Solution. 5 bushels will cost 5 times as much as 1 bushel. Now×5=5, or 21; that is, 5 times halves, equal to 2 and 1 half are 5 Ans. 2 dollars. 17. If a pound of tea cost 6 shillings, how much will of a pound cost? Solution. Multiplying by, is taking 1 third of the multiplicand twice. (Art. 132.) Now 1 third of 6 is the same as 6 thirds of 1, or; and 2 thirds of 6 must be 2 times as much; that is, x2, or 12; and 12=4. Ans. Note. Since the product of any two numbers will be the same, whichever is taken for the multiplier, (Art. 47,) the fraction may be taken for the multiplicand, and the whole number for the multiplier, when it is more convenient. Thus, ×6=12, or 4, which is the same result as before. 18. Multiply 12 by . Ans. 3. 19. Multiply 10 by 20. Multiply 15 by . 21. Multiply by 2. Ans. x2=0, or 14. Suggestion. Dividing the denominator of a fraction by any number, multiplies the value of the fraction by that number. (Art. 114.) Now, if we divide the denominator 8 by 2, the fraction will become, which is equal to 14, the same as before. Hence, 133. To multiply a fraction and a whole number together. Multiply the numerator of the fraction by the whole number, and write the product over the denominator. Or, divide the denominator by the whole number, when this can be done without a remainder. (Art. 114.) QUEST.-133. How multiply a fraction and a whole number together? Obs. 1. A fraction is multiplied into a number equal to its denominator by canceling the denominator. (Art. 89.) Thus, X7=4. 2. On the same principle, a fraction is multiplied into any factor in its denominator, by canceling that factor. (Arts. 91, 114.) Thus, 3X3-3. 15 22. Multiply by 5. 23. Multiply 4 by 9. 25. Multiply 36 by 13. 27. Multiply 35 by 25. 29. Multiply 9 by 5. Operation. 94 5 Ans. 47 25 5 times are, which are equal to 2 and §. Set down the . 5 times 9 are 45, and 2 (which arose from the fraction) make 47. Hence, 134. To multiply a mixed number by a whole one. Multiply the fractional part and the whole number separately, and unite the products. 30. Multiply 15 by 7. Ans. 1101. 31. Multiply 25 by 10. 32. Multiply 4818 by 8. 33. Multiply 24 by 34. Operation. 2)24 3 72 12 Ans. 84 We first multiply 24 by 3, then by, and the sum of the products is 84. Multiplying by is taking one half of the multiplicand once. (Art. 131.) But to find a half of any number, we divide the number by 2. (Art. 104. Obs.) Hence, 134. a. To multiply a whole by a mixed number. Multiply first by the integer, then by the fraction, and add the products together. QUEST. Obs. How is a fraction multiplied by any factor in its denominator? How by a number equal to its denominator? 134. How is a mixed number multiplied by a whole one? 134. a. How is a whole number multiplied by a mixed number? Ans. 90. 36. Multiply 75 by 123. 34. Multiply 27 by 31. 35. Multiply 63 by 103. CASE II. 37. A man owning of a ship, sold of what he owned. What part of the ship did he sell? Analysis of is; for, multiplying the denominator by any number, divides the value of the fraction. (Art. 113.) Now 2 thirds of is twice as much; that is, 12=5, which, reduced to its lowest terms, is 3 6 Again, the example may be performed thus: Ans. Since he owned 3, and sold of what he owned, it is evident he sold of of the ship. Now of is a compound fraction, whose value is found by multiplying the numerators together for a new numerator, and the denominators for a new denominator. (Art. 123.) 6 Operation. x, or 3. Ans. Hence, 135. To multiply a fraction by a fraction. Multiply the numerators together for a new numerator, and the denominators together for a new denominator. OBS. It will be seen that the process of multiplying one fraction by another, is precisely the same as that of reducing compound fractions to simple ones. 38. Multiply by 3. 39. Multiply by 3. 41. Multiply by §. Ans.. 40. Multiply by §. 136. In multiplication of fractions, the operation may often be shortened by canceling the factors common both to the numerators and denominators, as in reducing compound fractions to simple ones. (Art. 123.) QUEST.-135. How is a fraction multiplied by a fraction? Obs. To what is the process of multiplying one fraction by another similar? 136. How may the operation in multiplication of fractions be shortened? OBS. 1. This, in effect, is dividing the product of the numerators and that of the denominators by the same number, and therefore does not alter the value of the answer. (Art. 116.) 2. Care must be taken that the factors canceled in the numerators are exactly equal to those canceled in the denominators. 43. Multiply by 4. Operation. 2 1 6 4 2 The factor 3 is common to the numerator of the first fraction and the denominator of the second. Also the factor 4 is common to the denominator of the first, and the numerator of the second fraction. The product of the remaining factors in the numerator is 2; that of the remaining factors in the denominator is 9. Ans. 3. 3 3 Solution. 7, reduced to an improper fraction, becomes 15, and 3 becomes 10. Now 15×10-150, or 25. Ans. 6 137. Hence, when the multiplier and multiplicand are both mixed numbers, they should be reduced to improper fractions, and then be multiplied according to the rule above. EXAMPLES FOR PRACTICE. 1. What will 12 apples cost, at of a cent apiece? 2. If a bushel of wheat weighs of a hundred weight, how much will 10 bushels weigh? 3. If a man earns of a dollar per day, how much can he earn in 12 days? QUEST. Obs. Does this alter the answer? Why not? What is necessary to be observed with regard to canceling factors? 137. When the multiplier and multiplicand are mixed numbers, how proceed? 4. If a family consume of a barrel of flour in a week, how much will they consume in 15 weeks? 5. If I burn of a cord of wood in a month, how much shall I burn in 12 months? 6. If a man can reap of an acre of grain in a day, how many acres can he reap in 9 days? 7. If a pound of powder cost 6 shillings, how much will of a pound cost? 8. If a gallon of oil cost 7 shillings, how much will of a gallon cost? 9. When beef is 10 dollars a barrel, how much will § of a barrel cost? 10. What will of a firkin of butter cost, at 15 dollars a firkin? 11. At § of a dollar a cord, how much will the sawing of 20 cords of wood amount to? 12. What will 16 pounds of cheese cost, at 83 cents per pound? 13. What cost 9 dozen of eggs, at 12 cents per dozen? 14. What cost 15 yards of cambric, at 15 pence per yard? 15. What cost 83 cords of wood, at 2 dollars per cord? 16. At 12 cents a pound, what cost 23 pounds of per? рер 17. At 5 shillings a pound, what cost 123 pounds of tea? 18. What will 6 pounds of starch come to, at 122 cents per pound? 19. What will 18 ounces of nutmegs come to, at 6 cents an ounce ? 20. At 12 cents a yard, what will 17 yards of cotton come to ? 21. At 3 dollars a yard, what cost 16 yards of broadcloth? 22. What cost 15 yards of ribbon, at 10 cents per yard? 23. What cost 22 pocket handkerchiefs, at of a dollar a piece? 24. At a dollar a yard, what will of a yard of lace cost? |