2. I have paid for a knife $ , for a Common School Arithmetic $ 1, for a slate $ }, and for stationery $ &; what did I pay for the whole ? 3. R. Howland travelled one day 20 17 miles, another day 194 miles, and a third day 22 16 miles ; what was the whole distance travelled ? Ans. 621 miles. 4. I have bought 61 tons of anthracite coal, 194 tons of Cumberland coal, and 31 tons of cannel coal; what is the whole quantity purchased ? Ans. 300 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 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, 94 yards, and 3 yards; how much remains ? 8. From a cask of molasses containing 843 gallons, there were drawn at one time 44 gallons, at another time 11 gallons; at a third time 264 gallons were drawn, and į of 71 gallons returned to the cask; and at a fourth time 138 gallons were drawn, and 34 gallons of it returned to the cask. How much then remained in the cask ? Ans. 3521 .gal. 9. A merchant had 3 pieces of cloth, containing, respectively, 19} 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, FIRST OPERATION. SECOND OPERATION. FIRST OPERATION. SECOND OPERATION. Ex. 1. Multiply 1g by 9. Ans. fl = 31. It is evident that the fraction fi = 1 = 34 Ans. iis 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 Is X 9 = 1 = 34 Ans. To 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 š To multiply by } is to take of the x}= Ans. multiplicand, . 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 denominators, the operation may be shortened by cancelling those factors. of 14 Ans. OPERATION. EXAMPLES 4. Multiply by ti. Ans. OPERATION. Ans. $ 7 11 1 Х 3 5. Multiply 12 by 4. Ans. 84. 6. Multiply by 12. 7. Multiply 14 by 12. Ans. 40 8. Multiply by 18. Ans. 9. Multiply by a Ans. uz 10. Multiply 72 by it. Ans. 14. 11. Multiply I by 4. 12. Multiply by tt. Ans. 44 13. Multiply 19 by 13. Ans. 12 14. Multiply % by 14. 15. Multiply 13 by 4. Ans. 74. Ans. 218 16. Multiply 16 by i Ans. 64. 17. Multiply 11 by 4. 18. Multiply 1% by 14. Ans. 168. 19. Multiply : by 19. Ans. . 20. Multiply 11 by 24. Ans. is 21. Multiply by 15. Ans. 23 22. Multiply by 3%. 23. Multiply } by ii Ans. Ido 24. Multiply g'g by 1987. Ans. 124. 25. Multiply of 11 of 11 by 100. Ans. 3 26. Multiply 1 of of } by 11. 27. What cost ta of a ton of hay, at $ 17 per ton ? Ans. $913 28. What cost of an acre of land, at $37 per acre ? 29. At of a dollar per foot, what cost 7 cords of wood ? 30. Multiply 161; } by 1935 Ans. 313643. Ans. 34 31. Multiply by 83. Ans. 15 4 32. Multiply by 171. 33. Multiply s by 716. Ans. 6375. Ans. 78%. 34. Multiply of 94 by of 17. 35. Multiply so of 7 by 1 of 873. Ans. 63. 36. Multiply 8 hy č. to A. 37. Multiply 12 by 4. Ans. 84. 38. Multiply 15 by it: Ans. 84 39. A merchant owning / of a ship sells t1 of his share What part is that of the whole ship? 40. Multiply 37 by 104. Ans. 398. 41. Multiply of 71 by of 114. Ans. 49131. 42. Multiply 4 of 9 by of 17. Ans. 2635 43. Multiply 4 of 87by 4 of 94. Ans. 25245 236. When one of the factors is a whole number, and the other a mixed number, we may Multiply the fractional part and the whole number separately, and add together the products. EXAMPLES. 1. Multiply 73 by 9. 2. Multiply 12 by 31. x 9 of 12 12 9 33 - 9 12 x 3 36 45 Ans. 3. Multiply 83 by 7. Ans. 60+ 4. Multiply 17 by 31. 5. Multiply 13 by 8%. Ans. 1094 6. Multiply 37 by 1341. Ans. 507 11 7. Multiply 114 by 8. Ans. 94$. 8. What cost 7.6 lb. of beef at 5 cents per pound ? 9. What cost 2373bbl. of flour at $ 6 per barrel ? Ans. $ 1411. 10. What cost 8Žyd. of cloth at $ 5 per yard ? Ans. $ 41%. 11. What cost 9 barrels of vinegar at $ 63 per barrel ? Ans. $ 57 3. 12. What cost 12 cords of wood at $ 6.371 per cord ? Ans. $ 76.50. 13. What cost 11cwt. of sugar at $ 93 per cwt. ? 14. What cost 43 bushels of rye at $ 1.75 per bushel ? Ans. $ 7.653. 15. What cost 7 tons of hay at $ 113 per ton ? Ans. $ 831 16. What cost 9 dozen of adzes at $ 10% per dozen ? 17. What cost 5 tons of timber at $3$ per ton? Ans. $ 15. 18. What cost 15cwt. of rice at $ 7.621 per cwt. ? Ans. $ 114.373. 19. What cost 40 tons of coal at $8.374 per ton ? Ans. $ 335. DIVISION OF COMMON FRACTIONS. 237. Division of Fractions is the process of dividing when the divisor or dividend, or both, are fractional numbers. NOTE. — If the divisor is less than 1, the quotient arising from the division will be as many times the dividend as the divisor is contained times in 1. Therefore, the quotient arising from dividing a whole or mixed number by a proper fraction will always be larger than the dividend. 238. The reciprocal of a fraction is the number resulting from taking its numerator as denominator, and its denominator as numerator, since any two numbers, whose product is 1, are the reciprocals of each other. Thus, the reciprocal of nQ is that fraction inverted, or , since 10 X =1. 239. To divide when the divisor or dividend, or both, are fractions. Ex. 1. Divide 14 by 7. It is evident that the fraction 14 is dii* :7= Ans. vided by 7 by dividing its numerator by 7, since the size of the parts, as denoted by the denominator, remains the same, while the number of parts taken is only & as large as before. It is evident the fraction is also di14:7= 14 = Ans. vided by 7 by multiplying its denomi nator by 7, since the number of parts taken, as denoted by the numerator, remains the same, while the size of the parts is only as large as before. Therefore, Dividing the numerator or multiplying the denominator of a fraction by any number divides the fraction by that number (Art. 217). 2. Divide 13 by 143 Ans. 2. Since the fractional units of the two 1 ; fed . = 2 Ans. fractions are of the same kind, it is evident that 12 thirteenths contain 6 Ans. ist FIRST OPERATION. SECOND OPERATION. OPERATION |