14. From a hogshead of wine there leaked out 70 gallons; what quantity remained ? Ans. 55 i gal. 15. A man engaged to labor 30 days, but was absent 572 days; how many days did he work? 16. From 144 pounds of sugar there were taken at one time 175 pounds, and at another 2813 pounds; what quantity remains ? Ans. 973 lb. 17. A man sells 95 yards from a piece of cloth containing 34 yards ; how many yards remain ? Ans. 24gyd. 18. The distance from Boston to Providence is 40 miles. A, having set out from Boston, has travelled 14 of the distance ; and B, having set out at the same time from Providence, has gone 11 of the distance; how far is A from B ? Ans. 2814 m. 19. From of a square yard take ţ of a yard square. 233. To subtract one fraction from another, when their numerators are alike. Ex. 1. From take 7. Ans. . • OPERATION. 7. - 3 4, difference of the denominators. 7 x 3 2 1, product of the denominators. We first find the product of the denominators, which is 21, and then their difference, which is 4, and write the former for the denominator of the required fraction, and the latter for the numerator. By this process the fractions are reduced to a common denominator, and their difference found. Hence, to subtract one fraction from another, whose numerators are a unit, we may Write the difference of the denominators over their product. 2. Take & from . Ans. 1 OPERATION. Difference of the denominators mul- , Ans. Product of the denominators, 3 x 7 21 We multiply the difference of the denominators by one of the numerators for a new numerator, and the denominators together for a new denominator, by which process the fractions are reduced to a common denominator, and the difference of their numerators is found. Hence, when the given fractions have their numerators alike and greater than a unit, we may Write the product of the difference of the given denominators, by one of the numerators, over the product of the denominators. ܕܘܶ EXAMPLES 3. Take from 1, 3, , , , ; I'y from 1', 16, I's. 4. Take from, 4, 5; I's from , 3, 5, 7, $. 5. Take } from }, }, }; } from }, }, }, ž. 6. Take it from to, , }, 4, 5, 5, 5, 5, $. 7. Take } from }; 1 from 1, $ ; f from š. 8. Take from 1, , , , , , , , Ib, IT: 9. Take 1 from 1, 1, , }, , , , , , i. 10. Take To from 1, 5, 4, 5, 6, , 5, s. 11. From take ji is take to ; I's take 16; I'y take zł. 12. Take z from s; from $; 1 from t; it from š. 13. Take from #; t from ; t from #; from . 14. Take from ; f from ; Á from $; A from po. 15. Take i from $; i from si fos from fi is from f. 16. Take fs from ; t from *; from $; from tt. 17. Take A from $; i from $; from ; i from f. 18. Take from ; * from g; Í from ş; from $. 19. Take 1: from 42; 18 from 14; Ifrom 1$; 14 from 1t: 20. Take 12 from 12; 12 from 1; from ; tfrom 42. MISCELLANEOUS EXAMPLES IN ADDITION AND SUB TRACTION OF FRACTIONS. 1. A benevolent man has given to one poor family is of a cord of wood, to another } of a cord, and to a third of a cord; how much has he given to them all ? Ans. 2cords. 2. I have paid for a knife $ ;, for a Common School Arithmetic $ 1, for a slate $ }, and for stationery $ Ğ ; what did I pay for the whole ? 3. R. Howland travelled one day 2017 miles, another day 197 miles, and a third day 22 16 miles ; what was the whole distance travelled ? Ans. 623 miles. 4. I have bought 6.1 tons of anthracite coal, 194 tons of Cumberland coal, and 3 tons of cannel coal; what is the whole quantity purchased ? Ans. 303-5 tons. 5. There is a pole standing in the mud, in the water, and the remainder above the water; what portion of it is above the water? 6. F. Adams, having a lot of sheep, sold at one time of them, and at another time d of the remainder; what portion of the original number had he then left ? 7. From a piece of calico containing 314 yards there have been sold 11% yards, 9 yards, and 38 yards; how much remains ? 8. From a cask of molasses containing 843 gallons, there were drawn at one time 47 gallons, at another time 11 gallons ; at a third time 26 gallons were drawn, and } of 77 gallons returned to the cask; and at a fourth time 130 gallons were drawn, and 34 gallons of it returned to the cask. How much then remained in the cask ? Ans. 3531 Egal. 9. A merchant had 3 pieces of cloth, containing, respectively, 193 yards, 364 yards, and 333 yards. After selling several yards from each piece, he found he had left in the aggregate 71% yards. How many yards had he sold? Ans. 183. Ans. 1 MULTIPLICATION OF COMMON FRACTIONS. 234. MULTIPLICATION of Fractions is the process of multiplying when the multiplier, or multiplicand, or both, are fractional numbers. NOTE. — If the multiplier is less than 1, only such a part of the multiplicand is taken as the multiplier is of 1. Therefore, the product resulting from multiplying a number by a proper fraction is not larger, but less, than the multiplicand. 235. To multiply when one or both of the factors are fractions. Ans. 3 FIRST OPERATION. SECOND OPERATION. FIRST OPERATION. SECOND OPERATION. Ex. 1. Multiply 18 by 9. 31. It is evident that the fraction is X9 = f = 1 = 34 Ans. is multiplied by 9 by multi plying its numerator by 9, since the parts taken, 63, are 9 times as many as before, while the parts into which the unit of the fraction is divided remain the same. It is evident, also, that the fraction 18 X 9 = 1 = 34 Ans. is multiplied by 9 by dividing its denominator by 9, since the parts into which the unit of the fraction is divided are only as many, and consequently 9 times as large, as before, while the parts taken remain the same. Therefore, Multiplying the numerator or dividing the denominator of a fraction by any number multiplies the fraction by that number (Art. 217). 2. Multiply 14 by 4. Ans. 6. By dividing the whole number, 14, by 7) 14 7, the denominator of the fraction, we obtain of 14 = 2, which multiplied by 2 X 3 6 Ans. 3, the numerator of the fraction, gives of 14 6. By multiplying the whole number, 14, 14 by 3, the numerator of the fraction, we 3 obtain 42, a product 7 times as large as it should be, as the multiplier was not 3, 42 ; 7 = 6 Ans. a whole number, but $, or 3 ; 7; hence, we divide the 42 by 7; and thus obtain 6, as before. Therefore, Multiplying by a fraction is taking the part of the multiplicand denoted by the multiplier. 3. Multiply by Ans. 30 To multiply } by f is to take of the } x = ft Ans. multiplicand, Ž. Now, to obtain of }, we multiply the numerators together for a new numerator, and the denominators together for a new denominator (Art. 226). Therefore, Multiplying one fraction by another is the same as reducing compound fractions to simple ones. When either of the factors is not a fraction, as in examples first and second, it may be reduced to a fractional form, and then the operation may be like that in the last example. Hence the general Rule. — Reduce whole or mixed numbers, if any, to improper fractions. Multiply the numerators together for a new numerator, and the denominators together for a new denominator. NOTE. - When there are common factors in the numerators and denomina tors, the operation may be shortened by cancelling those factors. of 14 OPERATION. EXAMPLES. 4. Multiply 11 by di. Ans. . OPERATION. Ans. Ans. 170 Ans. 35 Ans. 195: 7 1 3 5. Multiply 12 by 4. Ans. 84. 6. Multiply & by 12. 7. Multiply 14 by 13. 8. Multiply by 18. 9. Multiply by Ir. 10. Multiply yz by it. 11. Multiply i by 4. Ans. 74. 12. Multiply by 18. 13. Multiply 1, by 11. Ans. 44 14. Multiply 1 by 14. Ans. 123. 15. Multiply 13 by $. Ans. 73. 16. Multiply 16 by is Ans. 219 17. Multiply 11 by 4. Ans. 64. 18. Multiply 10 by 14. 19. Multiply s by 19. Ans. 16. 20. Multiply 1 by 24. Ans. 3. 21. Multiply by 18. 22. Multiply s by 39. 23. Multiply } by 11. 24. Multiply g'g by 10%. 25. Multiply 4 of 11 of' }by 100. Ans. 127. 26. Multiply off of by 11. Ans. 33 27. What cost Iz of a ton of hay, at $ 17 per ton ? Ans. $95. 28. What cost zo of an acre of land, at $ 37 per acre ? 29. At of a dollar per foot, what cost 7 cords of wood ? 30. Multiply 161; } by 1935. Ans. 313643 31. Multiply by 83. Ans. 33 32. Multiply by 1781. Ans. 1561 33. Multiply s by 714. Ans. 6335 34. Multiply of 94 by of 17. Ans. 78%. 35. Multiply o of 7 by 1 of 8711 36. Multiply 8 hy š. Ans. 63. Ans. 8 185. Ans. Ps. Ans. Ito |