CASE II. 176. To multiply an integer by a fraction. MENTAL EXERCISES, 1. Multiply 12 by . SOLUTION.-12 multiplied by one equals 12, hence 12 multiplied by equals of 1%, which is 3, and 12 multiplied by t equals 3 times 3, which are 9. 2. 16 by : 6. 30 by 3. 3, 25 by . 7. 40 by 21. 4. 35 by 4. 8. 50 by 3. 5. 26 by $. 9. 60 by 46. 10. In multiplying 12 by $, by what do we divide 12, and by what do we multiply the quotient ? 11. How then may we multiply a whole number by a fraction ? OPERATION. WBITTEN EXERCISES. 1. Multiply 7 by A. SOLUTON.—7 multiplied by one equals 7, hence ? multiplied by t equals 1 of 7, which is 7, and 7 multi 7X=4 plied by $ equals 4 times }, wbich equals 4. Therefore, etc. 2. Multiply 28 by 7. OPERATION. SOLUTION.-Multiplying 28 by 3 and dividing by 5, 28 we have 28x } equals 164; and then multiplying 28 by 7, we have 196; adding 196 to 163, we have 2124. 5)84 Therefore, etc. 164 196 2124 Ans. NOTE.—This method of multiplying by a mixed number is more convenient than the one usually presented. Rule.--Multiply the integer by the numerator of the mul tiplier, and divide the product by the denominator ; or divide first and then multiply. Multiply 8. 9 by 1. Ans. 6. 9. 144 by 11 . Ans. 102. 4. 10 by Ans. 8. 10. 284 by 48 Ans. 2761 5. 18 by . Ans. 16. 11. 256 by 78. Ans 2032. 6. 27 by 4. Ans. 194. 12. 180 by 2544 Ans 46274 7. 28 br 34. Ans. 100. 13. 196 by is of Ans. 1101 8. 180 by 154. Ans. 2844. 14. 648 by Hof H. Ans. 3704 CASE II. 177. To multiply a fraction by a fraction. MENTAL EXERCISES. 1. What is the product of 4 by f? SOLUTION.- multiplied by one equals $, hence & multiplied by quale } of $, which is , and f multiplied by equals 3 times of, which is H. Therefore, etc. What is the product 2. Off by ? 6. Of 21 by 811 3. Of } by it? 7. Of 7? by ? 4. Of Lo by 15? 8. Of 131 by H? 5. Of 74 by 1%? 9. Of 5} by 41? 10. In multiplying f by $, what do we multiply together for the aumerator of the product and what for the denominator ? 11. How then do we multiply a fraction by a fraction ? WRITTEN EXERCISES, 1. What is the product off by /? SULUTION.multiplied by one equals }, hence & OPERATION. multiplied by } equals of t, which is us, and mul ** tiplied by } equals 7 times , which is . Therefore, etc. SOLUTION 20.-times equals of 5 times }; 5 times 7 equals y (Prin. 1); and f of equals (Prin. 3). Rule.- Multiply the numerators together and the denomi nators together, cancelling common factors. NOTR.-Practically this case is the same as finding a fractional part of a fraction, but theoretically the two o cases are entirely distinct. What is the product of 21 2. fby ? 63 Ans. 8. Hx4xH? Ans. 141 . 3. 14 by 45? Ans. 9. 4*** x 1? Ans. 1. t. H by 4? Ans. 74. 10. 121 81? Ans. 1045 6. $*$*$$? Ans. H. 11. 174 x 164? Ans. 290. 8. HX28? Ans. 18. 12. 27 x 215? Ans. 5954 7. * < 19 x ? Ans. 78. 13. 394 x 33%. Ans. 1320, 14. Multiply of by 14 of 29. Ans. 15. Multiply Hof H by Hof 11 Ans. 16. Multiply 1 of 2 by H4 of 1728 Ans. MENTAL EXERCISES. 1. How much will 44 yards of muslin cost, at 6 cents & yard ? 2. If you give to 8 boys each 23 apples, how many apples will it take ? 3. How many dollars does that man give away who gives to each of 10 beggars of a dollar? 4. How much will 15 inkstands cost at the rate of 2 inkstands for } of & dollar? 5. What will 12 Letter-writers cost at the rate of 8 for % of a dol. 6. If a man can dig a rod of ditch in f of a day, how many roda can he dig in 45 days? 7. Multiplying by 4 is the same as taking what part of a number? 8. Show that a fractional part of an integer equals the product of the integer by the fraction. 9. What will be the cost of 2} yards of cloth at $37 & yard ? 10. What will7+ bushels of shelled corn cost at 25 cents a bushel ? 11. A lady bought 44 dozen spoons at $6% a dozen; what did they cost her? WRITTEN EXERCISES. Required the cost of 1. 96 lb. of paper, at 2% cts. per pound ? Ans. $2.72. 2. 475 lb. of beef, at 64 cts. per pound ? Ans. $32.067. 8. 130 yd. of muslin, at 184 cts. a yard ? Ans. $24.37}. 4. 387 sheep, if 1 sheep cost 8f dollars ? Ans. $33281 5. 180 tons of coal, at $10.62; per ton? Ans. $19121. 6. 7294 lb. of sugar, at 6 cts. a pound ? Ans. $43.78. 7. 87 barrels of flour, at $6% a barrel ? Ans. $56. 8. 6f yards of cloth, at $5% a yard ? Ans. $33/t. 9. 34 bushels of apples, at $f a bushel ? Ans. $221. 10. 24 handkerchiefs, at 874 cents apiece ? Ans. $21. 11. 6f tons of hay, at $67 a ton ? Ans. $464). 12. 8f cords of wood, at $45 a cord ? Ans. $3813. 13. 15 barrels of vinegar, at $45 a barrel ? Ans. $714. 14. 164 quarts of beans, at $0.18% a quart? Ans. $3.094. 16. What is the value of jfXHFXXXIf ? Ans. fig. 16. What is the value of tXmX41X? Ans. str. 17. Valae of HiXWX444X 7044 ? Ans. stila NOTR.-In examples like 15, 16, 17, and 18, resolve the numerators and denominators into their factors and then cancel 415 18. Value of xH xHfx874x194? Ans. H 19. Value of (1+) $+(14+8)x3? Ans. 31. 20. Value of (++4)*(+3)x 74 x 198? Ans. 3%. 21. Value of 77X4+(41–37)(59+6)? Ans. 23448 22. Find the value of 73211-(33}+371) diminished by (64+61) x 498. Ans. 2005 23. Mr. Fish rented a house at $424 a month, taking a lease for 5 years; but disposed of the lease at the end of 31 years; how much rent did he pay? Ans. $16674 24. A bill of books at retail amounts to $376%, but I get a. reduction of f for wholesale and % for cash; what was the exact amount of the bill? Ans. $23547 25. Mr. Alden bought 524 bushels of wheat at 624 cts. per bushel, and sold } at 694 cts., and the remainder at 714 cts. per bushel; what did he clear by the transaction ? Ans. $4.431. DIVISION OF FRACTIONS. 178. Division of Fractions is the process of dividing when one or both of the terms are fractional. 179. There are Three Cases: 1st. A fraction by an integer; 2d. An integer by a fraction; 3d. A fraction by a fraction. CASE I. 180. To divide a fraction by an integer. MENTAL EXERCISES. 1. How many times is 4 contained in ? SOLUTION.-One is contained in it, & times; and if 1 is contained in the A times, 4 ls contained in Art of I times, or 4 times. Therefore, etc. How many times is 4 2. 3 contained in ? 6. 4 contained in 88? 3, 5 contained in ? 7. 6 contained in 12%? 4. 6 contained in ? 8. 7 contained in 6% ? 5. 7 contained in 1? 9. 9 contained in 1716? 10. Since 1:45h, how may we divide a fraction by a number that will divide the numerator? 11. Since 14--4=4 how may we divide a fraction by a number that will not divide the numerator? WRITTEN EXERCISES. OPERATION. OPERATION. OPERATION. 1. Divide H by 5, and also by 6. SOLUTION.—Dividing the numerator of If by 5, we have f:5 equals (Prin. 2). 11+5= SOLUTION 2D.-Multiplying the denominator of H4 by 6, we have 1f:-6 equals 7%, or j's (Prin. 3). 14:6=*=sh 2. Divide 8224 by 4. SOLUTION.-Dividing 822 by 4, we have 205 and a remainder of 2; 2 equals , which, added to $, equals 4)8223 ; divided by 4 equals 17; hence the quotient is 20511 20617 Rule.—Divide the numerator or multiply the denominator of the dividend by the divisor. NOTE.-Reduce & mixed number to a fraction; or divide the hoteger, unite the remainder with the fraction, and divide the result. Divide 3. by 3. Ans. %. 10. 785% by 7. Ans. 1123 4. A by 4. Ans. 1. 11. 287 by 12. Ans. 2334 5. H by 36. Ans. 6. 12. 1298 by 16. Ans. 86 6. of 1 by 7. Ans. 1. 13. 878 by 15. Ans. 588 7. f of it by 9. Ans. † 14. 1284 by 26. Ans. 4933 8. f of 74 by 18. Ans. 15. 16. 13474 by 30. Ans. 4444 9. 241 by }of 34. Ans. 15. 16. 28927 by 36. Ans. 80417 888 CASE II. 181. To divide an integer by a fraction. MENTAL EXERCISES, 1. How many times is contained in 3 ? SOLUTION.- One is contained in 3, 3 times; and if 1 18 contained in 8, 8 times, 1 is contained in 3, 4 times 3 times, which are 12 times; and is contained in 3, $ of 12 times, or 4 times. Therefore, etc. How many times is 2. f contained in 69 6. ij contained in 39 3. $ contained in 8? 7. 14 contained in 7? 4. / contained in 7? 8. 21% contained in 46? 5. & contained in 11? 9. 1% contained in 30? 10. In dividing 12 by f, by what do we multiply 12, and by what do we divide the product ? 11. How then may we divide a whole number by a fraction witb. out the analysis ? |