21 21 -w. =w. =w. = W- =w. 18. Express 12, 15, 23 in the form of fractions, each having for denominators 12, 15, 23, respectively. 12 12 12 15 15 -w. -w. -W- =W- -w- 15 23 23 23 -w. -w. -w. =W- TEACHERS EDITION. 135 Exercise XX. 1. Reduce to lowest terms iff, 1 2 _ 1 5 _ 5 Anc 2. Reduce to lowest terms |f f. 3. Reduce to lowest terms xWcJ- 928 _llfi yljio T32(7 — T65- ^'^S- 4. Reduce to lowest terms ||f f . 1728_216_108_12 /l^iQ 6. Reduce to lowest terms |f l^. 23]0_231_21_3 A^<. 3 80 — 3 8 — 'IJ — ¥• -^^S- 7. Reduce to lowest terms |^f ff. 8. Reduce to 1 owest terms ^2%%% 9. Reduce to lowest terms -j|^f . 10. Reduce to lowest terms -^^^j. 924 _231_77_11 A-nt 10^2 — 273 — ¥T - T3- -^^S- 11. Reduce to lowest terms |f |^. lift = f f = f • ^ns. 12. Reduce to lowest terms x^A- 13 Reduce to lowest terms f fo I- 6732 _ 1 683 _ 153 _ 1 7 Ay.^ 14. Reduce to lowest terms -oV/ifo • 6840 _171_19_1 J^a ■Z7360 — ■5'84 — 76 — ¥• •^^^• 15. Reduce to lowest terms f^f§. 5760 _ 576 _ 144 /I.,, 7170 — 70 — TYZ- ■^^''^■ 16. Reduce to lowest terms y^^f-jf. 17. Reduce to lowest terms ff"-!- 18. Reduce to lowest terms x^li- 1 01 5 _ 35 /l^o 19. Reduce to lowest terms 2X^7- 516 _ 1 2 /J^a ^T07 — ?9- ^^S. 20. Reduce to lowest terms -g^aVoV 3 JL7 2 _ _3 5 2 _ 3 2 72807 — 8¥37 — 767- 21. Reduce to lowest terms ||m. 78473)94653(1 78473 10 16180 1618 809)78473(97 7281 5663 5663 G. C. M. = 809. 136 ARITHMETIC. 22. Reduce to lowest terms ^ff If. 4|17o06 4399)26145(5 21995 4150 415 83)17596(212 1J86 99 83_ 166 166 G.C.M. = 83. 23. Re<] vice to lowest terms |f ^ff. 44323)61087(1 44323 4 1^764 3 4191 1 1397 127)44323(349 381 622 508 1143 1143 G.C.M. = 127. 24. Reduce to lowest terms i'jW. 3 1 339 113)1243(11 1243 G. CM. -113. 25. Reduce to lowest terms ^if?- 11 11177 107)2675(25 214 535 G. CM. = 107. ^ 26. Reduce to lowest terms ji^H. 3815 .763 G. CM. = 109)5123(47 436 763 763 27. Reduce to lowest terms i-^ J f ^. 14141)16289(1 14141 12 |2148 179)14141(79 1253 1611 G. CM. = 179. 1^ 28. Reduce tolowesttermsff^^lf. 29. Reduce to lowest terms uWjTffV Divide both terms Ijy 1001. 30. Reducetolowestterms^lf^^ Divide both terms by 142857. TEACHERS EDITION. 137 Exercise XXI, 1. Find the product of f x 2. §X^ = -=U. Ans. 2 2. Find the product of f X 9. 3. Find the product of 10 X |. 2 ^X- = 4. Ans. 1 ^ 4. Find the product of 15 x |. 1^ X ^ = 15. Ans. 1 3 5. Find the product of ^\ X 7. 3 -^X- = 3. ^ns. ^; 1 3 6. Find the product of 16 X |. 2 2^ X - = 10. Ans. 1 ^ 7. Find the product of f X 2. I^l^l^li. Ans. 4 8. Find the product of ^^ X 5. |. Ans. ^X^ ;^ 1 3 Find the product of 27 X f . 3 ^ X - = 15. .4ns. 1 9 10. Find the product of |f x 2. ^P 1 10 ^^ 10 11. Find the product of ^§ X 3. 20 1 20 '' 12. Find the product of |f X 4. L3x^ = l^ = 2|. 4ns. 20 1 5 ^ 13. Find the product of 5 x \^. ^xl^ = ^ = 3i4ns. 1 ^0 4 ^ 4 14. Find the product of 6 X |f . Ll^ = ^ = 3^. 4ns. 1 ^^ 10 ^*^ 10 138 ARITHMETIC. 15. Find the product of 7 X H- 1 20 20 ^^ 16. Find the product of 8 x f|. 5 17. Find the product of 1^ X 10. Exercise 18. Find the product of ^ X 12. gxLf = :,.». 5 19. Find the product of ^ X 15. 3 13 I^ 39 o'. A 4 20. Find the product of i^ x 20. I^xf = 13.^n3. XXII. 1. Simplify f of ^. 2. Simplify ^ of 2^^. 3 ^ X 2 1 - ^ y ^^ - ^ Ans 3. Simplify f of f . ?X^ = A ^n, 7 ^ 21 3 4. Simplify 2f X 2^. 6 2| X 2^ = ^ X I = G. Ans. 5. Simplify 4| X 2f 4fx2| = |x^^ ^ = lOf Am. 6. Simplify 4| x 9f 14 -^-451. .In*. 7. Simplify ^ off of 10. lx?xf.2.^n. teachers' edition. 139 8. Simplify | of f of f . - X - X - = -. Ans. 9. Simplify fxfXf of 4i. 2 3 ^X^X^xa-^X^X^X^-^-^-r Ans 5X-X-X4i--X-X-X--^-l^. Ans. % 10. Simplify \ of ^\. 2 11. Simplify f of ^^ of f of f of | of 15f . % 9 ^x ^ x^x^x 1x153 -^x ^ x^x^x^x^^-^'^-l^ An, 5 12. Simplify 5f X 8f . 2 13. Simplify | X f X jV X 7^ -X-X-X7^--X-X-X----l3. ^r^s. 14. Simplify f of |^ of ^% of 8f 5 ^x^^x ^ x8i-^x^^x ^ x^^-^ ^ns 3 2 15. Simplify x«T X M X If X 2ij 11 21 48 '' 11 ^/ ^^ 19 627 ^^^ 3 140 ARITHMETIC. 16. Simplify If XtV^xI^^it- 9 43X — Xl^^^--X^^^X^^^-^^^. ^m. n m 22 17. Simplify f X iM X H X 17. ^ ;?; ^^ ; 11 ^^ 11 ;7 18. Simplify If X^fX MX Ifi ;^ 4 3 38x^X^Xl|i = ^X^X^X^ = i = li ^n. 39 57 86 '^ 3^ ^7 ^^ )J3 3 ^ ;3 3 ;z 19. Simplify i of | of | of f of f of f of | of | of 1% of 10. Iv^X^X^X^X^X^X^X^X^-l Ans 20. Simplify ^ of ^^ of 30. 5 21. Simplify Hf X ^ X H X If. 3 il^x-^^vl^xU-^^^X-^X^X^-^ Ans 71 ;z 5 2 22. Simplify | x f X /r X f of | of | of 8. ^x^xl-x^x^x^x^-'*^ 23. Simplify tV of Ji; of /jiV- 3 ;3 5 3 teachers' edition. 141 24. Simplify j\ X y'a X f f X 48. xx^'x^^^^ X 25. Simplify \l of i^ of || of 12. 3 ?2x^X^X^-^ ^ns ^0><^^^^^^l-4^"'- ^ ^ ;^ 4 ? 26. Simplify If of4|of f. 27. Simplify 2fx If XllfX 8. 2 4 2tXlfXll|X8 = |x|x|xf = l|^ = 52A. ^n. 3 28. Simplify 3f of 2\ of l^^^ of l^V ? 3 3fx2^Xl^Xlx\ = f x|x|xl| = ^ = 20A. ^ns 29. Simplify H X 5^ X 4^ X A X 5. Iix5,x4,x|x5 = f|x|xixlxf = f=S2,... 30. Simplify f of ^ X 8f X ^^ of li|. 2 2 7 ^x-^x82x-^xP7_^x ^ x^^y ^x^^--"^ Ans 5 9 142 ARITHMETIC. 31. Simplify HxHxHf. 9 i% ^% 38 m 7ti 2 n 17 — . Ans. 32. Simplify Iff Xf If X^V^. 4 xn 8 m m im 405' l^^ 9 9 5 33. Simplify \m of ,% of i§|f • 2 ;zT^^ ;2;^^ im 243' 9 9 ;;2 3 Exercise XXIII. 1. Divide ff by 6. 24,6 = 1x^^4 ^n. 35 ^ 35 35 2. Divide |? by 5. 3. Divide f by 8. 3 o 1 3 3 . 7 8 7 56 4. Divide 18f by 7. 8 184 + 7 = 1 X ^ = 22 An^ ^ T 3 ' 5. Divide \ by f . 5 3 ^^5 n . --^- = -x- = -. Am, 8 4 3 ^ ti 6. Divide \\ by f . 16 8 ^ n i 7. Divide 1} by 3f ia oi 7 10 Q 7 Ol teachers' edition. 143 8. Divide 5^ by 4f . 7 9. Divide 8f by 4^. 2 «»-^ = f-f = f.x¥--- 10. Divide 71 by 4f. 6 ^ ^5 7 30 5 25 '^ 11. Divide 6f by 9^. 63.91 = 27^19^1^27^27^^^^ ^ ' 4 2 19 ^ 38 2 12. Divide 8f by 4|. 13 8.2-^41 = ^^^ = 1x^ = ^ = 15- Ans ' ' 3-3 ;^^ 3 7 '• ^ns. 13. Divide 3| by if ^ '^ 9 27 ;^ ^ 2 2 14. Divide 4f by 6f. 43.6f = ^.^2^1x^ = ^. ^n. ^ ^7 9 j?^ 7 14 2 15. Divide 5 by 4f . 6 16. Divide 3f of 2i by li of 2^. 34of2i-liof2^ = l^of?^|ofl^ = ^X?X?X^ = 3. Ans. 2 2 9 ^ ;i 3 ;^ 144 ARITHMETIC. 17. Divide 2f by 3^ of 1^. 2 18. Divide 2^^ of 5^ by 7f . 19. Divide 5f of 8^ of 1^ by 2^^ of 5|. .|of8^oflf^2^ofr>| = |of|of^^?lof| = 1x^x11x^^x1 = ^^ = 6^. ^n.. ^ "^ ^ "^ J ^^ ^0 7 7 ? Exercise XXIV. 1. Express with least common denominator ^, ^, ^. L.C.D. = 2x3x5 = 30. 1 2 5 15 12 25 2 5 6 30 Ans. 2. Express with least common denominator |, f , |, -j^^. L.C.D. = 23x3'»x5 = 360. 2 5 7 9 240 200 315 324 3' 9' 8' 10 360 Ans. 8. Express with least common denominator ^, ^, ^^^i-, ^f. L.C.D. = 2'x3x5x7 = 840. 5 1 5 19 700 105 200 V^a 6' 8' 21' 35 840 Ans. 4. Express with least common denominator •^, ^'^, ,^, ^. L. C. D. = 22 X 3> X 5» = 800. 2 7 3 8 120 315 108 160 15' 20* 25' 45 800 Am. teachers' edition. 145 5. Express with least common denominator |f, ij, -^f, }f. L. C. D. = 2=5 X 3 X 52 = 600. 12 17 13 19 288 255 130 152 25' 40' 60' 75 600 Ans. 6. Express with least common denominator |, -^^, /^, ^^, |f. L. C. D. = 23 X 3 X 5 X 7 = 840. 3 7 4 3 19 315 196 96 90 665 8' 30' 35' 28' 24 840 -. Ans. 7. Express with least common denominator |-^, -^j, |f . ff , ||. L. C. D. = 2* X 33 X 5 = 2160. 11 7 13 23 17 1485 840 1404 1()56 680 16' 18' 20' 30' 54 2160 8. Which is the greater, if or ^| ? | or | ? | or j^^ ? Ans. L.C.D. = 22x52 = 100. L.C.D.-2x32 = = 18. L.C.D. = 5xl2 = 6a 13 65 20 100' 5_15 6 18' 3_36 5 60' 17 68 28 100' 7_14 9 18' 7 _35 12 60' .-. Il" is the greater. .-. f is the greater. .•. 1 is the greater. 9. Arrange the fractions j\, j-^, ^f in order of magnitude. L. C. D. = 2^ X 3'^ = 72. L = ^ 11 = ^ 1^ = ^. 1^ 1 11 Ans 12 72' 18 72' 24 72* 24' 12' 18' 10. Arrange the fractions j\, j%, x*r, j\ in order of magnitude. L. C. D. = 22 X 32 X 5 X 11 = 1980. A A ± i _ H25 1056 720 770 12' 15' 11' 18 1980 A, 1, A, A. Ans. 11 18 12 15 146 ARITHMETIC. Exercise XXV. 1. Find the sum of J + f . 1 + 3^^ = 2. ^n. 2 2 2 2. Find the sum of i + f + i 3 3 3 3 ^ 3. Find the sum of ^ + ^ + f . l+i + § = 5 = U. Ans. 4 4 4 4^ 4. Find the sum of 1^ + 2^. H + 2i = 3lJJ^ = 4. ^ns. 5. Find the sum of 1| + 2f . H+2f- = 3iJi = 4. ^m. 6. Find the sum of 3^ + f . 3^ + 1 = 31^ = 4. Ans. 7. Find the sum of 2| + 3|. 2! + 3| = 5^ = 6f ^ns. 8. Find the sum of 1| + f . 9. Find the sum of ^j + ^j + {^ + H- 17 17 17 17 17 ^^ 10. Find the sum of 8^^ + 6^j + 5\^ + }f 8i^T + 6^7 + 5|^ + 1^ = 19f| = 21^7. Ans. 11. Find the sura of | + f 5 6 30 ^ 12. Find the sum of | + |. 3 + 7^6 + 7^1j.^n. 4 8 8 ^ 13. Find the sum of i + i- 3 + 1 2 1 + 1 2 G 6 Ans. 14. Find the sum of ^ + ^. L. C. D. = 2'' X 3 X 5 = 60. ^^^''-'l.Ans. 60 60 15. Find the sum of /j + ^\. L.C.D.=.2*X3 = 48. 15 + 22 37 48 48 -. Ans. 16. Find the sum of 12» + 7^. 121 + 7A- 191^4 -19H.^»w 17. Find the sum of 85,5j X 27|J. 85/j X 27H - n2H1P - 113^- ^^' TEACHERS EDITION. 147 18. Find the sura of ^ + ^ 4- ^ + i. L. C. D. = 22 X 3 X 5 = GO. 30 + 20 + 15 + 12 _ rz _ -, 1 7 ~60~ ''■ 60 19. Find the sum of i + | + f + |. L.C.D. = 2'^X 3x5 = 60. 30 + 40 + 45 + 48 163 Ayis. 60 60 m ^ns. 20. Find the sum of | + j^ + j% + ^\ + ^l L. C. D. = 22 X 3 X 5 = 60. 50 + 55 + 32 + 21 + 26 _ 184 ^ 3 1 60 60 ^^ 21. Find the sum of 5^} + HM L. C. D. = 2 Ans. ^ + 3% + 17tV + 14 + n^v ¥0' 22. Find the sum of 94 ^ 151^ + 163^1 + 111 ^ ify^ L. C. D. = 22 X 32 X 7 = 252. 1984|f = 199||. Ans. 23. Find the sum of 31 + 41 + 11 + 2. L.C.D. = 2 + 3 + 5 = 30. 24. Find the sum of h\ + 2-A + 5/^ + j%. L. G. D. = 22 X 3 X 52 = 300. 25. Find the sum of f + lf + 2 + 3f + 4^3j. L. C. D. = 23 X 33 X 7 = 504. 10HI = llfM- ^n^- h24 X 3 X 52 = 600. 85/o%. Ans. 26. Find the sum of 4i + 3f + 2f- + li + ,?j. L. C. D. = 23 X 32 X 7 = 504. 10fM=lli§i. ^m. 27. Find the sum of H + ^V + lO + ff. L.C.D. = 23x3x5x7 = 840. 10||f. Ans. 28. Find the sum of 29 L. C. D. = 2* X 32 X 52 = 3600. UU = nU- Ans. . Find the sum of 2 + 1 + If + 4| + 5i|. L.C.D.= = 23x32=72. 12W = = 14f i. Ans. 148 ARITHMETIC. 30. Find the sum of L.C.D. = 2*x 5x11 = 880. IG-VsV—IStI^. ^^«- 31. Find the sum of L. C. D. = 23 X 32 X 5 = 360. CIM = 81|. Arts. 32. Find the sum of ^j + j'i + n- L.C.D. = 2x7x11 = 154. 33. Find the sum of 20A + ll^V + 5i + 305. L. C. D. = 23 X 3 X 5 = 120. 34. Find the sum of H + i^ + H- L. C. D. = 2--' X 3 X 19 = 228. 35. Find the sum of iV + H + M + if- L.C.D. = 2-^x3xl7 = 204. 36. Find the sum of 3l7| + 17^r + 4r% + /3 + 6| + ^. L.C.D. = 2x3x5xl7 = 510. 344-W^ = 346|H. Am. 37. Find the sum of 4A + 82\ + 4i\ + 5f + 5| + f. L.C.D. = 3x5x7x11 = 1155. 26f!-H = 2%\V ^^• 38. Find the sum of H + ^^ + ^^h + n + hih- L. C. D. = 2880. 17tm = 18HH- ^^«- 39. Find the sum of 4A + 73\ + 5H + 275^3/, + 2§f L.C.D. = 2-^x3x7xl3 = 1092. 293Hfl = 294^1. ^^• 40. Find the sum of H + Vi + 6^ + 400^ + 51H. L.C.D. = 23x7x3x11 = 1848. 464J3|i = 465^H- ^^• Exercise XXVI. 1. Find the vahio of 52J 52^-46 = 0^. Ans. 46. 2. Find the value of ^ - f 6 3 6-3 3 1 9 9 9 3 Am. 3. Find the value of f - f 3_2_9^ 4 -8-1. Am. 12 12 4. Find the value of ^^ — •^. A _ A 32 - 25 2. 15 12" 60 "60' .^714. TEACHERS EDITION. 149 5. Find the value of {^ — j\. 11 3 ^ 77-27 ^ 50 __ 25 18 14 126 126 63 Ans. 6. Find the value of 4 — |. 4-i = 3i. Ans. 7. Find the value of 7 - f . 7_| = 6f Ans. 8. Find the value of 3 — |. 9. Find the product of 8 — f . 10. Find the product of 5 — f . 5-4 = 4^. Ans. 11. Find the value of 5 - |. 5 — I = 4f . Ans. 12. Find the value of 6^ - 5^. 61-5^ = 1-2-^ = 1^ Ans. 13. Find the value of 4f - 3f-. 4|-3f=ll-V5- = M- ^^«- 14. Find the value of 7^ - 2tV. ^ 1 9 3— f: 1 0-9 _ K 1 A„^ 15. Find the value of 7f — 4|. 7| _ 4| = 3J-\^0- = 2||. Ans. 16. Find the value of 6| - 2|. 6f - 2| = 4-^-/ = 3H- ^ns. 17. Find the value of 9f - 4f . 94 _ 4| = 524^-^25 ^ 429.. ^^3. 18. Find the value of 4f - ^. -^^ - 4^. Ans. 19. Find the value of 6| - 4f . 6|-4| = 2-V2'- = 2xV 20. Find the value of 7J - 2|. 71 _ 2J = 5-^f^ = 4|. Ans. 21. Find the value of 8^ - 4f 8^-4t = 4^F = 3H. 22. Find the value of 85-2^ - 27H- 85,V - 27H = 58-^V~ = 57111 = 57|§. Ans. 24. Find the value of 10 - 3|. 10 -31- = 61. Ans. 23. Find the value of 8yV - m- 8tV-2H = 6^-V(P = 6^V^^s- 25. Find the value of 120|| - llOif . 120fi - llOif = 10^^//^ = 101^. Ans. 26. Find the value of 5i| - 11 150 ARITHMETIC. 28. Find the value of 2^f J - If «3. 9151 _ 1 103 _ 16.0.4- 8J_5 _ 749 >!„, 29. Find the value of 4 - l^Ul 4 - imi = 2^VKru¥^ = -mi ^^«- 30. Find the valuo of 1473 - 279H. 1473- 279 j-|=1193tV Ans. 31. Find the value of 1473^5^ _ 279j^. 1473/j - 279i^ = ii94^^^lg^ =-- 1193^. Ans. 32. Find the value of 1473^V - 279^- 1473^^5 - 279H = 1194J-\-/^ = 1193^|. Ans. 33. Find the value of 278 j| - 30^^. 278H - 30-i5j = 248^^f^ = 248||. Ans. 34. Find the value of 125/j - 10^. 125A = lOU = 115^^^ = 1 14tf Ans. 35. Find the value of 118/r - 17^^. 36. Find the value of 94^^ - 91|f. 9'h«^r-91|| = 3i^VV^ = 2,Vr- Ans. 37. Find the value of 7^r - 2\l 7^ - 2H = 5^^ = m Ans. 38. Find the value of ^ - |f 235 _ 13 235 - 91 48 . 357 51 " 357 " 119* 39. Find the value of ^|- -j\\- 17 29 204 - 203 63 108 756 40. Find the value of /^ - -AV 9 38 43 209 99-86 418 teachers' edition. 151 = . Ans. 756 13 A = . Ans. 418 41. Find the value of iff - fff . 146 268 __ 1032 - 804 _ 218 273 637 1911 1911' Ans. 359 199 _ 1795- 1791 _ 1 ^^^^ 360 200 1800 450' ^'^* Exercise XXVII. 1. Simplify 3f - 2| + 4^^^ + 1| - 5^^. Sum of plus terms = 9f § Sum of minus terms = S-^i^ Difference = l|i;. Ans. 2. Simplify lA - H + n- - 2i - m Sum of plus terms = 8|| Sum of minus terms = 4}^ Difference = 3if-^-. Ans 3. Simplify 12 - 3f - 1^3^ - 4^V + 2M - 4|. Sum of plus terms = 14|^ Sum of minus terms == IS^Mjc Difference = l^.Ans. 4. Simplify 433-V - H - Ifi - 1|| - 2^1 - 2/^ - 2f f - 3,^. Sum of plus terms = 43y'3- Sum of minus terms == 16/5- Difference = 27f ^. Ans. 152 ARITHMETIC. 5. Simplify ^ + ^ + 7/^ + 8^1 + 7i + 8^^^ + 4^^ - 36^^- Sum of plus terms = 36^^ Sum of minus terms = 865^1^ Difference = 0. Ans. 6. Simplify (S^j, + H^ + 17^ + 40) - (30^ + 1 1 H)- Sum of plus terms = 66j§f Sum of minus terms = 41| Difference = 2,5r^^g. Ans. 7. Simplify (172}f + QSyi^^) + 172|| - 93^^^- (17211 + 93TV7) + (172f|-933^,\) = 172|| + 93tVt + 172^1 - 93tVt = 172|| + 1 72|f = 3441^^^ = 344^^ A,us. 8. Simplify (172|f + 93x^A) - (172f| - 93 AV)- (17211 + 93i»A)-(172|f-93TV\) = 172|f + 93^T-V- 17211 + 93^ = 93^tV + 93^ = 186-iVt. ^^■ 9. Simplify (t3j_^) + (^j + ^|^). U3 39/ \,78 156/ 13 39 78 156 36-8 + 10 + 7 15 . = 156 = 52-^"^- 10. Simplify ^ - ^ _ 2| + 3f + 7/j - If - 3^. Sum of plus terms = lljf Sum of minus terms = 4y ^ Difference = 6ff|. Ans. 11. Simplify T^^-y^^-T^^-T-^^^^. _3^_ J 9___5 3000 - 700 - 90 - 5 ^ 441 10 100 1000 10000" 10000 "200U 12. Simplify 9|- 7- f-f. 9J-7-i-^-2li4|-iil=.l2V. Ans. .ins. teachers' edition. 153 13. Simplify 5| + 8f - If - 4|. Sum of plus terms =- 14j\ Difference = 8j|^. Ans. Sum of minus terms = 6f ^ 14. Simplify 6| - 5f + 4| - 4/j. Sum of plus terms = llg^ Sum of minus terms = lOj^ Difference = 1^-^. Atis. 15. Simplify 14tV + 9| - 6| - 12| - 3f . Sum of plus terms = 23|| Ans. 2f-9f + 10x%-14TV Sum of plus terms = 30|f Sum of minus terras = 26|f Difference = 43^%. Ajis. 17. Simplify 95f - 9/^ - 8f - 14^% + 74f . Sum of plus terms = 169| Sum of minus terms = 32 A Difference = 137.^||. Ans. 18. Simplify 12| + 23| - {4j% + 12f + Tif). Sum of plus Sum of minus terms Difference = llx^^jj. Ans. + ^^\ + I'iA). Terms outside pareij thesis = MjW^ Terms inside parenthesis = 28|5-§ Difference = 5/'X. Ans. 154 ARITHMETIC. 20. Simplify 97f - (20 + 9| + 18^ + 24f^). Terms outeide parenthesis = 97 J Terms inside parenthesis = 72t'\^ Difference = 25^\y. Am. 21. Simplify 2}i + 3Jf - {l^ + IH + II), Terms outside parenthesis = 6^ Terms inside parenthesis = ij^^^n Difference = l^^iVj. Ans. 22. Simplify ill + flH - m^- 143 2471 100 1000 82643 143000 + 247100 - 82»i43 100000 100000 307457 100000 Ans, Exercise XXVIII, 1. Simplify ^ Multiply by 44 100 165 33 2. Simplify 3 Multiply by 8. 24 57 -l.An.. 3. Simplify Hi. 131 Multiply by 21. 360. -If. Am. 4. Simplify -L Multiply by 27. 225 1. ^n. 5, Simplify —^. Multiply by 99. 253 11 . 4r4-r8-^'"- 6. Simplify liil?i. Hof^ 2 2 r7'';<3'^y 2l TEACHERS EDITION. 155 7. Simplify 1^. Multiply by 72. 180-112^68^^ ^^^^ 132-117 15 '^ 8. Simplify l^iiiy. Multiply by 280. 2912-480 ^2432^^82 1995-861 1134 ^^^ 9. Simplify iA^. 10. Simplify ?i:ii4. ^ ^ 21 + 13 Multiply by 84 567-114 453 182 + 120 302 1|-. Ans. 1. Simplify M. 9 8^_;2^^ 39_351_ = lOJW If 35 ^^ 35 -4ns. 12. Simplify ^ 4 ;^ 3 ^ 8 12 24* Ans. 13. Simplify iiof?i. = -. Ans. 7 14.. Simplify 5i m m ^=lAns. 175 5 15. Simplify ?i±?l. ^=^lJl^^.Ans. 2 16. Simplify 3^^. n^ 9 n^ 7_189_22« 2 ^m. 17. Simplify ^^ + T7 + A + 7, Multiply by 60. 51 + 44 + 42 + 48 51-44 + 42 41 — 91 18. Simplify ^^^—^5 185. Ans. Multiply by 28. 116 - 63 53 182-60 122 Ans. 156 ARITHMETIC. 19. Simplify ?l^^>— i- Multiply by 280. 749 - 1280 + 875 344 8 . = -. Aru. 1640-1365 + 112 387 9 20. Simplify U X 1» + j of 2^- H X 2 . H + f-4! _ 45-f-21-26 if + f-H 26 + 21-45 ^ = 20. ^n,. 2 21.S.n.pUfy2ix^i^ix,JA_- 9 516 4 297 + 368 236 38 "^140 H 7 • 5 22. Simplify J j - 7f + 5f -4| ^ 7455 - 6600 + 4900 - 4032 1723 8316 - 7448 + 6615 - 5760 ^ 1723 1, Aiu. Exercise XXIX. 1. What fraction of 8 is 3? 3 ;. An». 2. What fraction of 3 is 8 ? |-2f Am. 3. What fraction of 9 is 7? 7 9' I. Ant, 4. What fraction of 7 is 9? 7 5. What fraction of 8 is 12? 12 = H. Ana. 6. What fraction of 12 is 8 ? 8 2 . — =■-. Am. 12 3 7. What fraction of 2^ is |. a What fraction of | is 2^? ^ - 3f . Afu, TEACHERS EDITION. 157 9. What fraction of 2| is 1^? 2| 11 10. What fraction of 1^ is 2f ? 21 U 2|. ^ns. 11. What fraction of 2^ is 7f 2i ^^ 13. What fraction of 7^ is 2| ? —5^ = . Ans. 71 176 13. What fraction of 31 is 8|? 14. What fraction of $ 2 is $ 1^ ? 3 $2 4 m = -. Ans. 15. What fraction of $ 2^ is $ 5 ? 1^=2. Ans. 16. What fraction of $f is fp $1 3 17. What fraction of $f is $|? $1 10 18. What fraction of |2f is ||? $2| 33 19. What fraction of $^ is l^^^ ? I^ = -- ^ns. $i 5 20. What fraction of $ 1 is $ | ? 11 = 7 $1 8' = X Ans. 21. What fraction of $ 10 is | f ? 1 10 15 22. What fraction of$100 is $6? $ 100 50 23. What fraction of $100 is $4^? ML =^. Ans. $100 200 24. What fraction of $4 is $25 ? $25 $4 95. What fraction of 100| is 8f ? ^==^.Ans. lOOf 905 26. What fraction of 21isif of3|? mii^lAns. 21 7 158 ABITHMETIC. 27. What fraction of 18iHi8|of33|? 28. What fraction of 3^i8fxH? H 15 29. Wliat fraction of 3^x52>7i8l720? 1720 3iVx5A llOHf Am. 30 What fraction of 3ixfof^i8lf? ^^ ^ im 3^ X f X t 10 31. What part of ||x|fi8ix4xf? ^-■^- 32. What part of ISfXlX/^isIoflflofli? ixillxii.l. ^,, 131 X|X^^ 33. What part oi U+H+^js + i is U-H + tV - I? Multipl -44 + 42-48^_l_, ^,, + 44 + 42 + 48 185 Multiply by 60. 51-44 + 42-48 _ 1 51 '• '- 34. What part of 4| - 2^ is 6^ - 2^ ? 6i_:^^182-60^12_2_^^ ^^ 4^-2i 116-63 53 ^' 35. What part of 17f - 12f is 5 - ^ - ,^ - 2^ ? 171 -12f • Multiply by 6825. 34125 - 525 - 700- 273 32627 120575-87750 32825 Ans. 86. What part of 24 - 17,^ is 7 + ^^ - ^ - Ji? .24-17A * Multiply by 5266. 36856 + 702 - 325 - 1 287 35945 , , . , . 1263ti0- 91126 35235 TEACHERS EDITION. 159 fx2TV ;/ 3 3 3^ 9 38. What part of ("-ihYiih-")- V4^~63|)-^ (737 + 41:1) _ (14 - ft) ^ {^P- - 13) _ -W X A (If -1)^(11 + 1) 226 V 45 TTl -^ 53 i^f. ^ns. Exercise XXX. 1. Keduce to common fractions in their lowest terms 0.125. 0.125 = tV¥o = h ^^s- 2. Reduce to common fractions in their lowest terms 0.625. 0.625 = T«j,V^ = f. Ans. 3. Reduce to common fractions in their lowest terms 0.675. 0.675 = tW, Ans. 4. Reduce to common fractions in their lowest terms 10.864. 10.864 = 10xVo% = lOitf • ^^s- 5. Reduce to common fractions in their lowest terms 50.84. 50.84 = 50xV% = 50|i. Ans. 6. Reduce to common fractions in their lowest terms 3.00025. 3.00025 = 3^ 7. Reduce to common fractions in their lowest terms 8.1075. 8.1075 = 8^1^^^^ =84§^. Ans. 8. Reduce to common fractions in their lowest terms 35.01024. 35.01024 35T^§M^ = 35^ff^. Ans. 9. Reduce to common fractions in their lowest terms 7.015625. 10. Reduce to common fractions in their lowest terms 20.100256. 20.100256 = 20j^^o_o^%%= 20/xWi7. Ans. 160 ARITHMETIC. 11. Reduce to common fractions in their lowest terms 10.012575. 10.012575 = 10t^^^^^ = 10j^M7- ^^«- 12. Reduce to common fractions in their lowest terms 104.236. 104.235 = 104.,^ = 1042^. Am. 13. Reduce to common fractions in their lowest terms 50.0004. 50.0004 = 50tt^ = 50^^. Am. 14. Reduce to common fractions in their lowest terms lOO.OOl. 100.001 = lOOy^. Am. 16. Reduce to common fractions in their lowest terms 8.00725. 8.00725 = SjMj^j, = 8jH7r- ^ns. 16. Reduce to common fractions in their lowest terms 20.018375. 20.018375 = 20jmih = ^0^^. Am. 17. Reduce to common fractions in their loweet terms 125.6048. 125.6048 = 125T^^^=125fH. Am. 18. Reduce to common fractions in their lowest terms 0.128. 0.128 = tV^ = ^. Am. 19. Reduce to common fractions in their lowest terms 0.73125. 0.73125 = tWi^ = IH- ^^• 20. Reduce to common fractions in their lowest terms 1.1876. 1.1875 -l^Wj^^lt*,. Am. 21. Reduce to common fractions in their lowest terms 0.603125. 0.603125 -VWWW- iff ^^• 22. Reduce to common fractions in their lowest terms 6.03125. 6.03125 -6yHf^-6iV. ^ns. 23. Reduce to common fractions in their lowest terms 60.3126. 60.3125 -60AVW-60A. Am. 24. Reduce to common fractions in their lowest terms 7.0316. 7.0315 -7x*i^-7,Sb. Am, TEACHERS EDITION. 161 Exercise XXXL 1. Reduce to decimals |. 0.875 8)7.000 2. Reduce to decimals \^. 0.9375 16)15.0000 3. Reduce to decimals -j^^, 0.28125 32)9.00000 4. Reduce to decimals ^\. 0.36 25)9.00 5. Reduce to decimals ^\. 0.078125 64)5.000000 6. Reduce to decimals 4^V(7- 0.01375 8)0.11000 4.01375. Ans. 7. Reduce to decimals ^-^^ioi^. 0.00015625 32)0-00500000 5.00015625. Ans. 8. Reduce to decimals 9^^|f (j. 0.0048046875 256)1.2300000000 9.0048046875. Ans. 9. Reduce to decimals ll^f^. 0.00475 4)0.01900 11.00475. Ans. 10. Reduce to decimals yf^. 0.072 125)9.000 11. Reduce to decimals ^^^^. 0.00425 4)0.01700 12. Reduce to decimals ^^f. 0.9296875 128)119.0000000 13. Reduce to decimals -^\^\. 0.0208 625)13.0000 14. Reduce to decimals ^^j. 0.04296875 256)11.00000000 15. Reduce to decimals yf^. 0.01875 16)0.30000 16. Reduce to decimals Jj^. 7.75 16)124.00 162 AEITHMETIC. 17. Reduce to decimals | of If. 19. Reduce to decimals 3f of 4f 3 2x2 = ^. ^^5 5 1.2 5)6.0 2 X?^37_74 d"" 9 5' 14.8 5)74.0 18. Reduce to decimals foffof^. 4^8^X0 64 2 20. Reduce to decimals f f of f f . 29 49 1421 32 64 2048' 0.328125 0.69384765625 64)21.000000 2048)1421.00000000000 Exercise XXXII. 1. In like manner simplify 7^ + 4f + 9^ + llff . 7^ + 4| + 9^^ + 11|| , 7.4 + 4.625 + 9.65 + 11.90625 = 33.58125. (1) n+n+m-^ hh = ^mi = 33^ = 33.58125. (2) 2. In like manner simplify 84^ + 19^ - f J. 84^ + 19H + f* = 84.65 + 19.523809^| + 0.82 = 104.993809^. (1) 84H + 19ii + fi - I03 l8ggtim-fl7 22 =. 10311^ = 104fW - 104.993809ii. (2) 3. In like manner simplify 4»| + 13^ + 42f^ + 2^ + li 4rj + 13 J^ + 42|^ + m + li = 4.421875 + 13.85 + 42.74 + 2.8125 + 1.5 - 65.324375. (1) m + mi + 42H + 2i| + H - 62 87g-M8g0^}i f itl80Qt800 - ^^m - ^W^ - 65.324375. (2) 4. In like manner simplify 5J + 13f + 19^ + 7^. 5J + 13f + 19^ + 7^" 5.876 + 13.8 + 19.4375 + 7.15 - 46.2626. (1) 6H 13f + 19^ + 7A = 44lJi±Ai^-*iP±ia - 44W - 46|^ - 46.2626. (2) teachers' edition. 163 5. In like manner simplify 5^^ + f of If + | of 2^ + f of f . 5T'(T + fXl| + |x2f + fxf = 5.5 + 0.666f X 1.8 + 0.875 x 2.285714f + 0.75 x 0.625 = 5.5 + 1 .2 + 2 H- 0.46875 = 9.16875. (1) 5A + |Xlf + |x2f + -|xf:=5i + li + 2 + M-8''ig r ^' = 811^ = ^7 = 9.16875. (2) 6. In like manner simplify 1^^ of 2f . lyi X 2| = 1.4166f X 2.625 = 3.71875. (1) 7 1AX2| = 1^X| = ^^ = 311 = 3.71875. (2) 4 7. In like manner simplify Sy^ + 2|-f , h% + 2M = 3.3125 + 2.95 = 6.2625. (1) h% + 2M = 5^W- = 5-W- = 6f ^ = 6.2625. (2) 8. In like manner simplify 7f — 4f . 7f - 4f = 7.4 - 4.625 = 2.775. (1) 7f-4f = 3J-VTp = 2fi = 2.775. (2) 9. In like manner simplify 82^ — 37||^. 82^ - 37H = 82.2 - 37.6875 = 44.5125. (1) 82^ - 37H = 45J-%-/-5- = 44|^ = 44.5125. (2) m- 10. In like manner simplify 100 — 17|i 100 - 17iif = 100 - 17.1808 = 82.8192. (1) 100 - 17HI = 82f if = 82.8192. (2) 11. In like manner simplify 5^ — 1| of 1||. 5^ - 1^ X HI = 5.5 - 1 .5 X 1.5416f = 5.5 - 2.3125 = 3.1875. (1) 5^ - 1| X HI = 5^ - 2j\ = 3^ = 3.1875. (2) 164 ARITHMETIC. 12. In like manner simplify i| — ^\. ^ _ ^1 _ 0.56 - 0.171875 = 0.388125. H - H - " i VoV^ =- im - 0.388125. (1) (2) 13. In like manner simplify 8^ — 1 J x t^. 8^ _ IJ X T^ = 8.2 - 1.5 X 0.1875 = 8.2 - 0.28125 = 7.91875. (1) 8i - H X A - Si - /j = 8^^ = 7||* = 7.91875. (2) 14. In like manner simplify ^| X 1000. j^f X 1000 = 0.29H875 x 1000 = 296.875. j^j X -i^V^ - ^«^ = 296f = 296.875. (1) (2) Exercise 1. Reduce to decimals f . 0.5 2. Reduce to decimals ^. 0.45 lll5!00 3. Reduce to decimals 3^. 0.416 12)5.000 3t»jj- 3.416. Am. 4. Reduce to decimals ^. 0.183 6)1.100 6. Reduce to decimals 3^. 0.35416 48)17.00000 3H - 3.35416. Afu. XXXIII. 6. Reduce to decimals 2/y. o.iss 37)5.000 2/7. = 2.1 35. Ans. 7. Reduce to decimals y^^. 0.00081 37)0.03000 8. Reduce to decimals ll^J. 0.13095238 84)11.00000000 11^=1113095238. Ans. 9. Reduce to decimals 9^^^. 0.10185 108)11.00000 9iVr - 9.10185. Ant. TEACHERS EDITION. 165 10. Reduce to decimals 11 35. 0.1 142857 35)4.0000000 1X3% = 11.1142857. Ans. 11. Reduce to decimals |^f . 0.267857142 56)15.000000000 12. Reduce to decimals ^*y. 0.380952 21)8.000000 13. Reduce to decimals |f. 0.39 33)13.00 14. Reduce to decimals f-J. 0.5285714 7)3.7000000 15. Reduce to decimals 2^-^^. 0.22745098039215686 255)58 .00000000000000000 2^5/^ = 2.22745098039215686. 16. Reduce to decimals 5^\. 0.230769 13)3.000000 5^6^ = 5_3_ ^ 5.230769. Ans. 17. If 117 be expressed as a decimal, the quotient will contain how many decimal places? As 7 is the highest power of 2 or 5 in the denominator, and as there are no otl^r factors than 2 or 5, there will be seven decimal places in the quotient. 18. If 119 be expressed as a decimal, how many decimal places 25 X 13 will precede the recurring period ? As 5 is the highest power of 2 or 5 in the denominator, and as there is another factor than 2 or 5, five decimal places will precede the repetend. 19. If 57 5=^x7 be reduced to a decimal, how many decimal places will precede the recurring period ? As 2 is the highest power of 2 or 5 in the denominator, and as there is another factor than 2 or 5, two decimal places will precede the repetend. 166 ARITHMETIC. Exercise XXXIV. 1. Reduce to common fractions in their lowest terms 0.245. 0.245 = ^-tV^. Ans. 2. Reduce to common fractions in their lowest terms 0.425. 3. Reduce to common fractions in their lowest terms 53.00243. 53.00243 53^^ =• 53yT%^. Ans. 4. Reduce to common fractions in their lowest terms 7.2011. 7.26li = ?§§&§. Am. 6. Reduce to common fractions in their lowest terms 2.5306. 2.5306 -2iM*-2iHi ^^• 6. Reduce to common fractions in their lowest terms 0.00426. 0.00426 - ^mn = lUh- ^'M. 7. Reduce to common fractions in their lowest terms 31.203. 31.203 = 31^^ = 31^. Am. 8. Reduce to common fractions iu their lowest terms 0.35i. 0.35i = fH = if Am. 9. Reduce to common fractions in their lowest terms 1.416. 1.416 = lH^ = ltV. Am. 10. Reduce to common frac- tions in their lowest terms 0.5575. 0.5575 - 1^^ = \%\. Am. 11. Reduce to common fractions in their lowest terms 2.081. 2.08 U 2^ -2^. Am. 18. Reduce to common fractions in their lowest terms 5.12297. 5.12297 = 5iHM = 5/iV ^^«- 18. Reduce to common fractions in their lowest terms 0.3590 0.3590- If e-T^. Am. 14. Reduce to common fractions in their lowest terms 4.3162. 4.3162 -4HH-4iH- ^'W. teachers' edition. 167 15. Reduce to common fractions in their lowest terms 0.7283. 0.7283 = |f|f = f Iff. Ans. 16. Reduce to common fractions in their lowest terms 5.142857. 5.142857 = 5ifffM = 55VW^V Ans. 17. Reduce to common fractions in their lowest terms 0.2368. 0.2368 = ff If = Hff- Ans. 18. Reduce to common fractions in their lowest terms 1.136. 1.136 = lift =:1tV8 = 1A- Ans. 19. Reduce to common fractions in their lowest terms 1.53i. 1.53i = lf|i = l,-5,-V Ans. 20. Reduce to common fractions in their lowest terms 3.28963. 3.28963 = 3f If If =3/2\V Ans. 21. Reduce to common fractions in their lowest terms 5.8783. - 5.8783 = 5Un-^U- Ans. 22. Reduce to common fractions in their lowest terms 1.69408. 1.69408 = If f Iff = Hif If Ans. 23. Reduce to common fractions in their lowest terms 0.48324. 0.48324 = fff^f = ff|. Ans. 24. Reduce to common fractions in their lowest terms 0.00i2213. O.OOi2213 = ^iffif^ = TT¥tVVo- Ans. 168 ARITHMETIC. Exercise XXXV. 1. Find the G.C.M. and L.C.M. H-i G.C.M. of 7, 14,2 -1. L.C.M. of 9. 27, 5 -105. .-. G. C. M. of fractions = y^^. L. C. M. of 7, 14, 2 = 14. G.C.M. of 9, 27, 5 ^^ 1. .*. L. C. M. of fractions = 14. 2. Find the G.C.M. and L.C.M. of2i2i:^. 2f-^,2f-V,:^ = ^. G. C. M. of 20, 12, 1 =1. L. C. M. of 9, 5, 10 = 90. .'. G. C. M. of fractions = ^. L. C. M. of 20, 12, 1 = 60. G.C.M. of 9,5, 10 =1. .-. L. C. M. of fractions = 60. 3. Find the G.C.M. and L.C.M of 33f , 50f 33^ = ifi, oOf = ^^, G.C.M. of 234. 405 =9. L.C.M. of 7, 8 =56. .-. G. C. M. of fractions = ^g. L.C.M. of 234, 405 =10,530. G.C.M. of 7, 8 =1. .-. L. C. M. of fractions - 10,530. 4. Find the G.C.M. and L.C.M. G. C. M. of 7, 35, 49 = 7. L.C.M. of 24, 36, 60 =-360. .•. G. C. M. of fractions = y||y. L. C. M. of 7, 35, 49 = 245. G.C.M. of 24, 36, 60 =12. .-. L. C. M. of fractions = Y/ 5. Find the G. C. M. and L. C. M. of 5^, 7^, 8^, 4f , 9^ Q^j. 5i, 7i 8 J, 4f , 9i 6A - Jjji, V. ^, V. ¥. H- G.C. M. of 11, 22, 33, 44, 55, 77 = 11. L.C.M. of 2, 3, 4, 9, 6, 12 =33. .-. G. C. M. of fractions = ^i. L. C. M. of 11, 22, 33, 44, 55, 77 = 4620. G. C. M. of 2, 3, 4, 9, 6, 12 -1. .-. L. C. M. of fractions - 4620. 6. Find the G. C. M. and L. C. M. of f f ^, |, ^ ^, ^. G.C.M. of 1,1, 1.1, 1,1, 1 -1. L. C. M. of 2, 3, 4, 5, 6, 10, 12 - 60. .'. G. C. M. of fractions — ^. L.C.M. ofl, 1.1, 1,1, 1,1 =1. G.C.M. of2, 3. 4. 6, 6, 10, 12 -1. .-. L. C. M. of fractions - 1. teachers' edition. 169 7. Find the G. C. M. and L. C. M. of 50^, 67^ 44|, 841 707. 50|, 67i 44f , 84|, 707 = ifi, ^-p, ^^, ^^, 707. G. C. M. of 101, 202, 404, 505, 707 = 101. L.C.M. of 2, 3, 9, 6, 1 = 18. .'. G. C. M. of fractions = V/ = Hi- L. C. M. of 101, 202, 404, 505, 707 = 14, 140. G.C.M. of 2, 3, .9. 6, 1=1. .-. L. C. M. of fractions = 14, 140. 8. Find the G. C. M. and L. C. M. of |, |, f , |, f , j%. G. C. M. of 4, 5, 6, 7,8, 9 = 1. L. C. M. of 5, 6, 7, 8, 9, 10 = 2520. .•. G. C. M. of fractions = -^^-^^jj. L. C. M. of 4, 5, 6, 7, 8, 9 = 2520. G.C.M. of 5, 6, 7, 8, 9, 10=1. .'. L. C. M. of fractions = 2520. 9. Find the G. C. M. of 1^\, l^f , ^, f f lTV.H!.4f.|f = iiM,-^^,|f. G. C. M. of 15, 40, 30, 25 = 5. L.C.M. of 14, 21, 7,42=42. .-. G. C. M. of fractions = ^%. L. C. M. of 15, 40, 30, 25 = 600. G. C. M. of 14, 21, 7, 42 = 7. .-. L. C. M. of fractions = -6fo = 85f. 10. Find the G. C. M. and L. C. M. of 18f , 57| 18| = -%\ 57^ = ^i^- G. C. M. of 92, 115 = 23. L.C.M. of 5, 2 = 10. .-. G. C. M. of fractions = H- ^\%. L.C.M. of 92, 115 = 460. G. C. M. of 5, 2 = 1. .-. L.C.M. of fractions = 460. 170 ARITHMETIC. 11. Find the G. C. M. and L. C. M. of 134f , 128^, 115^. 134f, 128^, 115^ = ^. ^^, -4^. G.C.M. of 539, 385, 231 = 77. L.C.M. of4, 3, 2 =12. .-. G. C. M. of fractions = H = ^i- L. C. M. of 539, 385, 231 = 8085. G.C.M. of 4, 3, 2, =1. .-. L. C. M. of fractions = 8085. 12. Ftnd the G. C. M. and L. C. M. of 2^, If |, ^^V G.C.M. of 72. 112, 63 =1. L. C. M. of 25, 75, 100 = 300. .'. G. C. M. of fractions = y^^. L.C.M. of 72, 112, 63 =1008. G.C.M. of 25, 75, 100 =25. .♦. L. C. M. of fractions = J^ = 40^. 13. A, B, and C start together and travel round a circular island, in the same direction. It takes A 2\ days, B 2f , C 2| days to walk round the island. They travel until they all meet at the point of starting. In how many days will they be together at the point of starting? 2i2|,2J = l.-V,^. L.C.M. of 7, 17, 23 = 2737. G. C. M. of 3, 6, 8 = 1. /.L.C.M. =2737. 2737 days. Am. 14. If the step of a man be 2^ft., and that of a horse be 2fft., find tlio smallest number of feet which is an exact number of man-paces and of horse-paces. L.C.M. of 7, 11 = 77. G.C.M. of 3, 4 -1. .-.L.C.M. -77, 77 ft. Ans. teachers' edition. 171 15. Find the largest number that is contained without remainder in 2f , 6tV, Hi and 19^ 2f , 6tV, Hi. I9i = ¥. W. ¥. H^' G. CM. of 23, 115,23, 115 = 23. L.C.M. of9, 18, 2, 6 =18. .-. G.C.M. =ff = lTV Exercise XXXVI. 1. Simplify iui nuh zm-h, im- 2709 _^ 43785 _ 973 2436 ^ 203 4087 ^ 67^ 6966 is' 56835 1263* 567216 47268* 5063 83* 2. Which is greater, and how much, | or |f ? 7 19 56, 57 19 . , u 1 -, — = — ! -. .-. — IS greater by — 9 24 72 24 ^ -^ 72 3. Find the sum of 3|, 2t*t, 5^ 7tV, l^V- H + 2A + 5H 7^^ + 1^\ = 18^+^0^, 5 +7 7+15 ^ 182 3 1 = 20tV. Ans. 4. Simplify 5i-3f + 2T%- If. Sum of plus terms = 8f . Sum of minus terms = 5-^^. Difference = 3^|. 5. Simplify If + 3f - 2^^ + 4.^% - 3^^. Sum of plus terms = 9f^. Sum of minus terms = 6^^^. Difference = 3^-^. 6. Simplify ii+ii. 3H3t^ 42 + 46 ^88^^_, ^ 4i-2^^ 52-31 21 ^2T- ^^«. 172 ' ARITHMETIC. 7. Simplify the expressions: 7^2|; ^- ^; 15-j-f; ^^ 7A^9; 43i^37i; f^ ; 5| ^ 4f ; l-^l^ ; 106 -.- 8f ; i]-- 10^ t X f ^7 ?5i_ 11x151 = 2101 ^11^^ 5f.4t = Axf = | = H; 2 3 ,5 45 „„. t2fli = ?x|x^x2=9 S^Jj- 95 2 11^ ^^■*-5 = I^T = f ==22i; |xf ?";2- 7' 3 14 1ft ^ Iff 7jL*9-lx«-9 il-l2xH = 52? = 3S* 8. Simplify the expressions ; 7J| X 8 ; 43Ji x 6| ; 6^ -.- 8^ ; 5^ X 51 HofM; HofAofJoffoa; HoffW; ix|xAx4xf 7Hx8 = ||xf = f = 60t. 4 573 43Hx6| = 2|2x|-5p = 286i; ■«**«i-^xf = -^ ' ^4 ;y ^z 11* 33 11 lx?x^x?x? = l. 3 3 6lV X51 '> 1 ■258; }|^ ,11^ ^13" _121. "l56' 52 r i xf- 'k- teachers' edition. 173 9. By what must |- be multiplied to obtain | ? | to obtain f ? ^- to obtain | ? f to obtain | ? | to obtain | ? 2 6 1 ^ ' 3 3 ' 6 13 ' 2 7. 3_5 7_35_.,i 8 • 5~3^8~24~^^- 10. By what must ^ be divided to obtain |? | to obtain ^? f to obtain f ? | to obtain f ? f to obtain |? 8 to obtain 7^1? 6"21^3' 8"548 32 ^" 2 37_8^3 24 2^1 = ^X^ = 4. 5"^8~7''5-35 = 3 6 13' 6^6"^''^-^' ^'^^^^-243''l~243-^^^^- 11. What number exceeds 5f by 4| ? 5| + 4| = 91-^2^-^ = 9|f = 10/3. 12. From what must 6f be subtracted to leave | of 3^ ? 14 2''^^-^^¥-¥-^^' 6| + lf = 7^%2_5 = 7|| = 8^V 13. What fraction falls short of ^V by -i^ ? 7 3 _ 35 - 9 ^ 26 ^ 13 12 20 60 60 30' 14. What fraction is that to which -^^^ must be added to give ^ ? 11 5 ^ 44-15 ^ 29 57 76 228 228' 174 ARITHMETIC. 15. Convert into decimals i; i; i; f: ii ii; iV.A.A. i^.^.H.H.H; if; f: 1; A: ^• 2)1.0 8)7.000 0.0625 6)1.00 0.5 0.875 16)1.0000 0.0625 9 0.16 4)1.00 0.25 0.5625 6)5.00 3)1.0 0.0625 0.0625 11 0.83 0.3 3 0.6875 7)3.000000 4)3.00 0.1875 0.42857i 0.75 0.0625 0.0625 9)5.0 8)1.000 5 13 0.5 12.^ 0.8125 V. 1 L^J 8)3.000 0.3125 11)3.00 0.375 0.0625 0.0625 0.27 8)5.000 7 15 4)0.700 0.626 0.4375 0.9376 0.175 16. Convert into common fractions: 0.16; 0.016; 0.125; 0.13; 0.725; 0.625; 0.00625; 0.8125 ; 0.03125 ; 0.08 ; 0.54 ; 0.016 ; 0.5437 ; 0.027; 0.277; 0.68494; 1.345. 0.16 = iW = A; 0.03125 =.Vytfy^ = ^ 0.016 = T*fT7 = Th; 0.08 =Tk = 3^ 0.126 -iVW = i; 0.54 =M = A 0.13 = tV^: 0.016 =^ = 7^ 0.725 -i^ -H; 0.5437 ^\m 0.027 = ^V = TfTr; 0.625 -im -*; 0.277 = \%% = A; 0.00626 -TlW^ r = Tb; 0.68494 =»H|^^ = IMH; 0.8126 -iVrfW = 11; 1..345 = \\\l -iH- 17. Simplify '^ X2.27 1.136 M>V 5 M ?? 6 ^ teachers' edition. 175 ^ 18. Multiply 6.954 by 5.303, and express the result as a whole number and common fraction. 51 6.954 = 6fi; 1^^175^8925^33213 5.303 = 5if;' 22 33 242 '^^" 19. Simplify li of 2f + 6| -f- 2f and reduce the result to a decimal. Hx2fH-6|-^2| = fx-V*- + T\X-V- = 4i + 2^ = 6^5_6/^ = 6.7. 20. From what number can 4^| be taken 9 times and leave no remainder ? 4Hx9 = :^xf = l|-l = 40i 4 21. Of what fraction is 17| the 7th part ? 17ix7 = ^x| = ^ = 12U. 22. Add I 0.35, f, f, 0.il2, 45.28. f + 0.35 + I + f + 0.112 + 45.28 = 0.8 + 0.35 + 0.625 + 0.75 + 0.112 + 45.28 = 47.917. 23. Convert into decimals H; A; J%< H; 11; A; ^V 0.86 0.27 0.1 142857 0.283 15)13. ll}3^ 35)4.0000000 60)l7.000 0.736842105263157894 0.384615 ' 0.1320754716981 19) 14.000000000000000000 13)5.000000 53)7.0000000000000 24. What part of }f is j^\j 1 1241 ' 73 ;^ JLUl 85' 5 17 25. Divide 0.0015 by 0.012, and express the result as a common fraction in lowest terms. 0.125 12)1.500 0.125 = \. 176 ARITHMETIC. 26. Convert into decimals : s\ ; jy^^ ; j| ; \. 0.09375 0.00009375 0.2297 0.141857 32)3.00000 32)0.00300000 74)17.0000 7)1.000000 27. If the product of two factors is |, and one factor is 1^, find the other factor, 8 * ^^ 2 28. If the dividend is |J and the quotient 6^, find the divisor. 12 ^ 13 ;;z 78 6 29. The dividend is 12||, quotient 3, remainder l^^ ; find the divisor. (12H-lA)-^3 = 10M-H3 = ixW = m = 3Mi 30. Find the G. C. M. and the L. C. M of 833, 1127, 1421, 343. 7 1833 1127 1421 343 7 1119 161 203 49 17 23 29 7 G. C. M. = 7 X 7 = 49. L. C. M. - 73 X 17 X 23 X 29 = 3,889,277. 31. Arrange in order of magnitude ^j, f f , if, ^, f §. .'. the order of magnitude is ^, ^, |f ^f , f^. 32. Find the L. C. M. of {^ |f , ^j. L. C. M. of 15, 26, 65 = 390. G.C.M. ofl7, 51, 102-17. .'. L.C.M. of fractions = ^7^ = 22|f . 33. Find the G. C. M. of f f , b^, f ^, and 6^. G.C.M. of 65, 39, 91, 13-13. L.C.M. of 68, 2,64, 2 =.1088. .'. G. C. M. of fractions - yif,. teachers' edition. 177 34. Convert into common fractions in lowest terras: 7.2011; .954; 5.303; 21.396. 7.201 i = 7|»ff. 6.954 = 6|ff = 6|i. 5.303 =5f|f = 5if. 21.396 = 21f 11=213*3-4.. 3^ "'"''"'•' 51-71^28^ + 1 nxlj\ + ^,\-3j% 4 + 4tV-3A 4ff 7 5i-7|-^28^V + i 5i-T\ + i 51 8 36 Simplify ^t^^^^X 2^ -7i 3^ + 21-4^^ 6f + 5ix3}-7i 6| + 17f-7i ^ + ^-^T\ 3^ + 2|-4tV _ 945 + 2420 - 1015 2350 ^^ , ., 448 + 350-574 224 ^'^' 37. -»"^^ff?^i- 2f-li + 9T-V 616-330 + 2000 2286 2 41-2^+13/j- 924-495 + 3000 3429 3 38. Simplify (3-71- 1-^08) X7.03_ 2.2 -3^% (3.71 - 1.908) X 7 03 1.802x7.03 12.66806 , ,,,^0 2.2-,^ 2|-^ 2 "■""^""' 39. Simplify ^t^t ^.^fHof4i ^ ^ Hof-l-^lOi"^ 13|of5i 5f-^f ^2.fHof4i l^off-5-10| ' 13|of5| = i^x^x^x'^x^x^x^x^^x ^ x^ -2^^-423 ^ '^2''^'^g'< 3 '^^''^'^y'^?;r;'^i6~"64 ~ ''^• 2 3 178 ARITHMETIC. 40. Simplify H of 2^ + 6| i- 2| + (s^ + ^:^^±^\ I .', of 2U 6J -H 2ii f ['^\ s ^A2i±0:^\ =. 4 1 + 2^ + 5^ + M V 2.2-0.G4/ If = 4i + 2i + 5| + Mi = ll^iii^i^^W^±^ = llHH=12Hf 41. Simplify 0.9 of f of f of 15|. 0.9o£Joffofl5i = ^^x2xfxf = fl = 5A. 2 2 42. What part of f 18 i- 12 2 4 43. What part ofO.390625 is 0.05? 16 0.05 ^_^_^^.. 1_16 0.390625 fl 25 ?0 125 5 44. 0.09 is what fraction of 0.2045? 0.09 _ -h ^tV_4 0.2045 \m ^? 9' 46. Convert into decimals |f ; |f ; f|. 0.731343283582089552238805970149253 67)49.000000000000000000000000000000000 0.378 0.84931506 37)14.000 73)62.00000000 46. Tlio Tf . 0. M. of three numbers is 15, and their L. C. M. is 450 What are tho numbers? 450-(5x3)x2x3x5 5X3-16-G.C.M. 15x2-30. 15 X 3 - 46. 16x5-76. TEACHERS EDITION. 179 EXEECISE XXXVII. 1. Reduce 3 yds. 2 ft. to inches. 5. Reduce 82,976,432 in. to 3 yds. 2 ft. miles. X3 12 82976432 in. 9 2 3 5i 6914702 ft. 8 in. 2304900 yds. 2 ft. 11 ft. Xl2 132 in. 2 11 320 4609800 [yds. 419072 rds. 8 half-yds. = 4 1309 mi. 192 rds. 2. Reduce 4 mi. 124 rds. to feet. 4 mi. 124 rds. 1309 mi. 192 rds. 4 yds. 2 ft. 8 in. X320 6. Reduce 7 mi. 3^ yds. to 1280 inches. 124 7 mi. 3| yds. 1404 rds. X1760 Xl6i 12320 23166 ft. H 3. Reduce 27 rds. 4|yds. to 123231 yds. inches. X36 27 rds. ^ yds. X5^ 443646 in. 148i X4| 7. Reduce 27 mi. 222 rds. to inches. 152^ yds. • 27 mi. 222 rds. * X36 X320 5499 in. 8640 4. Reduce 290 leagues to feet. 222 290 leagues. 8862 rds. X3 X5i 870 knots. 48741 yds. X6086 X36 5294820 ft. 1754676 in. 180 ARITUMETIC. 8. Eedace 712 mi. to inches. 712 mi. X5280 3759360 ft. Xl2 45112320 in. 9. Reduce 540,451 ft. to miles. 31540451ft. 5i|i80150yd8. 1ft. 2 11 320 3 60300 32754 rds. 6 half-yards 3 yds. 102 mi. 114 rds. 102 mi. 114 rds. 3 yds. 1 ft. 10. Reduce 271,256 in. to miles. 12 3 271256 in. 22604 ft. . . 7534 yds. . 2 *. . 8 in. . . 2 ft. 11 15068 320 1369 rds.. . . . 9 half-yards = 4 mi . . 89 rds 4 mi. 89 rds. 4^ yds. 2 ft. 8 in i yd. = 1 ft. 6 in 4| yds 4 mi. 89 rds. 5 yds. 1 ft. 2 in. 11. Reduce 723,964 ft. to miles. 31723964 ft. 5^1241321 yds. 1ft. 2 11 320 482642 43876 rds. 6 half-yards - 3 yds. 137 mi. 36 rds. 137 mi. 36 rds. 3 yds. 1 ft. TEACHERS EDITION. 181 12. Reduce 233,205 in. to miles. 12 233205 in. 3 19433 ft 9 in. ^ 6477 yds. . . . 2 2 ft. 11 12954 320 1177 rds. . . . 7 half-yards = 3^ yds. 3 mi. ... 217 rds. 3 mi. 217 rds. 3^ yds. 2 ft. 9 in. ^ yd. - 1 ft. 6 in. 3 mi. 217 rds. 4 yds. 1 ft. 3 in. 13. Reduce 10 chains to inches. 15. If the height of a horse 10 ch. be 16 hands, how many feet is X4 his height ? 16 hands 40 rds. X4 Xl6^ 12)_64 in. 660 ft. 5 ft. 4 in. Xl2 7920 in. 16. If a train move 40 ft. in a second, what is its rate in miles 14. Reduce 233,185 in. to per hour ? (One hour = 3600 fathoms. seconds.) 12)233185 in. 3600 40 ft. 6)19432 ft. 1 in. 5280)144000 ft. 27ift§ mi. 3238 fath. 4 ft. 3238 fath. 4 ft. 1 in. = 27Ami. Exercise XXXVIII. 1. Reduce 92,638 sq. yds. to 2. Reduce 1,223,527 sq. in. to inches. Q2638 sq. yds. yards. ^ X9 144)1223527 sq. in. 833742 sq. ft. 9)8496 sq. ft. 103 sq. in. Xl44 944 sq. yds. 1200588- t8 sq. in. 944 sq. yds. 103 sq. in. 182 ARITHMETIC. 3. Reduce 721 sq. mi. to rods, 721 sq. mi. _X_640 461440 A. Xl60 73830400 sq. rds. 4. Reduce 34,729 sq. yds. to rods. 30})34729 sq. yds. 4 121 )138916 1148 sq. rds. 8 quarter- sq. yds. = 2 sq. yds. 1148 Bq. rds. 2 sq. yds. 6. Reduce 1 A. to feet. 1 A. 160 160 sq. rds. X30^ 4840 sq. yds. X9 43560 sq. ft. 5. Reduce 3 A. 107 sq. rds. 27 sq. yds. 7 sq. ft. 23 sq. in. to inches. 3 A. Xl60 480 107 587 sq. rds. X S0\ 17756f 27 177831 sq. yds. X9 1600531 7_ 1600601 sq. ft. Xl44 23048748 23 23048771 sq. in. 7. Reduce 99,894,712 sq. in. to acres. 144 )99894712 sq. in. 9)693713 sq. ft. 40 sq. in. 30J)77079 sq. yds. 2 sq. ft. 4 121 )308316 160 )2548 sq. rds. 8 quarter-yds. = 2 sq. yds. 15 A. 148 sq. rds. 15 A. 148 sq. rds. 2 sq. yds. 2 sq. ft. 40 sq. in. TEACHERS EDITION. 183 8. Reduce 15,376 sq. yds. to acres. 30^)15376 sq. yds. 4 121 )61504 16 0)508 sq. rds. 36 quarter-sq. yds = 9 sq. yds. 3 A. 28 sq. rds. 3 A. 28 sq. rds, 9 sq. yds. 9. Reduce 562,934 sq. in. to rods. 144 )562934 sq. in. 9)3909 sq. ft. 38 sq. in. 30^)434 sq. yds. 3 sq. ft. 4 121 )1736 14 sq. rds. 42 quarter-sq. yds. = 10|^ sq. yds 14 sq. rds. 10|- sq. yds. 3 sq. ft. 38 sq. in. = 14 sq. rds. 10 sq. yds. 7 sq. ft. 110 sq. in. Exercise XXXIX. 1. Reduce 7 cu. yds. 13 cu. ft. to cubic feet. 7 cu. yds. 13 cu. ft. X27 189 _13 202 cu. ft. 2. Reduce 25 cu. yds. 5 cu, ft, 143 cu, in. to cubic inches. 25 cu. yds. 5 cu. ft. 143 cu. in. X27 675 5 680 cu. X1728 ft. 1175040 143 1175183 cu. in. 3. Reduce 74,325 cu. in, to cubic feet, 1728 )74325 cu, in. 43 cu. ft, 21 cu. in, 43 cu. ft. 21 cu, in, 4. Reduce 439,000 cu. in. to cubic yards. 1728 )439000 cu, in. 27)254 cu. ft. 88 cu, in. 9 cu. yds, 11 cu. ft, 9 cu. yds. 11 cu. ft, 88 cu. in. 5. Reduce 921,730 cu, in, to cubic yards, 1728 )921730 cu. in. 27)533 cu. ft. 706 cu. in. 19 cu. yds. 20 cu. ft. 19 cu. yds. 20 cu. ft. 706 cu. in. 184 ARITHMETIC. 6. Wood cut in lengths of 4 ft. is piled to a height of 3J ft. How long must the pile be to contain a cord ? S^ft. Xj ft. 14 sq. ft. 9fft 14)l28 126 Exercise 1. Reduce 3 pks. 5 qts. 1 pt. to pints. 3 pks. 5 qts. 1 pt. X8 24 5 29 qts. X_2 58 1 59 pts. 2. lieduce 4234 pts. to bushels. 214234 pts. 2117 qts. 264 pks. 5 qts. 7. A pile of wood 127 ft. long, 4 ft. wide, and 3 ft. 8 in. high is sold for $ 7 a cord. How much money is received for it ? 127 ft. X4ft. 508 sq. ft. X3f 128 )1862^ 143| cd. X$7 flOlff = $101.86. XL. 3. Reduce 24 gals. 2 qte. 1 pt. 2 gi. to gills. 24 gal. 2 qts. 1 pt. 2 gi. X4 96 2 98 qts. X2 196 I 197 pts. Xj 788 2 66 bu. 790 gi. 66 bu. 5 qts. 4. Reduce 272 liquid quarts to dry quarta. 17 11 1 67i 1 4 W 4 - 233=1 qts. TEACHERS EDITION. 185 5. 'Reduce 400 dry quarts to liquid quarts. 16 80 m 57| X400- 11 5120 11 465^5^ qts. 6. Express a bushel in cubic feet, carrying the decimal to three places. 1.244 1728)2150.420 7. Express a cubic foot as the decimal fraction of a bushel. 0.8036 215042)172800.0000 8. Reduce 1715| bushels to pints. 1715 |bu. X64 109792 pts. 9. 3047 gals, to barrels. 3047 3047 2 X — = = ^094 = 96pbbl. 3H 1 63 63 ^" Exercise XLI. 1. Reduce 27,587 grs. to pounds 3. Reduce 136,851 oz. to tons. troy. 16)136851 oz. 24)27587 grs. 100)8553 lbs. 3 oz. 20)1149 dwts. 11 grs 20]85 cwt. 53 lbs. 12)57 oz. 9 dwt. 4 t. 5 cwt. 4 lbs. 9 oz. 4 t. 5 cwt. 53 lbs. 3 oz. 4 lbs. 9 oz. 9 dwt. 11 grs. 4. Reduce 864,205 grs. (troy) 2. Reduce 34,652 lbs. to ong to pounds. tons. 24)864205 grs. 112)34652 lbs. 20)36008 dwts. 13 grs. 20)309 1. cwt. 44 lbs. 12)1880 oz. 8 dwts. 151. t. 9 1. cwt. 150 lbs. 15 1. t. 9 1. cwt. 44 lbs. 150 lbs. 8 dwts. 13 grs. 186 ARITHMETIC. 6. Reduce 864,205 grs. (apoth.) to pounds avoirdupois. m^m lbs. 7000)864205 123fH^ = 123T^lb8. -123 lbs. 7oz. 5.2dr8. 6. Reduce 5 lbs. 7 oz. 6 dwts. 12 grs. to grains. 51bs. 7oz.6dwtl2gr8. Xl2 60 _7 67 oz. X20 1340 6 1346 dwt. X24 32304 12 32316 grs. 7. Reduce 745 lbs. avoirdupois to troy weight. 175 »X^lb8. W0 1 144 = 905 lbs. 4 oz. 11 dwt. 16 grs. 8. Reduce 745 lbs. troy to avoirdupois weight. 144 149 xm 1 35 - 613,>5 lbs. = 613 lbs. 7^ drs. 9. Reduce 23,047,125 drs. to tons. 16 )23047125 drs. 16 )1440445 oz. 5 drs. 100 )90027 lbs. 13 oz. 20)900 cwt. 27 lbs. 45 t. 45 t. 27 lbs. 13 oz. 5 drs. 10. Reduce 90,252,381 drs. to tons. 16 )90252381 drs. 16 )5640773 oz. 13 drs. 100 )332548 lbs. 5 oz. 2 0)3525 cwt. 48 lbs. 176 t. 5 cwt. 176 t. 5 cwt. 48 lbs. 5 oz. 13 drs. 11. Reduce 1 pint to minims. 1 fl. oz. xvj. 16 16 fl. drm. viij. 8 128 nt Ix. 60 7680 nt. 12. Reduce 8000 jti to ounces. 60 )8000 ni, 8 )135 m Ix. 20 m- 16 fl. drm. viij. 5 n^ l^- 16 fl. drm. viij. 5 rrj, Ix. 20 n^^. TEACHERS EDITION. 187 Exercise XLII. 1. Reduce 6 hrs. 17 min. 25 5. F md the number of days, sec. to seconds. reckoning from noon of the one 6 hrs. 17 min. 25 sec. to noon of the other, between the X60 following days in the year 1880: 360 July 4 and December 2 Febru- 17 ary 1 and May 29; January 3 377 min. X 60 and October 15; also, between December 25. 1880 and May 25, 22620 1881. 25 22645 sec. 27 d. 28 d. 28 d. 6d. 31 d. 31 d. 29 d. 31 d. 2. Reduce 1 yr. 13 dys. 4 min. to minutes. 30 d. 30 d. 31 d. 28 d. 1 yr. 13 d. 4 min. 31 d. 29 d. 30 d. 31 d. X365 30 d. 118 d. ^^ ^• 30 d. 365 2d. 30 d. 25 d. 13 151 d. 31 d. 151 d. 378 d. aid. X24 30 d. 9072 h. 15 d. X60 544320 4 544324 min. 286 d. 6. How many miautes are 3. Reduce 48,567 min. to days. there from midnight of March 7 60)48567 min. to midnight of June 20 ? 24)809 hrs. 27 min. 24 d. 33 d. 17 hrs. 30 d. 33 d. 17 hrs. 27 min. 31 d. 20 d. 4. Reduce 742,392sec. to days. 105 d. 60)742392 sec. X 24 60)12373 min. 12 sec. 24)206 hrs. 13 min. 2520 hrs. 8 d. 14 hrs. X60 8 d. 14 hrs. 13 min. 12 sec. 151200 min. 188 ARITHMETIC. 7. Find the number of seconds 8. Which of the years 1600, from eight o'clock Monday morn- 1656, 1700, 1734, 1800, 1818, ing till six o'clock the next Sat- 1880, 1900, 1924, 2000 are leap urday evening. • years ? 16 hrs. 24 hrs. 24 hrs. 24 hrs. 1600 (divisible by 400). 1656 " •' 4). 1880 " " 4). 24 hrs. 1924 " 4). 18 hrs. 2000 " 400). 130 hrs. X60 7800 min. X60 468000 sec. Exercise XLIII. 1. Reduce 2° 30^ 25^^ to seconds. 2° 30' 25^' X60 120 _30 150' X60 9000 25 9025'' 2. Reduce 15° 3' 22" to seconds. 15° 3' 22". X_60 900 _3 903' X60 54180 22 54202'^ 3. Reduce 56,760" to degrees. 6 0)56760 " 60)946' 15° 46^ 4. Reduce 212,221" to degrees. 60)2^2221" 6 0)3537 ' 1" 58° 57' 58° 57' 1". 6. The hour and minute handa of a watch form an angle of how many degrees at 3 o'clock? at 4 o'clock? at 6 o'clock? at 7^ o'clock? at 11 o'clock? at 12 o'clock ? TEACHERS EDITION. 189 12 ;;2 2 8 4 45^ t\ = 1 = 120°. 0°. 6. How many geographical miles in the width of the torrid zone (46° 550 ? How many statute miles ? ^ 46° 55^ 46° 55^ = 46|f ° = 46.91|. 2760 55 2815^ = 2815 geog. mi. 46.911 X 69.16 3244.75661 = 3244.751 Stat. mi. Exercise XLIV. 1. Reduce £583 6 s. 8 d. to pence. £583 6 s. 8 d X20 11660 11666 8. Xl2 139992 140000 d. 2. Reduce £79 18 s. 11^ d. to farthings. £79 18 s. ll^d. X20 1580 18 1598 s. Xl2 19176 19187^ d. X4 76750 far. 3. Reduce 28,572 d. to pounds. 12 )28572 d. 20 )2381 s. £119 18. 4. Reduce 272,191 d to half- sovereigns. 1 2)272191 d. 10 )22682 s. 7 d. 2268 half-sov. 2 s. ' 2268 half-sov. 2 s. 7 c?. 5. Reduce 27,281 crowns to guineas. 27281 half-crowns. ^5 21 )136405 s. 6495^. 10 5. 190 ARITHMETIC. 6. Reduce 1,716,114 guineas 9. Reduce 286,347 far. to to pounds. 1716114 ^r. X21 crowns. 4)286347 far. 12)71586 d 3 far. 5)5965 8. 6d. 20)36038394 «. £1801919 14 s. 1193 crowns. 1193 crowns 6 d3far. 7. Reduce 291,374 far. to 10. Reduce 20 francs to dollars. pounds. $0,193 4)291374 far. 12)72843 (f. 2 far. 20)6070 8. 3d £303 108. X20 $3.86 11. Reduce 20 marks to dollars. £303 108. 3d 2 far. $0,238 X20 8. Reduce 709,126d to guineas. $4.76 12)709126 d 12. Reduce 5 roubles to dollars. 21)59093 8. 10 d 2813^. 20 8. $0,734 X5 2813^. 20 8. 10 d $3.67 13. Find the whole sum of money in a box containing 35 sover- eigns, 27 half-sovereigns, 13 crowns, 41 half-crowns, and 85 shillings. 35 sovereigns =700s. 2 0)1222^ 8. 27half-8ov. =2708. £612^-8. 13 crowns = 65 a. 41 half-crowns = 102 J «. 85 shillings - 858. =.£61 28. 6d 1222^8. Exercise XLV. 1. Express 59° F. in Centigrade scale ; in Reaumur's scale. 59° F. is 27° above freezing-point. 3 19 15° C. 3 19 12° R. teachers' edition. 191 CZ 2. Express 77° F. in Centigrade scale ; in Reaumur's scale. 77° F. = 45° above freezing-point. 5 5 ^ X - C. = 25° C. - ^ X - R. = 20° R. 1 ^ IF 3. Express 950° F. in Centigrade scale ; in Reaumur's scale. 950° F. = 918° above freezing-point. 102 102 ® X - C. = 510° C. &x-B.. = 408° R. 4. Express — 40° F. in Centigrade scale ; in Reaumur's scale. — 40° F. = 72° below freezing-point, f of- 72° C. = - 40° C f of- 72° R. = - 320° R. 5. Express — 4° F. in Centigrade scale ; in Reaumur's scale. — 4° F. = 36° below freezing-point. ^ of- 36° C. = - 20° C. I of- 36° R. = - 16° R. 6. Express 10° C. in Fahrenheit's scale ; in Reaumur's scale. 7. Express 22° C. in Fahrenheit's scale ; in Reaumur's scale. 22° C. = I of 22° + 32° F. = 71f° F. f of 22° R. = 17f° R. 8. Express — 30° C. in Fahrenheit's scale ; in Reaumur's scale. - 30° C. = f X - 30° + 32° F. = - 22° F. f x - 30° R. = - 24° R. 9. Express — llf° C. in Fahrenheit's scale ; in Reaumur's scale. - llf° C. == f X - llf° + 32° F. = llf° F. I X - llf° R. = - 9|° R. 10. Express 4° R. in Fahrenheit's scale ; in Centigrade scale. 4° R. - f of 4° + 32° F. = 41° F. f of 4° C. = 5° C. 11. Express 12° R. in Fahrenheit's scale ; in Centigrade scale. 12° R. = f of 12° + 32° F. = 59° F. f of 12° C. = 15° C. 192 ARITHMETIC. 12. Express — 20° R. in Fahrenheit's scale ; in Centigrade scale. 20° R. = f of- 20° + 32° F. = -13° F. ^ of-20° C. = -25° C. 13. Express 4° C. in Fahrenheit's scale ; in Reaumur's scale. 4° C. = f of 4° + 32° F. = 39^° F. f of 4° R. = 3|° R. 14. Express 0° F. in Centigrade scale ; in R6aumur's scale. 0° F. =. 32° below freezing-point. ^ of- 32° C. = - 17^° C. I of- 32° R. - - 14f° R. Exercise XLVI. 1. Add: hn. 14 17 9 12 22 mln. 21 13 47 53 17 37 32 43 54 50 3dys. 4 34 36 2. Add: on. ydi. ou. ft. cu. In. 130 5 820 56 20 304 37 4 86 8 10 129 12 19 175 245 4 1514 3. Add: t. ■. d. 35 2 6f 18 6 4 27 3 10 12 5 12 1 3 far. 4. Add: ml. rd«. jiB. ft. iB 6 120 3 2 2 18 15 1 1 6 3 215 2 2 3 7 95 1 1 8 35 126 3i 1 7 h = 1 6 35 126 5. Add: A. 8q. rds. iq. yds. sq.ft. sq. in, 74 21 5 4 100 23 37 13 5 83 12 106 17 8 7 41 50 23 24 151 55 29J 151 55 29 70 108 34 6. Add 5 bu. 3 pks. 6 qts. 1 pt. ; 6 bu. 2 pks. 7 qta. ; 7 bu. 1 pk. TEACHERS EDITION. 193 1 qt. 1 pt. ; 1 pk. 7 qts. ; 2 bu. 3 pks. 1 pt. bu. 5 6 7 2 pks. 3 2 1 1 3 23 6 7. Add 48 t. 13 cwt. 75 lbs. 6 oz. 4 drms. ; 25 t. 12 cwt. 27 lbs. 8 oz. 13 drms. ; 51 t. 10 cwt. 44 lbs. 15 drms. ; 80 t. 5 cwt. 6 oz. ; 19 cwt. 27 lbs. ; 25 lbs. 8 oz. 10 drms. ; 5 t. 5 cwt. 5 oz. t. cwt. lbs. oz. drs. 48 13 75 6 4 25 12 27 8 13 51 10 44 15 80 5 6 19 27 25 8 10 5 5 5 212 6 10 &■■■ gall qts. pts. gi- 8. Add 50 gals. 3 qts. 1 pt. 3 12 gal. 1 qt. 1 pt. 1 gi. ; 5 . 2 qts. 1 pt. 2 gi. ; 75 gal. 3 1 pt. 3 gi. ; 80 gals. 3 qts. 1 gi. ; 17 gals. 1 qt. 1 pt. 3 50 12 5 75 80 17 243 9. Add 13 lbs. 4 oz. 8 dwt. 6 grs. ; 25 lbs. 8 oz. 13 dwt. 20 grs. ; 8 lbs. 11 oz. 14 grs. ; 20 lbs. 16 dwt. 8 grs. ; 15 lbs. 9 oz. 12 dwt. ; 4 oz. 3 dwt. 13 25 8 20 15 11 13 16 12 3 20 14 8 84 14 10. 3 gals. qts. ; 14 gals. 1^ pts. qts. 1 pt. Add 4 gals. 3 qts. 1 pt. ; 2 qts. li pts.; 12 gals. 3 5 gals. 2 gals. 4 3 12' 14 5 41 J n U 1 1 11. Add 60° 50^ 50^/ ; 20° 41^ 52^^ ; 30° 25^ 20'^ ; 20° 32^ 43^^. 60 50 50 20 41 52 30 25 20 20 32 43 132 30 45 194 ARITHMETIC. Exercise XLVII. 1. Subtract 23 lbs. 8 oz. 19 dwt. 10 grs. from 58 lbs. 6 oz. 17 dwt. 21 grs. lb«. oz. dwt. grs. 58 6 17 21 23 8 19 10 34 18 11 2. Subtract 5 bu. I pk. n qts. 1 pt. from 5 bu, 3 pks. 3 qts. bn. pka. 5 3 5 1 qU. 3 3. Subtract 32 cu. yds. 13 cu. ft. 1600 cu. in. from 39 cu. yds. 17 cu. ft. 1400 cu. in. on. ydf . ca. ft. cu. in. 39 32 1400 1600 7 3 1528 4. Subtract £92 15 «. 1^ d. from £120 13 «. 4 d £. 120 92 •. d. 13 4 15 1^ 27 18 2 3 far. 5. Subtract 22 gals from 30 gals. 2 qta. 3 qta Ipt. gala. 30 qu. 2 pt.. 22 3 1 6. Subtract 17 1. 7 cwt. 17 lbs. 6 oz. 10 drs. from 25 t. 13 cwt. 15 lbs. 12 oz. 5 drs. t. cwt. lbs. 25 13 15 17 7 17 drs. 5 10 98 11 7. Subtract 13 A. 150 sq. rds. 98 sq. ft. 10 sq. in. from 20 A. A. sq. rds. sq. ft. sq. in. 20 13 150 98 10 "e 9 173^ 134 i= 36 6 9 174 26 8. Subtract 58° 33' 36'^ from 90° 11' 21''. 90 58 11 33 21 36 31 37 45 9. Subtract 2 yrs. 213 dys. 17 hrs. from 3 yrs. 147 dys. 14 hrs. yrs. dys. hrs. 3 147 14 2 213 17 298 21 10. Subtract 3 mi. 217 rds. 4 yda. 1 ft. 3 in. from 4 mi. 100 rds. 3 yds. ml. rds. yds. ft. In. 4 100 3 3 217 4 1 3 202 3i 1 202 4 TEACHERS EDITION. 195 Exercise XLVIII. 1. Multiply £31 2 s. 6^ d. by 31 249 4 2. Multiply 19 gals. 3 qts. 1 pt. by 70. gals. qts. pts. 19 3 1 70 1391 1 3. Multiply 3 lbs. 4 oz. 8 dwt. 10 grs. by 10. lbs. oz. dwt. grs. 3 4 8 10 10 33 8 4 4 4. Multiply 5 t. 10 cwt. 67 lbs. by 10. t. cwt. lbs. 5 10 67 10 55 6 70 5. Multiply 43 bu. 2 pks. by G3 ba. 43 304 2740 pks. 2 6. Multiply 15 wks. 3 dys. 5 hrs. 12 min. by 7. wks. dys. hrs. min. 15 3 5 12 7 108 1 12 24 7. Multiply 5 cu. yds. IG oa. ft. 371 cu. in. by 6. cu. yds. cu. ft. cu. in. 5 10 371 6 32 498 8. Multiply 27 A. 76 sq ids. 22 sq. yds. 5 sq. ft. by 90. A. sq. rds. sq. yds. sq. ft. 27 76 22 5 9 247 50 21| 247 50 21 4| 10 3 553 27 ^ sq.in \=2 36 3 553 27 3 2 36 196 ARITHMETIC. 9. Multiply 32 rds, 3 yds. 1 ft. by 57. 57 57 57 Xj X3 X32 3)57 171 1824 19 yds. J9 _34 5^) 190 320 ) 1858 _2 5 mi. 258 rds. 11)380 34 ... 6 half-yds. = 3 yds. 5 mi. 258 rds. 3 yds. Ana. 10. Multiply 34 dy8. 10 hrs. 13 min. 12 sec. by 108. 108 108 X 12 X 13 60 1296 1404 216 _11 21 . . . T% min. = 36 sec. 60 )1425 23 ... 45 min. 108 108 XlO X34 1080 3672 23 45 24)1103 365 )3717 45 ... 23 hrs. 10 . . . 67 dyi 10 yrs. 67 dys. 23 hrs. 45 min. 36 sec. Am. 11. Multiply 5 mi. 126 rds. 19 yds. 6 in. by 7125. 7125 X6 . 6 in. .1ft. 7125 X 19 2 42750 135375 3 3662. 1187 1187 . . 5i) 136562 2 11)273124 24829 |-2iyd8. TEACHERS EDITION. 197 7125 X126 320 )897750 2883 19 rds. 7125 X5 35625 2883 38508 mi. 38,508 mi. 19 rds. 2|- yds. 1 ft. 6 in. = 38,508 mi. 19 rds. 3 yds. Ans. 12. Multiply 11 5 5 32 3 11 grs. by 2197. 2197 Xll 20 ) 24167 1208 2197 X5 10985 1867 8 )12852 1606 7 grs. 13. 43 2197 X2 4394 1208 3 ) 5602 1867 2197 Xll 24167 1606 12 )25773 2147 lbs. 9 5. 2147 lbs. 9543137 grs. Ans. Exercise XLIX. 1. Divide 54 mi. 124 rds. 1 yd. 2 ft. 6 in. by 33. 33)54 rd8. 124 1- ft. 2 in. 6 (1 33 33)724(2 21 66 X320 6i 6720 X3 124 2U 33)6844(207 Xl2 66 258 244 6 231 33)264(8 13 264 X_5i 72^ 1 mi, , 207 rds. 2 yds. 198 ARITHMETIC. 2. Divide 5 cu. yds. 1 cu 84 cu . in by 1716 cu. in. on. yd*, cu. ft. 5 1 X27 135 1 cu. S4 136 X 1728 235008 84 1716)235092(137. Ans. 3. Divide 8426 wks. 6 dys. 21 hrs. 10 min. 21 sec. by 1029. wka. Ay: hri. min. sec. 1029)8426 6 21 10 21(8 8232 194 x7(+6) 1029)1364(1 1029 335 X24 8040 21 1029)8061(7 7203 858 X60 1029)51490(50 5145 40 60 2400 21 1029) 2421 (2TVW=-2iH- 2058 363 8 wks. Idy. 7 hrs. 50 min. 2|f^8ec. Ana. 4. Divide £394 2 8. 10^ d. by £5 28. 4H- £394 2 s. 10^ d. = 378,378 far. £5 2 8. 4^ d = 4914 far. 77. Ans. 4914)378378 5. Divide 22 wks. 2 dys. by 11 hrs. 31 min. 12 sec. 22 wks. 2 dys. = 13,478,400 sec' llhr8.31min. 12sec.=41,4728ec. 325. ^718. 41472)13478400 6. Divide 74,128 sq. mi. 517 A. 80 sq. rds. by 10,000. Bq. mi. 10000)74128 70000 4128 X640 A. sq. rda. 517 80(7 2641920 517 10000)2642437(264 2640000 2437 Xl60 389920 80 10000)390000(39 390000 7 sq. mi. 264 A. 39 sq. rds. A 71 s. 7. Divide 38° 37' 42'^ by 5*= 3V 6'^ 38° 37' 42'' = 139,062". 5° 31' 6" = 19,866". 19866)139062(7. Ans. 139062 TEACHERS EDITION. 199 Exercise L. 1. Find the value of f of a mile. I mi. = f of 320 rds. = 256 rds. Ans. 2. Find the value of j\ of an acre. j\ A. = j\ of 160 sq. rds. = 30 sq. rds. Ans. 3. Find the value of f of a hundredweight. f cwt. = f of 100 lbs. = 62i lbs. ^ lb. = I of 16 oz. = 8 oz. 62 lbs. 8 oz. Ans. 4. Find the value of f of a pound sterling. £f-f of 20s. = 13|s. is. = iof 12d = 4d 13 s. 4 d. Ans. 5. Find the value of -^j of a mile. t\ mi. - j\ of 320 rds. - 261^^1 rds. t\ rds. = /t of 51 yds. = 41 yds. i yd. = 1 of 3 ft. = 11 ft. ^ ft. = 1 of 12 in. = 6 in. 261 rds. 4 yds. 1 ft. 6 in. Ans. 6. Find the value of -^j of an acre. TT A. = i?T of 160 sq. rds. = 101/^- sq. rds. ^ sq. rds. = j\ of 30^ sq. yds. = 24| sq. yds. f sq. yds. = | of 9 sq. ft. = 6f sq. ft. f sq. ft. = f of 144 sq. in. = 108 sq. in. 101 sq. rds. 24 sq. yds. 6 sq. ft. 108 sq. in. Ans. 7. Find the value of f of a degree. f° = |of60^ =26f^ Y = I of 60^^ = 40^^. 26' W. Ans. 8. Find the value of |^ of a year. I yr. = ^ of 365 dys. = 121| dys. f dy. = I of 24 hrs. = 16 hrs. 121 dys. 16 hrs. Ans. 200 ARITHMETIC. 9. Find the value of 0.15625 of a bushel. 0.15625 bu. Xj 0.625 pks. X8 5 qts. Am. 10. Find the value of 0.625 of a gallon. 0.625 gal. X4 2.5 qte. X2 Ipt. 2 qts. 1 pt. Ans. 11. Find the value of 0.875 of a leap-year. 0.875 X366 320.25 dys. X24 6hr8. 320 dys. 6 hrs. 12. Find the value of 0.325 of a pound troy. 0.325 lbs. Xl2 3.9 oz. X20 18 dwt. 3 oz. 18 dwt. Ana. 1. Find the value of Exercise LI. f of 3 A. 101^ sq. rds. sq. rdi. lOli X2^ 5 )6 202f 1 72 16 sq. yds. 1 sq. ft. 28| sq. in. A. sq. rds. 3 lOH X6 21 1 128 72 •q. yd«. 16 ■q. ft. 1 sq. in. 28^ 23 40 16 2. Find the value of If of 7 hrs. 21 min. 27 sec. 7 21 27 X3 7)22 4 21 3 7 9 21 11^ 27 10 30 38^ 1 28t 3. Find the value of 10.0175 of 1 dy. 13 hrs. 10.0175 X 37 hrs. 370.6475 X60 38.85 min. X60 51 sec. 370 hrs. 38 min. 51 sec. =- 15 dys. 10 hrs. 38 min. 51 seu. Ant, TEACHEES EDITION. 201 4. Find the value of 17tV of 10 ydb. 2 ft. 31 in. yds. ft. in. 10 2 x7 12)75 m 6 9il 10 ft. 2 X in. 1?^ 182 6 2 6f 9H 189 4t\ 34 rds. 2 yds. ft. ij% in. Ans. 5. Find the value of 0.01284 of 14 mi. 0.01284 X 14 mi. 0.17976 ] X320 Tii. 57.5232 rds. X5i 2.8776 yds. X3 2.6328 ft. X 12 7.5936 in. 57 rds. 2 yds. 2 ft. 7.5936 m. Ans. 6. Find the value of 0.42776 of 12 t. 10 cwt. 10 cwt. = ^ t. 0.42776 Xl21 t. 5.347 t. X20 6.94 cwt. xioo 94 lbs. 5 t. 6 cwt. 94 lbs. Ans. + 31 oz. + 5 1 dwt. 4| oz. + 3| oz. = 8^ oz. J^ oz. - ^V of 20 dwt. = f dwt. f dwt. + 5f dwt. = 6i dwt. ^ dwt. = ^ of 24 grs. = 2f grs. 8 oz. 6 dwt. 2| grs. Ans. 8. Find the value of 0.35 of 4 lbs. 5 oz. 6 dwt. 16 grs. lbs, oz. dwt. grs. 16 X7 20)31 13 16 8 Ans. 9. Find the value of 3.726 mi. — 33.57 rds. 3.726 mi. X320 1192.32 rds. 33.57 1158.75 rds. = 3 mi. 198 rds. 4 yds. 4^ in. Ans. 10. Find the value of y\ of a year + -^-^ of a week + ^j of an hour. ■^\ yr. = 7^ of 365 dys. = 15 dys. Awk. = ^^of 7 dys. =lidys. I dy. = ^ of 24 hrs. = 3 hrs. x^^jhr. =/jof 60 min. = 35 min. 15 dys. Idy. 3 hrs. 35 min. 16 dys. 3 hrs. 35 min. 2 wks. 2 dys. 3 hrs. 35 min. Ans. 202 AEITHMETIC. 11. Find the value of 5.268 of 2 dys. + 2.829 of 16 hrs. + 0.9528 of 25 min. 6.268 X 2 dys. 2.829 X 16 hrs. 10.536 dys. X24 45.264 hrs. 12.864 hrs. 0.9528 X 25 min. 23.82 min. 7.68 min. 12.864 hrs. 31.5 min. X60 30 sec. 58.128 hrs. X60 7.68 min. 10 dys. 58 hrs. 31 min. 30 sec. = 12 dys. 10 hrs. 31 min. 30 sec. Arts. 12. Find the value of ^^ of a mile + f of 40 rds. + f of a yd. ^5 mi. = j% of 320 rds. = 60 rds. f of 40 rds. = 26| rds. 60 rds. + 262 rds. I rds. = I of 5^ yds f yd. + f yd. 1^^ yds. + 3 yds. 86 rds. 4 yds. 1^ in. Ans. = 86 1 rds. = 3 1 yds. = l,Vyd8. = 4^Vyd8. 13. Find the value of f of 2 cwt. 84 lbs. + f of 5 cwt. 98 lbs. + f of 7i lbs. 2 cwt. 84 lbs. XlOO 4)284 lbs. 71 X3 100)213 lbs. 2 cwt. 13 lbs. 5 cwt. 98 lbs. XlOO 698 lbs. X3 7 100 1794 25 56^ lbs. 2 cwt. 56 lbs. 4 oz. 9} drs. f of 7i lbs. = 3 lbs. 2 cwt. 13 lbs. 2 cwt. 56 lbs. 4 oz. 9f drs. 3 lbs. 4 cwt. 72 lbs. 4 oz. d\ drs. Am. teachers' edition. 203 Exercise LII. 1. Express a pound avoirdupois as the fraction of a pound troy. 1 lb. avoird. = 7000 troy grp. 1 lb. troy = 5760 troy grs. 5760 144 2. Express an ounce avoirdupois as the fraction of an ounce troy. 437- 1 grs. 16)7000 1 oz. troy = 480 grs. 437^^875^175 ^^ 480 960 192 3. Express 363 sq. yds. as the fraction of an acre. Xl60 4840 40 160 sq. rds. X 30| 4840 sq. yds. 4. Express ^ oi £2 I s. 3d. + j\ of £1 As. 9d. as the fraction of £2 14s. 14 £ «. 2 1 d. 3 £ 1 4 d. 9 £ 2 X20 X20 X20 41 8. 24 s. 54 s. Xl2 5)495d 99 Xl2 11^297 d. 27 X 12 648 cZ. X3 297 d X5 135^. 297 + 135 432 _2 Ans. 672 648 3 204 ARITHMETIC. 5. Express 2 mi. 138 rds. 1 yd. as the fraction of 3 mi. 265 rds. 3 yds. 1 ft. 6 in. 2 mi. 138 rds. 1 yd. 3 mi. 265 rds. 3 yds. 1 ft. 6 in. X320 X320 640 960 138 265 778 rds. 1225 rds. X5^ X5^ 4280 yds. 6740| yds. X36 X 3 154080 in. 20222^ ft. 1^4080^40^^^^ 24liin. 242676 63 6. Express f of 560 lbs. as the fraction of 5 long tons. I of 560 lbs. = 160 lbs. 2240 lbs. 160 1 X 5 11200 70 Ans. 11200 lbs. 7. Express f of 200 rds. as the fraction of 4 miles. I of 200 rds. = 133J rds. 320 rds. 5 X4 133± = _J_x^ = A. ^n«. — - 1280 Xm 3 48 1280 rds. 16 8. Express ^^ of 2 dys. 2 hrs. 24 min. as the fraction of 2 wks. 1 d. 2 dys. 2 hrs. 24 min. 2 wks. 1 dy. X24 X_7 60 hrs. 15 dys. X60 X_24 27 )3024 nvin. 360 hrs. 112 x60 X 10 21600 min. 1120 min. 21600 135 teachers' edition. 205 9. Express f of the difference between 3 yds. 2 ft. 11 in. and 10 yds. 7 in. as the fraction of 8 yds. yds. 10 ft. in. 7 3 2 11 6 8 X36 224 in. 8 yds. X36 X4 288 in. 5)896 179^ in. 179^ 1 288 m 28 5 _28 45' Am. 10. Express ^ of the difference between f of 7 hrs. and ^ of 15 min. as the fraction of 12 hrs. 18 min. 7 hrs. = 420 min. 12 hrs. 18 min. 105 60 5 1^^525. 733 ^i^^ ^ I 2 2 lx^ = — ^ 525^21^2583 3 15' 2 " 5 10 5 123 10 2382 -loo • 123 1 . ML X tiTJL = 123 mm. — - = -. Ans. ^X X^ 738 6 11. Express f pt. as the fraction of a gallon. lgal. = 8pte. 1 = 1x2 = 1. ^«., 4 206 ARITHMETIC. 12. What part of 4 lbs. 1 oz. 8 dwt. 15 grs. is 1 lb. 1 oz. 9 dwt. 15 grs. 4 lbs. 1 oz. 8 dwt. 15 grs. : 1 ib. 1 oz. 9 dwt. 15 grs.: 15grs. =^|dwt. =|dwt. 15 grs. = ^1 dwt. = i dwt. 8fdwt.=|oz. =,fi^oz. 9f dwt. =^oz. = l'«VOZ. WTrOZ.=^lb8. = ^^^lb8. li^oz.=iA^lb8, . = /^lbB. 4^Viy lbs. V^lbs. =n-^-- 11 13. What part of 2 mi. is | of 6 rds. 3 yds. 2 in. ? 6 rds. 3 yds. 2 in.: SI yds. = ^ rds. = I rd. 6| rds. I of 6| rds. = W rds. iMs lof 2 2 mi. = 59 4320 59 8640" i. Ana. '.. What part of a Dushel is 1 pk. 1 pk. 2 qts. 1 pt. = i qt- 2 qts Ipt.: . Ipt.? 2iqt8.=|pk8.= = 1^.^ l^a. l^ pks. = lA bu. = f i bu. Ans, 15. What part of 20 A. are 19 A. 3.5 sq. ch. 19. A. 3.6 eq. ch. : 3.5 8q.ch.-MA. = AA. 1M = 387 1^387 ^^ 20 20 ^ 20 400" ^ TEACHERS EDITION. 207 16. What part of 5 tons are 3 t. 240 lbs. ? 3 t. 240 lbs. : 2401bs. = ^%V7t. = ^3^t. 3& = lx^ 5 5 25 . Ans. 125 17. 38 sq. rds. 194 sq. ft. 108 sq. in = what part of an acre? 38 sq. rds. 194 sq. ft. 108 sq. in. : 108 sq. in. = ^f f sq. ft. = f sq. ft. 194^ 194| sq. ft. = iHll sq. rds. = ^y/^ sq. rds. 3835,V^sq.rds. = ^^A. = 42161 174240 A. Ans. Exercise LIII. 1. Express 16 s. 3|d decimal of a pound. the 3.75 d. 16.3125 s. £0.815625. Ans. 2. Express 233 rds. 9 ft. 10.8 in. as the decimal of a mile. 12 10.8 in. 3 9.9 ft. 5i 3.3 yds. 320 233.6 rds. 0.73 mi. Ans. 3. Express 71 sq. rds. 54 sq. ft. 64.8 sq. in. as the decimal of an acre. 144 64.8 sq. in. 9 54.45 sq.ft. 30| 6.05 sq. yds. 160 71.20 sq. rds. 0.445 A. Ans. 4. Express 15hrs. 14 min. 6 sec. i the decimal of 2 days. 6.000 sec. 14.100 min. 15.235 hrs. 0.6348 2 0.6348 dys. = 0.3174. Ans. 5. Express 38 sq. rds. 21 sq. yds. 5 sq. ft. 108 sq. in. as the decimal of an acre. 144 108.000 sq. in. 9 5.750 sq. ft. 30i 21.638 sq. yds. 160 38.715 sq. rds. 0.242 A. Am 208 ARITHMETIC. 6. Express 3 mi. 242 rds. 2 yds. 2 ft. 3 in. as the decimal of 7 mi. 160 rds. 12 3 320 3.00 in. 2.25 ft. 2.75 yds. 242.5 rds. 3201160 7.5 mi. 3.7578 mi. 3.7578 7.5 0.501. Am. 7. Express 5 hrs, 13 min. 30 sec. as the decimal of a week. 30.000 sec. 13.500 min. 5.225 hrs. 0.2177 dy. 0.0311 wk. Ans. 8. Express 27° 14^ 45'^ as the decimal of 90°. 45.00^/ 14.75^ 27.246° 27.246 90 = 0.303. Am. 9. Express 54 dys. 2 hrs. 40 min. as the decimal of 365 J dys. 54.1 365^ 40.0 min. 2.6 hrs. 54.1 dys. = 0.i48. Am, 10. Express 2 lbs. avoirdupois as the decimal of 10 lbs troy. ' 2 lbs. av. = 14,000 grs. troy. 10 lbs. troy = 57,600 grs. troy. 14000 57600 0.243. .4715. 11. Express 44,920.9025 hrs. as the decimal of a year. 1 yr. = 8760 hrs. 44920.9025 8760 = 5.128. Am. 12. Express 1 drm. avoirdupois as the decimal of 1 dwt. troy. 1 drm. avoird. = ^1^ of 7000 troy grs. = 27.344 troy grs. 1 dwt. = 24 troy grs. 27.344 24 1.139. Am. 13. Express 10 milligrams as the decimal of a grain, if a kilogram equals 2 lbs. 8 oz. 3 dwt. 1 gr. 2 lbs. 8 oz. 3 dwt. 1 gr. = 15,433 grs. 1 kg. = 100,000 X 10 mg. 15433 100000 - 0.15433. Am. TEACHERS EDITION. 209 14. Express 14.52 sq. yds. as the decimal of a square chain. 1 sq. eh. = 16 sq. rds. = 484 sq. yds. 14.52 484 0.03. Ans. 15. Express 8 cwt. 77 1' oz. as the decimal of a ton. 16 100 20 9.600 oz. 77.600 lbs. 8.776 cwt. 0.4388 t. Ans. 9.6 Exercise LIV. 1. Find the dilBference in longitude between two places, if the difiference in time be 1 hr. 15 min. 1 hr. 15 min. - 75 min. = ^ (75°) = 18° 45^ Ans. 2. Find the difference in longitude between two places, if the difference in time be 2 hrs. 11 min. 2 hrs. 11 min. - 131 min. = ^ (131°) = 32° 45^ Ans. 3. Find the difference in longitude between two places, if the difference in time be 5 hrs. 10 min. 10 sec. 5 hrs. 10 min. 10 sec. = 310 min. 10 sec. = ^ (310° 10^ = 77° 32^ 30^^. Ans. 4. Find the difference in longitude between two places, if the difference in time be 3 hrs. 25 min. 35 sec. 3 hrs. 25 min. 35 sec = 205 min. 35 sec. = | (205° 35^) = 51° 23^ 45^^. Ans. 5. Find the difference in longitude between two places, if the difference in time be 6 hrs. 12 min. 30 sec. 6 hrs. 12 min. 30 sec. = 372 min. 30 sec. (372° 300 93= 30^^ Ans. 6. Find the difference in longitude between two places, if the difference in time be 4 hrs. 8 min. 12 sec. 4 hrs. 8 min. 12 sec. = 248 min. 12 sec. = ^ (248° 12^ = 62° 3^ Ans. 7. Find the difference in longitude between two places, if the difference in time be 18 hrs. 10 min. 18 hrs. 10 min. = 1090 min. = ^ (1090°) = 272° 30^ Ans. 210 ARITHMETIC. 8. Find the difference in longitude between two places, if the difference in time be 15 hrs. 15 min. 15 sec. 15 hrs. 15 min. 15 sec. = 915 rain. 15 sec. = | (915° 15^ = 228° 48^ 45'^ Ans. 9. Find the difference in time between two places, if the differ- ence in longitude be 9° 20''. 9° 20^ = 4 X (9 min. 20 sec.) = 37 min. 20 sec. Ans. 10. Find the difference in time between two places, if the differ- ence in longitude be 70° 30''. 70° 30^ = 4 X (70 min. 30 sec.) = 4 hrs. 42 min. Ans. 11. Find the difference in time between two places, if the differ- ence in longitude be 56° 36^ 12^^. 56° 36' 12^' = 56° 36.2' = 4 x (56 min. 36.2 sec.) = 3 hrs. 46 min. 24.8 sec. Ans. 12. Find the difference in time between two places, if the differ- ence in longitude be 108° 32' 36". 108° 32' 36" = 108° 32.6' = 4 x (108 min. 32.6 sec.) = 7 hrs. 14 min. 10.4. sec. Ans. 13. Find the difference in time between two places, if the differ- ence in longitude be 120° 14' 30". 120° 14' 30" = 120° 14.5' = 4 x (120 min. 14.5 sec.) = 8 hrs. 58 sec. Ans. 14. Find the difference in time between two places, if the differ- ence in longitude be 100° 45' 54". 100° 45' 54" = 100° 45.9' = 4 x (100 min. 45.9 sec.) = 6 hrs. 4^ min. 3.6 sec. Ans. 'i 15. Find the difference in time between two places, if the differ- ence in longitude be 2° 2' 2". 2° 2' 2" = 2° 23V^ = 4 X (2 min. 2^ sec.) = 8 min. 8^^ sec. Am. 16. Find the difference in time between two places, if the differ- ence in longitude be 75° 10'. 75° 10' =. 4 X (75 min. 10 sec.) = 5 hrs. 40 sec. Ans. teachers' edition. 211 Exercise LV. The longitude of some public building in : (1) Berlin is 13° 23^ 43^^ E. (7) Jerusalem, 35° 32^ E. (2) Rome, 12° 2V W^ E. (8) Bombay, 72° 54^ E. (3) Constantinople, 28° 59^ E, (9) Calcutta, 88° 19^ 2^^ E. (4) Pekin, 116° 23^ W E. (10) Chicago, 87° 35^ W. (5) SanFrancisco,122°26a5^'W. (11) New York, 74° 0^ V^ W. (6) St. Louis, 90° 15^ 15^^ W. (12) Montreal, 73° 25^ W. 1. When it is noon at Greenwich, what is the clock-time at each of the above places ? (1) (5) 13° 23^ 43^^ 122° 26^ lb'' = 13°23|r = 122° 26^^ = 4x(13min. 23f|8ec.) = 4 X (122 min. 26^ sec.) = 53 min. 34if sec. p.m. = 8 hrs. 9 min. 45 sec. = 12 hrs. 53 min. 34|| sec. P.M. 12 hrs. Ans. 8 hrs. 9 min. 45 sec. (2) 3 hrs. 50 min. 15 sec. a.m. Ans. 12° 27^ 14^^ (6) 90° 15^ 15^^ = 12° 27^^ = 4 X (12 min. 27^ sec.) = 90° IbY = 49 min. 48|f sec. p.m. = 4 X (90 min. lb\ sec. ) = 12 hrs. 49 min. 48^f sec. P.M. = 6 hrs. 1 min. 1 sec. Ans. 12 hrs. M. (3) 6 hrs. 1 min. 1 sec. 28° 59^ 6 hrs. 58 min. 59 sec. a.m. Ans. = 4 X (28 min. 59 sec.) (7) 35° 32^ = 1 hi. 55 min. 56 sec. p.m. Ans. = 4 X (35 min. 32 sec.) (4) = 2 hrs. 22 min. 8 sec. P.M. Ans. 116° 23^ 45^^ (8) 72° 54^ = 116° 23f ^ = 4 X (116 min. 23f sec.) = 4 X (72 min. 54 sec.) = 7 hrs. 45 min. 35 sec. p.m .Ans. = 4 hrs. 51 min. 36 sec. p.m. Ans. 212 ARITHMETIC. (9) 88° 19' 1" = 88° 19^' = 4 X (88 min. 19^ sec.) » 5 hrs. 53 min. IGj^ sec. Am. (10) 87° 35' = 4 X (87 min. 35 sec.) = 5 hrs. 50 min. 20 sec. 12 hrs. M. 5 hrs. 50 min. 20 sec. 6 hrs. 9 min. 40 sec. a.m. Am. (11) 74° 0' V = 74°^' = 4 X (74 mm. -^ sec.) = 4 hrs. 56 min. \ sec. 12 hrs. M. 4 hrs. 56 min. \ sec. 7 hrs. 3 min. 59f 86c. A.M. Am. (12) 73° 25' = 4 X (73 min. 25 sec.) = 4 hrs. 53 min. 40 sec. 12 hrs. M. 4 hrs. 53 min. 40 sec. 7 hrs. 6 min. 20 sec. a.m. Ath. 2. When it is half-past four p.m. at Chicago, what is the clock- time at each of the above places ? • (1) 87° 35' W. 13° 23' 43" E. 100° 58' 43" - 100° 58ff' = 4 X (100 min. 58f ^ sec. = 6 hrs. 43 min. 54}f sec. 4 hrs. 30 min. p.m. 11 hrs. 13 min. 54|| sec. p.m. .4.718. (2) 87° 35' W. 12° 27' 14" E. 100° 2' 14" - 100° 2^' = 4x(100min. 2^sec.) — 6 hrs. 40 min. 8j^ sec. 4 hrs. 30 min. p.m. 11 hrs. 10 min. 8j| sec. p.m. Afn, (3) 87° 35' W. 28° 59' E. 116° 34' - 4 X (116 min. 34 sec.) = 7 hrs. 46 min. 16 sec. 4 hrs. 80 min. p.m. 12 hrs. 16 min. 16 sec. A.M. Ati^. (4) 87° 35' W. 116° 23' 45" E. 203° 58' 45" 360° 203° 58' 45" 156° 1'15" -156° 156° U' 4x(l56 min. 1^ sec.) 10 hrs. 24 min. 5 sec. 4 hra. 30 min. p.m. 1 hrs. 24 min. 5 sec . 6 hrs. 5 min. 55 sec. A.M. Am. teachers' edition. 213 (5) 122° 26i' W. 87° 35' W. (9) 88° 19' 2" E. 87° 35' W. 34° 51^' = 4 X (34 min. 51^ sec.) = 2 hrs. 19 min. 25 sec. 4 hrs. 30 min. p.m. 2 hrs. 19 min. 25 sec. 2 hrs. 10 min. 35 sec. p.m. Ans. (6) 90° 15^' W. 87° 35' W. 175° 54' 2" = 175° 543V. = 4 X (175 min. 54^\y sec.) = 11 hrs. 43 min. 363-2^ sec. 4 hrs. 30 min. p.m. 16 hrs. 13 min. 363-2^ sec. = 4 hrs. 13 min. S&j^^ sec. a.m. Ans. 2° 40^' = 4 X (2 min. 40^ sec.) = 10 min. 41 sec. (10) (4 hrs. 30 min. p.m.). Ans. 4 hrs. 30 min. p.m. 10 min. 41 sec. (11) 87° 35' W. 4 hrs. 19 min. 19 sec. p.m. Ans. (7) 87° 35' W. 35° 32' E. 74° 0' 3" 13° 34' 57" 123° 7' = 4 X (123 min. 7 sec.) = 8 hrs. 12 min. 28 sec. 4 hrs. 30 min. p.m. = 13° 341§' = 4 X (13 min. 34|^ sec.) = 54 min. 19f sec. 4 hrs. 30 min. p.m. 12 hrs. 42 min. 28 sec. a.m. 5 hrs. 24 min. 19f sec. p.m. Ans. Ans. (8) 72° 54' E. 87° 35' W. 160° 29' = 4 X (160 min. 29 sec.) = 10 hrs. 41 min. §6 sec. 4 hrs. 30 min. p.m. 15 hrs. 11 min. 56 sec. = 3 hrs. 11 min. 56 sec. a.m. Ans. (12) 87' > 35' W. 73' ' 25' W. 14' »10' = 4x(14 min. 10 sec •) :56 min. 40 sec. 4 hrs. 30 min. P.M. 5 hrs. 26 min. 40 sec. ] P.M. Ans. 214 ARITHMETIC. 3. When it is eight o'clock a.m. at Constantinople, what is th* clock-time at each of the above places ? (1) (5) 28° 59' E. • 28° 59' E. 13°_23M3^E. 122° 26 J' W. 15° 35' 17" 151° 25^' = 15° 35^' _ 4 ^ (151 jnin. 25^ sec.) = 4 X (15 ram. 35^^ sec.) ^ ^q ^^^ 5 ^^^^ 4^ ^^^^ = 1 hr. 2 rain. 21f^ sec. 8 hrs. A.M 1 hr. 2 min. 21t^ sec. 8 hrs. A.M. 10 hrs. 5 min. 41 sec. ^ , p^ . _„- „ . 9 hrs. 54 min. 19 sec. p.m 6 hrs. 57 mm. 38|f sec. a.m. Am. Atik (2) (6) 28° 59' E. 12° 27' 14" E. 16° 31' 46" 28° 59' E. 90° 15' 15" W. 119° 14' 15'-' = 16°31§§' =nqoi4i/ = 4x(16min. 31|f sec.) . ,iin -..1 ,, ^^ . ^ /*^ ' = 4 X (119 mm. 14i sec.) = 1 hr. 6 mm. 7^^ sec. m -, tzn • c^ ^ ^ =7 hrs. 56 mm. 57 sec. 8 hrs. A.M. Ihr. 6 min. 7,1^ sec. 8 hrs. a.m. 7 hrs. 56 mm. 57 sec. 6 hrs. 53 min. 52}| sec. a.m. Ans. ] 3 min. 3 sec. a.m. (3) (8 hrs. A.M.). Am. = 12 hrs. 3 min. 3 sec. a.m. Ans. (7) (4) 35° 32' E. 116° 23f ' E. 28° 59' E. 28° 59' E. 6° 33' ^'^°^^V -4 X (6 min. 33 sec.) - 4 X (87 min. 24| sec.) „ 26 min. 12 sec. «= 5 hrs. 49 min. 39 sec. 3 i^^s. a.m. 8 hrs. A.M. 8 hrs. 26 min. 12 sec. a.m. 1 hr. 49 min. 39 sec. p.m. Am. Aru. teachers' edition. 215 (8) 72° 5V E. 28° 59^ E. (10) 87° 35^ W. 28° 59^ E. 43° 55^ 116° 34^ = 4 X (43 min. 55 sec.) = 2 hrs. 55 min. 40 sec. = 4 X (116 min. 34 sec.) = 7 hrs. 46 min. 16 sec. 8 hrs. A.M. 8 hrs. A.M. 10 hrs. 55 min. 40 sec. A.M. 7 hrs. 46 min. 16 sec. Arts. 13 min, 44 sec. a.m. = 12 hrs. 13 min. 44 sec. a.m. Ans. (9) 88° 19{ 2^^ E. (11) 74° O^S^^W. 28° 59^ E. 28° 59^ E. 59° 20^ 2'-' 102° 59^ 3^^ 59°20^V =102°59^V 4 X (59 min. 20^V sec.) = 4 X (102 min. 59^V sec.) 3 hrs. 57 min. 20^2^ sec. = 6 hrs. 51 min. 56^ sec. 8 hrs. A.M. 8 hrs. a.m. llhrs.57min.20Asec.A.M. 6 hrs. 51 min. 56^ sec. Ans. 1 hr. 8 min. 3f sec. a.m. Ans. (12) 73° 25^ W. 28° 59^ E. 102° 24' = 4 X (102 min. 24 sec.) = 6 hrs. 49 min. 36 sec. 8 hrs. A.M. 6 hrs. 49 min. 36 sec. 1 hr. 10 min. 24 sec. a.m. Ans, 216 ARITHMETIC. Exercise LVI. When it is noon at Greenwich the time at (1) Boston, Mass., is 7 hrs. 15 min. 46 sec. a.m. (2) Augusta, Me., 7 hrs. 20 min. 40 sec. a.m. (3) Columbia, S.C., 6 hrs. 35 min. 32 sec. a.m. (4) Little Rock, Ark., 5 hrs. 51 min. 12 sec. a.m. (5) Salt Lake, 4 hrs. 30 min. a.m. (6) Albany, N.Y., 7 hrs. 5 min. 1 sec. a.m. (7) Columbus, 0., 6 hrs. 27 min. 48 sec. a.m. (8) Harrisburg, Penn., 6 hrs. 52 min. 40 sec. a.m. (9) New Orleans, La., 6 hrs. a.m. (10) Springfield, 111., 6 hrs. 1 min. 48 sec. A.M. (11) Washington, D.C., 6 hrs. 51 min. 44 sec. A.M. 1. What is the longitude of each of the above places ? (1) (4) hra. min. seo. hn. min. seo. 12 12 7 5 46 5 51 12 4 44 14 6 8 48 = 284 min. 14 sec. = J of 284° 14' - 71° 3' 30^' W. Am. = 368 min. 48 sec. = i of 368° 48' = 92° 12' W. Am. (2) hn. min. mo. 12 7 20 40 (6) hn. min. 12 4 30 4 39 20 7 30 = 279 min. 20 sec. = \ of 279° 20' - 69° 50' W. Am = 450 min. = iof450° = 112° 30' W. Am. (3) hri. min. aeo. 12 6 35 32 (6) hn. min. aeo. 12 7 5 1 5 24 . 28 4 54 59 - 324 rain. 28 sec. = 294 min. 59 sec. - 1 of 324° 28' .= ;^ of 294° 59' - 81° 7' W. Am. - 73° 44' 45" W. Ans. teachers' edition. 217 (7) (9) hrs. min. seo. hrs. 12 12 6 27 48 6 5 32 12 6 = 332 min. 12 sec. = 360 min. = i of 332° 12^ = iof360° = 83° 3^ W. Ans. = 90 ° W. Ans. (8) (10) hrs. min. seo. hrs. min. sec. 12 12 6 52 40 6 1 48 5 7 , 20 5 58 12 = 307 min. 20 sec. = 358 min. 12 sec. = \ of 307° 20^ = ^ of 358° 12^ = 76° 50^ W. Ans. ^(11) = 89° 33^ W. Ans. hrs. min. sec. 12 6 51 44 T 8 16 = 308 min. 16 sec. = i of 308° 16^ = 77° 4^ W. Ans. Exercise LVII. - 1. Reduce 7 gals. 3 qts. ] L pt. to gallons and decimal of a gallon. 2 l.OpO pt. 4 3.500 qts. 7.875 gal. Ans. 2. Reduce £4.375 to pounds, shillings, , and pence. £4.375 20 7.58. 12 ed £4 7s. 6d Am. 218 ARITHMETIC. 3. Reduce 7.6875 gals, to gallons, quarts, and pints. 7.6875 gal. 4 2.75 qts. 2 1.5 7 gals. 2 qts. 1.5 pts. Ana. 4. Reduce to pounds, shillings, and pence f 5.875 ; $ 7.38 ; $ 17.85 ; $21.75 ; if $4.85 be equal to a pound. (1) (3) £'i.2H £3|H 485)587.5 485)1785 0.2H = i¥i- m=u- ^^^ of 20 s. = 4^8. ff of208. = 13f?«. If of 12cf. = 2^^rf. ^ of 12 d. = 1^d. £14s. 2|^d Ans. i £3 13 8. 7ffd Am. (4) (2) £4fM ^iHf 485)2175 485)738 m=n- JIf of208. = 10f^«. f|of20s. = 9fj«. tf ofl2d = 5^^d ff of 12c?. = 8ffd £1108.5^^^ ^7i«. £4 9s.8|fd ilrw. 5. How many square yards in 6. If 2 qts. of linseed oil be 3.7156 acres? mixed with ^ pt. spirits of turpen- 3.7156 A. tine. what fraction of the mix- - Xl60 ture is turpentine? How much turpentine in one pint of the 594.496 sq. rds. mixture ? xso\ 2 qts. = 4 pts. 17983.504 sq. yds. Am. 4pts. + ipt. = 4ipts. ± = 2x1-1. (1) 4i 9^? 9 ^^^ i of 1 pt. = i pt. (2) TEACHERS EDITION. 219 7. Reduce 5.1732 mi. to yards, feet, and inches. 5.1732 mi. X 1760 9104.832 yds. X3 2.496 ft. Xl2 5.952 iu. 9104 yds. 2 ft. 5.952 in . An?,. 8. If a man walk 88 mi. in 26 hrs., how many feet does he walk each second ? 22 44 »^x^^x 1 xi- 9^8 ft. 1 '' 1 '' nn %^ 195 15 13 = 4Hf ft. Am. 9. Of a mixture of sand and lime 0.27 of the weight is lime. How many ounces of lime in a pound of the mixture? How many troy grains of lime in an avoirdupois pound of the mix- ture? 16 oz. 0.27 X 0.27 X 7000 4.32 oz. Am. 1890 troy grs. 10. A gill of water is put into a quart measure, and the measure filled with milk. What part of the mixture is water ? 8 gi. = 1 qt. .-. 1 gi. = i X 1 qt. = \ qt. .•. \ is water. 11. Reduce 555 ft. to the deci- mal of a mile. 0.1051136 mi. Am. 528)55.5000000 12. Reduce 1 mi. 13 rds. 2 yds. 2 ft. 6 in. to inches. 1 mi. X320 320 13 333 rds. X51 18331 yds. X3 5502| ft. Xl2 66036 in. Am 13. How many cubic inches in 1\ cubic feet ? 1728 cu. in. X2 | 4320 cu. in. Ans. 14. How many pounds avoir- dupois does a cubic yard of water weigh if a cubic foot weigh 1000 ounces ? 27 X 1000 oz. 1 6) 27000 oz. 1687i lbs. Am. 220 ARITHMETIC. 15. Express the weight of a cubic yard of water as the deci- mal of a ton. 1687} =, 675 _ 21 SOOOO 800 ~" 32' 0.84375 t. Ans. 32)27.00000 16. What is the weight 7 bu. 3^ pks. of poL^toes? 3^ pks. = M bu. = I bu. 60 lbs. x7| 472^ lbs. ^rw. of 17. A farmer sowed 5 bu. 1 pk. 1 qt. of seed, and harvested from it 103 bu. 3 pks. 5 qts. How much did he raise from a bushel of seed? 5 bu. 1 pk. 1 qt 1 qt. = i pk. lipk. = ^bu. = /^ 5/,, bu. : 103 bu. 3 pks. 5 qts. : 5 qts. = f pk. bu. 3f pks. = ^ bu. = f f bu. 103f I bu. 5/j 169 ^^^ lit of 4 pks. = 2H| pks. HI of 8 qts. =5^VVqts. ' 19 bu. 2 pks. 5.6 qts. Ans. 18. How many bushels in 5 t. of oats? 2000 lbs. • 5 10000 lbs. 312ibu. ^718. 32)10000 19. How many bottles, each holding 1 pt. 3 gi., can be filled from a barrel of cider ? 1 pt. 3 gi. : 3gi.-fpt. Hpt.-^gal.="Agal. 8 16 9 144. Am. 20. If a steamer make 13 mi. 6 rds. an hour, how far will she go between 6 a.m. and 6 p.m.? How many hours will she re- quire to make 113 miles? mi. 13 rd». 6 12 156 72. Ans. 13 mi. 6 rds. 6 rds. 113 if^mi. ^hh mi. _ 160 113 2083 1 18080 2083 Smilira TEACHERS EDITION. 221 21. If a locomotive run at the rate of 111 rds. a minute, how many hours will it require to run from Boston to Buffalo, 498 miles ? 498 mi. X320 159360 rds. 1435ff 111)159360 6 )143.5|f 24 nearly. Ans. 22. What is the cost of 12 A. 146 sq. rds. land at 1 16.25 an acre? 146 sq. rds. = |f A. 13 12Hx^l6i=l^X^ 16 _|13429_ 1 209.83. Ans. 64 * 23. What is the cost of 8 t. 3 cwt. 27 lbs. of coal at |5.75 a ton? 100127 lbs. 20 1 3.27 cw t. 8.1635 t. X$5| 146.94. Ans. 24. What is the cost of 7 t. 1560 lbs. of hay at 1 15.50 a ton ? 1560 lbs. = iM^t. = xVVt. 7.78 X.$15^ $120.59. Ans. 25. What is the cost of a car- load of wheat weighing 20,000 lbs., at $1.05 a bushel? 6 )2000 3331 X$1.05 1 350. Ans. 26. Reduce 5 rds. 4 yds. 2| ft. to the decimal of a mile. ft. yds. rds. 0.0184 mi. A71S. 27. Reduce 9 sq. ch. 11.25 sq. rds. to the decimal of an acre. 3 2.5 ^ 4.83 320 5.87 11.25 . 9.703125 0.9703125 A. Ans. 28. Reduce 0.09375 bu. to quarts. 0.09375 bu. X32 3 qts. Ans. 29. Reduce 7560 chains to miles. 7560 ch. 4 30240 rds. 94.5 mi. Ans. 32)3024.0 222 ARITHMETIC. \. How many gross are 2000 pens? 13|. Am. 144)2000 31. Find the cost of 27.248 A. at 193.75 an acre. 27.248 X$93| $2554.50. Am. 32. Which is the greater, 2.8 of 3 ft. 11 in. or 3.11 of 2 ft. 8 in., and by how much ? 3 ft. 11 in. 2 ft. 8 in. Xl2 47 in. X2.8 131.6 in. 99.52 Xl2 32 in. X3.ll 99.52 in. 12 )32.08 in. 2 ft. 8.08 in. The former by 2 ft. 8.08 in. Am. 33. Reduce 171 lbs. 6 oz. troy to the decimal of a ton avoir- dupois. 7)0.1715 0.0245 X5760 100 20 141.12 lbs. avoird. 141.12 1.4112 0.07056 t. Am. 34. Express 14.52 sq. yds. as the decimal of a square chain. 30^)14.52 4_ 121)58.08(0.48 sq. rd. 0.03 sq. ch. Am 16)0.48 35. If a sovereign be equal to 25.22 francs, or to $4.85, what decimal of a dollar is a franc? $0,192. Am. 2522)485.000 36. Express 2.805 florins — 1.89 half-crowns as the decimal of £0.472. 2.805 1.89 28. X2.0 8. 5.618. 4.725 «. 4.725 20 )08858. 0.04425 0.09375. Am. 472)44. 1'oOOO 37. If 0.327 of some work be done in 3 hrs. 38 min., how lon^ will the whole work require? 3 hrs. 38 min. 60 218 min. 666.6 327)218000.0 0.6 = f = *. 666.6 min. = 666f min. = 11 hrs. 6| min. =-^11 hrs. 6 min. 40 sec. Am. TEACHERS EDITION. 223 38. A can ran a mile in 7.68 min. ; B can run at the rate of 7.68 mi. an hour. Which is the faster runner ? 7.81 768)6000.00 .•. A is the faster runner. 39. How many miles an hour does a person walk who takes 2 steps a second and 1900 steps in a mile? 60 _><2 120 X60 7200 steps. 3|-f mi. Ans. 19)72 40. If an ounce troy of gold be worth $20, what is the value of a pound avoirdupois ? $20 $240 per lb. troy. 175 5 W Ans. 41. Two stars cross the merid- ian at 6 hrs. 4 min. 42.3 sec. and 7 hrs. 2 min. 57.21 sec, respec- tively. What is the interval be- tween the observations ? hrs. min. sec. 7 2 57.21 6 4 42.3 58 14.91 A71S. 42. How long will it tak 3 to fill ^ of a cistern, when the whole requires 6 hrs. 10 min 6 hrs. 10 min. ? IC min. = U lir. = 6i hrs. = ^-hr. 6ixM-¥x M = m = 3^ hrs. tV? of 60 min. = 12f min. I of 60 sec. = 24 sec. 3 hrs. 12 min. 24 sec. Ans. 43. The circumference of a circle is 6 yds. 1 ft. 5.1 in., and is divided into 360 degrees. What is the length of 55 degrees? 6 yds. 1 ft. 5.1 in. X_3 19 ft. X_12 228 X5.1 233.1 in. 3.2375 72)233.1000 3.2375 in. Xll 12) 35.6125 in. 2 ft. 11.6125 in. Ans. 44. Multiply 2 t. 16 cwt. 63| lbs. by If. t. 2 cwt. 16 lbs. 63f X4 )11 6 53f 1 2 5 16 63| 80tV. Ans. 224 ARITHMETIC. 45. Into how many shares 46. If \\ of one line be equal has £120 been divided when to f of another line, which is the each share is £3 8fi. 6fd.? greater ? and what fraction of it is the less ? 6fd = ||«. = ^. 13 8 39, 40 15' 9 45 8^.=£| = £f. .'. the former is the greater. Atii. 3 120 ^x^'^? S5.Ans. f = fx;^ = |-- ■^ U 1 5 47. Multiply 5 mi. 206 rds. 2 ft. 2 in. by r86. 2 in. 2 ft. 206 X 786 X 786 X786 12) 1572 in. 1572 161916 131 ft. 131 103 3 1703 320)162019 5i 567 .. 2 ft. 506.. 99 rds. X2 5 11)1134 X 786 103 .. ^ yd 3930 606 4436 mi. 99 rds. \ yd .2 ft. 4436 mi. = 4436 m I. 99 rds. 1 yd . 6 in. Am. 48. The returns of a gold mine are 241 t. of ore yielding 2 oz. 1 dwt. 15 grs. of fine gold a ton, and 193 t. yielding 1 oz. 12 dwt. 9 grs. a ton. Find the value of the whole yield, at $19.45 an ounca dwt. 1 15 X241 41 9 11 15 OI. dWU gr*. 1 12 9 nw. Xl93 26 8 9 41 9 11 15 67 X_12 804 10 814 oz. 10 119.45 X814 $15,832.30 ilrw. teachers' edition. 225 49. Divide 93 long tons 56 lbs. by 23 lbs. 5 oz. 93 1. t. 56 lbs. 23 lbs. 5 oz. X^20 X 16 1860 1. cwt. 368 Xll2 5 208320 37^ oz. 56 208376 lbs. Xl6 8938iff Ans. 3334016 oz. 373)3334016 50. Telegraph poles on railroads are generally erected at intervals of 88 yds. Show that if a passenger count the number of poles which the train passes in three minutes, that number will express the number of miles an hour the train is going. 1760 yds. = 1 mi. 60 min. = 1 hr. 88 yds. = ^ mi. 3 min. = ^^ hr. 51. If Greenwich time be 5 hrs. 8 min. 16 sec. later than Wash- ington time, and Chicago be 87° 35'' W., what is the difference between Washington and- Chicago time? 87° 35^ = 4 X (87 min. 35 sec.) = 5 hrs. 50 min. 20 sec. 5 hrs. 8 min. 16 sec. 42 min. 4 sec. Ans. Exercise LVIII. 1. A train from New York to Philadelphia, 90 miles, makes the whole distance in 2 hrs. 5 min. What is its rate ? 2 hrs. 5 min. = 2^^ h^^- 90 mi. H- 2^2 = 43| mi. Ans. 2. Winlock, in 1869, found that electricity went through 7200 miles of wire in f of a second. What was its rate per second ? 226 ARITHMETIC. 3. If the time required for a signal to pass through the cable from Brest to Duxbury, 3799 miles, be 0.816 of a second, what is the rate per second ? 3799 mi. ^ 0.816 = 4655.637 mi. Ans. 4. If the report of a gun 1\ miles distant is heard in 5| seconds after the flash is seen, what is the velocity of sound, in feet, per second ? 1| mi. -J- 5f = f mi. - ll73i ft. Ans. 5. If a man walk 3| miles in 46 minutes, what is his rate per hour? 3| mi. -*- f ^ = 4/j mi. Ans. 6. If a horse go 47^ miles in 10 hrs. 40 min., what is his average rate per hour ? 10 hrs. 40 min. = lOf hrs. 47^ mi. -^ lOf = 4f I mi. Ans. 7. If a stone on a glacier move 95^ feet in 188 days, what is its rate, in inches, per day ? 951 ft. ^ 188 = m ft. = 6^\ in. Ans. 8. If a horse trot f of a mile in 2^ minutes, in what time can he trot a mile ' 2| min. -5- f = 2f min. Ans. 9. If a train run 18 miles in 39 minutes, how long does it take to run one mile ? 39 min. -4- 18 = 2^ min. Ans. 10. If sound travel 1125 feet a second, how long will it take to travel one mile ? 1 mi. = 5280 ft. 5280 + 1125 = 4.7 sec. Am. 11. If a train require 3 hours to travel 104 J miles, find its aver- age time for travelling a mile. 104} mi. + 3 = 34| mi. 60 min. + 84| = \\^ min. = 1 min. 43^ sec. Ans. teachers' edition. ^ 227 12. If a mower cut 7| acres of grass in 3^ days, what part of a day will it take him to cut one acre ? If a day consist of 10 work- ing-hours, what part of an acre does he cut in an hour? 3idys.-^7i = x'3dy. Ans. SI dys. of 10 hrs. = 35 hrs., and 7| A. ^ 35 = y3__ ^ ^,^5^ 13. If a mower cut 3| square rods in ^ of an hour, how many acres can he cut in a day of 10 hours? 3| sq. rds. -J- 1- = 28 sq. rds. 10 X 28 sq. rds. = 280 sq. rds. = If A. Ans. 14. If a fountain yield 117|- gallons in | of an hour, at what rate per hour is it flowing? 117| gals. ^ f = 156| gals. Ans. 15. If a merchant's profits be $3147 in 7^ months, what are his profits for a year ? 7irao.= j|yr. = |-yr. $3147^1 = 15035.20. Ans. 16. If a wheel turn 17° 30^ in 35 minutes, in how many hours does it make a complete revolution ? 17° 30^ - 35 = 30^ = 1° in one min. 360 ^ I = 720 min. = 12 hrs. Ans. 17. If a man's expenditures be $4358 in 13^ months, what is his yearly rate of expenditure ? 13^ i3i "^- = -^ yr. = -V- yr. $4358-^1^ = $3922.20. Ans. 18. If a cistern lose by leakage 7 gals. 1 pt, in 49 hrs. 40 min., what is its hourly rate of loss ? 49 hrs. 40 min. = 49f hrs. 7 gals. 1 pt. = 57 pts. 57 pts. ^ 49f = 1^^\ pts. 228 ARITHMETIC. 19. If the circumference of the earth at the equator be 24,900 miles, at what rate per hour is a person there carried round, one whole rotation being made in 23 hrs. 56 min. ? 23 hrs. 56 min. = 23}f hrs. 24,900 mi. -s- 23^ = 1040^f ^ mi. Am. 20. If a man travel ^ miles m 7^ minutes, how many miles will he travel in 50 minutes ? and how long will he take to travel 50 miles ? 7\ min. -4- 3f = 2^^ min. 50 mi. -f- 2^ = 24 mi. Ans. 3f mi. ^ 7^ = H rai. 50 -5- H = 104^ min. = 1 hr. 44 min. 10 sec. Ans. 21. If A can mow a certain meadow in 4 days, and B in 3 days, how long will it take both ? If A can mow it in 4 days, in one day he can mow ^ of it. If B can mow it in 3 days, in one day he can mow -^ of it. Both together can mow ^ + ^ = ^7^ of it in one day. .-. both together can mow the whole in -^^ days, or l^days. Ans. 22. If A can lay a certain wall in 4| days, and B in 5^ days, how long will it take both ? If A can do it in i^ days, in one day he can do — = f of it. If B can do it in 5J days, in one day he can do — = A- of it. Both together can do f + ^ = f § of it in one day. .-. both together can do the whole in |^ days, or 2^ days. Ans. 23. If a pipe will fill a vessel in 4^ hours, and another in 3 J hours, how long will it take both to fill the vessel? If one pipe will fill it in 4J hrs.. in one hr. it will fill — — f of it If another pipe will fill it in 3 J.hr8., in one hr. it will fill — = ^ of it Both pipeB together will fill f + f - ^f of it in one hour. .-. both pipes together will fill it in f4= Ifi hrs. = 1 hr. 58 min. 7i sec. Ans. teachers' edition. 229 24. If A can go from Boston to Albany in 9| hours, and B from Albany to Boston in 11^ hours, and they start at the same time, in how many hours will they meet? If A can go in 9| hrs., in 1 hour he can go — = /y of the distance. If B can go in 11| hrs., in 1 hour he can go — = ^\ of the distance. Both together can go /y + ^\ = j^^y^ of the distance in 1 hour. .-. the 25. A requires 4 days, B 3 days, and C 4| days, to do a certain piece of work. How long will it take all three working together ? If A can do it in 4 days, in one day he can do ^ of it. If B can do it in 3 days, in one day he can do ^ of it. If C can do it in 4| days, in one day he can do — = f of it. All together can do | + |- + | = ff of it in one day. .•. it will take them, all working together, || days = IgV- -4?2s. 26. A can mow f of a field in 3 days ; B can mow | of it in 4 days. How long will it take both to mow the field ? 3 days -^ f = 5| days, and 4 days -4- | = 6 days. If A can mow it in 5f days, in one day he can mow — = /y of it. If B can mow it in 6 days, in one day he can mow | of it. Both together can mow y\ + i = if of it in one day. .•. both together can niow it in ff days = 2|f days. Ans. 27. One pipe can fill a cistern half full in f of an hour, and another can fill it three-quarters full in | an hour. How long will it take both pipes to fill the cistern ? f hr. -r- 1 = 1|^ hrs., and | hr. -=- 1- = f hr. If one pipe fills it in 1| hrs., in one hour it will fill — = f of it. If another pipe fills it in f hr., in one hour it will fill - = | of it. Both together will fill | + f = -y- of it in one hour. .-. both together will fill it in 6 ^ 13 = j% hr. Ans. 230 ARITHMETIC. 28. A ci-stern which holds 100 gallons can be filled from a pipe in 25 minutes, and emptied by a waste-pipe in 45 minutes. If both are opened together, how long will it take to fill the cistern, and how much water will be wasted ? The water-pipe fills ^ every minute. The waste-pipe empties -^ every minute. When both are open, -^^ — ;i^ = ^f^ is gained every minute. .•. the whole will be filled in -^f^ = 56^ min. Ans. If -^^ of the cistern is wasted every minute, in 56 J minutes 66^ x ^^ would be wasted. Now, as the cistern holds 100 gals., the number of gallons wasted would be 56^ X :^ X 100 gals. 5 25 bQ\ X ^V X 100 = 2?f X :^ X ^?^ = 125 gals. Ans. 29. A pipe can fill a cistern one-third full in \ of an hour ; a waste-pipe can empty \ of the cistern in 20 minutes. If both pipes are opened, in what time will the cistern be filled? ^ hr. X 3 = f hr. = 45 min., and 20 min. x 4 = 80 min. The water-pipe fills -^-^ every minute. The waste-pipe empties -^j^ every minute. When both are open, ^ — ^i^ = ^^^ is gained every minute. .-. the whole will be filled in ifa = 102^ min. =» 1 hr. 42 min. 51f sec. Ans. 80. If one pipe runs into a cist6rn at the rate of 2 gallons in 3 minutes, and another at the rate of 5 gallons in 4 minutes, while the water is running out of a third pipe at the rate of 4 gallons in 5 minutes, how long will it take to gain 71 gallons in the cistern ? 2 gals. -1-3 = 1 gals., 5 gals, -i- 4 = J gals., and 4 gals. -«- 5 = f gala. If one pipe pours in | gals, per minute, another pours in \ gals, per minute, and another empties | gals, per minute, the cistern gains I + J — I = 1^ gals, per minute. .'. it will take as many minutes to gain 71 gals, as 71 -»- f^ = 63}f min. = 1 hr. 3 min. 34f f sec. teachers' edition. 231 31. A and B can do a piece of work in 2i days ; A and C in Si- days ; B and C in 4|- days. Required the time in which all three, working together, can do the work, and in which each can do it alone. If A and B can do it in 2i days, they can do — = f of it in one day. If A and C can do it in 3| days, they can do — = ^^ of it in one day. If B and C can do it in 4| days, they can do — = -^^ of it in one day. All can do f + x% + tV = xft ^"^ 2 days, or \\% of it in one day. .-. all can do it in \\% = 2j2j2_ days. An8. If A, B, and C can do \\%, and B and C j*y of it in one day, A can do \\% - tV - 3T0 of it in one day. ,'. A can do the whole in ^f^- = 4ff days. An%. If A, B, and C can do \\%, and A and C y% of it in one day, B can do \\% - A = sVo of it in one day. .'. B can do the whole in ^^ = 5f f days. Ans. If A, B, and C can do ^|f , and A and B f of it in one day, C can do iH - 1 = 3 ¥% of it in one day. .'. C can do the whole in -\*^ = 14|f days. Ans. 32. Sampson & Reed sold f of a lot of wheat to one party, | of the remainder to another, and had 93 bushels left. How much had they at first ? After selling f of the wheat they had f left. After selling f of f of the wheat they had ^ of f = -^^ left. Then 93 bush. = ^\ of the lot. .'. the whole lot = 93 bush. -^ j\ = 992 bush. Ans. 33. In a certain school -^^ of the scholars are girls, f of the boys are over IG years old, and 6 boys are under 16. How many girls, and how many scholars in all ? After the girls, or y\ of the school, are taken out, there remain y% of the school, or the boys. After |- of j^^ are taken out, f of j^^ = j\ are left. .-. the whole number of scholars is 6 -^ y\ ^ 32 scholars. Ans. ■^^ of 32 scholars = 18, number of girls. Ans. 232 ARITHMETIC. 34. In a certain school || are boys ; ^j of the girls are under 16, and 13 girls are over 16. How many boys and how many girls in the school ? After the boys, or ^| of the school, are taken out, there remains ^^ of the school, or the girls. After /j of H are taken out, |f of ^\ = if are left. .-. the whole number of scholars = 13 -j- ^f = 48 scholars, ii of 48 = 22 girls. Ans. ^f of 48 = 26 boys. Ans. 35. If from a certain number | of it be subtracted, then ^ of the remainder, then j- of that remainder, and 6 still remain, what is the number? After f of it is subtracted, ^ is left. After ^ of I is subtracted, | of | = | is left. After } of I is subtracted, f of | = s% is left. .-. the number = 6 -s- /^ = 35. Ans. 36. 20 is f of f of f of what number ? f of ^ of ^ = -• .-. the number = 20 -^ 4 = 120. Ans. ^^36 ^ 2 37. 6 is f of ^ of i of what number ? - of ^ of - = — . .-. the number = 6 -^ A = 35. Ans. 7 5 ^ 35 ^ 38. Express ^^ of 1 lb. troy + ^ of 1 lb. avoirdupois as troy and as avoirdupois weights. 175 1 29 lb. av. = 29 W^ 144 175 4176 lbs. troy. 1 29 + 175^ 4176 144 + 175 4176 319 4176 -1^- tVi lbs. = tV* of 12 oz. = ji oz. = li of 20 dwt. = 18J dwt = 18 dwt. 8 grs. Ans. iilbs.troy-jyLof?^=lll,^^^ 144 ^ IH }fm 175 175 ^Yy lbs. — ^yy of 16 oz. => 1 jfy oz. av. Ans. teachers' edition. 233 39. The cargo of a ship, worth 1 45,000, belongs to three partners. A owns I of f of it, B's share is equal to 3^^ of f of A''s share, and C owns the remainder. What ought each to receive from the sale ? I of f = iV of ship. tV of $45,000 = $ 21,000, A's share. 3^3^ of f of tV = i of ship, i of $ 45,000 = $ 1 5,000, B's share. $45,000 - ($21,000 + $15,000) = $9000, C's share. 40. Find the largest number which is contained an integral num- ber of times in each of the following : 2|, 6j^^, 11^, 19^. 2f,6A,iH,i9i = -^/.W-.¥.H^. G. CM. of 23, 115,23, 115 = 23. L. CM. of 9, 18, 2,6 = 18. .-. G. C M. of fractions = ff = lyV ^^s- 41. A person bequeathed j\ of his property to A, ^ of it to B, | to C, I to D, and the remainder, $550, to E. What was the value of the whole property ? A + i + i + i = ff- After If is subtracted, y^^ remains. .-. whole money = $550 h- ^\ = $ 13,200. Ans. 42. Arrange in descending order of magnitude, |f , |f , |f . 13 15 16 _ 9126, 10125, 10400 25' 26' 27 17550 .•. the order of magnitude is |f, |f , ^f . Ans. 43. A bankrupt's debts are $ 2520, and the value of his property is $1890. How much can he pay on a dollar? 2520 4 ^ 44. A bankrupt's debts are $4264, and he pays 62^ cents on a dollar ; what are his assets ? 533 0.62| = -. - of ^-^^ = $ 2665. Ans. 8 8 1 234 ARITHMETIC. 45. If 15 yards of silk cost $ 18.75, how much will 20^ yards cost ? If 15 yds. cost $18|, 1 yd. costs ^^, 15 and 20J yds. cost 20^ x ^^• 15 20}xii|i = ^xf x^ = i|25 = |25.42. An.. 15 S Xp 4 12 46. If 3f pounds of tea cost f 3.80, how much can I buy for 1 21.87 ? If 3| lbs. cost $3.80, 1 lb. costs ^^^, and as much can be bought for 121.87 as 2187 ^ ^• 2187 ^'^ = i^x^hX^^ = -V^ = 19{m lbs. Am. 47. If x\ of a ton of coal cost $ 1.12, what is the price of 5J cwt. ? 5icwt. = |t.. = -Ht. If ^j t. cost $ 1.12, 1 t. costs ^^, and ^ t. costs ^ x ?^^. 11 |W 14_|21L56_. ^^"^ 1 "^3" 15 -*1-^-^^- 5 48. If fV of a i)iece of work be done in 25 days, how much will be done in Uf days? If ^ can be done in 25 days, the whole can be done in — days, "A" and as much can be done in 11 1 days as 11| + — • 7 * 118^25 2 ^ 1 ^?? 14 . teachers' edition. 235 49. A man walks 18 mi. 106 rds. 3f yards in 5| hours. How long does he take to walk a mile and a half? 18 mi. 106 rds. 3f yds. : ^^, ^^ = lOSI _ J mi. ■^2 vd^ - M rd - ^ rd ^^^ ' ^ ~ 51 " ' 18 mi. 106 rds. 3f yds. = 181 mi. 1 81 If he walks 18|^ mi. in 5^ hrs., he will walk — ^ mi. in 1 hr,, and it will take him as long to walk l^ miles as 1^ -. ^ li^lM = ilx — X- = — hr.-27min. Ans. ' 5i 2 ^^ 2 20 o 50. When an ounce of gold is worth $19.46, what is the value of 0.04 of a j)0und ? $19.45 X 12 X 0.04 = $9,336. Ans. 51. If 9 horses can plow 46 acres in a certain time, how many acres can 12 horses plow in the same time ? Since 9 horses can plow 46 acres in a certain time, 1 horse can plow ^ of 46 acres in the same time, and 12 horses can plow 12 X ^ of 46 = 61^ acres. Ans. 52. If 12 men can reap a field in 4 days, in what time will 32 men reap it? Since 12 men can reap a field in 4 days, 1 man can reap it in 12 X 4 days, and 32 men can reap it in ^ days == 1^ days. Aiis. 53. If 72 men dig a trench in 63 days, in how many days will 42 men dig another trench three times as great? Since 72 men can dig a trench in 63 days, 1 man can dig it in 72 X 63 days. 1 man can dig one 3 times as large in 3 X 72 X 63 days, and 42 men can dig it in ^ X '^^ X ^^ days = 324 days. Ans. 23G ARITHMETIC. 54. If a rnan travels 540 miles in 24 days, walking 6 hours a day, how many miles can he travel in 3 days, walking 8 hours a day ? Since he can go 540 mi. in 24 X 6 hrs. = 144 hrs., in one hour he can go y^ of 540 mi., and in 3 X 8 hrs. = 24 hrs. he can go 24 X xt¥ ^^ ^"^^ ™^- = 90 mi. Atu. 55. If 15 men can perform a piece of work in 22 days, how many men will finish another piece of work four times as large in ^ of the time ? Since 15 men can do the work in 22 daySf it will take 4x15 men to do 4 times the work in 22 days, and to do 4 times the work in ^ of 22 days it will take 5 X 4 X 15 men = 300 men. Ans. 56. A garrison of 2100 has provisions for 9 months, but receives reinforcements of 600 men. How long will the provisions last? Since the provisions will last 2100 men 9 months, they will last 1 man 2100 X 9 months, and they will last 2100 + 600 = 2700 men ^1^><1 mos. = 7 mos. Ans. 2700 57. If a cubic foot of ice weigh 57f pounds, how many cubic feet of ice will weigh a ton ? Since 1 cubic foot of ice weighs 57| lbs. ; to weigh a ton it will take ^ cu. ft. = 34f|^ cu. ft. Ana. 57f 58. How many bushels of wheat will serve 72 people 8 days when 4 bushels serve 6 people 24 days? 72 people will eat twelve times as much as 6 people in the same time. And the same number of people will eat | as much in 8 days as in 24 days. Hence, 72 people in 8 days will eat 12 x | times as much as 6 people in 24 days. 12 X i X 4 bushels = 16 bushels. Ans. teachers' edition. 237 59. If 2 horses eat 8 bushels of oats in 16 days, how many horses will eat 3000 bushels in 24 days ? In 16 days 8 bu. can be eaten by 2 horses. In 1 day 8 bu. can be eaten by 16 x 2 horses. 16 X '' In 1 day 1 bu. can be eaten by — ^^^-^ horses. 8 1 6 V ^ In 24 days 1 bu. can be eaten by horses. ^ -^ 24 X 8 In 24 days 3000 bu. can be eaten by ^^^Q X 16 X 2 y^^^ ^ -^ 24x8 = 500 horses. Ans. 60. If a man travel 150 miles in 5 days, when the days are 12 hours long, in how many days of 10 hours each Avill he travel 500 miles ? He can go 150 miles in 5 days of 12 hours = 60 hours. He can go 1 mile in j%% hour. TT KAA -1 • 500 X 60 1 He can go 500 miles m — hours. loU He can so 500 miles in — — days of 10 hours, ^ 150x10 "^ = 20 days. Aiis. 61. If a regiment of 939 soldiers consume 351 bushels of wheat in 21 days, how many soldiers will consume 1404 bushels in 7 days ? 1404 bu. will last the same number of men four times as long as 351 bu. And the same amount will last three times the number of men for 7 days as for 21 days. 3 X 4 X 939 soldiers = 11,268 soldiers. Ans. 62. If 5 men can reap a field of 12i acres in 3|- days, working 16 hours a day, in what time can 7 men reap a field of 15 acres, working 12 hours a day ? 5 men can reap 12-^- acres in 3|- days of 16 hours = 56 hours. 1 man can reap 12^ acres in 5 x 56 hours. 1 1 • 5 X 56 T_ 1 man can reap 1 acre in -^ — hours. ^ 12i 238 ARITHMETIC. 15 X 5 X 56 1 man can reap 15 acres in -^—^ — — — hours. 12^ 15 X 5 X 56 7 men can reap 15 acres in -^-^ — ^-^-^ hours. 7x12^ 1 5 )^ 5 X 56 12 X 7 X 12J 15 ^^ 5 X 56 7 men can reap 15 acres in days of 12 hours, 4 days. Ans. 63. If 7 men mow 22 acres in 8 days, working 11 hours a day, in how many days, working 10 hours a day, will 11 men mow 360 acres ? 7 men can mow 22 acres in 8 days of 11 hours = 88 hours, 1 man can mow 22 acres in 7 X 88 hours. 7 X 88 1 man can mow 1 acre in —^ — hours. 22 7 X 88 12 men can mow 1 acre in hours. 12x22 12 men can mow 360 acres in — hours. 12x22 12 men can mow 360 acres in ^^Q X 7 X 88 ^ ^^ ^^ ^^ 10 X 12 x 22 -^ = 84 days. Ans. 64. If 44''cannon, firing 30 rounds an hour for 3 hours a day, consume 300 barrels of powder in 5 days, how long will 400 barrels last Q6 cannon, firing 40 rounds an hour for 5 hours a day ? 44 cannon firing 30 rounds for 3 hours consume 300 bbls. in 5 days. 44 cannon firing 30 rounds for 1 hour consume 300 bbls. in 3 X 5 days. 44 cannon firing 1 round for 1 hour consume 300 bbls. in 30 X 3 X 5 days. 1 cannon firing 1 round for 1 hour consumes 300 bbls. in 44 X 30 X 3 X 5 days. 1 cannon firing 1 round for 1 hour consumes 300 •' teachers' edition. 239 66 cannon firing 1 round for 1 hour consume 300 X 66 ^ 66 cannon firing 40 rounds for 1 hour consume 40 X 300 X QQ ^ 66 cannon firing 40 rounds for 5 hours consume Ibbhin 44X30X3X5 5 X 40 X 300 X 66 ^ 66 cannon firing 40 rounds for 5 hours consume 400 bbls. in 400x44x30x3x_r> ^ ^ ^ ^ ^^^ 5 X 40 X 300 X 66 ^ ^ 65. How many times will a wheel 2f feet in circumference turn round in travelling over 12f yards ? The wheel revolves once in going 2f- feet. .-. it will turn around as many times in going 12f yds. = 38f ft. as 38f ^ 2f- = 15. Ans. 66. How much ground will be travelled over by a wheel If yards in circumference, when it has made 4|- turns ? If the wheel turns once in going If yds., in making 4| turns it will go 4^ times If yds. = 6f yds. Ans. 67. Find the circumference of a wheel which makes 9 turns in travelling over 7^ yards. If it makes 9 turns in going 7^ yds., it will go 7-^ yds. -r- 9 = f yd. = 2f ft. in making one turn. .•. the circumference of the wheel is 2| feet. Ans. 68. If the circumference of a wheel be -^^^ of 1 yd. li ft., how many times will it turn in travelling 3f miles ? 1 yd. li ft. = 1-i yds. = If yds. 3f mi. = 3f X 1760 yds. o If the wheel makes 1 turn in going -V_ of If yds., it will make as many turns in going 3f X 1760 yds. as ^l^ ™^ = UdQj\. Ans. 240 ARITHMETIC. 69. If the wheel of a locomotive be 3| times 5.52 feet in circum- ference, how many times does it turn in a minute, when the locomo- tive is running at the rate of 13.34 miles an hour? 5.52 = 5^1, and 13.34 = 13^^. If it is going at the rate of 13^^ miles per hour, it is going at the rate of ^^ miles per minute, or at the rate of ^^H X ^^^^ feet 60 ^ 60 per minute. If it turns once in going 3| x 5^f feet, it will turn as many times . 13Ux5280. . 13Ux5280 /oi ., ki<.\ c-2 in going —^^ feet as —^^ (H X 5^1) = 6/f . Ans. 70. A can run /- of a mile in | of a minute, B can run /^ of a mile in | of a minute, and C -^^ of a mile in f of a minute. Which is the fastest runner? and if he can run a certain distance in 3 min. 10 sec, how much longer will each of the others take to run the same distance ? If A can run /^ mi. in f min., to run a mile it will take him | min. -^ "5^ = 8^ min. If B can run -^^ mi. in f min., to run a mile it will take him | min. ■^■i'z=Vj inin- If C can run -^ mi. in f min., to run a mile it will take him f min. -4- ^g min. = 6|^ min. .-. C is the fastest runner. 3 min. 10 sec. = 3^ min. If C can run a certain distance in 3^ min., and a mile in 6}^ min., the distance is that part of a mile which 3^ is of 6{^ = ^. If A can run a mile in 8J min., he can run -j^ mi. in ^ of 8^ min. == 3^^ min. = 3 min. 51 sec. But C can run it in 3 min 10 sec. .*. it takes A 41 sec. longer than C. If B can run a mile in 7^^ min., he can run ^ mi. in /y of 7-j^ min. = 3i§| = 3 min. 29^15 sec. But C can run it in 3 min. 10 sec. .*. it takes B 19j^ sec. longer than C. TEACHERS EDITION. 241 Find the amount of the following bills : 71. Mr. Richard Rowe, Boston, Nov. 23, 1880. To John Doe, Dr. To 125 lbs. sugar @ 10 cts. " 1 hag coffee, 115 lbs. @ 32 cts. " 25 gals, molasses @ 62 cts. " 8 lbs. Japan tea @ 92 cts. " 28 lbs. crackers @, 8 cts. 2 bbls. flour @ $7.50 $12 36 15 7 2 15 50 80 50 36 24 00 40 Received Payment, 72. John Doe. Mr. James Hardy, Boston, Feb. 29, 1888. To a H. Mills, Dr. To 275 bbls. flour @ $6.75 324 bbls. flour @ $6.25 300 bu. potatoes @ 48 cts 1578 lbs. butter @ 32 cts 2000 bbls. apples @ $1.25 1 car-load oats, 20,000 lbs., 625 bu. @ 42 cts 1 car-load corn, 28,575 lbs., 510.27 bu. (a), 55 cts $1856 2025 144 504 2500 262 280 25 00 00 96 00 50 65 Received Payment, 73. $7573 36 a IT. Mills. James Harlow, Boston, Jan. 1, 1888. To John Dike, Dr. To 12 bales Texas cotton, 5760 lbs. @ 91 cts. . . $532 80 " 7 bales upland cotton, 3514 lbs. @ 10| cts. . . 360 19 " 3 bales low middling, 1476 lbs. @ 9f cts. . . 143 91 " 8 bales good ordinary, 9220 lbs. @ 9 cts.. . 793 80 Received Payment, $1830 70 John Pike. 242 ARITHMETIC. Exercise LIX. 1. What length of board 15 in. wide will contain 11 sq. ft. 36 sq. in. ? 11 sq. ft. 36 sq. in. = llj sq. ft. 15 in. = IJ ft. llj -^ li = 9 ft. Ans. 2. What length of road 44 ft. wide will contain an acre ? 1 A. = 160 sq. rds. 44 ft. = 2| rds. 160 - 2i = 60 rds. Ans. 3. Find the area of a rectan- gular field 13.12 chains long, 10.35 chains broad. 13.12 ch. X 10.35 ch. 1 0)135.792 sq. ch. 13 A. 5.792 sq. ch. Ans. 4. A path 216 ft. long meas- ured 72 sq. yds. Find its breadth . 72 sq. yds. = 648 sq. ft. 648 + 216 = 3 ft. Ans. 5. A rectangular field of 21.66 acres is 250.8 yds. broad. Find its length. 1 A. = 4840 sq. yds. lM2^2L66.418yds.^ns. 250.8 ^ 6. What is the area of a table if length and breadth be 4 ft. 3^ in. and 2 ft. 9| in., respectively ? 4 ft. 3f in. = ^ ft. 2 ft. 9f in. = 2f ft. 4f X 2f = 12 sq. ft. Ans. 7. From each corner of a square, the side of which is 2 ft. 5 in., a square measuring 5 in. on a side is cut out. Find the area of the remainder of the figure. 2 ft. 5 in. = 2^5^ ft. 2Ax2/j = 51Hsq.ft = 5 sq. ft. 121 sq. in. 5 X 5 = 25 sq. in. 25 sq. in. X 4 = 100 sq. in. 5 sq. ft. 121 sq. in. = 100 sq. in. = 5 sq. ft. 21 sq. in. Ajis. 8. The length and breadth of a map are 4 J ft. and 3^ ft., re- spectively. If the map represent 77,760 sq. mi. of country, how many miles are there to a square inch? 4^X3^ = 15 sq.ft. = 2160 sq. in. 77,760 + 2160 = 36 sq. mi. Ans. 9. In rolling a grass plot 24 yds. long and containing 400 sq. yds., how many times must a roller 3 ft. 4 in. wide be drawn TEACHERS EDITION. 243 over it lengthwise so that the whole may be rolled? 400 - 24 = 16f yds. 10. How many sods, each 2 ft. 3|- in. long and 8|- in. broad, would be required to turf an acre of ground ? 2 ft. 3|in. = 27iin. 1 A. = 6,272,640 sq. in. 6272640 27i X 8^ = 27,648. Ans. 11. Find the area of a picture-frame 2^ in. broad and having an outside measurement of 4 ft. 6| in. in length and 2 ft. 8 in. in width. to, CC| ^■1 21 in. X 2 = 4i in. 4 ft. 6| in. - 41 in. = 4 ft. 2 in. 4 ft. 2 in. + 2 ft. 8 in. = 6 ft. 10 in 6 ft. 10 in. X 2 = 13 ft. 8 in. = 1 3f ft. 2iin. = fVft. P lb lb ^T% ^q- ft- = - ^c[. ft. 81 sq. in. Ans 4 ft. 2 in. 12. Find the expense of glazing four windows, each containing 12 panes, the panes being each a foot long and 10 in. wide, and the price of the glass 38 cents per square foot. 10 in. = f ft 2 n 1 X f = I- sq. ft. XiiX- = 10 sq. ft. 10 sq. ft. X 4 = 40 sq. ft. 40x10.38 = $15.20. Ans. 13. A garden 76 yds. long and 56 yds. broad, enclosed by a wall, has a border 4 ft. wide within the wall, and within this a ])ath 5 ft. wide, the middle being grass. Find the areas of the border, path, and grass, respectively. 4 ft. = li yds. 2 X 1^ yds. = 2f yds. 76 yds. - 2f yds. = 73i yds. 731 yds. + 56 yds. = 1291 yds. 1291 yds. X 2 = 258 1 yds. = perimeter. 258§ X li = H- X i = ^^^ = 344| sq. yds. (1) Ans. 244 ARITHMETIC. ■7^ y^s- 56 yds. - 2f yds. = 53^ yds. 731 yds. - 2 X If = 70 yds. 70 yds. + 53| yds. = 123^ yds. 123^ yds. X 2 = 246f yds. 246fxlf = ^Xi = ^^ = 411^Bq. yds. (2) Ans. 5 ft. + 4 ft. = 9 ft. = 3 yds. 76 yds. - (2 X 3 yds.) = 70 yds. 56 yds. - (2 X 3 yds.) = 50 yds. 70 X 50 = 3500 sq. yds. (3) Am. il Ur 73|- g en * 70 1 1 i i 1 14. Find the area of a circle which has a radius of 3 ft. 3 X 3 - 9 sq. ft. 3.1416 X 9 sq. ft. 28.27^^ sq. ft. Ans. 15. What is the area of a cir- cular field with a radius of 400 yards ? 400 X 400 = 160,000 sq. yds. 3.1416 X 160000 sq. yds. 502,656 sq. yds. Ans. 16. The radius of the rotunda of the Pantheon at Rome is 71 ft. 6 in. Find the area of the floor. 71 ft. 6 in. - 7H ft. 71ix71i = 5112^ sq.ft. 3.1416 X5112 ^ sq. ft. 16060.64;IP sq. ft. Ans. 17. The diameter of a cistern is 13 ft. What is the area of the bottom ? 13 ft. -^- 2 = Gl- ft. Radius. 6^X6^ = 42^ sq. ft. 3.1416 X42^ sq. ft. i32.'/32^ sq. ft. Ans. 18. The two dials of the clock of St. Paul's, London, are each 18} feet in diameter. What is the area of each in square feet ? 18} ft. + 2 = 9^j ft. 9i^X 9,^ = 82^^ sq.ft. 3.1416 _J<82^^ sq. ft. 258.52jr^ sq. ft. Ans. 19. How many square inches on the surface of a ball 3 in. in diameter ? 3 X 3 = 9 sq. in. 3.1416 X 9 sq. in. 28.27)1)1 sq. in. Ana. TEACHERS EDITION. 245 20. How many square inches of surface in a spherical black- board 12 in. in diameter? 12 X 12 = 144 sq. in. 3.1416 X 144 sq. in. 452.39^^ sq. in. 21. What is the interior sur- face of a hemispherical vase 20 in. in diameter? 20 X 20 = 400 sq. in. 400 sq. in. -J- 2 - 200 sq. in. 3.1416 X 200 sq. in. 628.32 sq. in. = 4 sq. ft. 52.32 sq. in. 22. How many yards of car- peting f of a yard wide will be required for a floor 26 ft. long, 15| ft. wide, if the strips run lengthwise? How many if the strips run across the room? How much will be turned under in each case ? 15f ft. = 51 yds. 5J -5- f = 7 strips. 26 ft. = 8| yds. 7 X 8f yds. = 60f yds. Ans. None to turn under. 8f-v-f=llf= 12 strips. 12 X 5| yds. = 63 yds. Ans. 4 3 1 ^ X - = - yd. to turn under, ^ 4 o 23. How many yards | of a yard wide will be required for a room 8|- yds. long and 17 ft. wide, if the strips run length- wise, and there is a waste of j^^- of a yard in each strip, in match- ing patterns ? 17 ft. = 5f yds. 5f - I = 6^f = 7 strips. 7 X 81 yds. - 59i yds. 59^- yds. + j\ yds. = 59{^jds. Ans. 24. How many square yards of oil-cloth will be required for a hall floor 6^ yds. long and 10 ft. wide ? 10 ft. = 31 yds. 5| X 3 1 = 17|sq.yds. Ans. 25. What will be the cost of carpet | of a yard wide for a room 28 1 ft. by 18| ft., if the strips run lengthwise, and the cost per yard is 92 cents ? 18f ft. = 6| yds. 6^ -^1=5 strips. 28|~ ft. = 9^ yds. 91 X 5 = 471 yds. 471 X $0.92 = $43.70. ^^s. 26. Find the cost of carpet 30 in. wide, at $1.25 per yd. for a room 18 ft. by 14 ft., if the strips run lengthwise ; if the strips run across the room. 246 ARITHMETIC. 30 in. = § yd. 18 ft. = 6 yds. 14 ft. = 4f yds. 4| -5- f = 5f = 6 strips. 6x6 yds. = 36 yds. 36x$U = $45. Ans. 6 - f = 7^ = 8 strips. 8x4f x$li = $46| =$46.67. Ans. 27. Find the cost of carpeting 27 inches wide, at $1.12^ per yard, for a room 29 ft. 9 in. by 23 ft. 6 in., if the strips run across the room. 27 in. = f yd. 29 ft. 9 in. = 29 J ft. = 9}^ yds. 23 ft. 6 in. = 23^ ft. = 7f yds. 9H-^f = 13f. 14 X 7f = 109| yds. = 14 strips. 109fx$ 1.125 = $123.38. Ans. 28. Find the cost of carpeting f of a yard wide, at $2.75 per yard, for a room 34 ft. 8 in. by 13 ft. 3 in., if the strips run lengthwise, and if there be a waste of ^ of a yard on each strip in matching the pattern. 34 ft. 8in. = 34Jft. = ll|yds. 13 ft. 3in. = 13jft. = 4-i5jyd8. 4fk -!- 1 = 5J = 6 strips. 6 X n fj = 69} yds. 6 X i yd. = 1^ yds. waste. 69i + H=.70tyd8. 70JX $2.75 = $194.79. Ans. 29. Which way must the strips of carpet J of a yard wide run in order to carpet most economically a room 20 ft. 6 in. long and 19 ft. 6 in. wide, if there be no waste for matching the pattern ? 20 ft. 6 in. = 20^ ft. = 6f yds. 19 ft. 6 in. = 19^ ft. = 6^ yds. 6| - f = 8f = 9 strips. 9 X 6f = 61^ yds. lengthwise. 6f -=- f = 9^ = 10 strips. 10 X 6| = 65 yds. across. .•. the strips must run lengthwise. 30. Find the number of yards of plastering in the walls of a room 2 If ft. long, 16^ ft. wide, and 11 ft. high, if 12 sq. yds. be allowed for doors, windows, and base-boards. 21| + 16^ = 38^ft. 2 X 38^ ft. == 76^ ft. 76^X11 = 841^ sq.ft. = 93^ sq. yds. 93^ sq. yds. — 12 sq. yds. = 81^ sq. yds. Ans. 31. How many square yards of plastering in the walls and ceiling of a room 30 ft. 8 in. long, 26 ft. 5 in. wide, 10 ft. 6 in. high, if 24 sq. yds. be allowed for doors, windows, and base-boards ? 30 ft. 8 in. = 30f ft. 26 ft. 5 in. = 261&J ft. 30J -f- 26t\ = 57tV ft. 2x57T»j = 114ift. 114J xlOi = 133^5 sq. yds. 30Jx2635j = 905VBq.yd8. 133,V + 90^ = 223j^. 223^-24=199^ sq. yds. Ans. TEACHERS EDITION. 247 32. What will be the cost of plastering the walls and ceiling of a room 27 ft. 4 in. long, 20 ft. wide, and 12 ft. 6 in. high, at 27 cents per square yard, if 20 sq. yds. be deducted for doors, win- dows, and base-board ? 27 ft. 4 in. = 27i ft. = 9^ yds. 20 ft. = 6f yds. 12 ft. 6 in. = 12i ft. = 4i yds. 9i + 6f = 15|yd8. 2xl5| = 31|yds. 31f X4i = 131if sq.yds. 9-1 X 6f = 60|^ sq. yds. 131|f X 60f ^ = 192f sq. yds. 192f-20=172|8q. yds. 172f x|0.27 = |46.50. Ans. 33. Find the cost of whitening the ceiling and walls of a room 14 ft. 4 in. wide, 15 ft. 6 in. long, 10 ft. 6 in. high, at 5 cents per square yard, allowing 9 sq. yds. for doors and windows. ft. 4 in. = 14-1- ft. = 4| yds. ft. 6 in. = 15i ft. = 5iyds. ft. 6 in. = 10| ft. = H yds. 4| + 5i = 9Hyd8. 2x9i|=19|yds. 19|x3i = 69iisq.yds. 4| X 51 = 24|| sq. yds. 69ii + 24f| = 94^Vsq.yds. 942V - 9 = 85387 sq. yds. 85^^ X ? 0.05 = $ 4.26. Ans. 34. Find the cost of plastering a room 21 ft. long, 15 ft. wide, 12 ft. high, at 40 cents per square yard, allowing for a door 7 ft. high, 3 I ft. wide; 3 windows, each 5 ft. high, 3 ft. wide ; and a dado 2 ft. 9 in. high around the room. 21 -f 15 = 36 ft. 2 X 36 ft. = 72 ft. 72x12 = 1179 sq.ft. 3 X 7 = 21 sq. ft. door. 5 X 3 X 3 = 45 sq. ft. windows. 2| X 72 = 198 sq. ft. dado. 21 -f 45 -f 198 = 264 sq. ft. 1179 - 264 = 915 sq. ft. 915 sq. ft. = 101f sq. yds. lOlfx $0.40 = 140.67. Ans. 35. Find the cost of papering a room 20 ft. 6 in. long, 17 ft. 4 in. wide, 9 ft. high, with paper 18 in. wide, 8 yards in a roll, at 75 cents a roll ; allowing for 2 doors, each 7 ft. high, 3 ft. wide, and for 3 windows, each 5 ft. 6 in. high and 3 ft. 3 in. wide. 20 ft. 6 in. = 20i ft. = 6f yds. 17 ft. 4 in. = 17i ft. = 51 yds. 9 ft. = 3 yds. ^ + 5l = l2\ljds. 2xl2ii = 25f. 2 x 7 X 3 = 42 sq. ft. = 4f sq. yds. doors. 248 ARITHMETIC. 3 X 5| X 3| = 53| sq. ft. = 5f f sq. yds. windows. 4f + 5||=10f sq. yds. 75f - lOf = 65^5 sq. yds. 18 in. = ^ yd. 8x^ = 4 8q. yds. 65^5 ^ 4 = 16f| = 17 rolls. 17 x!p0.75 = $ 12.75. Ans. 36. Find the cost of papering a room 32 ft. long, 22 ft. wide, 13 ft. high, with paper 18 in. wide, 8 yards in a roll, at $1.25 a roll, if 50 sq. yds. be allowed for doors, windows, and base-board. 32 + 22 = 54 ft. 2 X 54 = 108 ft. 108 X 13 = 1404 sq. ft. = 156 sq. yds. 156 - 50 = 106 sq. yds. 18 in. = ^ yd. 8x^ = 4 sq. yds. 106 -^ 4 = 26^ = 27 rolls. 27 X $1.25 = $33.75. Ans. 37. Find the cost of papering a room 26 ft. long, 21 ft. wide, 12 ft. high, with paper 20 in. wide, 8 yards in a roll, at $1.50 a roll, and a border at 25 cents per running foot; allowing for a fire-place 6 ft. 3 in. by 4 ft., a door 7 ft. by 4^ ft., and 3 windows, each 6 ft. by 3i ft. 26 ft. = 8f yds. 21 ft. = 7 yds. 12 ft. = 4 yds. 8f + 7 = 15|yd8. 2xl5f = 3Hyds. 3H X 4 = 125} sq. yds. 5 ft. 3 in. = 5^ ft. 5^ X 4 = 21 sq. ft. = 2J sq. yds. fire-place. 7 X 4i = 31} sq. ft. = 3} sq. yds. door. 3 X 6 X 3} = 63 sq. ft. = 7 sq. yds. windows. 12| sq. yds. to be deducted. 125}-12J = 112}8q. yds. 20 in. =. f yd. 8 x f = 4f sq. yds. 112i-4-4J = 25T!V = 26roll8. 26 x$ 1.50 = $39.00. Perimeter = 31} yds. = 94 ft. 94 X $0.25 = $23.50. $39.00 + $23.50 = $62.50. Am. teachers' edition. 249 How many feet board measure in : 38. A board 18 ft. long, 9 in. wide, | in. thick? A board 16 ft. long, 11 in. wide, 1 in. thick ? 9 in. - f ft. 18xf =13ift. Ans. 11 in. = |i ft. 16 X H - 14f ft. Ans. 39. Twenty boards averaging 14 ft. long, 10 in. wide, ^ in. thick? 10 in. = I ft. Mx^x22 = ^ = 233ift. ^ns. 1^13 ^ 3 40. Three joists 13 ft. long, 8 in. wide, 3 in. thick? 8 in. -f ft. Hence 1 joist = 3 boards 13 by f. ^x|xfxf=78ft. Ans. 1 p 1 1 41. A stick of timber 8 in. by 9 in. and 27 ft. long? 8 in. == I ft. Hence 1 stick = 9 boards 27 ft. by f ft. 9 ^x|x? = 162ft. Ans. 1 P 1 42. Two beams, each 6 in. by 9 in. and 23 ft. long ? 6 in. - i ft. Hence 1 beam = 9 boards 23 ft. by | ft. ^xix^X^-207ft. Ans. 1/11 250 ARITHMETIC. 43. Three joists, each 3 in. by 4 in. and 11 ft. long' Hence 1 joist = 4 boards 11 ft. by \ ft. i^xix^X? = 33ft. Ans. 1^11 44. Five joists, each 6 in. by 4 in. and 14 ft. long? 6 in. = I ft. Hence 1 joist = 4 boards 14 ft. by ^ ft. 2 i^ X J X f X ^ = 140 ft. Ans. 1 /S 1 1 45. A stick of timber 10 in. square and 36 ft. long ? 10 in. = f ft. Hence 1 stick = 10 boards 36 ft. by f ft. ^x|x^ = 300ft. Ans. 1 p 1 46. Ten planks, each 13 ft. long, 15 in. wide, 2 in. thick? 15 in. == f ft. Hence 1 plank = 2 boards 13 ft. by f ft ^X^x|x^ = 325ft. Ans. 14 11 Find the cost of: 47. Nine joists, each 15 ft. long, 3 J in. by 5 in., at fil2 per M. 5 in. - -jf^ ft. 1 12 per M. = -^ per sq. ft. Hence 1 joist =■ 3^ boards 15 ft. by -j^ ft. teachers' edition. 251 48. Thirty planks, each 12 ft. long, 11 in. wide, 3 in. thick, at $ 15 per M. llin. = ilft. |15perM.--^~ per sq.ft. Hence 1 plank = 3 boards 12 ft. by |^ ft. 12xlix3x30x^ = l|l = fU.85.^«. 49. Four sticks of timber, each 8 in. by 9 in. and 23 ft. long, at |18per M. 9in. = fft. $ 18 per M.= 11^ per sq.ft. Hence 1 stick = 8 boards 23 ft. by | ft. T>^4'^l>'l''l000-"T25~-^^-^^- ^^'- 50. A board 24 ft. long, 23 in. wide at one end and 17 in. at the other, and 1^ in. thick, at $30 per M. ^^"^•^'^ = 20 in. average width. 20 in. = If ft $30 per M. = ^ per sq.ft. Hence the board = 1^ boards 24 ft. by If ft, 24x-X-X-^ = ^ = $1.80. Ans. 3 2 1000 5 * 51. A stick of timber 29 ft. long, 10 in. by 12 in. at $ 13.50 per M. 12 in. = 1 ft. $ 13.50 per M. = tiM = 31L per ft. ^ 1000 2000^ Hence 1 stick = 10 boards 29 ft. by 1 ft. 29 X ;^ X 1 X ^^ = ^^ = $3.92. Ans. 200P 200 * 52. The flooring for two floors, each 23 ft. by 17 ft., each floor double, and of boards ^ in. thick ; the lower floor at $ 18, and the upper at $ 24, per M. 252 ARITHMETIC. 782 sq. ft. Boards | in. thick are reckoned as 1 in. Hence each floor, being double, will require 2 X 23 X 17 Average cost per floor = $ 21 per M. .'. Average cost both floors = f 42 per M. $42 Whole cost = 782 x 1000 $32.84. ^718. 17 ft. 53. The flooring timbers for a room 23 ft. by 17 ft. at $18 per M, if they are 2 in. by 10 in., 17 ft. long, and are placed on edge, two close to the walls, and the others with spaces of f^ of a foot between them. The room being 17 ft. wide and the timbers 17 ft. long, the timbers must run across the room. After a timber is placed against the wall at one end, the remaining distance to be occupied with timbers and spaces = 23 ft. - 2 in. = 22f ft. The distance occupied by a timber and a space .-. 22f remaining space. 20 137 ^f ^ = number of timbers required for 22-1-^ 120 ^ 20. 20 + 1 1 timber is supposed to have been placed. 21, the whole number of timbers required. 3 XPxixl7x $;^_$5i 1000 100 21 X $0.51 = $ 10.71, whole cost. = $0.51, cost of one timber. 54. A log 14 ft. long, 17 in. in diameter. 17^-2x17 = 289-34 = 255. H of i^ of 255 = 187 ft. Am. '5. A log 11 ft. long, 13 in. in diameter. 13' - 2 X 13 = 169 - 26 = 143. fi of H of 143 = 83 ft. Aii%. teachers' edition. 253 56. A log 16 ft. long, 20 in. in diameter. 202 - 2 X 20 == 400 _ 40 = 360. 21 of ^ of 360 = 302 ft. Ans. 57. A log 12 ft. long, 15 in. in diameter. 152 - 2 X 15 = 225 - 30 = 195. 2^ of if of 195 = 123 ft, Ans. Find the value, at $ 9 per M. of : 58. A log 17 ft. long, averaging 11 in. in diameter. 112-2x11 = 121-22 = 99. f ^ of \i of 99 = 88.3575 ft. 88.3575 X 10.009 = $0.80. Ans. 59. A log 18 ft. long, averaging 13 in. in diameter. 132-2x13 = 169-26 = 143. I^ofif of 143 = 135.135 ft. 135.135x10.009 = $1.22. Ans. 60. A log 13 ft. long, 16 in. in diameter. 162 - 2 X 16 = 256 - 32 = 224. 1^ of If of 224 = 153 ft. 153 X $0,009 = $1.38. Ans. ., 61. A log 14 ft. long, 12 in. in diameter. 122 - 2 X 12 = 144 - 24 = 120. f^ofif of 120 = 88ft. 88 X $0,009 = $0.79. Ans. 62. How many clapboards will be required to cover the front of a house 60 ft. long and 20 ft. high, if they are laid 4 in. to the weather, and if 120 sq. ft. be deducted for doors and windows ? 60 X 20 = 1200. 1200 - 120'= 1080. 4 X i = li sq. ft. 1080 H- 11 = I of 1080. = 810. Ans. 254 ARITHMETIC. 63. If one thousand shingles cover 120 sq. ft. of roof, what is the average width of a shingle ? T'(ftp7 = /:?8q-ft. = 17273sq. in. I of 16 in. - 5] in. 173;^-5i = 3/^in. Am. 64. Allowing one thousand shingles for 120 sq. ft., how many- thousand will be required to cover the pitched roof of a house 60 ft. long, if the width of each side of the roof be 24^ ft. ? 2^ X 60 = 1470 sq. ft. 1470 V 120 = 24i Ans. 1. Find the volume of a rectangular solid whose length, breadth, and thickness are 7 ft, 2 ft. 6 in., and 11 in. respectively. 7 X 2i X ii = le^j cu. ft. => 16 cu. ft. 72 cu. in. Ans. Exercise LX. 5. IIow many cubic feet of water does a cistern hold whose length, breadth, and height are 5 ft. 4 in., 3 ft. 6 in., 2 ft. 10 in., respectively ? 5| X 3i X 2f = 52| cu. ft. Ans. 2. How many cubic feet of air in a hall 54 ft. long, 33 ft. wide, 21 ft. 4 in. high ? 54 X 33 X 21| = 38,016 cu. ft. Ans. 3. Find the volume of a cube whose edge is 2]^ yds. 2i X 2J X 2i = 15f cu. yds. = 15 cu. yds. 16 cu. ft. 1512 cu. in. Ans. 4. A cellar is dug 21 ft. long, 17 ft. 3 in. wide, 9 ft. deep. How many cubic yards of earth are taken out ? 21X17^X9 27 120|cu.yd8. Ans. 6. If the dimensions of a brick be 8 in. by 3^ in. by 2^ in., find its volume. 8 X 3i X 2^ = 63 cu. in. Ans. 7. In a bar of iron 21 ft. long 3 in. wide, 2 in. thick, how many cubic inches are there ? 21 ft. = 252 in. 252 X 3 X 2 = 1512 cu. in. Ans. 8. What is the value of a bar of gold 8 in. long and | of an inch square, at |190 a cubic inch? 8xfXfX|190. = $855. Ans. TEACHERS EDITION. 255 9. A reservoir whose length and breadth are 15 yds. and 12 yds., respectively, holds 330 cu. yds. of water. What is its depth ? 330 15x12 1| yds. Ans. 10. What length must be cut off a beam 9 in. by 15 in. to con- tain 21 cu. ft. ? ^ foff 2fft. 2 ft. ) Ans. 11. How high should a room be made, if its length be 31 ft. 3 in. and breadth 24 ft., in order that it may contain 10,000 cu. ft. of air? 10000 = 13i ft. Ans. 12. A piece of wood 5 ft. long, 1 ft. broad, and 9 in. thick, is cut up into matches 2| in. long and 0. 1 of an inch square. How many will there be if no allowance be made for waste in cutting? 5 ft. = 60 in. ; 1 ft. = 12 in. 60 X 12 X 9 ^ X tV X tV = 259,200. Ans. 13. How long a wall 6 ft. high, 12J in. thick, could be built with the bricks forming a pile 17 ft. 6 in. long, 5 ft. wide, 4 ft. 3 in. high ? 12| in. = IxV ft. 17ix5x4 f = 581 ft. Ans. 14. Find the surface of a cube whose edge is 3 ft. 5| in. 5f in. = i| ft. 3Hx3Hx6=72^^sq.ft. = 72 sq. ft. 48f sq. in. Ans. 15. Find the surface of a rec- tangular block of stone 4 ft. long, 2| ft. broad, 1\ ft. thick. 4 X 2^ X 2 = 20. 2|xlix2 = 6i. 4 X H X 2 = 10. 20 -f- 6^ + 10 = 36^ sq. ft. = 36 8q. ft. 36 sq. in. Ans. 16. A lake whose area is 45 A. is covered with ice 3 in. thick. Find the weight of the ice in tons, if a cubic foot weigh 920 oz. avoirdupois. 1 A. = 43,560 sq. ft. 43560 X45 1960200 sq. ft. 3 in. = I: ft. 4)1960200 490050 cu. ft. X920 450846000 32,000 oz. = 1 t. 1408811 1. Ans. 32)450846 256 ARITHMETIC. 17. How many bricks will be required to build a wall 75 ft. long, 6 ft. high, and 16 in. thick, each brick being 8 in. long, 4 in. wide, 2J in. thick ? 75X6XH^14 4QQ^^ fxixA 18. Find the cost of making a road 110 yds. in length and 18 ft. wide, the soil being first re- moved to the depth of 1 ft. at a cost of 25 cents a cubic yard ; rubble being then laid 8 in. deep, at 25 cents a cubic yard, and gravel placed on top 9 in. thick, at 62J cents a cubic yard. 18 ft. - 6 yds. 110^6 1 $25^ 1 ^1^3^ 100~ 8 in. = f yd. 110^6 2 $25_ 1 ^1^9^100" $55. $36|. 9 in. iyd. 1 "^i^'i^'Tooo-^^^^*- Whole cost = $ 194.79. Ans. 19. A room whose length is 27 ft., breadth 24 ft., height 10 ft., is to have its ceiling raised so as to increase the space by 84 cu, yds. What will then be its height? 27 ft. = 9 yds. 24 ft. = 8 yds. 9x8 '^ 10 + 3| = 13^ ft. Ans. 3nt. 20. A block of wood 5 ft. 4.8 in. long, 1 ft. 9 in. wide and thick, weighs 7.56 cwt. Deter- mine the weight, in pounds, of a cubic foot, 5 ft. 4.8 in. = 5f ft. 7.56 cwt. = 756 lbs. 756 5f xlfxlf = 45^ lbs. Ans. 21. How many cords in a pile of wood 40 ft. long, 4 ft. wide, 5 ft. 4 in. high ? 40X4X5^^3 ^^^^ 8x4x4 * 22. A pile of wood containing 67* cords is 270 ft. long and 4 ft. wide. How high is it? 67^ X 128 270x4 8 ft. Ans. 23. What will be the cost of a pile of wood 25 ft. long, 4 ft. wide, 4 ft. 8 in. high, at $3.75 a cord? 25x4x4?x$3| ^^^3^^^^ 128 TEACHERS EDITION. 257 24. What must be the length of a load of wood 3| ft. high and 5 ft. wide, to contain a cord ? 128 ^1 28x2 5x3i 5x7 7Hft. Ans. 25. How high must manure be in a cart 6 ft. by 4 ft., in order to be J a cord. k>il^ = 22 ft. Ans. 6x4 ^ 26. Find the number of bush- els in a bin that is 8 ft. long, 4 ft. wide, 3 ft. deep. X4x3 96-4 of 96 - 19^ - 76f bu. Ans. 27. Find the number of bush- els in a bin 9 ft. long, 6 ft. 6 in. wide, 3 ft. 4 in. deep. 9 X 6i X 3i = 195. 195-^ of 195=195-39 = 156 bu. Ans. 28. Find the depth of a bin to hold 360 bu., if its length be 12 ft. and its width 6 ft. 360 - i of 360 = 450 cu. ft. 450 12x6 Q\it. Ans. 29. Find the length of a bm that is 6 ft. wide and 5 ft. deep, if it hold 400 bu. 400 + \oi 400 = 500 cu. ft. . ^^ = ^-^ = ^^ = W^itAns. 6x5 30 3 ^ 30. Find the number of bush- els that will fill a bin 8.5 ft. long, 4.5 ft. wide, 3.5 ft. deep. 8.5 X 4.5 X 3.5 = 133.875 cu. ft. 133.875-^ of 133.875 = 107.1 bu. Ans. 31. A bin 20 ft. long, 12 ft wide, and 6 ft. deep, is full of wheat. What is its value, at 11.25 a bushel? 20 X 12 X 6 =1440 cu. ft. 1440-1 of 1440 = 1152 bu. 1152x|li=$1440. Ans. 32. If a ton of coal occupy 40 cu. ft., how many tons will a bin hold that is 21 ft. long, 10 ft. wide, 5 ft. deep ? 21 X 10 X 5 40 26^ t. Ans. 33. If a ton of Lehigh coal occupy 35 cu. ft., how many tons will a bin hold that is 8 ft. long, 5 ft. 9 in. wide, 4 ft. 6 in. deep ? sxjfxii 35 m t. Ans. 258 ARITHMETIC. 34. Find the number of gal- lons that a cistern will hold that is 13 ft. long, 6 ft. wide, 7 ft. 4 in. deep. 13 X 6 X 7J = 572 cu. ft. - 988,416 cu. in. 4278 f gals. Ans. 231)988416 35. Find the number of gal- lons that a tank will hold that is 4 ft. long, 2 ft. 8 in. wide, 1 ft. 8 in. deep. 4x2|xl§ = 17^cu. ft. 1728 XlT^ 23l)30720(l32|f gala. Ans. 36. Find the number of gal- lons in a cubic foot. 7^} gals. Ans. 23l)l728 37. Find the capacity of a cistern, in cubic feet, that will hold 200 barrels of water. 200 x 3H 7i 840 cu. ft. Am. 38. Find the number of gal- lons that a round cistern will hold that is 6 ft. in diameter and 7 ft. deep. Z^ = 9. 3.1416 X 9 X 7 X 7J = 1484.41 gals. Ans. 39. Find the number of gal- lons that a vessel will hold that is 12 in. in diameter and 10 in. 6in. = Ht. (J)» = J. 10 in. = 6 ft. Jxfx3.1416x7J = 4.91 gals. Ans. 40. How many quarts will a round vessel hold 5^ in. in diam- eter and 6 in. deep ? (5^)2 = 26M. 0.7854 X26f t 20.9658 X6 126.7948 cu. in. 30 qts. = 1 cu. ft. •• 1 qt. = jV of 1728 cu. in. = 57.6 cu. in. 2.18 qts. Ans. 576)1257.95 41. Find the number of cubic inches in a sphere 11 in. in diam- eter. 11 X 11 X 11 X 0.5236 — 696.9 cu. in. Ans. TEACHERS EDITION. 259 42. How many quarts will a sphere hold that is 12 in. in diam- eter ? 13=1. 0.5236 X 1 - 0.5236 cu. ft. 30 qts. = 1 cu. ft. 0.5236 X 30 = 15.708 qts. Ans. 43. What part of a bushel will a hemispherical bowl hold that is 13 in. in diameter ? 13^ = 13x13x13 = 2197. 0.5236 X2197 2 )1150.3492 cu. in. 575.1746 cu. in. 1 bu. = 2150.42 cu. in. 0.267. Ans. 215042)57517.460 44. If a cubical box 2 ft. on an edge contain a solid sphere 2 ft. in diameter, how many gal- lons of water can be poured into the box ? 2x2x2 = 8. 8x0.5236 = 4.1888. 8.0000 4.1888 3.8112 cu. ft. X1728 6585.5536 cu. in. 28.51 gals. Ans. 231)6585.55 45. If 64 qts. of water be poured into a vessel that will hold 2 bu. of wheat, what part of the vessel will be filled ? 2 bu. = 64 dry qts. 64 dry qts. = 64 X 67^ cu. in. 64 liquid qts. = 64 X 57f cu. in. 6i>» = 4.89ha. Ans. 6. Reduce 10 cords to sters. 10 X 3.624«t = 36.24^'. Ans. 7. Reduce 4 cwt. 24 lbs. to kilograms. 4 cwt. 24 lbs. = 424 lbs. 424 X 0.454'^K = 192.496kg. Ans. 8. Reduce 25 bu. 2 pks. to hektoliters. 25 bu. 2 pks. = 816 qts. 816 X 1.101 qts. = 898.416> = S.dSm^K Ans. 10. Reduce to the common system 3ha. 2.471 A. X3 7.413 A. X160 66.08 sq. rds. X30^ 2sq. yds nearly. 7 A. 66 ) sq. rds. 2 sq yds nearly. Ans. 11. Reduce to the common system 12.125ci>m. 1.308 cu. yds. X121 15.8295 cu. yds. X27 23.3965 = 23.4 cu. ft. 15 cu. yds. 23.4 cu. ft. Ans. 9. Reduce to the common sys- tem IS""*. 0.621 Xl5 12. Reduce to the common system 101.251. 1.0567 liquid qts. xioii 9.315 mi. X320 106.9908 = 107 liquid qts. nearly. Ans. 100.8 rds. 0.908 dry qts. X5i xioii 4 yds. nearly. 91.935 9 mi. 100 rds. 4 yds. nearly. Ans. = 92 dry qts. nearly. Ans. 264 ARITHMETIC. 13. Reduce to the common system 20. 25". 20.25" = 20251 1.0567 liquid qts. X2025 4 )2139.8175 qts. 535 gals, nearly. Am. 0.908 dry qts. X2025 32 )1838.7 57 bu. nearly. Ans. 14. Reduce to the common system (troy weight) 5^«. 5kg = 50008. 15.432 grs. X5000 24 77160 grs. 20 3215 dwt. 12 160 oz. 15 dwt. 13 lbs. 4 oz. 13 lbs. 4 oz. 15 dwt. Ans. 15. Reduce to the common system 24»». 0.276° X24 6.624 Xl28 79.872 =- 80 cu. ft. nearly. 6 c. 80 cu. ft. nearly. Ana. 16. Reduce to the common system 62.5i". 1.196 sq. yds. X62| 74.75 sq. yds. Ana. 17. Reduce to the common sys- tem (avoirdupois weight) 1001^8. 2.205 lbs. XlOOl 2207.205 lbs. 100 20 2207 lbs. 22 cwt. 7 lbs. 1 t. 2 cwt. 1 t. 2 cwt. 7 lbs. nearly. Ana. 18. Find in acres, etc., the area of a field if its length be 100°» and breadth 75°'. 100 X 75 = 75001"'. 1.196 sq. yds. X7500 160 8970 sq. yds. 296 sq.rd8.168q.yds. 1 A. 136 sq. rds. 1 A. 136 sq. rds. 16 sq. yds. Ana. 19. Determine the number of cubic meters in a box 2 yds. long, 3 ft. wide, 2^ ft. deep. 2 yds. = 6 ft. 6 X 3 X 2J = 45 cu. ft. = 1 J cu. yds. If X 0.765«»>'» = 1.275«'»>'». Am. TEACHERS EDITION. 265 20. Determine the number of 0.9461 cubic yards in a box 2"" long, Xl7 75«" wide, 50"=* deep. 16.0821 2 X 0.75 X 0.50 = 0.75«^'». 0.75 X 1.308 cu. yds. 16.082»^« = 0.981 cu. yds. Ans. » X 2.205 lbs. 35.461 lbs. 21. If a man walk 75™ 'a min- X 1.841 ute, what is his rate in miles per 65.284 lbs. hour? 75 X 60 = 4500'" = 4.5'^'^. 65.284 4.5 X 0.621 mi. = 2.795 mi. Ans. X$0.02i $1.47. Ans. 22. If cast-iron weigh 7.113« per cubic centimeter, how many pounds does a cubic foot weigh ? As the weight of the iron is 7.113^ per cubic centimeter, the specific gravity is 7.113. 7.113 X 62.5 lbs. = 444.5625 lbs. Ans. 23. How many steps 2 ft. 6 in. long will a man take in walking a kilometer ? 0.621 mi. X5280 32788.8 ft. 1311.5 25)32788.8 1311.5 steps = 1312 steps. Ans. 24. Find the value of a car- boy (17 qts.) of sulphuric acid, of 1.841 specific gravity, at 2^ cents a pound. 25. Find the value of a car- boy (17|i) of nitric acid, of 1.451 specific gravity, at 15 cents a pound. 2.205 lbs. 38.588 lbs. X 1.451 55.971 lbs. X 10.15 $8.40. Ans. 26. Find the weight in pounds and in kilograms of 31 1 gals, of the best alcohol, specific gravity 0.792. 31 1 gals. - 124f qts. 0.9461 Xl24 f 117.9351 of water. 266 ARITHMETIC. X 0.792 93.405"!?. (1) Ans. 93.405 2.205 lbs. 205.958 lbs. (2) Ans. 27. If the specific gravity of Bea- water be 1.026, and that of olive-oil be 0.915, what will be the weight of a hektoliter of each in pounds and in kilograms? iw = 1001 = lOO^K. 1.026 X 100^8 102.6''8. (1) Ans. X 2.205 lbs 226.233 lbs. (2) Ans. 0.915 X lOO^^K 91.5''8. (3) Ans. X 2.205 lbs. 201.76 lbs. (4) Ans. 28. Find the weight in pounds and in kilograms of the air, spe- cific gravity 0.00129206, in a room 7™ by 5", and 3.5™ high, 7 X 5 X 3J = 122i'"»». 122i«">» of water = 122,500k«. 0.00129206 X 122500''« 158.277JJ^«. (1) Ans. 158.277 X 2.205 lbs. 349 lbs. (2) Ans. 29. Find the weight in pounds and in kilograms of the air, spe- cific gravity 0.00129206, in a room 23 ft. long, 16 ft. wide, and 10 ft. high. 23 X 16 X 10 = 3680 cu. ft. 3680 X 62J lbs. 230000 lbs. X 0.00129206 297.1738 lbs. (1) Ans. 297.1738 X 0.454kK 134.9169''8. (2) Ans. 30. If a balloon weigh 2^«, and contain 10,000* of hydrogen gas, specific gravity 0.00008929, what is its lifting force in kilo- grams and in pounds when the air has a specific gravity of 0.00129206? 0.00129206 0.00008929 0.00120277 XlOOOO** 12.0277^ 2. 10.0277''«. (I) Am. TEACHERS EDITION. 267 10.0277 X 2.205 lbs. 22.111 lbs. (2) Ans. 31. If a pile of wood be 1.2™ wide, 7"* long, and 2™ high, how much is it worth, at $4.50 a cord? 1.2 X 7 X 2 = 16.8«t. 16.8 X 0.276° = 4.6368<'. 4.6368 x|4^ = $20.87. Ans. 32. How many miles will be travelled in 1 hr. 28 min. 21 !., at the rate of 50'^°' an hour? sec 1 hr. 28 min. 21 sec. = 1.4725 hr. 50^^™ = 50 X 0.621 mi. = 31.05 mi. 1.4725 X 31.05 mi. = 45.721 mi. Ans. 33. Find the time of travel- ling 31 mi. 180 yds. at 1 min. 25 sec. per kilometer. V^ = 0.621 mi. 1 min. 25 sec. = 1y\ min. 31 mi. 180 yds. = 31.1023 mi. 31102.3 ^ 621 = 50.08 50.08 X 1^^ min. = 70.95 min. = 1 hr. 11 min., nearly. Ans. 34. What is the weight of 12 cu. yds. 16 cu. ft. 720 cu. in. of earth of which a cubic meter weighs 1 1. 17 cwt. ? 12 cu. yds. 16 cu. ft. 720 cu. in. = 12.608 cu. yds. icbm _ 1,308 cu. yds. 12608 ^ 1308 = 9.639. 1 1. 17 cwt. = 37 cwt. 9.639 X 37 cwt. = 356.643 cwt. = 17 t, 16 cwt. 64 lbs. Ans. 35. Find the weight in grams of a liter of mercury, of which a cubic inch weighs 0.4925 of a pound avoirdupois, 11 = 61.03 cu. in. 0.4925 lbs. X 61.03 = 30.057 lbs. 30.057 X 453.598 = 13633.55«. 36. How many yards of cloth, at $3.12^^ a meter, should be given in exchange for 15*" at $ 2.75 a yard ? 1 yd. = 0.914°'. 0.914 X $ 3^ = $ 2.856 per yd. 15 X 1.0936 yds. = 16.404 yds. 16.404 x$2|- = $45,111. 45111 -J- 2856 = 15.79 yds. Ans. 37. If a wine merchant buy 3" of wine for 1600 francs, at what rate. United States money, does he pay a gallon, reckoning 25 francs equal to $4.85? 3" = 3001. 1600 francs -^ 300 = 5i francs. $4,850 ^25 = $0,194. $0,194x5^ = $1,035 perl. 11 = 1.0567 qts. = 0.264 gals. 1035 ^264 = $3.92. Ans. 268 ARITHMETIC. 38. A mill-wheel is turned by a stream of water running at the rate of a yard per second in a channel 5 ft. wide and 9 in. deep. Determine the weight of water in metric tons, supplied in 12 hrs., if a cubic foot of water weigh 1000 oz. 3x5x|=lUcu.ft. 12 hrs. = 43,200 sec. 43200 X Hi cu. ft. = 486000 cu. ft. 486000 X 1000 oz. = 486000000 oz. = 30375000 lbs. 0.00045359 m. t. x 30375000 = 13777.796 m. t. Ans. Exercise LXII. 1. Which is the greater ratio, 6:8 or 6:9? 5:8 = f = M. .-. 6:9 is greater. 2. Which is the greater ratio, 7: 10 or 9: 12? 7:10 = ^ = M. 9:12 = ^3. = | = M. .'. 9 : 12 is greater. 3. Which is the greater ratio, 8: 9 or 10: 12? 8:9 = f =if. 10:12 = M = f = if. .'. 8 : 9 is greater. 4. Which is the greater ratio, 6: 12 or 8: 14? 6:12 = t% = ^ = /t. 8:14 = ^. .". 8 : 14 is greater. 6. Which is the greater ratio, 10 cwt. : 15 cwt. or $ 7 : |9? 10 cwt. 10 cwt. : 15 cwt. = 15 cwt. I = f- 17 |7:$9 = p = i. .-. $7 :|9 is greater. 6. Which is the greater ratio, 5 dys. : 7 dys. or 8 ft. : 11 ft.? 5dy8.:7dys.=|iZ!:»^ = |^. 8 ft. : 11 ft. = AiL ^^=.^. .-. 8 ft. : 11 ft. is greater. 11 It. teachers' edition. 269 7. Which 9 yds. is the greater ratio, 9 yds . : 6 yds. or 5 : 3? 5:3 = ■i = ¥- .5:3 iE ! greater. 8. Which fib.: is the greater ratio, 1 lb.: i lb. or fyd. :fyd.? f yd. : I yd. = f^ = |. .-. f yd. : | yd. is greater. Exercise LXIII. 1. If 24 men can finish some work in 14 days, how long will it take 21 men to do it ? 21:24::14dys. :what? 8 2 2^^^ dys. = 16 dys. Ans. 2. A well is dug in 13 days of 9 hours each. How many days of 10 hours each would it have taken ? 10:9:: ISdys. : what? L><13 dys. = 112 = IIX dys. Ans. 10 ^ 10 ^^ ^ 3. A man who steps 2 ft. 5 in. takes 2480 steps in walking a cer- tain distance. How many steps of 2 ft. 7 in. will be required for the same distance ? 2 ft. 5 in. = 29 in. 2 ft. 7 in. - 31 in. 31:29: : 2480: what? 80 ^^^^^^^ = 2320. 2320 steps. Ans. pX 270 ARITHMETIC. 4. If T^j ton cost $6, what will 7f cwt. cost, at the same rate? .7fcwt. = ||t. = Ht. ^:H::|6:what? ^ ^^X5 75 ^ 15 5. If 42 yds. of carpet 2 ft. 3 in. wide are required for a room, how many yards 2 ft. 4 in. wide will be required ? 2 ft. 3 in. = 27 in. 2 ft. 4 in. = 28 in. 28:27::42yds. : what? 3 2ZAl^ = |yd8. = 40iyds. ^n.. 6. A court was paved with 950 stones, each If sq. ft., and is re- paved with 836 stones of a uniform size. Find the size of each. 836:950: : If sq. ft. : what? 25 950X11 ^WxiI=2,Vsq. ft. ^n.. 836 mxQ ^ ^ 2 7. If a train, at the rate of ^j of a mile per minute, take 3^ hra. to reach a station, how long will it take at the rate of f^ of a mile a minute ? 1^5 :t«V: -SI lira: what? ^ 7 ;3 4 28 ^^ 8. If a post 4 ft. 8 in. high cast a shadow 7 ft. 3 in. long, how long a shadow will a post 11 ft. high cast? 4 ft. 8 in. = 4f ft. 7 ft. 3 in. = 7J ft. 4J:ll::7ift. :what? 11 X71^ _ 3x 11x29 ,^.f. ikV, ,,. , ~~W~ 14x1 ^ " ^* teachers' edition. 271 9. When a shadow 8 ft. 5 in. long i? cast by a post 5 ft. 7 in. high, how high is a steeple that casts a shadow of 211 ft. at the same time ? 8^^:211::5TVft. :what? ^^^ ^ ^tV = ;;^X 211x67 _ -^39 9 8 ^^ _ J39 ^^ ^.^5 j^. 8t\ 101 x;^ '^' ^"^ 10. If 4 men can mow a certain field in 10 hrs., how many men will it take to mow it in 5 hrs. ? 2 5 : 10 : : 4 men : what? ^ men = 8 men. Ans. 11. A tap discharging 4 gals, a minute empties a cistern in 3 hrs. How long will it take a tap discharging 7 gals, a minute to empty it ? 7 : 4 : : 3 hrs. : what ? ^-^ hrs. = If hrs. Ans. 7 ^ 12. A pipe discharging 3 gals. 1 pt. a minute fills a tub in 4 min. 20 sec. How long will it take a pipe discharging 83 qts. a minute to fill it? 3 gals. 1 pt. = 25 pts. 83 qts. = 166 pts. 4 min. 20 sec. = 260 sec. 130 166 : 25 : : 260 sec. : what ? ^^ ^ ^^^ = 39f | sec. Ans. 83 13. If both pipes of Ex. 12 discharge at the same time into the tub, how long will it take to fill it? 4-J- min. = 260 sec. 95^ : 12| : : 260 sec. : what ? 12^X260 ^ % 25 260 95J 191 ^ 1 14. How long will it take to fill a cistern of 165 gals, by a pipe that fills one of 120 gals, in 7 min. 16 sec. ? 16 sec. = j% min. 120: 165: : 7/^ min.: what? 11 165x^ ^ WXJ09 ^ „ ^ ^ 9 ^.^ 59^ ^^^ ^,^^_ 120 120 x;^ 272 ARITHMETIC. 15. A ship has sailed 1800 mi, m a fortnight. How long, at the same rate, will it take for a voyage of 5000 mi. ? 1800: 5000:: 2 wks.: what? 25 x2 = 50_5^^^g^ xm 9 9 5f wks. = 5 wks. 4 dys. nearly. Ans. 16. The wheels of a carriage are 6 ft. 9 in. and 9 ft. 6 in. in cir- cumference. How many times will the larger turn while the smaller turns 3762 times ? 6 ft. 9 in. = 6f ft. 9 ft. 6 in. = 9^ ft. 9J:6|:: 3762: what? 99 6|_><3762 ^ ^X27xm^ ^ 2673. Ans. 17. If ^\ of a ship be worth $2167, what is the value of -^^ of it? ^^Vit^t:: $2167: what? ^ X $2167 ^ 25 X 7 X 2167 ^ $379225 ^ ^^^3^ ^g ^^ 18. What will be the weight of 18 cu. ft. 432 cu. in. of stone of which 10 cu. ft. 864 cu. in. weigh 14 cwt. 7 lbs. ? 10J:18J:: 1407 lbs.: what? 67 18^Xl407^;Zx73x.I^^7^4891^^^^ ^^^^ lOi %lx^ 2 ^ 2 = 1 t. 4 cwt. 45J lbs. Am. 19. If 280 lbs. of flour make 360 lbs. of bread, how many four pound loaves can be made from 1 cwt. of flour ? 280 : 100 :: 360 lbs. : what? 9 ^^^]^^ = ^ - 128f lbs. 128i+ 4 = 321. Am. teachers' edition. 273 20. If a column of mercury 27.93 in. high weigh 0.76 of a pound, what will be the weight of a column of the same diameter 29.4 in. high? 27.93 : 29.4 : : 0.76 lb. : what ? ^i^ lb. =0.8 lb. ^m. 0.19 21. How many francs will pay a bill of £100, when £42 10 s. 8d is equivalent to 1090.98 francs ? £42 10s. 8^. = £42xV 42y8^ : 100 : : 1090.98 francs : what? 1090^8X100 ,^^^^^ ^]^^l]m^m = 2565 franc. 42tV m m 1 22- What will be the weight of a cube whose edge is 2 ft. 2 in., when a cube of the same material whose edge is 1 ft. 4 in. weighs 537.6 lbs. ? 2 ft. 2 in. = 2^ ft. 1 ft. 4 in. = li ft. (H)' : {2i)^ - 537.6 lbs. : what? It : -¥tV - 537.6 lbs. : what? 21 m P 2m^m^mi lbs. = 2306.85 lbs. Ans. ^X^ 10 20 2 8 23. If a square field measuring 50 yds. lOf in. on each side be worth |2710}f, what is the value of a square field 62 yds. 1 ft. each way ? 50 yds. lOf in. == 50f yds. 52 yds. 1 ft. = 62^ yds. (50f )2 : (62i)2 . . 1 2710if : what ? 123904 34969 .. |46080 . j^^^.^ 49 9 '''' 17 -^ ^ 17 5 _49_ ^3»Ex«-^ = 14165. ^ns. m 274 ARITHMETIC. 24. A gains 4 yds. on B in running 30 yds. How much will he gain while B is running 97^ yds. ? 30:971: : 4 yds. : what? 13 ;2 ^^^X^ yds. = 13 yds. Ans. 25. If 10 cu. in. of gold weigh as much as 193 cu. in. of water, what is the size of a nugget weighing as much as a cubic foot of water? 193 : 1728 : : 10 cu. in. : what? 1728 X 10 17280 193 193 89||f cu. in. Ans. 26. If a garrison of 1500 men have provisions for 13 months, how long will the provisions last if it be increased by 700 men ? 1500 + 700 = 2200. 2200: 1500: : 13 mo. : what ? 15 IM^<2^ mo. = W mo. = 8M mo. Ans. 22 27. If a tree 38 ft. high be represented by a drawing \\ in. high, what, on the same scale, will represent the height of a house 45 ft. high? 38:45::l^in. :what? ||^in. = Win. = lf|in. ^m. 28. If a country 630 mi. long be represented on a raised map by a length of 5J ft., by what height ought a mountain of 15,750 ft. be represented on the map ? 630 mi. = 3,326,400 ft. h\ ft. = 66 in. 3326400 : 15750 : : 66 in. : what? 5 mm 16 teachers' edition. 275 29. A train travels ^ of a mile in 18 sec. How many miles an hour does it travel ? 1 hr. = 3600 sec. 18 : 3600 : : ^ mi. : what ? 200 M2^^ = 50mi.^ns. 30. If 4J tons of coal fill a bin 9 ft. long, 5 ft. broad, 5 ft. high, how many cubic feet will be required for the coal of a steamer carry- ing 3 weeks' consumption at 20 tons a day ? 9 X 5 X 5 = 225 cu. ft. 3 wks. = 21 dys. 21 X 20 t. = 420 t. 4^ : 420 : : 225 cu. ft. : what ? 25 420XMX_2 eu ft. = 21.000 cu. ft. ^ns. 31. If 2 lbs. of rosin be melted with 5 oz. of mutton tallow, to make a grafting* wax, how many ounces of tallow will 20 oz. of the wax contain ? 2 lbs. + 5 oz. = 2 lbs. 5 oz. = 37 oz. 37 : 20 : : 5 oz. : what ? 20x5 100 o2fi A — ^^^- = oz. = 2 If oz. Ans. 37 37 '^ Exercise LXIV. 1. How many days 8 hours long will 60 men take to finish some work which 24 men can do in 15 days, working 10 hours a day ? ^1 ^^:: 15 dys. -.what? 60 I 24 ^ 5 3 ^ X ^P 2 276 ARITHMETIC. 2. What will be the expense of covering a roorn with drugget 4 ft. wide, at 91 1 cts. a yard, when carpet 2 ft. 3 in. wide for the room costs $ 70.50. at $ 1 .37^ a yard ? $0.91| = |H- $1.37i = $lf. 4 2i If H Lllxlilxlxi = ^^ = $26.44.^n,. $70}: what? n 4 i 11 16 3. If 4418 tons of iron ore produce $36,190 worth of metal, when iron is at $37.50 a ton, what will be the value of the iron from 2275 tons of ore, at $47 a ton ? 37} I 47 4418 I 2275 $36,190: what? 91 335 2 X ^T X m^ X mX9^ ^ $70070 ^ ^23.356.67 Ans n X im 3 3 H 4. If a bar of iron 3^ ft. long. 3 in. wide, 2| in. thick weigh 93 lbs., what will be the weight of a bar 3f ft. long, 4 in. wide, and 2^ in. thick ? 93 1bs. : what? 31 3 3f 4: 2| 2i 31 xfx^x|-X:^ = 124 lbs. Atu. ^ I 1)3 Jl 6. If 40 bu. of wheat can be grown on the same area as 48 bu. of barley, and 28 acres produce 840 bu. of wheat, how much barley will be obtained from 38 acres? 40 840bu. : what? 6 6 30 ^><^8iiM!»i368bu. TEACHERS EDITION. 277 6. If 18 men can dig a trench 150 ft. long, 6 ft. broad, and 4 ft. 6 in. deep in 12 days, how long will 16 men take for a trench 210 ft. long, 5 ft. broad, and 4 ft. deep ? 12dys. : what? i 3^ 16 18 150 210 6 5 4i 4 ^ 7 7. In the reprint of a book consisting of 810 pages, 50 lines are contained in a page, instead of 40, and 72 letters in a line, instead of 60. Of how many pages will the new edition consist ? 810: what? 10 6 J ^0 X ^0 X ^ mxn = 540. Ans. 8. If 3280 42-lb. shot cost $3000, how many 32-lb. shot can be bought for $4200? 3000 I 4200 32 42 3280: what? 7 21 ^m xnx ?>m X n 41 6027. 9. "What must be the rate of wages, that 12 men may earn in 10 days the same amount that 9 men earn in 14 days, at $ 1.50 a day ? 12 19 10 I 14 : $1.50: what? 0.05 7 O.I^ Qx^^xW^ = $M5^cpi.575. Ans. 1%XX^ 2 2 278 ARITHMETIC. 10. A reservoir 15 yds. long and 4 ft. deep holds 32,500 gals. Determine the quantity of water it will hold when it has been increased in length by 18 ft. and in depth 1 ft. 15 21 4 7 8125 :: 32,500 gals. : what? ;^x^ gals. = 56,875 gals. Ans. 11. How far can A, who takes 3.1 ft. each step, run, while B, who takes 2.3 ft. each step, runs 220 yds., if A takes 7 steps while Stakes 11? 220yd8. : what' 2.3 ^•^••22( 11 7 '-^^^ 20 3.1 X 7 X m 2 .3x;; yds. 434 2.3 yds. = 188^1 yds. Am. 12. If 6 hours be required for travelling a given distance at a given rate, how long will be required when the distance is diminished by one-fourth and the rate is increased by one-half? 6 hrs. : what ? X ^ hrs. 3 hrs. Ans. 13. IIow many hours a day must 5 men work to mow the same quantity of grass in 8 days that 7 men can mow in 6 days, working 10 hours a day ? 3 g 10 hrs. : what ? '^^Xfx^^ hrs. = ^ hre. - lOi hrs. ?X? i 2 6 14. If a bar 10 ft. 6} in. long, 3f in. broad, 3J in. thick weigh 4 cwt. 8.23 lbs., what length must be taken to weigh a long ton when the breadth and thickness are 4f in. and 4J in. respectively? TEACHERS EDITION. 279 1 1. t. = - 2240 lbs. 40823 224000 4f 3f :: 7000 xm0 4 cwt. 8.23 lbs. = 408.23 lOMft. : what? 23 40823 X ;t^ X 33 X ^ X ^ X ^^ ^^^^^ • 2 XX 3 38 ft. 1.5 in. 15. If 27 men, in 28 days of 10 hours each, dig a trench 126 yds. long, 2^ yds. broad, 1} yds. deep, how long a trench 2| yds. broad, If yds. deep, will 56 men dig in 25 days of S^ hours ? 27 56 10 28 8i 25: :126 yds.: what? 2| If 2J 1^ 6 % % ^ n ^ X ^ X ^^ X 33 X 25 X ^ X 3 X ; ^^ „^ „ XI x7x^Jx;px^x^^x^x;2 '^^' 150 yds. An%. 16. What must be the length of a bar of silver f in. square, that it may weigh the same as a bar of gold \ in. square and 6| in. lonpr, if the weight of a cubic inch of silver have to that of a cubic inch of gold the ratio 47 : 88 ? (f )^ i (^)' : : 6| in, : what ' 47 88 ^ 47 what? 4 22 3 i^L> 3 X 1.84 = 5.52 9.972 ') 12.52' 1.256. Am. 8. If 4' of water and 1' of sul- phuric acid, specific gravity, 1.842, when mixed shrink \ of 1%, what is the specific gravity of the mixture ? 4x1' = 4.000> 1x1.842= 1.842 4.9875 ') 5.842» 1.171* 4 + 1 = 5. 5-0.01^ = 4.9875. An$. teachers' edition. 365 9. In what proportions must tin of specific gravity 7.29 and lead of specific gravity 11.35 be mixed to make a solder of specific gravity 10.21, if no allowance be made for expansion or condensation ? (Give the proportions in bulk.) The specific gravity of tin lacks 2.92 of the required specific gravity ; and the specific gravity of lead is 1.14 above the required specific gravity. Therefore, the tin is to the lead in the inverse ratio of 292 to 114. That is. Tin : lead = 114 : 292 = 57 : 146. Ans. 10. In what proportions must oils worth $1.25 and 80 cents a gallon be mixed to make a mixture worth $1 a gallon? (Test the answer.) The cost of the better oil is $0.25 above the required cost; and the cost of the worse oil lacks $ 0.20 of the required cost. Therefore, the better is to the worse in the inverse ratio of $0.25 to $0.20. That is, The better : the worse = 20 : 25 = 4 : 5. Ans. 11. In what proportion may oils worth $1.20, 80 cents, and 60 cents a gallon be mixed so that the mixture shall be worth 70 cents a gallon ? When the 80-cent oil alone is taken, in what ratio to the 60-cent must it be used? When the $1.20 oil alone is taken, in what ratio to the 60-cent oil must it be used? When the $1.20 and 80-cent oils are mixed gallon for gallon, how much 60-cent oil must be added ? When 1 gal. of the $1.20 oil and 3 of the 80-cent oil are taken, how much 60-cent oil must be added ? If three-fourths of the mixture consist of the 60-cent oil, what per cent of each of the other two must be taken ? (1) The cost of the $1.20 is $0.50 above the required cost; the cost of the $0.80 oil is $0.10 above the required cost; and the cost of the $0.60 oil lacks $0.10 of the required cost. Therefore, the $1.20 oil is to the $0.80 oil is to the $0.60 oil in the inverse ratio of 50 to 10 to 10. That is. The $ 1.20 oil : the $0.80 oil . the $0.60 oil = 10 : 10 : 50 = 1 : 1 :^ L 366 AEITHMETIC. (2) The cost of the $0.80 oil is $0.10 above the required cost; the cost of the $0.60 oil lacks $0.10 of the required cost. Therefore, the $0.80 oil is to the $0.60 oil in the inverse ratio of 10 to 10. That is. The $0.80 oil : the $0.60 oil = 10 : 10 = 1 : 1. Am. (3) The cost of the $1.20 oil is $0.50 above the required cost; and the cost of the $0.60 oil lacks $0.10 of the required cost. Therefore, the $1.20 oil is to the $0.60 oil in the inverse ratio of 50 to 10. That is, The $ 1.20 oil : the $0.60 oil = 10 : 50 = 1 : 5. Ans. (4) When 1 gal. of the $1.20 oil is mixed with 1 gal. of the $0.80 oil, the mixture costs $l'20 + $0.80 _ ^j qq p^^ ^^-^^^^ A The cost of the $1.00 oil is $0.30 above the required cost; the cost of the $0.60 oil lacks $0.10 of the required cost. Therefore, the $1.00 oil is to the $0.60 oil in the inverse ratio of 30 : 10. That is, The $ 1.00 oil is to the $0.60 oil = 10 : 30 = 1 : 3. 1:3: : 2 gals. : what? 1 : 3 : : 2 gals. : 6 gals. Ans. (5) When 1 gal. of the $1.20 oil is mixed with 3 gals, of the $0.80 oil, the mixture costs 1^-20 + 3 x $0.80 _ ^q^ ^^^ ^^ The cost of the $0.90 oil is $0.20 above the required cost; the cost of the $0.60 oil lacks $0.10 of the required cost. Therefore, the $0.90 oil is to the $0.60 oil in the inverse ratio of 20 : 10. That is. The $0.90 oil 1 : 2 : : 4 gals. 1 : 2 : : 4 eals. the $0.60 oil = 10: 20= 1:2. what? 8 gals. Ans. (6) Each of the others will be "" ^ = J of the whole, or 12J %. Ans. TEACHERS EDITION. 367 12. A solder composed of tin and lead, specific gravities 7.29 and 11.35, has a specific gravity of 10.44. What is the weight of each metal in a kilogram of solder ? The specific gravity of the tin lacks 3.15 of the required specific gravity ; and the specific gravity of the lead is 0.91 above the required specific gravity. Therefore, the tin is to the lead in the inverse ratio of 3.15 to 0.91. That is, The tin : the lead = 91 : 315. 91 + 315 = 406. 5%V of 10008 = 224/^« tin. fif of 1000« = 775|f« lead. Ans. 13. Find the equated time for the payment of |250 due in 3 mos., $400 due in 6 mos., $700 due in 8 mos. $250x0 = $400x3 = 11200 $700x5 = $3500 $1350 )$4700 3if 3|f mos. = 3 mos. 14 dys. 3 mos. 14 dys. after 3 mos. mos. 14 dys. 14. Find the equated time for the payment of $300 due in 30 dys., $500 due in 60 dys., and $ 200 due in 90 dys. $300x00 = $500x30 = $15,000 $200x60 = $12,000 $1000 .27,000 27 Hence, the equated time is 27 dys. after 30 dys. = 57 dys. 15. Find the equated time for the payment of $325 due now, $200 due in 30 dys., $460 due in 60 dys., and $ 150 due in 90 dys. $325x00 = $200x30 = $ 6,000 $460x60 = $27,600 $ 150 X 90 = $ 13,500 $1135 )$47,100 41 Hence, the equated time is 41 dys. 16. Find the equated time for the payment of $240 due May 10, $420 due July 2, $310 due Sept. 14, and $600 due Oct. 1. $240 X 00 = $420x 53 = $22,260 $310x127 = $39,370 $600x144 = $86,400 $1570 )$ 148.030 94 Hence, the equated time is 94 dys. after May 10 = Aug. 12. 368 ARITHMETIC. 17. Find the equated time for the payment of |275 due June 21, $175 due July 16, $200 due Aug. 6, and $150 due Sept. 3. $275x00 = $175x25 = $ 4,375 $200x46 = $ 9,200 $150x74 = $11,100 $800 )$ 24.675 31 Hence, the equated time is 31 dys. after June 21 = July 22, 18. Find the equated time for the payment of $ 112.30 due July 6, $115.25 due July 30, $282.15 due Sept. 4, and $102.36 due Oct. 1. $112.30x00- $115.25x24 = $ 2,766.00 $232.15x60 = $13,929.00 $102.36x87 = $ 8,905.32 $562.06 )$ 25.600, 32 46 Hence, the equated time is 46 dys. after July 6 = Aug. 21. 19. A owed B $2000 payable in 4 mos., but at the end of 1 mo. he paid him $500, at the end of 2 mos. $500, and at the end of 3 mos. $500. In how many months is the balance due ? $500x3 = $1500 $500x2 = $1000 $500x1=$ 500 $1500 $3000 Therefore, he is entitled to keep the balance ($500) -3^^ mos. = 6 mos. after its maturity. 20. A merchant bought, Feb. 11, 1881, a bill of goods amounting to $1700, on 4 months' credit; but he paid March 22, $400, April 20. $220, May 10, $300. When is the balance due ? 1881 dyi. 11 1881 6 11 = June 11. 1881. $400x81 = $32,400 $220x52 = $11,440 $300 X 32 = $ 9,600 $920 $53,440 Therefore, he is entitled to keep the balance ($780) *?f^* dys 69 dys. after its maturity, June 11, 1881, = Aug. 19, 1881. TEACHERS EDITION. Find the equated time of maturity of each of the following bills, and the amount due at settlement, including interest at 6%: 21. James Pkice, to John Bates, Dr. 1881. Apr. 5. To mdse. on 4 mos. credit ... $ 120.50 Apr. 15. " " 3 " " ... 87.33 May 7. " " 3 " " ... 218.17 May 21. " " 4 " " ... 317.00 1743.00 Paid Oct. 18, 1881. $ 87.33 X 00 = From Aug. 23 to Oct. $120.50x21 = $ 2,530.50 18 is 56 dys. $218.17x23 = $ 5,017.91 56 dys. = O.OOQi $317.00x68 = $21,556.00 $743 $743.00 )$ 29,104.41 ^'^^^^ 39 $6.93 Hence the equated time is 39 dys. after '__ Julyl5, 1881 = Aug. 23,1881. $749.93 22. Hall & Co. bought of Boles & Co. 1881. Feb. 11. To mdse. on 30 dys $250.00 Apr. 20. " " 2 mos 500.00 May 31. " " 3 mos 150.00 July 6. " " 60 dys 1000.00 Paid Nov. 10, 1881. 30 dys. after Feb. 11 = Mar. 13. Hence, the equated 2 mos. after Apr. 20 = June 20. time is 132dys. after Mar. 3 mos. after May 31 = Aug. 31. 13, 1881 = July 23, 1881. 60 dys. after July 6 = Sept. 4. From July 23 to Nov. $ 250 x 00= 10 is 110 dys. $ 500 X 99 = $ 49,500 110 dys. = 0.018i $ 150x171 = $ 25,650 _$1900 $1000x175 = $175,000 $34.83 $ 1900 )$ 250,150 i^QQQQ 132 $1934.83 370 ARITHMETIC. 23. Find the equated time of maturity of each side of the following account : Adams & Co. in account vdth Bacon & Co. Db. Ce. 1881. Jan. 3. To mdse. 90 dys. $250 Apr. 11. By cash, $200 Mar. 7. " 60 " 150 Apr. 30. " 100 May 3. 60 " 325 May 30. (t 125 June 7. " 30 " 175 July 2. " 400 90 dys. after Jan. 3 = April 3. 60 dys. after Mar. 7 = May 6. 60 dys. after May 3 = July 2. 30 dys. after June 7 = July 7. De. $250x00 = $150x33 = $ 4,950 $325x90 = $29,250 $175x95 = $16,625 $900 )$ 50,825 56 Hence, the equated time is 56 days after April 3 -- May 29. Ce $200 X $100 X $125 X $400 X 00 = 19^ 49 = 82 = = $ 1,900 = $ 6,125 = $32,800 $825 )$40,825 49 Hence, the equated time is 49 days after April 11 = May 30. Find the time for paying the balance in the following equated bills Average due. Dr. 24. May 17, 1881 . . . $950 25. Apr. 12, 1881 . . . $950 26. May 30, 1881 . . . $1000 27. July 6, 1881 . . . $500 Average due. Apr. 12, 1881 . Ce. $1000 May 17. 1881 .... $1000 June 23, 1881 .... $920 Apr. 14, 1881 .... $480 teachers' edition. 371 24. Differences in equated time = 35 dys. Balance of account = 1 1000 - $ 950 = 1 50. If the account were settled at the later date, May 17, the $1000 on the Cr. side would have been on interest 35 dys., and this is equivalent to having the balance, 1 50, on interest -f ^- of 35 dys. = 700 dys. Hence, the balance should begin to draw interest 700 dys. before May 17, 1881 ; that is, June 17, 1879. 25. The difference in equated time = 35 dys. Balance of account = 1 1000 - $ 950 = 1 50. If the account were settled at the later date. May 17, the $950 would have been on interest 35 dys., which is equivalent to having the balance, 1 50, on interest ^-^^^- of 35 dys. = 665 dys. Hence, to increase the Cr. side by an equal amount of interest, the balance should remain unpaid 665 dys. ; that is, the bal- ance is due March 13, 1883. 26. The difference in equated time == 24 dys. Balance of account = 1 1000 - |920 = $80. If the account were settled at the later date, June 23, the $ 1000 on the Dr. side would have been on interest 24 dys., and this is equivalent to having the balance, $80, on interest ^f^^ of 24 dys. = 300 dys. Hence, the balance should begin to draw interest 300 dys. be- fore June 23, 1881 ; that is, Aug. 27, 1880. 27. The difference in equated time is 83 dys. Balance of account = $ 20. If the account were settled at the later date, July 6, the $480 would have been on interest 83 dys., which is equivalent to having the balance, $20, on interest -W- of 83 dys. = 1992 dys. Hence, to increase the Dr. side by an equal amount of interest, the balance should remairi unpaid 1992 dys. ; that is, the bal- ance is due Dec. 19, 1886. 372 ARITHMETIC. Find (by either method) the cash balance in the following bills, reckoning interest at 6%: Qft 1881. Dr. Int. 1881. Cr. Int. Apr. 5. To mdse. $250 $3.17 Apr. 20. Bycafih, $200 $2.03 " 27. " 610 5.49 " 30. " 500 4.25 June 1. " 200 0.63 June 4. " 400 1.07 " 20. Tobal.acc. 40 " 20. Bybal. int. 1.94 $1100 $9.29 $1100 $9.29 Hence, cash balance = $40 - $1.94 = $38.06. Ans. 29. 1881. Dr. Int. 1881. Cr. Int. Apr. 15. To mdse. $250.00 $7.42 Apr. 26. By cash, $150.00 $4.18 May 25. " 98.50 2.27 May 17. " 150.00 3.65 June 7. " 300.00 6.25 July 7. •' 200.00 Oct. 10. Bybal. ace. 148.50 3.17 •' 10. " int. 4.94 $648.50 $15.94 $648.50 $15.94 Hence, cash balance = $148.50 + $4.94 = $153.44. Ans. 30. 1881. Dr. Int. 1881. Cr. Int. Feb. 2. To mdse. $100 $3.02 Feb. 25. By cash, $100 $2.63 Apr. 7. " 200 3.90 Mar. 22. " 150 3.33 June 2. •• 95 0.97 June 20. " 200 1.43 " 9. 150 1.35 Aug. 2. Bybal. ace. 95 " 2. " int. 1.85 $545 $9.24 $545 $9.24 Hence, cash balance = $95 + $1.85 = $96.85. Ans. 31. 1881. Dr. Int. 1881. Cr. Int. Apr. 5. To mdse. $250 $6.21 Apr. 20. By cash, $200 $4.47 •' 27. " 670 14.18 " 30. •• 500 10.33 June 4. " 200 2.97 June 1. •• 400 Sept. 1. By bal. ace. 20 6.13 '• 1. •• int. $1120 2.43 $1120 $23.36 $23.36 Hence, cash balance = .$20 + J2.43 = $22.43. Am. TEACHERS EDITION. 373 1881. Dk. Mar. 10. To mdse. |580 Apr. 20. " 200 May 5. " 150 " 17. " 325 32. Int. $8.31 1.50 0.75 0.98 1881. Ce. Mar. 15. By cash, $500 Apr. 15. " 300 " 25. " 120 May 20. " 225 June 4. By bal. ace. 110 " 4. " int. Int. $6.75 2.50 0.80 0.56 0.93 $1255 $11.54 $1255 $11.54 Hence, cash balance = $110 + $0.93 = $110.93. Ans. Exercise LXXVII. 1. Find the square root of 2916. 4. Find the square root of 20,164. 29^6(54 25 104)416 416 2^01^64(142 1 24)101 96 282)564 2. Find the square root of 7921. 79^21(89 64 169) 1521 564 5. Find the square root of 3,345,241. 1521 3^34^52^41 (1829 1 3. Find the square root of 494,209. 28)234 224 49^42^09(703 49 362) 1052 724 1403)4209 4209 3649)32841 32841 374 ARITHMETIC. 6. Find the square root of 125,457.64. 8. Find the square root of 21,609. 2^6^09(147 1 24)116 96 287)2009 2009 9. Find the square root of 53.7289. 12^54^57.64(354.2 9 65)354 325 704)2957 2816 53.72^89(7.33 49 143)472 429 1463)4389 4389 7082) 14164 14164 7. Find the square root of 47,320,641. 47^32^06^41(6879 36 128)1132 1024 10. Find the square root of 883.2784. 8^83.27^84(29.72 4 49)483 441 587)4227 4109 1367)10806 9569 5942)11884 11884 13749) 123741 123741 11. Find the square root of 1.97262025. 1.97'26'20^25(1.4045 1 24)97 96 2804)12620 11216 28085) 140425 140425 TEACHERS EDITION. 375 12. Find the square root of 0.0002090916. 0:00^02^09''09a6 (0.01446 1 24) 109 96 284)1309 1136 2886)17316 17316 13. Find the square root of 2. 2.00^00^00(1.414213 24)100 96 281) 400 281 2824) 11900 11296 2828)6040 5656 3840 2828 10120 8484 14. Find the square root of 5 5.00^00^00(2.236067 42)100 84 44:5) 1600 13 29 4466)27100 26 796 4472)"30400 26832 35680 31304 15. Find the square root of 0.3. 0.30^00^00^00(0.547722 25 104) 500 416 1087)8400 7609 10947) 79100 76229 10954) 24710 21908 28020 21908 16. Find the square root of 3^. 3.25^00^00(1.802775 1 28) 225 224 3602) 10000 7204 3604)27960 25228 27320 25228 20920 18020 17. Find the square root of 8|. 8.83^33^33(2.972092 4_ 49)483 441 587)4233 4109 6942)12433 118S4 54933 53496 14373 11888 376 ARITHMETIC. 18. Find the square root of 0.9, 0.90^00^00^00(0.948683 81 184) 900 736 1888)16400 15104 18966)129600 113796 18972) 158040 151776 62640 56916 19. Find in yards the side of a square field containing 20 acres. 20 A. = 96,800 sq. yds. 9^68^00.00^00 (311.12 9 61)68 61 621)700 621 6221) 7900 6221 62222) 167900 124444 20. Find the side of a square the area of which is 150 sq. ft. 9 sq. in. 150 sq. ft. 9 sq. in. = 21,609 sq. in. 2^6^09(147 24)116 96 287) 2009 2009 147 in. = 12 ft. 3 in. 21. Find the side of a square the area of which is 8 sq. yds. 7 sq. ft. 73 sq. in. 8 sq. yds. 7 sq. ft. 73 sq. in. = 11,449 sq. in. 1'14'49(107 1 207) 1449 1449 107 in. = 8 ft. 11 in. 22. Find to six places of deci- mals the square roots of | ; f ; ^; I; f; f; f; I (1) V| = I = 0.666667. (2) f = 0.5. 0.55^55'55'55^55^55 (0.745355 49 144)655 576 1485) 7955 7425 14903)53055 44709 83465 74515 89505 74515 TEACHERS EDITION. 377 (3) 1 = 0.5. 0.50^00^00^00(0.707106 49 1407) 10000 9849 14141)15100 14141 14142)95900 84852 (4) 0.60^00^00^00(0.774596 49 147)1100 1029 1544) 7100 6176 15485)92400 77425 15490)149750 139410 103400 92940 (5) f = 0.714285714285. 0.71^42^85^7F42'85 (0.845154 64 164) 742 656 1685)8685 8425 16901) 26071 16901 169025)917042 845125 1690304)7191785 6761216 (6) f = 0.75. 0.75^00^00^00(0.866025 64 6)1100 996 1726) 10400 10356 17320)44000 34640 93600 86600 (7) f = 0.66666666. 0.66^66^66^66(0.816496 64 161)266 161 1626) 10566 9756 16324)81066 65296 16328) 157706 146952 107546 97968 (8) f = 0.833333. 0.83^33^33(0.91287: 81 181)233 181 1822)5233 3644 1824) 15893 14592 13013 1 2768 2453 1824 378 ARITHMETIC. 23. Fin u = i^r of log 3 = x'r of -.4771 = 0.390-1. Am. TEACHERS EDITION. 395 19. log 7^ = I of log 7= I of 0.8451 = 2.9579. ^w«. 20. logSt = I of log 3- -f of 0.4771 - 0.6361. Ans. 21. logSt = I of log 5= I of 0.6990 = 2.4465. Ans, 22. log 2'T = -If of log 2 = -If of 0.3010 - 0.4730. Ans. 23. log 51 = f of log 5= f of 0.6990 = 0.5243. ^ns. 24. log 7-V- = If of log 7 = -V- of 0.8451 = 1.3280. yins. 26. log 2li = I of log 21 = I of log (3 x 7) = I of (0.4771 + 0.8451) = I of 1.3222 = 1.1569. Ans. Exercise LXXXII. ^3 = 0.4771; log 5 = 0.6990; log 7 Given : log 2 = 0.3010 ; = 0.8451. Find logarithms for the following quotients : ^og f ^ ^°g 2-|-colog5— 10. log 2 = 0.3010 colog5-10 = 9.3010-10 9.6020-10. Ans. 2. log f = log 2+colog 7-10. log 2 = 0.3010 colog7-10 = 9.1549-10 9.4559-10. Ans. log I = log 3 4-colog 5 - 10. log 3 = 0.4771 colog5-10 = 9.3010-10 9.7781 - 10. Am. log f = log 3 -Fcolog 7-10. log 3 = 0.4771 colog 7 - 10 = 9.1549 - 10 9.6320-10. Ans. 5. log f = log5-i-colog7— 10. log 5 = 0.6990 colog 7 -10 = 9.1549 -10 9.8539 - 10. Ans. log ^ = log 7 +colog 5 - 10. log 7 = 0.8451 cologS- 10 = 9.3010 -10 0.1461. Ans. 396 ARITHMETIC. log f == log 5 + colog 3—10. log 5-0.6990 colog 3 - 10 = 9.5229 - 10 0.2219. Am. 8. log I = log 5 + colog 2—10. log 5 = 0.6990 colog 2 - 10 = 9.6990 - 10 0.3980. Ans. 9. log I = log 7 + colog 3-10. log 7 = 0.8451 colog 3 - 10 = 9.5229 - 10 0.3680. Am. 10. log 1 = log 7 + colog 2-10. log 7 = 0.8451 colog 2 -10 = 9.8990 -10 0.5441. Am. 11. log f = log 3 + colog 2-10. log 3 = 0.4771 colog 2 - 10 = 9.6990 - 10 0.1761. Am. 12. log -^ = Iog7+colog0.5-10. log 7= 0.8451 colog0.5-10= 10.3010 -10 1.1461. Am. log 0.05 3 log 0.05 = colog 3 - 10 = 13. = log 0.05 + colog 3 -10. 8.6990 - 10 9.5229 - 10 log 0.005 8.2219 - 10. Am. 14. log 0.005 + colog 2 -10. log 0.005 = 7.6990 - 10 colog 2 - 10 = 9.6990 - 10 7.3980 - 10. Am. log 0.07 5 16 log 0.07 + colog 5 -10. log 0.07 = 8.8451 - 10 colog 5 - 10 = 9.3010 - 10 8.1461 - 10. Am. teachers' edition. 397 16. log ^ = log 5 + colog 0.07 - 10. log 5= 0.6990 colog 0.07-10 = 11.1549-10 1.8539. Ans. 17. log — ^ = log 3 + colog 007 - 10. ° 0.007 log 3= 0.4771 colog 0.007 - 10 = 12.1549 - 10 2.6320. Ans. 18. log ^^ = log 0.003 + colog 7 - 10. log 0.003 = 7.4771 - 10 colog 7 - 10 = 9.1549 - 10 6.6320 - 10. Ans. 19. log -^:^ = log 0.05 + colog 0.003 - 10. ^ 0.003 6^6 log 0.05= 8.6990-10 colog 0.003 - 10 = 12.5229 - 10 1.2219. Ans. 20. log M^ = log 0.007 + colog 0.02 - 10. log 0.007= 7.8451-10 colog 0.02 - 10 = 11.6990 - 10 9.5441 - 10. Ans. 398 ARITHMETIC. 21. log -M2 = log 0.02 + colog 0.007 - 10. ^ 0.007 ^ ^ log 0.02= 8.3010-10 colog 0.007 - 10 = 12.1549-10 0.4559. Am. 22. log 2i^ = log 0.005 + colog 0.07 - 10. log 0.005= 7.6990-10 colog 0.07-10=11.1549-10 8.8539-10. Am. 23. log 2:^ = log 0.03 + colog 7 - 10, log 0.03 = 8.4771 - 10 colog 7 - 10 = 9.1549 - 10 7.6320-10. Ans. 24. log ^^991 = log 0.0007 + colog 0.2 - 10. log 0.0007 = 6.8451-10 colog 0.2 - 10 = 10.6990 - 10 7.5441 - 10. Am. 25. log 2:^ - log 0.02« + colog 3» - 10. o log 0.02'^ = 6.6020 - 10 colog 3» - 10 = 8.5687 - 10 5.1707 - 10. Am. teachers' edition. 399 26. log ^ = log33 + colog0.02^-10. log 33= 1.4313 colog 0.022 - 10 = 13.3980 - 10 4.8293. Ans. 27. log ^ = log 73 + colog 0.022 _ 10. log 73= 2.5353 colog 0.022-10=13.3980-10 5.9333. Ans. 28. log -Mli _ log 0.073 + colog 0.0033 _ 10. ^ 0.0033 & -r & log 0.073= 6.5353-10 colog 0.0033 - 10 = 17.5687 - 10 4.1040. Ans. 29. log 2:^ = log 0.0052 ^ colog 73 - 10. log 0.0052 _ 5 3980 _ 10 colog 73 - 10 = 7.4647 - 10 2.8627 - 10. Ans. 30. log -^ = log 73 + colog 0.0052 _ 10. ^ 0.0052 & & log 73= 2.5353 colog 0.0052 - 10 = 14.6020 - 10 7.1373. Ans. 400 ARITHMETIC. Exercise LXXXIII. Find logarithms of the following numbers. 1. log 70 = 1.8451. Am. 2. log 101 = 2.0043. Ans. 3. log 333 = 2.5224. Am. 4. log 3491 = 3.5428 + (yV of 13) = 3.5429. Am. 5. log 1866 = 3.2695 + {^ of 23) = 3.2709. Am. 6. log 6897 = 3.8382 + (tV of 6) =3.8386. Am. 7. log 9901 = 3.9956 + (yV of 5) = 3.9957. Am. 8. log 4389 = 3.6415 + {j% of 10) = 3.6424. Am. 9. log nil = 3.0453 + (tV of 39) = 3.0457. Avs. 10. log 58343 = 4.7657 + {j%% of 7) = 4.7660. Am. 11. log 77860 = 4,8910 + {^% of 5) = 4.8913. Am. 12. log 30127 = 4.4786 + {^%\ of 14) = 4.4790. Am. 13. log 730.84 = 2.8633 + {^Ss of 6) - 2.8638. Am, teachers' edition. 401 14. log 0.008765 = 7.9425 + {j% of 5) - 10 = 7.9428 - 10. Am. 15. log 8.0808 = 0.9074 + (yf „ of 5) = 0.9074. Am. 16. log 5.0009 = 0.6990 + {j^ of 8) = 0.6991. Am. 17. log 0.3769 = 9.5752 + {j\ of 11) - 10 = 9.5762 - 10. Am. 18. log 0.070707 = 8.8494 + (j^^ of 6) - 10 = 8.8494 - 10. Am. 19. log 0.03723 = 8.5705 + {j% of 12) - 10 = 8.5709 - 10. Am. 20. log 98.871 = 1.9948 + (^5^^ of 4) = 1.9951. Am. 21. Find antilog 3.9017. The number corresponding to the mantissa 9015 is 7970. The number corresponding to the mantissa 9020 is 7980. The difference between these numbers is 10, and 7970 + f of 10 = 7974. Am. 22. Find antilog 1.2076. The number corresponding to the mantissa 2068 is 1610. The number corresponding to the mantissa 2095 is 1620. The difference between these numbers is 10, and 1610 + ^\ of 10 = 1613. Therefore, the number required is 16.13. Ans. 23. Find antilog 0.4442. The number corresponding to the mantissa 4440 is 2780. The number corresponding to the mantissa 4456 is 2790. The difference between these numbers is 10, and 2780 + y2^ of 10 = 2781. Therefore, the number required is 2.781. Am. 402 ARITHMETIC. 24. Find antilog 1.0090. The number corre8{)onding to the mantifsa 0086 is 1020. The number corresponding to the mantissa 0128 i» lOlJO. The difference between these numbers is 10, and 1020 + 5\ of 10 = 1021. Thwefore, the number required is 10.21. Ans. 25. Find antilog 2.9850. The number corresponding to the mantissa 9850 is 9660. Therefore, the number required is 966. Ans. 26. Find antilog 4.5388. The number corresponding to the mantissa 5378 is 3450. The number corresponding to the mantipsa 5391 is 3460. The difference between the numbers is 10, and 3450 + |f of 10 = 3458. Therefore, the number required is 34,580. Ans. 27. Find antilog 0.8550. The number corresponding to the mantissa 8549 is 7160. The number corresponding to the mantissa 8555 is 7170. The difference between these number is 10, and 7160 + ^ of 10 = 7162. Therefore, the number required is 7.162. Aiu, 28. Find antilog 9.9992 - 10. The number corresponding to the mantissa 9991 is 9980. The number corresponding to the mantissa 9996 is 9990. The difference between these numbers is 10, and 9980 + ^ of 10 = 9982. Therefore, the number required is 0.9982. Ans. 29. Find antilog 8.7324 - 10. The number corrosj)onding to the mantissa 7324 is 5400, Therefore, the number required is 0.0540. Am. teachers' edition. 403 30. Find antilog 9.5555 - 10. Tlio number corresponding to the mantissa 5551 is 3590. The number corresponding to the mantissa 5563 is 3600. The difference between these numbers is 10, and 3590 + ^^ of 10 = 3593. Therefore, the number required is 0.3593, Ans. 31. Find antilog 6.0216 - 10. The number corresponding to the mantissa 0212 is 1050. The number corresponding to the mantissa 0253 is 1060. The difference between these numbers is 10, and 1050 + 5^ of 10 = 1051. Therefore, the number required is 0.0001051. Ans. 32. Find antilog 7.0080 - 10. The number corresponding to the mantissa 0043 is 1010. The number corresponding to the mantissa 0086 is 1020. The difference between these numbers is 10, and 1010 + 11 of 10 = 1019. Therefore, the number required is 0.001019. Ans. 33. Find by logarithms the product 948.22 x 0.4387. log 948.22 = 2.9769 log 0.4387 = 9.6422 - 10 2.6191 = log 416. Ans. 34. Find by logarithms the product 1.9704 x 0.0786. log 1.9704 = 0.2946 log 0.0786 = 8.8954-10 9. 1900 -10 = log 0.1 549. ^ns. 35. Find by logarithms the product 380.25 x 0.00673. log 380.25 = 2.5801 log 0.00673 = 7.8280 -10 0.4081 = log 2.559. Am. 404 ARITHMETIC. 36. Find by logarithms the product 216.21 X 0.76312. log 216.21 = 2.3349 log 0.76312 = 9.8826 - 10 2.2175 = log 165. Ans. 37. Find by logarithms the product 0.56127 X 1.2312. log 0.56127 = 9.7492 -10 log 1.2312 = 0.0903 9.8395 - 10 = log 0.691 . Ans. 38. Find by logarithms the product 0.86311 X 56.371. Iog0.86311 = 9.9361 -10 log 56.371 = 1.7511 1.6872 = log 48.67. Ans. 39. Find by logarithms the product 59.795 x 0.7955. log 59.795 = 1.7767 log 0.7955 = 9.9007-10 1.6774 =47.58. Ans. 40. Find by logarithms the product 270.05 X 0.0087. log 270.05 = 2.4315 log 0.0087 = 7.9395-10 0.3710 = log 2.349. Ans. 41. Find by logarithms the product 11.163 x 0.3333. log 11.163 = 1.0478 log 0.3333 = 9.5228-10 0.5706 = log 3.721. Am. 42. Find by logarithms the product 777.78 x 0.0787. log 777.78 = 2.8909 log 0.0787 = 8.8960 - 10 1.7869 = log 61.21. Ans. teachers' edition. 405 43. Find by logarithms the product 2.6537 X 0.2313. log 2.6537 = 0.4238 log 0.2313 = 9.3642 - 10 " 9.7880 - 10 = log 0.6137. Ans. 44. Find by logarithms the product 37.587 X 12.371. log 37.587 = 1.5750 log 12.371=1.0924 2.6674 = log 464.9. Ans. 45. Find by logarithms the product 89.313 x 2.3781. log 89.313 = 1.9510 log 2.3781 = 0.3762 2.3272 = log 212.4. ^ns. 46. Find by logarithms the product 9.1765 X 0.00089. X log 9.1765 = 0.9627 log 0.00089 = 6.9494 -10 7.9121 - e quotient ot^fi-JOT. log 56.407=1.7513 colog 13.045 = 8.8846 - -10 = log 0.008168. Ans. -10 0.6359 ,. , .857.06 e Quotient oi = log 4.324. Ans. 3079.8 log 857.06 = 2.9330 colog 3079.8 = 6.5114 -10 49. Find the quotient of 9.4444 - 10 = log 0.2783. Ans. 0.9387 598.6 log 0.9387 = 9.9726 - 10 colog 598.6 = 7.2229-10 7.1955 - 10 = log 0.001569. Ans. 406 ARITHMETIC. 50. Find the quotient of -^2^. ^ 0.7891 log 3069= 3.4870 colog0.7891 = 10.1028 -10 3.5898 = log 3889. Am. 51. Find the quotient of ^^-^^XQQ^^^. ^ 93.08 X 98.071 log 75.46 = 1.8777 log 0.0765 = 8.8837 - 10 colog 93.08 = 8.0312-10 colog 98.071 = 8.0084 - 10 6.8010-10 = log 0.0006324. ^rw. 52. Fmd the quotient of 98x537x0.0079 , ^ 67309 X 0.0947 log 98= 1.9912 log 537= 2.7300 log 0.0079= 7.8976-10 colog 67309= 5.1719-10 colog 0.0947 = 11.0237 - 10 8.8144 - 10 = log 0.06523. Am. 53. Fmd the quotient of 314X7-18X8132 ^ H""^' '^ 519 X 827 X 3.215 log 314 = 2.4969 log 7.18 = 0.8561 log 8132 = 3.9102 colog 519 = 7.2848 - 10 colog 827 = 7.0825 - 10 colog 3.215 = 9.4928 - 10 1.1233 = log 13.28. Am. teachers' edition. 407 KA T?- A ,\. ^- , r 212x2.16x8002 54. Find the quotient of ^ ^ 536 X 351 X 7.256 log 212 = 2.3263 log 2.16 = 0.3345 log 8002 = 3.9032 colog 536 - 7.2708 - 10 colog 351 = 7.4547 - 10 dolog 7.256 = 9.1393-10 0.4288 = log 2.684. Ans. 55. Find by logarithms 5.06'. 3 log 5.06 = 3 X 0.7042 = 2.1126 2.1126 = log 129.6. Ans. 56. Find by logarithms 2.50P. 51og2.501 =5x0.3981= 1.9905 1.9905 = log 97.84. Ans. 57. Find by logarithms 1.7161 7 log 1.716 = 7 X 0.2345 = 1.6415 1.6415 = log 43.8. Ans. 58. Find by logarithms 1.178i». 10 log 1.178 = 10 X 0.0712 = 0.7120 0.7120 = log 5.153. Ans. 59. Find by logarithms 0.7685«. 6 log 0.7685 = 6 X (9.8857 - 10) = 9.3142 - 10 9.3142 - 10 = log 0.2061. Ans. 60. Find by logarithms 0.96118. 8 log 0.9611 = 8 X (9.9828 - 10) = 9.8624 - 10 9.8624 - 10 = log 0.7285. Ans. 408 ARITHMETIC. 61. Find by logarithms 0.02312. 2 log 0.0231 = 2 X (8.3636 - 10) = 6.7272 - 10 6.7272 - 10 = log 0.0005336. Ans. 62. Find by logarithms 0.8567^. 3 log 0.8567 = 3 X (9.9329 - 10) = 9.7987 - 10 9.7987 - 10 = log 0.629. Ans. 63. Find by logarithms (f|)*. 4 log 61 =4x1.7853 =7.1412 4 colog 73 = 4 X (8.1367 - 10) = 2.5468 - 10 9.6880 - 10 = log 0.4876. Ans. 64. Find by logarithms (ff)». 3 log 13 = 3x1.1139 =3.3417 3 colog 71 = 3 x (8.1487 - 10) = 4.4461 - 10 7.7878 - 10 = log 0.006134. Ans. 65. Find by logarithms {^f. 5 log 16 = 5x1.2041 =5.0205 5 colog 9 = 5 X (9.0458 -10) = 5.2290- 10 0.2495 = log 17.76. Ans. 66. Find by logarithms {^f. 3 log 35 = 3x1.5441 =4.6323 3 colog 4 = 3 X (9.3979 - 10) = 8.1937 - 10 2.8260 = log 699.9. Ans. 67. Find by logarithms (5^)*. 2 loj.^ 60 = 2 X 1.7782 = 3.5564 2 colog 1 1 - 2 X (8.9586 - 10) = 7 9172-10 1.4736 = log 29.76. Ans. TEACHERS* EDITION. 409 68. Find by logarithms (43^)^. 3 log 128 = 3 X 2.1072 = 6.3216 3 colog 31 = 3 X (8.5086 - 10) - 5.5258 - 10 1.8474 = log 70.37. Ans. 69. Find by logarithms {2^f. 5 log 103 = 5 X 2.0128 = 10.0640 5 colog 37 = 5 X (8.4318 - 10) = 2.1590-10 2.2230 = log 167.1. Ans. 70. Find by logarithms (f|})». 3 log 871 - 3 X 2.9400 = 8.8200 3 colog 711 = 3 X (7.1481 - 10) = 1.4443 - 10 0.2643 = log 1.838. Ans. 71. Find by logarithms 133. ilog 13= i of 1.1139 = 0.3713 = log 2.351. ^ns. 72. Find by logarithms 29^ I log 29 = I of 1.4624 = 0.2925 = log 1.961. Ans. 73. Find by logarithms 879tV yV log 879 = j\ of 2.9440 = 0.2944 = log 1.97. Ans. 74. Find by logarithms 0.609?. log 0.609 = 9.7846 - 10 30. - 30 4)39.7846 - 40 9.9462 - 10 = log 0.8834. Ans. 75. Find by logarithms 93.73^ I log 93.73 = I of 1.9719 = 0.9860 = log 9.683. Ans. 410 ARITHMETIC. 76. Find by logarithms 21.97^ ^ log 21.97 = f of 1.3418 = 1.1182 = log 13.13. Ans. 77. Find by logarithms 7.935^ f log 7.935 = f of 0.8996 = 0.6426 = log 4.391. Ans. 78. Find by logarithms 0.8151. log 0.815 =.9.9112 -10 3 29.7336 - 30 10 -10 4 )39.7336-40 9.9334 - 10 = log 0.8578. Ans. 79. What weight of sulphuric acid, specific gravity 1.841, will fill a silver sphere J 38"'"' in diameter? log 1383 _ 6.4197 log 0.52.'.6 = 9.7190 -10 log 1.841-0.2650 6.4037 = log 2534000. That is, 2534000<»""» = 2.534»'«. Ans. 80. What is the area of a circle 13.75 in. in diameter? log 6.875'* = 1.6746 log 3.1416 =0.4971 2.1717 = log 148.5. That is, 148.5 sq. in. Am. 81. Find the depth of a cubical bin that holds 75 bu. log 75 = 1.8751 log 2150.42 = 3.3325 3 )5.2076 1.7359 = log 54.44. Tliat is, 54.44 in. Ans. TEACHERS EDITION. 411 82. Find the diameter of a 24-lb. shot, specific gravity 7.6. log 24 = = 1.3802 log 1728 = = 3.2375 colog 7.6 = = 9.1192- -10 colog 62.5 = = 8.2041- -10 colog ( ).5236 = = 10.2810- -10 3)2.2220 0.7407 = = log log 5.504. That is, 5.504 in. Am. Exercise LXXXIV. What number is 3 less than its square ? Assume 2.3 and 2.4. 2.32 - 2.3 = 5.29 - 2.3 = 2.99, an error of - 0.01. 2.42 - 2.4 = 5.76 - 2.4 = 3.36, an error of + 0.36. The difference of the assumed numbers is 0.1, and the difference of the errors is 0.37. Hence, error of 2.3 : 0.1 : : 0.01 : 0.37, or, error of 2.3 : 0.1 : : 1 : 37. 1x0.1 Therefore, the error of 2.3 = 2.3-^0.0028 = 2.3028. Ans. 37 0.0028. 2. A flag-staff 50 ft. high broke, and the top falling over rested one end on the stump and the other 17 ft. from its base. How high was the stump ? Assume 22 ft. and 23 ft. 412 ARITHMETIC. 222 + 172 _ 484 + 289 = 773, an error of - 11. 23» + 172 = 529 + 289 = 818, an error of + 89. The difference of the assumed nnmbers is 1, and the difference of the errors is 100, Hence, the error of 22 : 1 : : 11 : 100. Therefore, the error of 22 = 12 1 8 '^:—i 1+ ' , =- . 1+ 1+^ 2, f, f, f, ft, \i. Am. 422 ARITHMETIC. 4. Find common fractions approximating to 0.382 ; 1.732 ; 0.6253. (1) 0.382 = ,3^V^ = i^. 1 191)500(2 382 -'-m 118)191(1 118 73)118(1 73 1 + 1 + 1 + 45)73(1 45 1 + 28)45(1 28 1 + 1 + 17)28(1 17 1 + 1 + 11)17(1 11 6)11(1 _6 5)6(1 5 1)5(5 5 2 + 1_1 2 2 1__1 3' 2 + 1 + 2 + 1 + 2 + 1 + 1 + 1 + 1 + 2 + 1 + 1 + 1 + 1 + TEAOHEES' EDITION. 423 1 ^13 1 ^21 1+ ^ . 1+ ^ i + -3-^ 1+ ' 1 + ^^ 1+ 1 1 + -!- 1+ 1 1 _34 1 + - ^ 1 + ^ l + -i 1 + -^ IH-^ ■^-^i i. h h i A. A. M. fi. ft- ■^^«- (2) 1.732 =H|f l^^)???^ .-. UM = 1+-^ 183 ••™ -^_^ 1 )183(2 ^ 1 134 2 + _ 1 1 49)67(1 1 + T 49 2 + —^ 18)49(2 1 + — ^ — r 36 2+—^ 13)18(1 1+-^ 13 1+1 5)13(2 2 3)5(1 3 2)3(1 2 1^2(2 2 424 ARITHMETIC. ,,., .._^_^ .n. 2 + -L 1 + J_-5. 1 + 1 2+i 2+ 1 1 1 + -L 2+ 1 1 + ^— =12. 1 + -!-, 2+ ' l+-i-. ^26. ,^ 1 l+-i-. 1^ 2+ 1 2 + -^ 1+ ' l.-A., 2.^ 2.1 1.1 2. i J. H. If. H. M. ¥/- ^ns. 0.6253 = ^<^^j. 6253)10000(1 . e253 1 6253 ••■iirtnny= j- 3747)6253(1 ^ + 1 3747 1 + - 2506)3747(1 1 + — ; 2506 2 + 7)17(2 1241)2506(2 51+ ^ 14 2482 1 I 1 3)7(2 24)1241(51 o , 1 6 1224 3 1)3(3 17)24(1 3 17 7 teachers' edition. 425 1 ^ 1 ^262 1+ — - 1 2 1+ ' 1 , J- ^ 1 1+T 24- i 1 1 '+ 1 51 +i 1 + i 1 781 1+— t- 1249 5 1 + ^ 8 1+. 1 ^-^^-T ^2. 1 2 257 1 + _i 411 1. i f. f . ffi. Hi AV -471S. 1 + 51 5. Find approximate values of ^ ; f^f ; |^^ ; f ||. (1) 171)457(2 . 1.1 1 342 " ^^^ ~ o 1 2 + 115)171(1 I4._i 115 1+ 1 56)115(2 2 + - 112 18+—^ 3)56(18 1 + ^ 54 ^ T 1 = 1. -3- = l 1)2(2 2 2 2 + i ^ 2 1 426 ARITHMETIC. 1^-i !.-•- 2 + _L 18 + 1 55 1 2 + -^ 147 1 + — L_ i i. I, T^^. i^A- ^^• 2 + — 18 (2) 613)757(1 . «ig 1 5_76 37)144(3 33 1 13 .-. m- 613 ••- ^^ 1 144)613(4 " ^ ^ 1 3 + -A m 1 + -^ 33)37(1 8 + ^ 4)33(8 32 l_j 1 ^4 1)4(4 1 • 1+1 ^' ^ 4 _J ^149 3 3+—!— 17 '4 1 + _L 21 3 + 1 teachers' edition. 427 (3) 1t¥ = f f t- 237)271(1 237 34)237(6 204 33)34(1 33 1)33(33 33 5)14(2 10 -■-m 1 , 1 6, 1 i_i 1 1 7 1 • 1 6 1, f, |. Am. (4) W = 8A. .,8AV = 8+^ 3+ 1 33)113(3 9 ^ 1 ^ 2+ 1 14)33(2 rri 28 1+5 8 = 8. 4)5(1 . 8+i=8i. ^)f 8+^ = 8f. 8 + -^ =8tV 8+-J— - =8^V 3+-^ 3+ ^ 2+1 2+ 1 2 2 + 1 8, 8i, 8f, 8tV, 8/j. ^m. 428 ARITHMETIC. 6. Find the proper fraction that, when reduced to a continued fraction, will have 2, 3, 5, 6, 7 as quotients. 1 V09 ^ 1_7. 1 43 . o . 1 1640 """" 6| 43 ' 5^ 222 ' '^ 1 ^ 1 999 1 709 2m 1640 5 + ^ 3^ 709' 6 + 1 7. Find a series of fractions approximating to the ratio of the pound troy (5760 grs.) to the pound avoirdupois (7000 grs.). nn=m- . i44 = __i i^= 144)175(1 1 + — - 144 4 + 31)144(4 1 + L 124 1+, 1 20)31(1 ._l 20 1 + 11)20(1 ^ + 2 n 9)11(1 1 = 1 ^ =5 -^ 1 ' 1 + ^- ' 2W , , 4^1 1)2(2 1 + 1 5 2 4 9 1 11 ^^ I 4+ ^ ^^I 1^^ -^1 1 + _1 17 4 4+ 1 l + _i_ 1. t. i A. if H. ^'w- 1 i i+i teachers' edition. 429 8. Find a series of fractions approximating to the ratio of the side of a square to its diagonal ; that ratio being 1 : 1.414214 nearly. 1 1000000 7071 1.414214 1414214 10000 7071)10000(1 . ,_ojLi_ _ 1 7071 Toooo -| 2929)7071(2 ^ + 1 5858 2 + 1213)2929(2 2 + — 2426 2 + — 503)1213(2 2 -f ^ 1 = 1. 1 2 3" 1 5 --'i 7 1 12 1 2 + ^ 17 1 006 2 I ^ 207)503(2 3 , _L_ 89)207(2 2 178 29)89(3 87 2)29(14 28 1)2(2 2 1 _29 2+ ' 2 + -A 2 + 1 1 ^70 2 + - 1 2 + -^ 2 + ^ -I 1. f , i {h ih U- ^ns. 430 ARITHMETIC. 9. Find a series of fractions approximating to the ratio of the ar to the square chain, from the equality 1 ar = 0.2471 of a square chain. 0.2471 = AW5. 2471)10000(4 _..^.^ ^ 0884 4 + 116)2471(21 2436 21 + 35)116(3 105 3 + 11)35(3 5 + 1 33 2 2)11(5 10 1)2(2 2 1 = 1 1 _2i 1 ^eA 4 4 4 . JL 85 4 + -^_ 259 214 10. Find a series of fractions approximating to the ratio of the 48- pound shot to the weight of the French shot of 2A^«. 48 lbs. = 48 X 0.45359 = 21.77232^«. 907)1000(1 21.77232 ^ 907 907 24 1000 ^^m^ ••• W.=-^ 837 '""" j_^ 1 70)93(1 9^_1 1 + 23)70(3 \^1 1)23(23 23 1 1 1 _ 9 1 _10 1 ^39 i' ' 1+1 !«■ 1+-1- 11' 1 + -1- ~'' ^ 9 + 1 9 + ^— 1 1+r 1. A. H. if. ^n,. 3 teachers' edition. 431 11. If the mean diameter of the Earth is reckoned at 7912 mi., and that of Mars 4189 mi., find a series of fractions approximating to the ratio of the mean diameters of these two planets. 4189)7912(1 . 4189 __J 4189 •• 7^ 3723)4189(1 ^ + i 3723 1+ . 466)3723(7 7 + - 3262 ^ 461)466(1 461 1=1. -^— =^. 1,1 2 ^' T5' XT' 1 + -^ 15 ^4 12. Find a series of fractions approximating to the ratio of a cubic yard to a cubic meter from the equality 1 cu. yd. = 0.76453 of a cubic meter. 0.76453 = tWAV 76453)100000(1 .,^5 3 1 76453 • • T^^^os 23547)76453(3 70641 5812)23547(4 23248 299)5812(19 5681 131)299(2 262 ~d7 432 ARITHMETIC. 1 = 1 1 250 1 ' ^ ^ t 327 J_ = 3 3 + — L- 1+1 ^ 4 + -1- ^+3 19 1 + -t 13. Find a series of fractions approximating to the ratio of the kilometer to the mile, from the equality 1" = 1.09362 yds. !"» = 1.09362 yds. P™ = 1093.62 yds. l""" = 0.621 mi. 0.621 =Tm- 621)1000(1 . g2i _ 1 621 • • TSVff - 1 379)621(1 ^ "^ ~ 379 242)379(1 1 + ^^^-T 242 1 + 137)242(1 1 + 1 137 ^ 105)137(1 105 32)105(3 __96 9)32(3 27 5 1 = 1 1 3 ^ 1. ^^rtl 1 + 12 1 1 + 2 1+ ' 1+1 1+1 TEACHERS EDITION. 433 1 + 1 + 18 29' 1 + 1 + 1 + 1 + 1 + 59 95* 1 + -I Exercise LXXXVI. 1. Find the seventh term of the series 3, 5, 7 3 + (6 X 2) = 3 + 12 = 15. Ans. 2. Find the fifteenth term of the series 2, 7, 12 2 + (14 X 5) = 2 + 70 = 72. Ans. 3. Find the sixth term of the series 2, 2f, 3f 2 + (5xf) = 2 + 3f = 5f Ans 4. Find the twentieth term of the series 2, 3^, 4| 2 + (H X 19) = 2 + 23f = 25|. 5. Find the seventh term of the series 21, 19, 17 21 -(6x2) = 21 -12 = 9. Ans. 6. Find the twelfth term of the series 18, 17^, 16f 18 - (11 X I) = 18 - 7i = 10|. 7. When the first terra of a series is 5, and the common dif- ference 2^, find the thirteenth and eighteenth terms. 5 + (12x2i) = 5 + 27 =32. (1) 5 + (17x2i) = 5+38i = 43^.(2) 8. Find the common difference in a series whose fourth term is 12 and seventh term 27. 27-12 = 5. Ans. 9. Find the common difference in a series whose first term is 20 and fourth term 40. 10. Find the common differ- ence in a series whose first term is 2 and eleventh term 20. 20-2 10 Ans. 434 ARITHMETIC. 11. Find the common differ- ence in a series whose third term is 7 and eighth term 12^. 121:^^ = 1.1. 5 Arts. 12. Find the common differ- ence in a series whose first term is 1 and fourth term 19. 19-1 6. Ans. Exercise LXXXVII. 1. Find the sum of 14-5 + 9 + 10 twenty terms. Z=l+(19x4) = 77 1 + 77 « = 20x = 780. Ans. 2. Find the sum of 4 + 5^ + 7 + to eight terms. Z = 4 + (7xH) = 14i «»8xi^tiii=.74. Ans. 3. Find the sum of 8 + 7f + 7^ + to sixteen terms. Z=8-(15xi) = 3. « = 16 X ^4^ == 88. Ans. 4. Find the sum of 20 + 18^ + 16^ + to seven terms. Z = 20-(6xl|) = 9i ,^7x20±9i =miAm. 5. Find the sum of the first twenty natural numbers. 1 + 20 «-20x 210. Am. 6. Find the sum of the natural numbers from 37 to 53 inclusive. 8=17x 37 + 53 765. Ans. 7. Find the sum of a series of thirty terms, of which the first is 21 and the last 59. « = 30x 21 + 59 1200. A71S. 8. Find the sum of the series whose first two terms are 3 and 9 and last 75. l = a + {n—l)d. 75 = 3 + 6n- 6. 6n=.78. n = 13. , = 13x^4^ = 507. Ans. 9. Find the sum of a series of twenty terms whose third and fifth terms are 10 and 15. 10 + 15 -2i Z=-5 + (19x2J) = 52J. ,=,20x^^^tM-575. Am. TEACHERS EDITION. 435 10. A stone, when dropped from a height, falls through 16.1 ft. in the first second, 48.3 in the next, 80.5 in the third, and so on, in arithmetical progression. How far will it fall in the seventh second? and how far in 7 sec. ? 7th term is 16.1 + (6 X 32.2) = 209.3 ft. (1) Ans. 7x 16.1 + 209.3 788.9 ft. (2) Ans. 11. A, who travels 8 mi. the first day, 11 the second, 14 the third, and so on, overtakes in 17 dys. B, who started at the same time, and travelled uniformly. What is B's rate per day ? Z = 8 + (16 X 3) = 56. 4- 56 s = 17x 544. -^V = 32mi. Ans. 12. On^ hundred stones lie in a straight line, 1 yd. apart. A boy starts at the first stone, brings each of the others in separately, and piles them with the first stone. How far does he travel ? Z = 2 + (98 X 2) = 198 yds. 8= 99 X 1(2 + 198) - 9900 yds. 9900 yds = 5f mi. Ans. Exercise LXXXVIII. 1. Find the eighth term of the series 2, 6, 18 2 X 3^ = 2 X 2187 = 4374. Ans. 2. Find the fifth term of the series 8, 4, 2 8 X ihf ^ 8 X tV = h ^^s- 3. Find the seventh term of 2 X (f )« = 2 X -W- = 22f |. Ans. 4. Find the sixth term of the series 4, 2f, 1^ 4x(f)^ = 4x^ = H«- ^ris. 436 ARITHMETIC. 5. Find the eighth term of the series 4, 10, 25 4x(fy = 4xi||F = 244m. 6. Write the first three terms of the series whose fifth and sixth terms are 112 and 224, respec- tively. 7. (1) Ans. r = 2. 112 2* 2x7 = 14. (2) Ans. 22 X 7 = 28. (3) Ans. 7. The seventh and ninth terms of a series are 100 and 144, respectively. Find the twelfth term. The 9th term = 7th term X r*. ••• r^ = W 12th term = 144 x (f)=» = 144 X iU = 248.832. Avs 8. A capital of $1000 is in- creased by Y^^ of itself each year. What will it be at the beginning of the fifth year ? f 1000 x(H)* = ? 1000 xiM^i = $1464.10. Am. 9. A capital of $1000 is in- creased by j^jj of itself each yea". What will it be at the beginning of the sixth year ? $1000 X (!§§)» = $ioooxHMM5HS = $1338.23. Am. Exercise LXXXIX. 1. Find the sum of 2 + 6 -|- 18 -h to six terms. 3«- 2x 3- 2. Find the sum «=« 1 x^^ 3. Find the sum 3»- 8 = 3x 2x^F = 728. Am. of 1 -I- 2 -f 4 -F to nine terms. = lX^P = 511. Am. of 3 -f 9 -f 27 + to five terms. ==3x^F = 363. An.<;. 4. Find the sum of 2 -h 3 -I- 4 J -I- to eight terms. a » 2 X ^D-::i = 2 X ^^^ = 98|?. Am. i-1 i teachers' edition. 437 5. Find the sum of 1 + I + ^ + to eight terms. s^lX V^ = ^ ~ r''^ = If f f X I = IMf f Ans. 6. Find the sum of 1 + |^ + | + to ten terms. 1 — ^i^lo 1 _ _!_ J- — ^ ^- 7. Find the sum of | + I + f + to eight terms. -•■ - 3 J 8. Find the sum of the first six terms of the series whose first term is 3 and ratio 5. s = 3 X ^^^ = 3 X i^P^ = 11718. Ans. 5-1 * 9. Find the sum of the first eight terms of the series whose first term is 3 and ratio I. s = 3x^^^' = 3x^^^f^ = 3xeMx| = 4fff. Ans. 10. A person saved in one year $64, and in each succeeding year, for 9 years more, 1^ times as much as in the preceding year. Find the whole amount saved. ( 3 \10 _ 1 ,F;fi 6 8 1 s = $ 64 X ^-^ = 1 64 X ^^^-2-^ = $ 7253.13. Ajis. f — 1 i Exercise XC. 1. Find the sum of the infi- 1 1 _-^_ = 2 = 1. Ans. 1-i i 2. Find the sum of the infi- nite series 1 + 1+27 + s = -^ =2=2. Ans. 4d8 ARITHMETIC. 3. Find the sum of the infi- 6. Find the sum of the infi- nite series j + i*^ + ^j + nite series 0.212121 -r^rh^-^"'- ^ 0.21 0.21 1 - 0.01 0.99 •■>l —.Am. 99 4. Find the sum of the infi- 7. Find the sum of the iufi- nite series i + ^V + jhz + nite series 0.9999 '=r^i=f=i-^'"- s_ 0.9 0.9 J 1 - 0.1 0.9 . Ans. 5. Find the sum of the infi- 8. Find the sura of the infi- nite series 0.171717. nite series 0.232323 s 017 _ 0.17 _ 17^^^ 1-0.01 0.99 99 0.23 0.23 _ 1 - 0.01 0.99 = 1- 9. Find the sum of the infinite series 0.36848484 0.0084 0.0084 84 * 1-0.01 0.99 9900" 36 84 3648 100 9900 9900' Ans. 10. Find the sum of the infinite series 0.15272727. ^ 0.0027 _ 0.0027 _ 27 1 - 0.01 0.99 9900 15 27 ^ 1512 100 9900 9900' Ans. Exercise XCL 1. A deposits $60 in a savings bank, and draws it out at the end of 8 yrs., with 4% compound interest. What does he receive? log ^ = log P -I- n X log (1 -f- r), log ^ = log 60 h 8 X log 1.04. log 60 = 1.7782 8 X log 1.04 = 0.1360 That is, $82.08. Ans. 1.9142 = log 82.08 2. What will $ 100 amount to in 7 years, interest at 8 % per annum, compounded semi-annually ? log il = log P -I- n X log (1 + r), log il = log 100 + 14 X log 1 .04. teachers' edition. 439 log 100 = 2.0000 14 X log 1.04 = 0.2380 = log 173. 2.2380 That is, 1 173. Ans. 3. In how many years will a sum of money double itself at 6%, compounded annually ? log ^ = log P + n X log (1 + r), log 2 =logl +nx log 1.06, log 2 -logl= nx log 1.06. log2 -logl ^^ log 1.06 ,. 0.3010 ^,^g^^_ 0.0253 That is, 11.8 yrs. Ans. 4. In how many years will a sum of money treble itself at 6%, compounded annually ? log J. = log P + n X log (1 + r), log 3 = log 1 +nx log 1.06, log 3 - log 1 = n X log 1.06, log3-logl ^^ log 1.06 ...0^771^ 18.9 = n. 0.0253 That is, 18.9 yrs. Ans. 5. In how many years will $ 87 amount to f 99 at 3 %, compounded annually ? log .A = log P + n X log (1 + r), log 99 = log 87 + n X log 1.03, log 99 - log 87 = n X log 1.03, log 99 -log 87 _^ log 1.03 0.0561 . oo „ .•. = 4.00 ^ n, 0.0128 That is, 4.38 yrs. Aiis. 440 ARITHMETIC. 6. In how many years will $100 amount to $175 at 4%, com- pounded annually ? log A = logP+nxlog(l -l-r), log 175 = log 100 + w X log 1.04, log 175 - log 100 = n X log 1.04, log 175 -log 100 ^ log 1.04 .•.^^i^= 14.29 = n. 0.00170 That is, 14.29 yrs. Ans. 7. At what rate per cent will a sum of money double itself in 12 years, compound interest? log ^ = log P + n X log (1 + r), log 2 = logl + 12xlog(l+r), log 2 - log 1 = 12 X log (1 + r), l^&l^i2^ = log(l+.). .-. ^^^ = 0.0251 = log (1 + r). .-. 1.0595 = 1 + r, or r = 0.0595. That is, 5.95%. Ans. 8. At what rate will a sum of money treble itself in 15 years, at compound interest ? logA = logP+wxlog(l +r), log 3 = log 1 + 15 X log (1 + r), log 3 -log l = 15x]og(l+r), i^S^^^ = log(H-r). .-. M221 = 0.0318 = log(l + r). 15 .-. 1 + r = 1.076, or r = 0.076. That is, 7.6%. Ans. 9. At what rate will $80 at compound interest amount to $110 in 8 yrs. ? log ^ = log P + n X log (1 + r), log 110 = log 80 + 8 X log (1 + r), log 110 -log 80-8xlog(l+r), teachers' edition. 441 log 110 -log 80 1 /I , N -^ ^ ^— = log (1 + ^)- .-. 2iM83 = 0.0173 = log(l+r). 8 .-. l+r= 1.041, or r = 0.041. That is, 4.1%. Ans. 10. What sum must be invested at 5%, compound interest, to amount to $1200 in 7 yrs. log ^ = logP+nXlog(l +r), log 1200 = log P + 7 X log 1.05, log P=logl200-7xlogl.05. log 1200 = 3.0792 7xlog 1.05 = 0.1484 2.9308 = log 852.8. That is, $852.80. Ans. 11. What sum must be invested at 4%, compound interest, to amount to $ 2000 in 10 yrs. ? To amount to $ 5000 in 8 yrs. ? log ^ = logP+nXlog(l +r), log 2000 = log P + 10 X log 1 .04, log P= log 2000- 10 X log 1.04. log 2000 = 3.3010 10 X log 1.04 = 0.1700 3.1310 = log 1352. That is, $1352. (1) Ans. log ^ = logP+nXlog(l +r), log 5000 = log P+ 8 X log 1.04, log P = log 5000 - 8 X log 1 .04. log 5000 = 3.6990 8 X log 1.04 = 0.1360 3.5630 = log 3656. That is, $3656. (2) Am. 442 ARITHMETIC. 12. If A puts $100 a year into a savings bank that pays 4% per annum, compound interest, what will he have in the bank at the end of 10 years ? From ? 438, ^ a(r»-l) _ 1 100 X 0.479 *^ r-1 * * 0.04 ^^ $100(1.04^° -1) ^_ $47.9 1.04-1 " * 0.04' ^^ $100(1.479- 1) s = 11197.50. Ans. 0.04 13. What will be the amount in the last problem if the bank pays 4t% per annum? _ a (r»-l) _ $ 100(1 ■045 ^° -1) _ $100(1.5525-1) r-1 1.045-1 0.045 ^ $100x05525 ^|55^^.j227 78 Am 0.045 0.045 14. What should be paid to-day for an annuity of $500 a year, for 12 years, if money is worth 3^%, compound interest? _ a(r»-l) _ 500(1.035^^-1) * r-1 0.035 Present worth of s is 12 log 1.035 = 0.1788 500 X 0.509 - log 1.509. 0.035 X 1.035"* log 500 = 2.6990 log 0.509 = 9.7067 -10 colog 0.035 = 1.4559 colog 1.035 = 9.8212 -10 3.6828 -log 4818. That is, $4818. Ans. 15. What should be paid to-day for an annuity of $300 a year, for 10 years, if money is worth 4%, compound interest? 300(1.04^0-1) * " 0.04 teachers' edition. 443 Present worth of s log 1.04 = 0.0170 300 X 0.479 ^ 0.04xl.04i«' 0.1700 log 300 = 2.4771 = log 1.479. log 0.479 = 9.6803 - 10 colog0.04 =1.3979 colog 1.04"' = 9.8300 - 10 3.3853 = log 2428. That is, $2428. Ans. 16. What should be paid to-day for the assurance that 5 years hence I shall begin to receive $500 a year, for 8 years, if money is worth 4|, compound interest? ^_; 500(1.0458-1) 0.045 Present worth of s is log 1.045 = 0.0191 500 X 0.422 ^ 0.045 X 1.04513' 0.1528 log 500 = 2.6990 = log 1.422. log 0.422 = 9.6253-10 colog 0.045 = 1.3468 colog 1.045^3 = 9.7517 -10 3.4228 = log 2647.50. That 18, $2647.50. Ans. 17. If interest is reckoned at 6 %, what sum of money must be paid .nually, beginning a year hence, to clear off a debt of $10,000 in 5 equal payments ? s = " ~ , the amount of $1 annually deposited at 6% for 5 yrs. 0.06 Each payment must be ^0^00^^-^^' ^ 10000 x 1.06^ x 0.06 ^ ^ ^ (1.06^-1) 0.338 0.06 ann 444 ARITHMETIC. log 10000 = 4.0000 log 1.06* = 0.1265 log 0.06 =8.7782-10 colog 0.338 = 0.4711 3.3758 = log 2376. log 1.06 = 0.0253 5 0.1265 = log 1.338. That is, $2376. Ans. 18. If interest is reckoned at 6%, what is the amount of each of 12 equal semi-annual payments, the first to be paid 6 months hence, required to clear off a debt of 1 24,000 ? CI 03^^ — 1) s = ^ ' — \ the amount of $1 annually deposited at 3% for yj.yJo 12 yrs. Each payment must be ^^QQ^X^^^" = 24000 x 1.03^' X 03. ^ ^ (1.03^"-!) 0.424 0.03 log 24000 = 4.3802 log 1.03^2 _ 0.1536 log 0.03 = 8.4771-10 colog 0.424 = 0.3726 3.3835 = log 2418. log 1.03 = 0.0128 12 0.1536 = log 1.424. That is, 12418. Am. Miscellaneous Problems. 1. Make six different numbers by logarithms, their co with the digits 1, 2, 3, and find product. their sum. 235 X 253 X 325 123 X 352 X 523 X 532. 132 log 235 = 2.3711 213 log 253 = 2.0431 231 log 325 =2.5119 312 log 352 = 2.5465 321 log 523 = 2.7185 1332. Ans, log532=. 2.7259 2. Make six different numbers with the digits 2, 3, 5, and find, continued 15.2770 = log 1,892,000,000,000,000. TEACHERS EDITION. 445 3, Make six different numbers with the digits 8, 7, 3, and find, by logarithms, their continued product. 873 X 837 X 783 . X 738x387x378. log 873 = 2.9410 log 837 = 2.9227 log 783 = 2.8938 log 738 = 2.8681 log 387 = 2.5877 log 378 = 2.5775 16.7908 = log 61,770,000,000,000,000. 4. Find, by logarithms, the missing term in each of the fol- lowing proportions : (1) 7.13 : 3.57 : : 4.18 : ? 3.57 X 4.18 ^ ^ 7.13 log 3.57 = 0.5527 log4.18 = 0.6212 colog 7.13 = 9.1469 - 10 0.3208 = log 2.093. Ans. 5.89 : 76.3 : : ? 5.89 X 38.7 38.7. = ? 76.3 log 5.89 = 0.7701 log38.7 = 1.5877 colog 76.3 = 8.1175 10 0.4753 = log 2.987. Ans. (3) 7.37 : ? : : 86.1 : 43.7. 7.37 X 43.7 ^ ^ 86.1 log.7.37 = 0.8675 log43.7 = 1.6405 colog 86.1 =8.0650-10 0.5730 = log 3.741. Ans. (4) ? : 69.7 : : 3.79 : 29.4. 69.7x3.79 ^^ 29.4 log 69.7 = 1.8432 log 3.79 = 0.5786 colog 29.4 = 8.5317 -10 0.9535 = log 8.984. Ans. 5. Find, by logarithms, the values of 0.08* ^log 0.08 = ^ of (8.9031 -10) = 9.6344 J log 2734 = i of 3.4368 Jlog21.97 = ^ of 1.3418 3.6 log 7 =3.6x0.8451 2734^; 21.97^; 7««. 10 = log 0.4309. 1.1456 = log 13.98. 04478 = log 2.801. 3.0424 = log 1103. 446 ARITHMETIC. 6. Find, by logarithms, the values of 9.71^ ; 7.935^ I log 9.71 = I X 0.9872 = 2.3035 = log 201.1. (1) Am. f log 7.935 = f X 0.8996 = 0.6426 = log 4.391. (2) Am. 7. What is the horizontal distance between two points, when the air-line distance is 1534 ft., and the difference of level 34 ft.? V15342 - 342 = V2352000 = 1533.623 ft. Am. 8. Find the horizontal distance when the road distance is 1 mile, and the rise 347 ft. V(5280 + 347)(5280 - 347) = V27757991 = 5268.585 ft. Am. 9. If the road distance is half a mile, and the horizontal distance 2513 ft., find the difference of level. V(2640 + 2513)(^40 - 2513) = V65443i = 808.97 ft. Am. 10. The diagonal of a rectangular floor is 34.6 ft., and the width is 17.8 ft. Find the length of the floor. V(34.6 + 17.8)(34.6 - 17.8) = V880.32 = 29.67 ft. Am. 11. The height of a tower on a river's bank is 55 ft., the length of a line from the top to the opposite bank is 78 ft. Find the breadth of the river. V(78 + 55)(78 - 55) = \/3059 = 55.31 ft. Am. 12. The number of seamen at Portsmouth is 800, at Charlestown 404, and at Brooklyn 756. A ship is commissioned whose comple- ment is 490 seamen. Determine the number to be drafted from each place in order to obtain a proportionate number from each. 800 + 404 + 756 = 1960. ^VW X ^p = 101, C. Wis X H^ = 200, P. ^^ X ^ = 189, B. 13. Show, without division, that 36,432 contains 8, 9, 11 as factors 432 = 54 X 8. 3 + 6 + 4 + 3 + 2=. 18. 3 + 4 + 2 = 6 + 3. (See § 222.) teachers' edition. 447 14. Find the smallest multiplier that will make 47,250 a perfect cube. 47,250 = 2 X 33 X 53 X 7. 22 X 72 = 4 X 49 = 196. Ans. 15. Find the proper fraction which, when reduced to a continued fraction, has for quotients, 1, 3, 5, 7, 2, 4. 1 1«99 Ans. 1 + _L^ 1443 5 + - ' ^-^-4 ^^i 16. If the meter is equal to 1.09362 yds., find a series of four frac- tions that will express more and more nearly the true ratio of the meter to the yard. 1.09362 = lTfM!Ty = 1/oVoV 4681)50000(10 ,. i4e8. =1+_1 46810 •• "5^0^^ 10 1 3190)4681(1 ^ ^ 1 3190 1 + 1491)3190(2 2 + - 2982 ' 208)1491(7 1456 35 1+1.11. 1+ 1 =2^7. 10 10 io + _JL_ 235 10 + - ^^ 2 + i 1 7 l + ^-r =i- ii. i^. ^, ^-Ans. 10 + _I_ ^2 10' 11' 32 235 ^4 448 ARITHMETIC. 17. Find the square factors contained in 33,075. 33075 = 33 X 5* X 72. 32 X 5» = 225, 32 = 9, 32x7^ = 441, 52 = 25, 52x72=1225, 72 = 49. 32x52x72=11025. 9, 25, 49, 225, 441, 1225, 11,025. 18. The top of St. Peter's, Rome, is yf ^ of a mile above the ground, and that of St. Paul's, London, is ^V? of a mile. By how many feet does the height of St. Peter's exceed that of St. Paul's ? 20 340 ft. 48 9 of ^^^^-432 ft. ;;p 1 m^^ 432 ft. -340 ft. = 92 ft. Ans. 19. How many days elapsed between the annular eclipse of May 15, 1836, and that of March 15, 1858 ? 1858 — 3 — 15 During the interval there were five leap years, 1836 — 5—15 and in the ten months from May 15 to March 15 01 _ iQ there are 304 days. 365 X 21 = 7665. ^665 + 304 + 5 = 7974 days. Ans. 20. In a gale, a flag-staff 60 ft. high snaps 28.8 ft. from the bot- tom ; and, not being wholly broken off, the top touches the ground. If the ground is level, how far is the top from the bottom? The distance = V(n72 + 28.8)(31.2 - 28.8) = V60 x 2.4 = \/l44"= 12 ft. Ans. 21. Seventeen trees are standing in a line, 20 yds. apart from each other ; a person walks from the first to the second and back, then to the third and back, and so on to the end. How far does he walk ? The total distance is the sum of an arithmetical series in which n = 16, rf = 40, a = 40. Z^ a + (n - l)(f = 40 + 15 X 40 = 640. « = - (a + = ¥ X (40 -h 640) = 5440 yds. = 3 mi. 160 yds. Am. teachers' edition. 449 22. A level reach in a canal is 14| mi. long and 48 ft. broad. At one end is a lock 80 ft. long, 12 ft. broad, and with a fall of 8 ft. 6 in. How many barges can pass through the lock before the water in the canal is lowered 1 in. ? The amount of water that can be drained off in lowering the level 1 in. is 14| x 5280 x 48 x jV = 311,520 cu. ft. The amount of water wasted each time a barge goes through the lock is 80 X 12 X 8^ = 8160 cu. ft. Hence, 311,520 ^ 8160 = 38 barges. Ans. 23. Find the capacity, in liters and in bushels, of a box 1.7™ long, g7cm wide, and 31««» deep. 1.7m_170cm_ V = 0.908 qt. 170 X 87 X 31 = 458,490«'™ = 458.491. (1) Am. 458.49 X 0.908 - 416.309 qts. 1 bu. = 32 qts. 416.309 ^ 32 = 13 bu. (2) Ans. 24. Find the number of kilograms of olive oil, specific gravity 0.915, to fill a vessel 2.3"" long, 1.8"" wide, and 74««» deep. 74cm ^ 0.74'". 2.3 X 1.8 X 0.74 = 3.0636«i'n» = 3063.6''k. 3063.6''« X 0.915 = 2803.194'^8. Ans. 25. How many tons in a block of marble 4 ft. long, 34 in. wide, 17.3 in. thick, if its specific gravity is 2.73? 4 ft. = 48 in. 48 X 34 X 17.3 = 28233.6 cu. in. 28233.6 ^ 1728 = 16.34 cu. ft. 16.34 X 62i lbs. = 1021.25 lbs. 1021.25 lbs. X 2.73 = 2788.0125 lbs. = 1.394 t. Ans. 26. Find the surface of a sphere 18.3 in. in diameter. The radius = 9.15 in. 9.15 X 9.15 X 3.1416 = 263.0226 sq. in. 263.0226 X 4 = 1052.09 sq. in. Ans. 450 ARITHMETIC. 27. Find the number of acres in a circular field 213 yds. 2 ft. across. 213 yds. 2 ft. = 04 1 ft. in diameter. The radius is 320.5 ft. 1 A. = 43560 sq. ft. 3.1416x320.52 Area = 43560 log 3.1416 = 0.4971 log 320.5"^ = 5.01 16 colog 43560 = 5.3609 - 10 0.8696 = log 7.407. 7.407 A. Am. 28. How many cubic inches in a 10-inch globe? in a 20-inch globe ? What is the ratio of their volumes ? The ratio of their volumes is lO^ : 20^ - l^ . 2^ = 1 : 8. Am. 10» X 0.5236 = 523.6 cu in. (1). Aiis 523.6 X 8 = 4188.8 cu. in. (2). Am. 29. How many balls 3 in. in diameter can be ca« ^ ^ ,,„. ^,, n 12 11 4 11 20 ¥^¥ 45. Find the area of a triangle whose sides are 12, 5, and 13 in. Observe that 13^ = 12-' + 5". Hence the triangle is a right tri- angle. Area = J^ of 12 X 5 = 30 sq. in. An&, 46. Find the area of a triangle whose sides are 73, 57, and 48 ft. 53 + 57 + 48 ^ gc^ Area = V80 x 16 x 32 x 41. log 89 =1.9494 log 16 -1.2041 log 32 -1.5051 log41-. 1.6128 2 )6.271 4 3.1357 = log 1367. 1367 sq. ft. An%. teachers' edition. 455 47. Find the number of hektars in a triangular field whose sides are 37.5°S 91.7°', and 78.9'". 3r^+^L7 + 7M^ 104.05. Area = V104.05 x 66.55 x 12.35 X 25.15. log 104.05 = 2.0172 log 66.55 = 1.8232 log 12.35 = 1.0917 log 25.15 = 1.4006 2)6.3327 3.1663 = log 1467. 1467i"' = 0.1467h^ Ans. 48. Find the number of hektars in a triangular field whose sides are 67.5°^, 81.2'", and 102.7'". 67.5 + 81.2 + 102.7 _ -j^g 7 Area = V125.7 X 58.2 x 44.5 x 23 log 125.7 = 2.0994 log 58.2=1.7649 log 44.5 = 1.6484 log 23 =1.3617 2 )6.8744 3.4372 = log 2736. 27361'° = 0.2736ha. Ans. 49. Find the number of acres in a triangular field whose sides are 227, 342, and 416 ft. 1 A. = 43,560 sq. ft. 227jl342Jl416^492 5. 2 Area = ^^^2.5 X 265.5 x 150.5 x 76~5 ^ 43560 456 ARITHMETIC. log 492.5 =- 2.6924 log 265.5 = 2.4241 log 150.5 = 2.1776 log 76.5 = 1.8837 2 )9.1778 4.5889 colog 43,560 = 5.3609 -10 9.9498 -10 = log 0.8908. 0.8908 A. Ans. 50. Find the number of acres in a triangular field whose sides are 79 chains 8 links, 57 chains 3 links, and 102 chains 19 links. 79.08 + 57.03 + 1 02.19 _ ^^g ^^ Area 2 Vl 19.15 X 40.07 X 62.12 X 16.96 10 log 119 15.= 2.0761 log 40.07 = 1.6028 log 62.12=1.7932 log 16.96 = 1.2294 2 )6.7015 3.3508 colog 10 = 9.0000 - 10 2.3508 = log 224.3. 224.3 A. Am. 51. Find the number of square rods in a triangle whose sides are 7 rds. 2 yds., 6 rds. 5 yds., and 9 rds. 4^ ft. 7 rds. 2 yds. = 121.5 ft. 6 rds. 5 yds. = 114 ft. 9 rds. 4i ft. = 153 ft. 1 rd. = 272.25 sq. ft. 121.5 + 114 + 153 ^g^og 2 Area = V1 94.25 x 72.'r5 X ft0.2f^ Y IT^ 272.25 ' teachers' edition. 457 log 194.25 = 2.2884 log 72.75 = 1.8618 log 80.25=1.9043 log 41.25 = 1.6155 2 ) 7.6700 * 3.8350 colog 272.25 = 7.5650 - 10 1.4000 = log 25.12. 25.12 sq. rds. Am. 52. Find the number of acres in a four-sided field, the sides of which are in order 361, 561, 443, and 357 ft. ; and the distance from the beginning of the first side to the end of the second side is 682 ft. 361 + 561 + 682 2 802. Area = ^§0^ X 441 x 241 x 120 ^ 43560 log 802= 2.9042 log 441= 2.6444 log 241= 2.3820 log 120= 2.0792 2 )10.0098 5.0049 colog 43560= 5.3609 - 10 0.3658 = log 2.322. 357 + 443 + 682 _ ^^^ 2 Area = V741 x 384 X 298 X 59 43560 log 741 = 2.8698 log 384 = 2.5843 log 298 = 2.4742 log 59 = 1.7709 2 )9.6992 4.8496 colog 43560 = 5.3609 - 10 0.2105 = 1.624 1.624 + 2.322 = 3.946 A. Ans. 458 ARITHMETIC. 53. Find the number of hektars in a field of three sides, one of wliich is 82.1™, and the distance from this side to the opposite corner is 47.3'». ^ (82.1 X 47.3) = 1941.671°' = 0.194167^'^ Aiis. 54. Find the number of acres in a triangular lot, one side of which is 313.6 ft., and the distance from this side to the opposite corner is 163.2 ft. J (343.6 X 163.2) = 28037.76 sq. ft. 28037.76 -5- 43560 = 0.6436 A. Am. 55. Find the altitude of a triangle, if each side is 1000 ft 1000 + 1000 + 1000 _ ^5QQ Area = V1500 x 500 x 500 x 500 = 250000 \/3 = 43301 2.5 sq. ft. 433012.5 ^ 500 = 866.025 ft. Ans. 56. Find the distances of the vertices from the opposite sides of a triangle, when these sides are 17.8"^'", 23.6"'", and 31.5°>"». 17.8 + 23.6 + 31.5 _ ^^^^ Area = V36.45 x 18.65 x 12.85 x 4.95. log 36.45 = 1.5617 log area= 2.3179 log 18.65 = 1.2707 colog 11.8 = 8.9281 - 10 log 12.85 =1.1089 JTTTT log 4.95 = a694_6 = log 17.62. 2)4.6359 17.62»». (2) Ans. log area = 2.3179 log ^rea = 2.3179 colog 8.9 - 9.0506 - 10 colog 15.75 = 8.8027 - 10 1.3685 1.1206 -= log 23.36. = log 13.2. 23.36""". (1) Ans. 13.2"'™. (3) Ans. teachers' edition. 459 57. If the four sides of a field measured in succession are 237, 253, 244, and 261 ft., and the diagonal measured from the end of the first side to the end of the third side is 351 ft. ; find its area. 237 + 261 + 351 _ ^^^ ^ 2 ' Area of triangle = V424.5 x 187.5 x 163.5 x 73.5. log 424.5 = 2.6279 log 424 = 2.6274 log 187.5 - 2.2730 log 171 = 2.2330 log 163.5 = 2.2135 log 180 = 2.2553 log 73.5 = 1.8663 log 73 = 1.8633 2 )8.9807 2 )8.9790 4.4904 4.4895 = log 30,925. _ log 30,860. 253 + 244 + 351 424. 30,860 + 30,925 = 61,785 sq. ft 2 , . . H Area of triangle ns. \/424x 171x180x73. 58. If the four sides of a field are 237, 253, 244, and 261 ft., taken in order, and if the corner formed by the second and third sides is a square corner ; find the diagonal from the beginning of the second side to the end of the third side, and also find the area of the field. Diagonal = \/253'^ + 244^ = VT23545 = 351.489. The area is the same as in the last problem. 59. Find the area of a circle that has a radius of 10 in. ; of a cir- cle that has a diameter of 10 ft. ; of a circle that has a circumference of 30 in. Area = irR' = 3.1416 x 100 = 314.16 sq. in. (1) Ans. Area = rri^^ = 3.1416 x 25 = 78.54 sq. ft. (2) Ans. Circumference = 2 irE. Z0 = 2irR. R = ^± IT Area = -jriS^ = i: x — = -^^ = 71.62 sq. in. (3) Ans. 1 TT^ 3.1416 ^ 460 ARITHMETIC. 60. A horse is tied by a rope 28.8™ long ; what part of a hektar can he graze? Area = ttE" = 3.1416 X 27.82 = 3.1416 x 772.84 = 2427.951'" = 0.2428'>». Ans. 61. How many square feet in a circle that has a diameter of 17f yds.? 17f yds. = 53 ft. Area = itB:^ = 3.1416 x 26.5'. log 26.52 =. 3.8464 log 3.1416 = 0.4971 3.3435 = log 2205.5 2205.5 sq. ft. Am. 62. How many square feet in a circle that has a circumference of 117 yds.? log 3512 = 5.0906 colog 3.1416 = 9.5029 -10 colog4 =9.3979-10 3.9914 = log 9804. 9804 sq. ft. Am. 117 yds. = 351 ft. Circumfei ence ==2-nR. 351 = 2nB. R _351 27r" »i22 35P 4 X 3.1416 63. How many square inches in the surface of a globe that has a radius of 12.37 in.? Area = 4 tt/?* = 47rx 12.37*. log 4 = 0.6021 log 3.1416 = 0.4971 log 12.37^=2.1848 3.2840 - log 1923. 1923 sq. in. Am. 64. Find the area of the sur- face of the largest globe that can be turned out (Voni a joist 4 in. by 6 in. Area = 4 7r22-16T = 1<; • :'..l \\r> = 50.Jtif> S^J. 111. .I/(N. 65. How many cubu- liuin-,-^ in a globe that has a diameter of 10 in.? Volume = 0.5236 x diam.» = 0.5236x1000 — 523.6 sq. in. Am. TEACHERS EDITION. 461 66. If a tree be round, and the girt is 17 ft. 6 in., find its diameter. Find the area of a cross-section, and find the num- ber of cubic feet in the largest sphere that can be cut from it. Diameter = - — ~ 3.1416 = 5.57 ft. (1) Ans. Area = Tri?^. log 3.1416 = 0.4971 log 5.572 =1.4918 colog 22 = 9.3979 - 10 log Area = 1.3868 = log 24.37. 24.37 sq. ft. (2) Ans. Volume = 0.5236 x diam.^ log 0.5236 = 9.7190 -10 log 5.573 = 2.2377 1.9567 = log 90.52. 90.52 cu. ft. (3) Ans. 67. Find the weight in kilo- grams and in pounds of an iron ball 21.5''™ in diameter, specific gravity 7.47 ; of a tin ball 13<=°» in diameter, specific gravity 7.29 ; of a lead ball 17.3'='» in diameter, specific gravity 11.35 ; of a silver ball 1.31°™ in diameter, specific gravity 10.47. Iron. V= 0.5236 X 21.53 log 0.5236 = 9.7190 -10 log 21.53 =3.9972 log 747 = 0.8733 colog 1000 = 7.0000-10 1.5895 = log 38.86. 38.86''8. (1) Ans. log 2.205 . = 0.3434 log 38.86 = 1.5895 1.9329 = log 85.68. 85.68 lbs. (2) Ans. Lead. log 17.33 = 3.7140 log 0.5236 = 9.7190 -10 log 11.35 = 1.0550 colog 1000 = 7.0000 - 10 1.4880 = log 30.76. 30.76''st.-(l) ^ns. log 2.205 = 0.3434 log 30.76 = 1.4880 1.8314 = leg 67.83. 67.83 lbs. (2) Ans. Tin. F= 0.5236 X 133. log 0.5236 = 9.7190 -10 log 133 = 3.3417 log 7.29 = 0.8627 colog 1000 = 7.0000-10 0.9234 = log 8.383. 8.383'^g. (1) Ans. log 2.205 = 0.3434 log 8.383 = 0.9234 1.2668 = log 18.48. 18.48 lbs. (2) Ans. Silver. log 1.313 = 0.3519 log0.5236 = 9.7190 -10 log 10.47 = 1.0199 colog 1000 = 7.0000 - 10 8.0908 - 10 = log 0.01233. 0.01233'^?. (1) Ans. log 2.205 log 0.01233 0.3434 8.0908 - 1 10 8.4342 = log 0.0271 7. 0.02717 lbs. (2) A71S. 462 ARITHMETIC. 68. A slab of cast-iron 4 ft. 2^ in. long, 17 in. wide, and 8^ in. thick, specific gravity 7.31, is cast into 2-lb. balls. If there is a loss of 5% in melting, how many balls are obtained, and what is the diameter of each ? The Blab will make 50.5x17x25x0.95x62.5x7.31 ^^^^ 2 X 3 X 1728 log 50.5 = 1.7033 The diameter will be ^ oI^J?o^o i/ 50.5x 17X25X0.95 log 25=1.3979 ^ 0.5236X3X698 log 0.95 = 9.9777 -10 log 62.5 = 1.7959 ^^g 50.5 = 1.7033 log 7.31 = 0.8639 ^ ^^g 17=1.2:m colog 2 = 9.6990-10 ^^8 25=1.3979 colog 3 = 9.5229-10 ^^g 0.95 = 9.9777-10 colog 1728 = 6.7625 - 10 ««^«g ^-^^236 = 0.2810 ^ colog 3 = 9.5229-10 2.9535 = log 898. ^olog 898 = 7.0467 - 10 898 balls. (1) Ans. 3)1.1 5v>9 0.3866 = log 2.436. 2.436 inches. (2) Arts, 69. How many pounds avoirdupois would a ball of such iron 30 in. in diameter weigh ? log 3a'' = 4.4313 log0.5236 = 9.7190 -10 log 62.5 = 1.7959 log 7.31 = 0.8639 colog 1728 = 6.7625-10 3.5726 = log 3732.5. 3732.5 lbs. Am. 70. If the specific gravity of ice is 0.921, find the weight and the surface of each of three spheres of ice whose diameters are l***, 10"™, and l"*. Which of these spheres would roll first on a plain, in a gradually-increasing wind? teachers' edition. 4G3 F= P X 0.5236 - 0.5236««'«. 0.5236«<='» = 523. e^K. 0.921 X 523.6™« = 482.24'ng (1) Ans. V= 103 X 0.5236 = 523.6««'». 523.6««°» = 523.68. 0.921 X 523.6s = 482.248. (2) Am. V= 13 X 0.5236 = 0.5236«*>m^ 0.5236«bm = 523.6»'8. 0.921 X 523.6''8 = 482.24'^8. (3) Ans. Area = AirE" = irD'' = 3.1416 x P = 3.14161'=™ (1) Ans. Area = 4 7r/?''' = irD' = 3.1416 x 10^ = 314.16» 3.1416* log 4 = 0.6021 log 231 = 2.3636 colog 3. 14 16 = 9.5029 - 10 3)2.4686 0.8229= log 6.651 6.65 in. (1) An&. V 4 Iqt. = -2|icu. in. Z^X = irZ)». D =;'23i. >'3.1416 n^'i 4000 3.1416 log 4000 = 3.6021 colog 3.1416 = 9.5029 3)3.1050 1.0350 = log 10.84. 10.84«». (3) Am. teachers' edition. 465 76. Find the volume of a triangular priam 11 in. long, the sides of the ends being 2, 3, and 4 in. long. Area of end = V4.5 x 2.5 x 1.5 x 0.5 = 2.9047. F= 11 X 2.9047 = 31.952 cu. in. Ans. 77. Find the capacity in bushels of a bin 6 ft. long, the end of which is a square measuring 3 ft. 3 in. on a side. F= 6 X 3^ X 31 = 6 X \' X -\« = -5-F = 63f cu. ft. 0.80356 bu. = 1 cu. ft. 0.80356 X 63^- = 50.93 bu. Ans. 78. Find the number of cubic yards in a square prism 200 ft. on a side, and 40 ft. long. J. 2002 y^ 40 V= — = 592o9^'y cu. yds, Ans. 79. How many cubic yards in a square pyramid 210 ft. on a side, and 123 ft. high ? 210 ft. = 70 yds. 123 ft. = 41 yds. V= i(702 X 41) = 669661 cu. yds. Ans. 80. Find the capacity of a cup, the mouth of which is a square 4 in. on a side, and the sides of which are four equilateral triangles. ^ the diagonal of base = |\/l6 + 16 = | V32. Altitude = V42-(i>/32)2 = Vl6^^ = VS = 2.8285. V= i of 16 X 2.8285 = 15.085 cu. in. Ans. 81. The largest of the Egyptian pyramids is 14 T*" high, with a base 231™ square. Find its volume in cubic meters. ^ of 23P X 147 = 2,614,689°'"". Ans. 82. The slant depth of a conically-shaped drinking-cup is 93°»"*, and the diameter at the top 8«°». What is its capacity ? 466 ARITHMETIC. 93min ^ 9_3cm_ Height = V9.32 - 4' - S.-SeSS^"*. F=|tX42x8.3958°"» = ^ of 3.1416 X 16 X 8.3958 = 140.67«'°' = 0.1406?. Ans. 83. Tlie volume of a cone is 1"**™ ; its height is equal to the radiiis of its base. Find the dimensions of the cone. y _ 7r B^h v=— E^ = ^ ^_ /3_x_1000000 log3,000,000 = 6.4771 colog 3.1416 = 9.5029 -10 3)5.9800 1.9933 = log 98.47. 98.47«". Ans. 84. Find the capacity of a wash-bowl 30*^ in diameter and 5*"' deep. i of 30» = J of 900 = 225. J of 52 = J of 25 = 8.33. 225 + 8.33 = 233.33. -»/ of 5 X 233.33 = 1833.31<'«»' = 1.833'. Am. 85. Find the capacity in liters of a boiler 89«™ in diameter and 31''"» deep. i of 892 = J of 7921 = 1980.25. iof3P = iof 961 = 320.33. 1980.25 + 320.33 = 2300.58. V of 31 X 2300.58 = 112,071. ll^"" - 112.071. Ans. 86. Find the capacity in quarts of a bowl 10 in. in diameter and 4 in. deep. } of 10» = i of 100-25. J of 4'» = iof 16 = 5.333. 25 + 5.333 = 30.333. V of 4 X 30.333 = 190.66 cu. in. 1 qt. — 57.75 cu. in. 190.66 + 57.75 =- 3.3 qt«. Ans. teachers' edition. 467 87. deep; of a bowl 7 in. across and 3 in. deep; of a bowl 8 in. across and 3j in. deep. 1 pt. = I of 231 cu. in. = 28.875 cu. in. Jof 62 = 1 of 36 = 9. i of (11)2--=^ of 1 = 0.75. 9 + 0.75 = 9.75. -If of 1.5 X 9.75 = 22.982 cu. in. 1 pt. = 28.875 cu. in. 22.982 -^ 28.875 = 0.8 pt. (1) Ans. J of 72 = i of 49 = 12.25. i of 32 = ^ of 9 = 3. 12.25 + 3 = 15.25. -V- of 3 X 15.25 = 71.893 cu. in. 71.893 - 28.875 = 2.5 pts. (2) Ans. J of 82 = J of 64 = = 16. Jofa)2 = iof-V- = = 4tV 16 + 4tV = = 20tV i n x?x24lx ^ -241 = = 3.8 pts. (3) Ans. 7 ^ n m 63 3 21 X^ V / • 88. How many gallons will a boiler 5 ft. in diameter and deep hold? ^ of 52 = ^ of 25 = iof22 = iof 4 = 6.25 + 1.333 = = 6.25. = 1.333. - 7.583. V of 2 X 7.583 = = 23.832 cu. ft. 23.832 X 1728 -^ 231 = = 178.3 gals. Ans. 2 ft. 89. How many gallons will a boiler 30 in. in diameter and 1 ft. deep hold ? \ of 302 _ ^ of 900 = 225. i of 122 = ^ of 144= 48. 225 + 48 = 273. -Vof 12x273 = 5148 cu. in. 5148 ^ 231 = 22.3 gals. Ans. 468 ARITHMETIC. 90. Find the capacity in pints of a cylinder 1.9375 in. in diameter, 2.4375 in. high ; of a cylinder 3^ in. in diameter, 3| in. high; of a cylinder 3|| in. in diameter, 5^^ in. high. 1 pt. = 28.875 cu. in. V= ^ilil^ X 0-9fi875g X 2.4375 "28.875 log 3.1416 = 0.4971 log 0.968752 = 9.9724 -10 log 2.4375 = 0.3870 colog 28.875 = 8.5394 9.3959 -10 = log 0.2488. 0.249 pt. (1) Ans. y^ 3.1416 X 1.5625' x 3.625 log 3.1416 = 0.4971 log 1.56252 = 0.3876 log 3.625 = 0.5593 colog 28.875 = 8.5394 - 10 9.9834 = log 0.9625. 0.963 pt. (2) Alls. 3.1416 X : 1.90625' X5.06 28.875 3.1416 = = 0.4971 F= log log 1.906252 = 0.5605 log 5.0625 = 0.7044 colog 28.875 = 8.5394 - 10 0.3014 = log 2.002. 2.002 pt. (3) Am. 91. Find the capacity in pecks of a cylinder 15.865 in. in diameter, 12.5 in. high ; of a cylinder 9.25 in. in diameter, 4.25 in. deep ; of a cylinder 18.5 in. in diamet<3r, 8 in. deep. y^ 0.7854 X 15.865' x 12.5 x 4 2150.42 teachers' edition. 469 log 0.7854 = 9.8950-10 log 15.8652 = 2.4009 log 12.5 = 1.0969 log 4 = 0.6021 colog 2150.42 = 6.6675 - 10 0.6624 = log 4.596. 4.596 pks. (1) Ans. y^ 0.7854 X 9.25^ x 4.25 X 4 2150.42 log 0.7854 = 9.8950-10 log 9.252 = 1.9322 log 4.25 = 0.6284 log 4 = 0.6021 colog 2150.42 = 6.6675 -10 9.7252 -10 = log 0.531. 0.531 pks. (2) A71S. y^^ 0.7854x 18.52x8x4 2150.42 log 0.7854 = 9.8950-10 log 18.52 _ 2.5344 log 8 = 0.9031 log 4 = 0.6021 colog 2150.42 = 6.6675 - 10 0.6021= log 4. 4 pks. (3) Ans. 92. What must be the diameter of a circle, in order that it may- contain 78.54 sq. ft. ? to contain 314.16 sq. ft. ? Area = 0. 7854 x D''. Area = 0. 7854 x D^. 78.54 = 0.7854 x D\ 314.16 = 0.7854 x D\ 7)2=100. i)2_4oo. i> = 10 ft. (1) Ans. i) = 20 ft. (2) Ans. 470 ARITHMETIC. 93. What must be the diameter of a circle to contain 1 A. ? to contain 9 A. ? 1 A. = 43,560 sq. ft. jy _ /43560 Area = 0.7854 X D^. \0.7854 log 43,560 = 4.6.391 colog 0.7854 = 0.1049 2)4.7440 2.3720 = log 235.5. 235.5 ft. (1) Am. V9 = 3. 3 X 235.5 = 706.5 ft. (2) Am. 94. What must be the diameter of a circle to contain l***? to con- tain 25h« •' l**"^ = 10,0001™. r^ . < 10000 .7854 log 10.000 = 4.0000 colog 0.7854 = 0.1049 2 )4.1049 2.0525 = log 112.8. 112.8"'. (1) Am. \/25 = 5. 5 X 112.8"' = 564"'. (2) Am. 95. Find the number that exceeds its square root by 20. On testing the square numbers exceeding 20, namely, 25, 36, etc., we see that 25 is the number ; and no process of approxi- mation is needed. 25, Am. 96. How much water will a hemispherical bowl hold that is 10 in. in diameter ? V= \ of 0.5236 X 2>» = 1 of 0.5236 x 1000 = 261.8 cu. in. Am. 97. What will it cost to gild a hemispherical dome 10 ft. in diam- eter, at 50 cents a square foot ? 8=-2x 0.7854 x 10» = 157.08 sq. ft. 157.08 X $0.50 = $78.54. Am. teachers' edition. 471 98. If the moon is a sphere 2170 miles in diameter, about how many million bushels would she hold if hollow ? and how many- yards of cloth a yard wide would it take to cover her ? 2170 mi. = 137,491,200 in. F= 0.5236 xZ>3. log 137,491,200=^ = 24.4149 log 0.5236= 9.7190-10 colog 2150.42= 6.6675-10 log V= 20.8014 V= 633,000,000,000,000,000,000 bu. (1) Ans. 2170 mi. = 3,819,200 yds. xog 3,819,2002 = 13.1640 log 3.1416= 0.4971 log >S'= 13.6611 S= 45,800,000,000,000 yds. (2) Ans. 99. If the earth is 7920 miles in diameter, and the air is 40 miles deep, how many cubic miles of air are there about the planet ' 7920 + 80 = 8000 log 7920^ = 11.6961 log 80003 = 11.7093 log 0.5236= 9.7190-10 log 0.5236 = 9.7190 - 10 UAlbl 11.4283 = log 260,100,000,000 = log 268,100,000,000. 268,100,000,000 260,100.000,000 8,000,000,000 cu. mi. Ans. 100. What is the difference between 2 feet square and 2 square feet ? between a foot square and a square foot ? between half a foot square and 6 in. square? 472 ARITHMETIC. " 2 feet square " means a square 2 ft. on a side ; " 2 square feet," any surface equivalent in area to two squares each 1 foot on a side. A " foot square " is a square ; while a square foot is an equivalent area in any shape. "Half a foot square" is am- biguous. Half "a foot square" is half a square foot, while "half a foot" square is 6 inches square ; that is, one-fourth a square foot. 101. Find the volume of a square frustum of which the base is 3 ft. square, top 2 ft. square, and height 4 ft. F= ^ X 4[>/3^^^r22 + 32 + 22] = ^(6 + 9 + 4) = 25^ cu. ft. Ans. 102. Find the capacity in liquid quarts of a tin pan 10 in. in diameter at top, 8 in. in diameter at bottom, and 4 in. deep. Volume of whole cone = | X 0.7854 D'^h = ^x 0.7854 X 100 X 20 = 523.6 cu. in. Volume of part cut off = } of 0.7854 D'% = ^x 0.7854 x 64 X 16 = 268.08 cu. in. 523.6 - 268.08 = 255.52 cu. in. 255.52 - Afi = 4 42 qts. Ans. 103. How many hektoliters will a circular vat hold 5°> in diam- eter at the top, 4.57™ at the bottom, and 1.17" deep? Area of top = 5» x 0.7854 = 19.635o«». Area of base = 4.57» X 0.7854 = 16.4030". V19.635 X 16.403 = 17.946. 1(19.635 + 16.403 + 17.946) x 1.17 = 21.054«''>» « 210.54'». Ans. 104. Find the area of an ellipse 8 in. by 11 in.; of an ellipse 15 in. by 21 in. 0.7854 X 11 X 8 = 69.115 sq. in. (1) Ans. 0.7854 X 15 X 21 - 247.401 sq. in. (2) Ans. TEACHERS EDITION. 473 105. The ends of a cord 100 ft. long are fastened to stakes placed 80 ft. apart on level ground. A ring, to which a kid is tied, plays freely on the \ 40 cord. How far from the straight line join- ing the stakes can the ring be pulled? What are the diameters of the ellipse which the kid can graze ? How many square feet in the ellipse ? The cord can be pulled from AB V502 - 402 _ V900 = 30 ft. (1) Ans. The diameters are 100 ft. and 60 ft. (2) Ans. 0.7854 X 60 X 100 = 4712.4 sq. ft. (3) Ans. 106. Using the same rope as in the last problem, but putting the stakes 25 ft. apart, how many per cent is the kid's pasturage in- creased ? The rope can now be pulled from AB a, distance of VSO'^ - 12.5^ = V2343.75 = 48.4 ft. Hence the diameters are 96.8 and 100. Area = 96.8 X 100 x 0.7854 = 7602.7. 7602.7 - 4712.4 Hence, 4712.4 of 100 = 61.3. 61.3%. Ajis. 107. A cylindrical log, 11 in. in diameter, is sawed off on such a slant that the pieces are 8 in. longer on the longest than on the shortest side. Find the dimensions of the ellipse thus made, and its area. The shorter diameter is evidently the diameter of the log, or 11 The longer diameter is VlP + 82 = V'121 + 64 = V\.6o Area = 13.6 x 11 X 0.7854. log 13.6 = 1.1335 log 11 = 1.0414 13.6 ft. log 0.7854 = 9.8951 2.0700 10 log 117.5 sq. in. Ans. 474 ARITHMETIC. 108. Find the length of a pendulum beating half-seconds ; of a pendulum beating quarter-seconds. The length of pendulums is inversely as the square of the num- ber of vibrations in a given time. Hence, a half-seconds pendulum will be ^ X ^ = } the length of a seconds pendulum, or \ of 39.138 in. = 9.784 in, (1) Ans. A quarter-seconds pendulum will ^e | X ^ = yV ^^^ length of the seconds pendulum, or -^ of 39.138 in. = 2.446 in. (2) Ans. 109. How man)' centimeters long is a pendulum swinging 80 times a minute ? A pendulum swinging 30 times a minute ? The first pendulum must be (f^)* of 39.138 in. = 22.014 in. 2.54x22.014 =55.92«'». (1) Ans The second pendulum must be (f^)^ of 39.138 in. = 156.552 in. 2.54 X 156.552 = 397.64«'". (2) Am. 110. If a cannon-ball be suspended by a fine wire 176 ft. long in th(! central well of the Bunker Hill Monument, how many times a minute will it swing? V2112 : V39l38 : : 60 : a;. ^_ 60x>/39.138 V2IT2 log 60 = 1.7782 log V39.13 8 = 0.7963 cologV2ll2 =8.3377-10 0.9122 = log 8.17. 8.17. ^rw. 111. Find the lifting- power of a hydraulic press, the plunger being l"" in diameter and driven with a force of lOO''*, if the lifting- piston is 1"' in diameter. Suppose the plunger and piston square : the one would press on a surface of li''"', the other on a surface of 10,000*»<'«. By driving the plunger in 10*"*', you force 10*»™ of water under the piston. But as this is spread under 10,000'>«'°, you raise the piston thereby only 0.001«=™; that is, only 0.0001 part of the way you move the plunger. Hence, by the principle of virtual velocity (what is lost in time is gained in power), you lift the piston with 10,000 times your force of 100*8 applied. The lifting power, in other words, is 1,000,000*«, or 1000*. teachers' edition. 475 112. If the pluuger is | in. in diameter, and is driven with a force of 1000 lbs., how much can it lift with a lifting-piston 4 ft. in diameter ? • i in. : 4 ft. = 1 : 96, 12 : 962 == 1 : 9216. 9216 X 1000 lbs. = 9,216,000 lbs. Ans. 113. If the plunger is 2 in. in diameter, and is driven with a force of 1000 lbs., how much can it lift with a lifting-piston 2 ft. in diameter ? 2 in. : 2 ft. : : 1 : 12, 12 : 122 = 1 : 144 144 X 1000 lbs. = 144,000 lbs. Ans. 114. The water stands in a fissure in a rock 10°^ high and 12™ long. What pressure is exerted to split the rock on the lowest meter's width ? on the highest meter's width ? in the whole fissure ? 1 X 12 X 9.5 = 114«'"^ = 114 t. (1) Ans. 1 X 12 X 0.5 = e^""* = 6 t. (2) Ans. 10 X 12 X 5 = 600«^°^ = 600 t. (3) Ans. 115. A dam is 100 ft. long and 10 ft. deep, and the water is just flowing over it. What pressure is exerted over the lowest two feet of the dam ? 2 X 9 X 100 = 1800 cu. ft. 1800 X 62^ lbs. = 112,500 lbs. 112,500 -^ 2000 = 56| t. Ans. 116. What velocity in meters a second will a cannon-ball acquire in falling three-quarters of a second ? in falling three and a quarter seconds ? f of 9.806'° = 7.3545'«. (1) Ans. 31 of 9.806'° = 31.869'". (2) Ans. 117. How long will it take a leaden ball, rolling off a table 29 in. high, to reach the floor? 29 in. = 0.7366'°. 4.903 : 0.7366 = 12 : x\ ^=a/4: 0.7366 903* 476 ARITHMETIC. log 0.7366 = 9.8673 -10 colog 4.903 =9.3095-10 9.1768 - 10 10.0000 - 10 2) 19.1768-20 0.3876 sec. Ans. 9.5884 - 10 = log 0.3876. 118. What velocity will a crowbar attain in falling endwise from a balloon 2000'n high ? How long will it be in coming down ? 4.903"' : 2000'" - 12 : a;2. X /2000 \4.903 log 2000 = 3.3010 colog 4.903 = 9.3095 - 2)2.6105 10 1.3053 = log 20.2. 20.2 sec. (2) Ans. 20.2 X 9.806"' =1 98.08 12". 198"° per sec. (1) Ans. 119. What velocity will a crowbar attain in falling endwise from a balloon one mile and a quarter high ? How long will it be com- ing down ? 4.903"' = 16.086 ft. 1^ mi. = 6600 ft. 16.086: 6600 = 'l»:r». ^600" V 16.086 log 6600 = 3.8195 colog 16.086 = 8.7936 - 10 2)2.6131 20.26 see. (2) Ans. 1.3066 - log 20.26. 2 X 20.26 X 16.086 ft. = 651.80 fl. 651.8 ft. per sec. (1) Aita. . teachers' edition. 477 120. If Carisbrook Well is 210 ft. deep, how long after a pebble is dropped will it be before it is heard to strike the bottom, if its velocity is reckoned at 32 ft. at the end of a second, and the velocity of sound is 1120 ft. a second? 16:210=P^ X = V-W- log 210 = 2.3222 colog 16 = 8.7959-10 2 )1.1181 0.5591 = log 3.623. 3.623 sec. in falling. yYa^V =" 0,188 sec. sound in rising. 3.623 + 0.188 = 3.811 sec. Ans. 121. On the same suppositions as in Ex. 120, how long after a pebble is dropped will it be heard to strike the bottom of a ventilating shaft 1600 ft. deep ? 16 : 1600 = 12 : a;2 X = V^ff^ = VlOO = 10 sec. 10 + 1.429 = 11.429 sec. Ans. 122. If a pebble is dropped over a precipice, and is heard to strike the bottom in 7^ sec, how far has it fallen, on the same suppositions asm Ex. 120? Assume 745 ft. and 750 ft. to be the distance. 16 : 745 = P : a;2 16:750 = P:a;'» x = VW x=V^~ log 745 = 2.8722 log 750 = 2.8751 colog 16 = 8.7959-10 colog 16 = 8.7959-10 2)1.6681 2)1.6710 0.8341 0.8355 = log 6.825. = log 6.847. ,7^Vtt = 0.665. ,7,a,o, = 0.670. 6.825 + 0.665 = 7.49 sec. 6.847 + 0.670 = 7.51 7 sec. An error of -0.01. An error of + 0.017. Difference between errors is 0.027. 478 ARITHMETIC. Error of 745 : 5 = 0.01 : 0.027. Error of 745 : 5 = 10 : 27. Error of 745 X 27 = 50. Error of 745 = f ^ = 1.8. 745 + 1.8 = 746.8 ft. 747 ft. Ans. 123. A pebble dropped down a shaft is heard to strike the bottom in 3 sec. after it begins to fall. Find the depth of the shaft. Assume 130 and 140. 16 : 130 = 12 : x\ 16 : 140 = P : a;2 X = V^^^. ^=vw. log 130 =2.1139 log 140 = 2.1461 2)0.9098 2)0.9420 0.4549 0.4710 colog 16 = 8.7959-10 colog 16 = 8.7959 - 10 = log 2.851. = log 2.958. t¥2'V = 0-116 sec. 1^17 = 0.125 sec. 5.851 +0.116 = 2.967 sec. 2.958 +0.125 = 3.083 sec. An error of - 0.033. An error of + 0.083. Difference between errors is 0.117. Error of 130 : 10 = 0.034 : 0.117. Error of 130 : 10 = 34: 117. Error of 130 = '\f^ 2.9 ft. 130 + 2.9 = 132.9 ft. 133 ft. Am. 124. How long will it take a ball, rolling off a table, to drop 1«"? 1 in.? 10<"«? 6 in.? 4 903m . icm _ 12 . ^7 4.903 : 0.01 = l:x\ ^=^V4.903' 4.903™ : 1 in. = 1« : x". 4.903 : 0.0254 = P : x\ ^ /0.0254 "~V4.903- log 0.01 =8.0000-10 colog4.903 = 9.3095 -10 log0.0254 = 8.4048 -10 colog 4.903 =9.3095-10 2)17.3095-20 8.6548-10 = log 0.04517. 0.04517 sec. (1) Ana. 2)17.7143-20 8.8572 - 10 = log 0.07198. 0.07198 sec. (2) Ans. teachers' edition. 479 4.903°» : 10«™ = l''^ : .t2. 4.903"' : 6 in. = 1^ : x\ 4.903 : 0.1 = P : x\ 4.903 : 0.1524 = 1^ : x^. \ 4.903 \ 4.903 log 0.1 = 9.0000 - 10 log 0.1524 = 9.1830 - 10 colog 4.903 = 9 3095 - 10 colog 4.903 = 9.3095 - 10 2 )18.3095-20 2 ) 18.4925 - 20 9.1548-10 9.2463 -JO = log 0.1428. = log 0.1763. 0.1428 sec. (3) Ans. 0.1763 sec. (4) Ans. 125. With what velocity will water flow through a hole 9 ft. below the surface? \/9 : Vl6 = 3:4. f of 32 = 24 ft. Ans. 126. With what velocity will water leave a fountain having free play, and a head of 25 ft.? a head of 100 ft. ? V25 : VT6 =5:4. I of 32 = 40 ft. (1) Ans. VIOO : Vl6 = 10 : 4 = 5 : 2. f of 32 = 80 ft. (2) Ans. 127. If a hole in the side of a cistern 4 ft. below the surface of the water is delivering 10 gals, an hour, how many gallons would it deliver with 5 ft. more head ? \/4 : V9 = 10 : x. 2 : 3 = 10 : a;. x = -%^- = 15 gals. Ans. 128. If a pipe 2 in. in diameter, and 1 ft. long, inserted in a dam, the head of water being kept constant, delivers 4 gals, a minute, how many gallons a minute may be expected when another pipe of the same length, but 2^ in. in diameter, is substituted for the two- inch pipe? 22 : (2^)2 = 4 : X. 4 : 6^ = 4 : a;. x = 6^ gals. Ans. 129. If a one-inch pipe, 20 in. long, is substituted for the two- inch pipe, 1 ft. long, in Ex. 128, and the flow is found to be 5 pts. a minute, what part of the diminution is due to the smaller area of the orifice, and what part to the increased friction on the sides of the longer pipe ? 22 : 12 = 4 : a;. 4 : 1 = 4 : a;, a; = 1 gal. 4-1=3 gals. (1) Ans. 1 gal. = 8 pts. 8-5 = 3 pts. (2) Ans. 480 ARITHMETIC. 130. A miller is using water flowing through the gate-way under 4 ft. head. How much more work could he do if the head was raised to 9 ft. ? how much more if the head was raised to 25 ft. ? V#: \/93 = 1 : X. Vl^: VW = 1 : X. V64 : V729 = 1 : a. V64 : V 15625 = 1 : x. 8: 27 = 1: a;. 8: 125=1: a;, a; = Y = 3|. (1) Ans. x = J-f^ = 15f . (2) Ans. 131. If a top 3 in. in diameter is making 200 revolutions a second, with what force does the outer layer pull away from the centre ? 1.227 X i X 200-^ = 6135. 6135 times the weight of the material. Am. 132. If a sling 30 in. long contains a stone, and is whirled round 80 times a minute, what is the force pulling on the string? 30 in. = 2\ ft. 1.227 X 2^ X {l\f = 5.453. 5,453 times weight when even with the hand. 5.453 — 1 = 4.453 times weight when highest. 5.453 + 1 = 6.453 times weight when lowest. 133. With what force does a locomotive running at 30 mi. an hour, on a curve of 800 ft. radius, bear against the outer rail ? 30 mi. per hr. = \ mi. per min. = y^^ mi. per sec. = 44 ft. per sec. Diameter is 1600 ft. 3.1416 X 1600 = 5026.56 ft. circumference. 44 Hence, the locomotive makes revolutions per sec. . 5026.56 *^ Force = 1.227 X 800 x \^5026.56y log 1.227 = 0.0889 log 800 = 2.9031 log 44» = 3.2870 colog 5026.56' =» 2.5974 - 10 log force = 8.8764 - 10 = log 0.07523. 0.07523 times weight. Am. TEACHEES' EDITION. 481 134. If washed wool is put wet into a wire basket 1.2'" in diam- eter, and the basket is set to spinning at the rate of 180 revolutions a minute, with what force is water wrung out of the wool ? 180 revolutions per min. = 3 revolutions per sec. 4.025x0.6x32 = 21.735. 21.735 times its weight. Ans. 135. If steel pens are revolved in a basket 32<"" in diameter, 17 revolutions a second, with what force is the oil drained from them? 4.025x0.16x172 = 186.116. 186.116 times weight. Ans. 136. How strong a horizontal pull on a chain, weighing half a pound to the yard, is required to make the lowest part curve with an 18-in. radius? with a 6-ft. radius? 18 in. = i yd. 6 ft. = 2 yds. I yd. weighs I lb. 2 yds. weigh 1 lb. .-. tension = I. (1) Ans. .'. tension = 1 lb. (2) Ans. 137. A |-in. rope, weighing ^ of a pound to the yard, is fastened at one end to a staple, and near the other end, on the same level, runs over a pulley, and has a 25-lb. weight hung to it. What is the radius of its curvature at the middle ? The horizontal tension is 25 lbs., which represents 100 yds. of rope. .•. radius = 100 yds. Ans. 138. A shower wets the rope of Ex. 137, and increases its weight 40%; what does its radius now become? The weight of the rope being |f^ = 1 of what it was, it takes only f of 100 yds. = 71f yds. Ans. 139. A steam-tug, in attempting to move a ship, straightened her hawser until the radius of the lowest point was 1980 ft. The rope was wet, and weighed 3 j lbs. to the yard. With what force was it stretched ? 1980 ft. = 660 yds., and 3^ x 660 = 2145 lbs. tension. .4??.«. 482 ARITHMETIC. 140. A chain 31 ft. long hangs between points on a level, and sags 4 ft. ? What is the radius at the lowest point ? ^ _ (^ch.XBag)ach.-sag) _ (1 5.5 + 4)(1 5.5-4) 2 sag 2x4 = ^^""^ ^ ^^-^ = 28.031 ft. Am. 141. The whole chain, in Ex. 140, weighs 18 lbs. What is the horizontal tension? What is the distance of the points apart? What is the slant, or batter, of the end of the chain ? T== weight of B. 1 ft. weighs || lb. Tension - if X 28.031 = 16.27 lbs. (1) Am. From Prop. III. log ^ span == log 19.5 + log 11.5 + log (1.29 - 1.0607) -t- colog4 + 0.0612. 1.29 - 1.0607 = 0.2293. log 19.5 = 1.2900 log 11.5 = 1.0607 log 0.2293 = 9.3604 -10 colog 4 = 9.3979 - 10 0.0612 1.1702 = log 14.79. 2x14.79 = 29.58. (2) Am. Batter = ^^ - 1.808. (3) Am. 15.5 142. A chain weighing l''^ to the meter is suspended from points on a level; the length of chain is 31", and it sags 1.3™. Find all the conditions, and find how much it falls below a level at 10°° from each end. i ch. - 15.5». sag = 1.3°. Radius - gch.-fsa^X^ch.-Bag) ^ 16.8 x 14.2 2 sag 2.6 = 91.75™ (1) Am. Tension - 91. 75''K. (2) Am. teachers' edition. 483 From Prop. III. log 16.8 = 1.2253 log 14.2 = 1.1523 log 0.0730 = 8.8633 -10 colog 1.3 = 9.8861 - 10 0.0612 1.1882 = log 15.42. 2 X 15.42 = 30.84 span. (3) Ans. ' Batter = ^^ = 5.92. (4) Ans. 15.5 ^ ^ For drop with hypotenuse of 10<=™ 6 : 10 : : 1 : a;. X = 1.667 depression. For drop with batter of 10, 5.92 : 10 : : 1 : «. a; =1.689. 1.689 depression. (5) Ans. 143, A chain 100" long, weighing 14 oz. to the foot, is suspended from points on a level 80'*' apart. What is the sag, the batter at the ends, and the horizontal tension? Assume that the sag is 26.5™. log I span = log (^ ch. + sag) + log (^ ch. — sag) + log [log(^ ch. + sag) — log (2 ch. — sag)] + colog sag + 0.0612 = log (50 + 26.5) + log (50 - 26.5) + log [log (50 + 26.5) - log (50 + 26.5)] + colog 26.5 + 0.061 2 = log 76.5 + log 23 5 + log (log 76.5 - log 23.5) + colog 26.5 + 0.0612. log 76.5 = 1.8837 log 23.5=1.3711 log0.5126 = 9.7098 -10 colog 26.5 = 8.5768 - 10 0.0612 ' 1.6026 = log 40.05. I span = 40.05. An error of + 0.05. 484 ARITHMETIC. Assume that the sag is 26.6"». log ^ span = log 76.6 + log 23.4 -f log (log 76.6 - log 23.4) + colog 26.6 + 0.0612. log 76.6 = = 1.8842 log 23.4 = = 1.3692 log 0.5150: = 9.7118 colog 26.6 = = 8.5751 0.0612 1.6015 = log 39.95. 1 span = : 39.95. Error of 26.5 : 0.1 = 0.05 :0.1 An error of - - 0.05. .-. error of 26.5 = 0.05. 1 sag = 26.5 + 0.05 = 26.55. Am. R-^ (50 + 26.55)(50 - 2e 53.1 1.55) 76.55x23.45 .^.^ ^^ 53.1 Batter = R 33.806 \ ch. 50 = 0.6761. Am. I" = 3.28 ft. 1 ft. weighs | lb. Tension = 33.806 x 3.28 x | lb. = 97 lbs. = 44kK Am. 144. Suppose the points of suspension in Ex. 143 to remain un- changed, and the chain to be shortened 5". What does the tension become ? Assume that the sag is 22.5. ^ log \ span = log 25 + log 70 + log (log 70 - log 25) + colog 22.5 + 0.0612. log 25 = 1.3979 log 70 = 1.8451 log 0.4472 = 9.6505 -10 colog 22.5 - 8.6478 - 10 0.0612 1.6025 = log 40.04. \ span = 40.04. An error of + 0.04. Assume that the sag is 22.6. log \ span = log 70.1 + log 24.9 + log (log 70 - log 24.9) + colog 22.6 + 0.0612. teachers' edition. 485 log 70.1 = 1.8457 log 24.9 = 1.3962 log 0.4489 = 9.6527 - 10 colog 22.6 = 8.6459 - 10 0.0612 1.6017 = log 39.96. I span = 39.96. Error of 22.5 : 0. 1 = 0.04 : 0.08. An error of — 0.04. Hence error of 22.5 is 0.05. .-. sag is 22.5 + 0.05 = 22.55°>. ^ _ 24.95 X 70.05 45.1 24.95 X 70.05 X 3.28087 X 14 45.1 X 16 log 24.95 = = 1.3971 log 70.05 = = 1.8454 log 3.28087 = = 0.5160 log 14 = = 1.1461 colog 45.1 = = 8.3458 - -10 colog 16 = = 8.7959 - -10 2.0463 = log 111.3 lbs. r= 111.3 lbs. Ans. 145. How long a rope is required between points 100 ft. apart to sag 30 ft.? 20 ft.? 10 ft.? Assume that the chain is 120.9 ft. log \ span = log (60.45 + 30) + log (60.45 - 30) + log [log (60.45 + 30) - log (60.45 - 30)] + log 30 -h 0.0612 = log 90.45 -f- log 30.45 -h log (log 90.45 - log 30.45) + colog 30 + 0.0612. log 90.45 = 1.9564 log 30.45 = 1.4836 log 0.4728 = 9.6747 -10 colog 30 = 8.5229 - 10 0.0612 1.6988 = log 49.99. \ span = 49.99. An error of - 0.01. 486 ARITHMETIC. Assume that the chain is 121 ft. log i span = log (60.5 + 30) + log (60.5 - 30) + log [log (60.5 + 30) - log (60.5 - 30)] + colog 30 + 0.0612 = log 90.5 + log 30.5 + log (log 90.5 - log 30.5) + colog 30 + 0.0612. log 90.5 = 1.9566 log 30.5 - 1.4843 log 0.4723 = 9.6741 -10 colog 30 = 8.5229 - 10 0.0612 1.6991 = log 50.01. ^ span = 50.01. An error of + 0.01. Error <>f 120.9 : 0.1 = 0.01 : 0.02. .-. error of 120.9 is 0.05. .-. chain is 120.9 + 0.05 = 120.95 ft. (1) Ans. Assume that the chain is 109.9 ft. log I span = log (54.95 + 20) + log (54.95 - 20) + log [log (54.95 + 20) - log (54.95 - 20) J + colog 20 + 0.0612 = log 74.95 + log 34.95 + log (log 74.95 - log 34.95) + colog 20 + 0.0612. log 74.95 = 1.8748 log 34.95 = 1.5435 log 0.3313 = 9.5202 -10 colog 20 = 8.6990 - 10 0.0612 1.6987 = log 49.97. i span = 49.97. An error of - 0.03. Assume that the chain is 110 ft. log i span = log (55 + 20) + log (55 - 20) + log [log (55 + 20) - log (55 - 20)] + colog 20 + 0.0612 = log 75 + log 35 + Iog(log75-log35)+colog20 r 0.0612 log 75 = 1.8751 log 35 = 1.5441 log 0.3310 = 9.5198 -10 colog 20 = 8.6990 - 10 0.0612 1.6992 -log 50.02. teachers' edition. 487 An error of + 0.02. Error of 109.9 : 0.1 = 0.03 : 0.05. Error of 109.9:0.1 = 3.5. .-. error of 109.9 = ^ = 0.06. 5 /. chain is 109.9 + 0.06 = 109.96. (2) Am. Assume that the cnain is 102.5 ft. log ^ span = log (51.25 + 10) + log (51.25- 10) + log [log (51.25 + 10) - log (51.25 - 10)] + colog 10 + 0.0612 = log 61.25 + log 41.25 + log (log 61.25 - log 41.25) + colog 10 + 0.0612. log 61.25 = 1.7872 log 41.25 = 1.6154 log 0.1718 = 9.2350 -10 colog 10 = 9.0000 - 10 0.0612 i.6988 = log 49.98. I span = 49.98. An error of - 0.02. Assume that the chain is 102.7 ft. log I span = log (51.35 + 10) + log (51.35 ~ 10) + log [log (51.35 + 10) - log (51.35 - 10)] + colog 10 + 0.0612 - log 61.35 + log 41 .35 + log (log 61.35 - log 41.35) + colog 10 + 0.0612. log 61.35 = 1.7879 log 41.35 = 1.6165 log 0.1714 = 9.2340 colog 10 = 9.0000 - 10 0.0612 1.6996 = log 50.07. ^ span = 50.07. An error of + 0.07. Error of 102.5 : 0.2 = 0.02 : 0.09. Error of 102.5 : 0.2 = 2 : 9. .-. error of 102.5 = f of 0.2 = 0.044. Hence the chain is 102.5 + 0.044 = 102.544 ft. (3) Ans. 488 ARITHMETIC. 146. If 4 cu, in. of iron weigh 1 lb. avoirdupois, what is the weight of 1 cu. in. in grains ? What is the specific gravity of the iron? 1 lb. = 7000 gr. .'. 1 cu. in. of the iron weighs \ of 7000 = 1750 grs. (1) Ans. (1728 -^ 4) X 16 = 6912 oz. 6912 ^ 1000 = 6.912. (2) Am. 147. If 4 cu. in. of iron weigh 1 lb., what is the diameter of a 6-lb. ball? ofa32-lb. ball? F= 4 X 6 = 24 cu. in. F- 32 x 4 = 128 cu. in. V = 0.5236 D^. V= 0.5236 D^. 24 = 0.5236 i>». 128 = 0.5236 D^. i>=;/: 2i_. i> = ;/-12L. 5236 ^0.5236 log 24 = 1 .3802 log 128 = 2.1072 colog 0.5236 = 0.2810 colog 0.5236 = 0.2810 3 ) 1.6612 3 )2.3882 0.5537 0.7961 = log 3.578. = log 6.253. 3.578 in. (1) Ans. 6.253 in. (2) Ans. 148. If a block of iron (with all its corners square) measures 17.36 in. by 8.7 in. by 1.76 in., what does it weigh at \ lb. to the cubic inch? and what would be the diameter if cast into a ball, if 11% is allowed for waste ? \ of 17.36 X 8.7 X 1.76 = 66.45 lbs. (1) Ans. 265.82 - 0. 1 1 of 265.82 = 236.58. log 236.58 = 2.3739 colog0.5236 = 0.2810 3 )2.6549 0.8850 =. log 7.673. 7.673 in. (2) Ans. 149. Answer the same questions as in the last examplo, for a block of which the dimensions are 71.4 in. by 8J in. by 3J in. 71.4 x3J=. 238. 238 X 8f = 2062.67. I of 2062.67 = 515.67 lbs. (I) Aiut. teachers' edition. 489 log 2062.67 = 3.3145 log 0.89 = 9.9494 - 10 colog 0.5236 = 0.2810 3 )3.5449 . 1.1816 = log 15.19. 15.19 in. (2) Ans. 150. What is the diameter of a cylinder 11 in. long that holds 2 gais. '!' 2 gals. = 462 cu. in. F=i7ri)2xA. 462 = 0.7854 Z)2x 11. / 462 . / 42 i>=V ^^^ =v ^0.7854x11 ^' 0.7854 log 42 = 1.6232 colog 0.7854 = 0.1049 2)1.7281 0.8641 = log 7.313. 7.313 in. Ans. 151. What is the diameter of a cylinder 9 in. long that holds 2 gals. ? 462 = 0.7854 i)2x 9. ^ 0.7854 '2 log 462 = 2.6646 colog 0.7854 = 0.1049 colog 9 = 9.0458 - 10 2)1.8153 0.9077 = log 8.086. 8.086 in. Ans. 490 ARITHMETIC. 152. What is the diameter of a cylinder SO'"* long that holds It? 10' = lOOOO^"™. 10000 = 0.7854 i)2x 30. ^^ /lOOO ^lO.?^ ■854 X 3 log 1000 = 3.0000 colog 0.7854 = 0.1049 colog 3 = 9.5229 - 10 2)2.6278 1.3139 = log 20.6. 20.6««». Am. 153. What is the circumference of a globe if the square centi- meters of its surface are three times the cubic centimeters of its vol- ume? F= 0.5236 i)». /S'= 3.1416 i)2. 3.14162)2- 3 X 0.5236 Z)». Divide both sides by 3 x 0.5236 D^. 2 = 1). Hence, circumference = 2 x 3.1416 = 6.2832<'™. Ans. 154. Find the diameter of a circle of which the number of inches in its circumference is equal to the square feet of its area. Area = ^ir D^ sq. ft. Circumference = ir i) ft. or 12irZ) in. 12irD = \irD\ Multiply each side by 4. 48TrI) = irD\ Divide each side by tD. 48 = i). .-. i> = 48 ft. Ans. 155. How many times does a carriage-wheel 3 ft. 2 in. in diame- ter turn in going a mile on a smooth road ? The circumference of the wheel is 3J X 3.1416 = 9.9484 ft. 1 mi. = 5280 ft. 5280 .-. the wheel turns - — - — = 530.7 timos. Ans. 9.9484 teachers' edition. 491 156. The top of a wheel is at each instant moving with twice the velocity of the carriage, and is moving in a curve whose centre, at the instant, is as far below ground as the point is above ground. What, then, is the force exerted to separate the mud from the top of a wheel 3 ft. 2 in. in diameter, when the carriage is moving at the rate of 10 mi. an hour ? The carriage going at 10 mi., the top of the wheel goes at 20 mi. per hour, or 29i ft. per second. The radius of the curve is 2 X 3| = 6|- ft., the total circumference of which would be 12f X 3.1416 = 39.7936 ft. .-. the force is 1.227 X 6i X l ^l'lll^ y= 4.224 times the weight. log 1.227 = 0.0889 log 6i = 0.8016 log (291)2 _ 2.9348 colog (39.7936)2 = 6.8004 - 10 0.6257 = log 4.224. Ans. 157. A point in the tire, as a spike-head, moves, while the wheel rolls over once, just four times the diameter of the wheel. How far does the point travel while the wheel, 3 ft. 2 in. in diameter, travels Imi.? In going one mile the wheel turns 530.7 times. .-. the point travels 3i x 4 X 530.7 = 6722.2 ft. Ans. 158. An oil-can is formed of two cylinders connected by a frustum of a cone. The upper cylinder, or neck, is 6"^ in diameter, and 75™°* high ; the lower cylinder is 13°™ in diameter, and 153™™ high ; the total length ©f the can is 30"™. Find its capacity in liters. A square shaft to enclose the neck would contain 6 X 6 X 7.5 = 270««™ ; one to enclose the body would contain 13 X 13 X 15.3 = 2585.7««'«. The frustum to enclose the remainder would contain (\/6 X 6 X 13 X 13 -f 169 -f 36) I of 7.2 = 679.2««™ (270 + 2585.7 + 679.2) x 0.7854 = 2776'«=™ = 2.776V Ans. 492 ARITHMETIC. 159. A common tunnel is formed of a frustum of a cone terminated with a cylinder. .The height of the frustum is 14*=™, and the diam- eters of the two bases are 175™°* and 16^'^ respectively. The cylinder is 8°™ long. Find the capacity of the tunnel in liters. IQmm ^ 1.6cm. The cylinder holds l.G^ x 8 X 0.7854 = 20.48 x 0.7854««. The frustum holds ^(17.5' + 1.6^ + 17.5 x 1.6) x 0.7854 = 1571.78 X 0.7854««''°. Both together hold (1571.78 + 20.48) x 0.7854<'«°' = 1592.20 x 0.7854«=™ = 1250«»» = 1.25i. Am. 160. A pan is in the form of a frustum of a cone. The interior measurement is 10"™ deep, 12°™ across the bottom, and 23*=™ across the top. Find the capacity of tlie pan in liters. Y(122 + 232 ^ 12 X 23) X 0.7854 = ^(144 + 529 + 276) x 0.7854 = -V- of 949x0.7854 = 3163^ X 0.7854 = 2484.5«'"> = 2.48451. ^^s, 161. A stove-pipe is 4™ long, 26"™ in diameter, and 1™™ thick. Find how many square centimeters of sheet-iron it has taken to make it, if the edges lap one centimeter ; and give the weight of the pipe, if the specific gravity of the sheet-iron is 7.8. 4m _ 4()Qcin . Imm _ Q 1cm S = TBh = (3.1416 X 26 + 1) X 400 = 33,072.64«»<^. (1) Ans. 0.1 X 33,072.64 = 3307.26''"™. 7.8 X 3307.26 = 25,796.659* = 25.8*«. (2) Ans. 162. A spherical bomb is 32*™ in diameter, and the sides 38™™ thick. The specific gravity of the metal of which it is made is 7.2. Find its weight and interior capacity. Inside diameter is 32 - 2 x 3.8 = 24.4<»». V" 0.5236 B^ = 0.5236 x 24.4». log 5236 = 9.7190 -10 log 24.48 = 4.1622 3.8812 = log 7607. 7607°™ = 7.60?. (1) Am. teachers' edition. 493 F= 0.5236 X 323. log 323 _ 4 5153 logO.5236-9.7J90- 10 4.2343 = log 17,150. 17 150ccm _ 17 151 7.2 X (17.15- 7.607) = 68.71''«. (2) Ans. 163. The diameters of a lamp-shade are 25*'™ and 7°™; its slant height IS 134™*". Give its curved surface in square centimeters. 134mm ^ 13.4cm 25 I 7 — ^^^- = 16, average diameter. 2 ^ 3.1416 X 16 X 13.4 = 673.6 high visible? 2000™ high? y/lb X 1000 = \/l5000 = 122.5'^'». (1) Ans. Vl5 X 2000 = V30000 = 173.2'^«'. (2) Ans. 174. How far can a man see from the shore, if he stands on a bluff that raises his eye ll"" above the sea? Vl5x 11 = Vl65 = 12.85J^°^. ^718. 175. If in steaming away from a mountainous island a sailor esti- mates his distance at 171^"", when the island disappears beneath the wave, how high shall he estimate the mountains ? 2 X log 171 = 4.4660 colog 152 == 8.8239 - 10 3.2899 = log 1950. 1950™. Ans. 496 ARITHMETIC. 176. The flash of a gun is seen 7J sec. before the report of the gun is heard ; there is no wind, and the thermometer is 73° F. How far off was the gun ? 73-57 = 16. 16 + 1114 = 1130 ft. per sec. 7i X 1130 = 8475 ft. Ans. 177. A meteor was seen to burst; the report followed in 4 min. 17 sec. What was its distance if the average temperature of the intervening air was 50° F. ? 4 min. 17 sec. = 257 sec. 57 - 50 = 7. 1114-7 = 1107 ft. per sec. 257 X 1107 ^ 5280 = 53.88 mi. Ans. 178. Find the distances of the last two examples by § 461. 73° F. = 22.8 C. 22.8 - 21 = 1.8. 1.8 X 0.53 = 0.95. 7^345 + 0.95) = 2594.6» = 2.5946^". (1) Ans. 50° F. = 10° C. 0.53 X (21 - 10) = 5.83. 257 X (345 - 5.83) = 87.166«» = 87.\G^«^. (2) Ans. 179. How long will it take for an explosion at the equator to be heard at the antipodes of the place, if the circumference of the earth at the equator be reckoned at 40,000'"*', and the average temperature at the equator at 23° C. ? 0.53x(23-21) = 1.06». 345 + 1.06 = 346.06"» = 0.34606^ 20,000 -*- 0.34606 = 57,793 sec. 57,793 sec. = 16 hrs. 3.2 min. Ans. 180. If an explosion at the equator occurs at sunset, and Ihe aver- age temperature east of the spot is 22° C, and that to the west 24° C, how far from the antipodes would the sounds meet? teachers' edition. 497 The velocity at 22° C. is 345.53«> per sec. The velocity at 24° C. is 346. 59°^ per sec. 692. 12°* per sec. is the velocity with which the two sounds are approaching each other. 692.12 : 346.59 : : 40,000 : X. ^^3_4a59x40000j^^_ g3^ 692.12 20,030.63 - 20,000 = 30.63>™. Ans. 181. How far off is the lightning when the thunder follows in 13 eec, the temperature being 76° F. ? 76 -57 + 1114 = 1133 ft. 13 X 1133 -^ 5280 = 2.79 mi. Am. 182. How long would it take sound to go through a whispering- tube 3 mi, long, temperature 61° F. ? 61 -57 + 1114 = 1118 ft. 3 mi. = 3 X 5280 = 15,840 ft. 15,804 ^ 1118 = 14.16 sec. Ans. 183. Sound travels in iron about 10| times as fast as in air. How long, then, after seeing the blow of a sledge-hammer given on the other end of an iron pipe 1| mi. long, may I expect to hear the sound by the iron ; and how long after, to hear the sound through the air in the pipe ; thermometer 63° F. ? 63 - 57 + 1114 = 1120 ft. per sec. li mi. = 7920 ft. 7920 -^ 1120 = 7.07 sec. through air. (1) Am. 1.01 -h 10.5 = 0.67 sec. through iron. (2) Ans. 184. Two gunners fire at each other simultaneously from forte 1^ mi. apart ; the wind, at 70° F., blows steadily from one fort to the other, at 11 mi. an hour. How soon will each hear the report of the other's gun ? Suppose one ball flies on the average 987 ft. a second, the other 818 ft. a second ; when will each receive the other's shot? 70 - 57 + 1114 = 1127 ft. per sec. li mi. = 7920 ft. 11 mi. per hr. = 16.1 ft. per sec 498 ARITHMETIC. The velocity of the sound with the wind will be 1127 + 16.1 = 1143.1. The velocity of the sound against the wind will be 1127-16.1 = 1110.9. 7920 -f- 1143.1 = 6.93 sec. (let sound). Am. 7920 -^ 1110.9 = 7.13 sec. (2d sound). Am. 7920 -^ 987 = 8.02 sec. (1st ball). Am. 7920 -5- 818 = 9.68 sec. (2d ball). Am. 186. Sound travels in water about 4.26 times as fast as in air. How many seconds sooner would the sound of a torpedo exploded under water 2 mi. off reach you by water than by air, at 68° F. ? 68 - 67 + 1114 = 1125 ft. per sec. 2 mi. = 10,560 ft. 10,560 -5- 1125 = 9.387 sec. in air. 9.387 -^ 4.26 - 2.203 sec. in water. 9.387 - 2.203 = 7.18 sec. Am. 186. Required the expense of painting the walls and ceiling of a room 22 ft. 6 in. long, 13 ft. 6 in. wide, and 10 ft. high, at 30 cents a square yard. 2(22^ + 13^) X 10 = 720 sq. ft., walls. 22^ X 13^ = 303.75 sq. ft., ceiling. ^(720 -f 303.75) x $0.30 - $34.13. Am. 187. In what time would a cistern be filled by three pipes whose diameters are \ in., f in., and 1 in., when the largest alone would till it in 40 min. ; the rates of flow being proportional to the squares of the diameters ? The smallest alone would fill it in (J)" of 40 min. = ItH) min. The other alone would fill it in {jY of 40 min. = 71^ min. Hence, in 1 min., the largest fills -^^ of the cistern, the smallest fills yj^ of the cistern, the other fills ^|^ of the cistern, and all three together fill f\j + j^j^ + ^ J^ = ^/^ of it. Hence, it will take 1 -*- ^^"^ = 22| min. Ans. teachers' edition. 499 188. How many gallons of water are contained in a length of of 50 yds. of a canal, if its width at the top is 8 yds. and at the bottom 7 yds., and its depth 5 ft. ? The average width is 7|- yds. = 270 in. ; 50 yds. = 1800 in. ; 5 ft. = 60 in. Volume = 1800X60X270 ^ ,26.233.8 gals. An.. 231 ^ 189. Find the weight supported by each of two men, A and B, who carry a hundred-weight suspended on a pole 6 ft. long, at a point 2 ft. 3 in. from A's end, if the pole weighs 16 lbs. 100 4- 16 = 116 lbs., total weight. 6 - 2.25 = 3.75 ft. from B's end. 6 : 3.75 = 100 : X. ^^100x^75 = 62.5 lbs. 6 62.5 + 8 = 70.5, A. (1) Ans. 116-70.5 = 45.5, B. {2) Ans. 190. A man who rows 4 mi. an hour in still water takes 1 hr. 12 min.^o row the same distance up a river. How long will it take him to row down again ? In still water, the man could row 4 X 1.2 = 4.8 miles in 1 hr. 12 rain. Hence, the stream bears him down 0.8 mi. in 1.2 hrs., or flows at the rate of | miles per hour. When he rows with the stream he will row 4f miles per hour, and row 4 miles in f hr. = 51f min. Ans. 191. How long a ladder will be required to reach a window 40 ft. from the ground, if the distance of the foot of the ladder from the wall is one-fifth of the length of the ladder. Let I be the length of the ladder. \l\a the distance of the foot from the wall. The height of the wall is ^Z^ - ^^Y- But 40 is the height of the wall. Therefore Jp-ILX= 40. 500 ARITHMETIC. Square both sides, f | ^ = 1600. Divide both sides by f^, P = 16 66.67. .-. I = Vl666.b7 = 40.8 ft. Arts. 192. If 3 oz. of gold 15 carats fine are mixed with 7 oz. 12 carats fine, what will be the fineness of the compound ? What must be the fineness of 11 oz. so that, when added to this compound, the whole may be 14 carats fine ? 3 X 15 = 45 7 X 12 = 84 10)129 12.9 carats. (1) Ans. 14 -12.9 = 1.1 carats. 1.1 X 10 - 11 carats. 11 -5- 11 + 14 = 15 carats. (2) Ans. 193. Find the surface of each face of a cube whose volume is 14 cu. ft. 705.088 cu. in. 14 cu. ft. 705.088 cu. in. = 24,897.088 cu. in. 24897.0881 29.2 8 3 X 202 = 1200 3(20x9)= 540 9»= 81 16897 508088 1821 [_16389 3 X 290» = 252300 3(290x2)= 1740 22= 4 508088 254044 (29.2)2 = 852.64 sq. in. Am. 194. What must be the length of spermaceti candles | of an inch in diameter, that 6 of them may weigh a pound, if the specific gravity of spermaceti is 0.943 ? teachers' edition. 501 V= 0.7854 X - X A X 6 cu. in. 1728 1 lb. is the weight of en. in. of spermaceti. ^ 0.943 X 62.5 ^ Hence 0.7854 x - X A X 6 ^^^^ 82 0.943 X 62.5 ^ _ 1728 X 82 0.7854 X 72 X 6 X 0.943 x 62.5 log 1728 = 3.2375 log 82 = 1.8062 colog 0.7854 = 0.1049 colog 72 = 8.3098 - 10 colog 6 = 9.2218 - 10 colog 0.943 = 0.0255 colog 62.5 == 8.2041 - 10 0.9098 = log 8.124. 8.124 in. Ans. 195. A cylinder 10 in. across and 10 in. high contains 0.3927 of a cubic foot of water. How many shot 0.1 in. in diameter must be poured in to raise the water to the top ? The volume of the cylinder is 0.785398 X 102 x 10 = 785.398 cu. in. 0.3927 cu. ft. = 678.5856 cu. in. 785.398 - 678.5856 = 106.8124 cu. in. 0.13 X 0.5236 = 0.0005236 cu. in., vol. of 1 shot. 106.8124 ^ 0.0005236 = 203,997. Ans. 196. Determine the depth of conical-shaped wine-glasses 2^ in. across the top, that 60 of them may hold a gallon. ■^jj of 231 = 3.85 cu. in., the capacity of a glass. 2.5 X 2.5 X 0.7854 = 4.90875, the area of the top. 3 X _M^ = 2.353 in. Ans. 4.90875 502 ARITHMETIC. 197. A train approaching a station at the rate of 25 mi. an hour passes over two signals placed half a mile apart. Find the interval between the times at which their reports are heard at the station, sound travelling at the rate of 1090.2 ft, per second. 25 mi. an hr. is ^ mi. in 1^ mm. = 72 sec. When the train reaches the second signal, the sound of the first signal is 72 X 1090.2 - 2640 ft. ahead of it ; and this distance, divided by 1090.2, is 72 sec. - 2.4 = 69.6 sec. A7is. 198. The apparent intensity of light varies inversely as the square of its distance. Find the point between two lamps 50 yds. apart at which one appears twice as bright as the other. The distances are in the proportion VT : V2 = 1 : 1.4142, and 2.4142 : 50 = 1 : a:. x= 20.71 yds., smaller distance. (1) Ans. 50 — 20.71 = 29.29 yds., greater distance. (2) Ans. 199. How deep may a round cistern 4 ft. across be made so as to be lined with the same lead as a cubical cistern 4 ft. each way? Compare their capacities. 4 X 4 X 4 = 64 cu. ft. in cubical cistern. 5 X 16 = 80 sq. ft. of lead to line it. 0.7854 X 16 - 12.5664 sq. ft. in bottom of round cistern. 80 - 12.5664 = 67.4336. ^^•^■'^^^ = 5.366. (1) Ans. 4x3.1416 16 X 0.7854 X 5.366 = 67.43 cu. ft. in round cistern. (2) An.. 200. The material for lining a cubical cistern cost $10; find tiie cost of material for lining two similar cisterns which shall each hold one-half as much. The cost is proportional to ( v^)' : ( v^)' = 1» : 0.7937« = 1 : 0.63. 1:0.63 = $10:0;. _ $10x0.63 1 2x16.30 = 112.60. Ans. = $6.30. TEACHERS EDITION. 503 201. What distance can be seen from the top of a mountain 4 mi. high? 4 mi. = 21,120 ft. |log 21,120 =2.1622 0.1215 192.2 mi. Ans. 2.2837 = log 192.2. 202. If 5 excavators sink a circular shaft 8 ft. in diameter and 125 fathoms deep in 100 dys. of 10 hrs. each, how many nights of 7 hrs. each will 4 excavators be in sinking a shaft 6 ft. in diameter and 75 fathoms deep ; if the difficulty of working by night is one-seventh greater than by day, and the hardness of the ground in the smaller shaft is to that in the larger shaft as 7 is to 5 ? 5 3 ^^ 100 : X. ^X6X6XJ^X10X^X8XTXZP^ ^X^x;^^xyx7x^ = 96f nights. Ans. 203. Find the length of a pendulum that oscillates 80 times a minute. . 802 : 602 :: 39.138 : ». 42 : 32 :: 39.138 : a;. 16 : 9 = 39.138 : x. 4 5 82 62 125 75 7 10 7 8 5 7 9x39.138 16 22.015 in. Ans. 204. A man whose weight is 160 lbs., wishing to raise a rock, leans with his whole weight on a horizontal crowbar 5 ft. long, which is propped at the distance of 4 in. from the end in contact with the rock. Find what force he exerts on the rock, and what pressure the prop has to sustain, if the weight of the crowbar is not reckoned. 4 : 56 = 160 : a;. 40 ^^^lii^ = 2240 lbs. {l)Ans. 2240 + 160 = 2400 lbs. (2) Ans. 504 ARITHMETIC. 205. A child weighing 56 lbs, is at one end of a plank 16 ft. long, and a child weighing 72 lbs. is at the other end. Find the distance of each child from the fulcrum when the plank is used for a see-saw. 56 : 72 = 7 : 9. 9 ft. and 7 ft. Ans. 206. In a pair of nut-crackers if the nut be placed at a distance of 1 in. from the hinge, and the hand presses at a distance of 8 in. from the hinge, find the pressure upon the nut for every ounce of pressure exerted by the hand. 1 : 8 = 1 oz. : what ? 8 oz. Ans. 207. A substance is weighed from both arms of a false balance, and its apparent weights are 9 lbs. and 4 lbs. Find the true weight. 4 : a: = a; : 9. a;^ = 36. a; = 6 lbs. Am. 208. A rope passes over a single pulley. How much force is re- quired to raise 180 lbs. attached to one end of a rope if 1% of the force applied is required to overcome friction ? -V^ of 180 = 181.82 lbs. Ans. 209. A cask of lamp-black weighing 160''k is attached to a rope wound on an axle 19°™ in diameter ; at one end of the axle is a wheel 175°™ in diameter. With what force must a man pull down on a rope passing over the wheel to raise the cask if 1 J% of the pull is required to overcome friction ? 175 : 19 = 160^<5 : X. 32 1^2