214. Multiplying by a fraction, we have seen, is taking a certain portion of the multiplicand as many times, as there are like portions of a unit in the multiplier. Hence, To multiply by: Divide the multiplicand by 2. To multiply by : Divide by 3, and multiply the quotient by 2. 215. Hence, to multiply a whole number by a fraction. Divide the multiplicand by the denominator, and multiply the quotient by the numerator. Or, multiply the given number by the numerator, and divide the product by the denominator. Obs. 1. When the given number cannot be divided by the denominator without a remainder, the latter method is generally preferred. 2. Since the product of any two numbers is the same, whichever is taken for the multiplier, (Art 83,) the fraction may be taken for the multiplicand, and the whole number for the multiplier, when it is more convenient. 22. If 1 ton of hay costs 21 dollars, how much will of a ton cost? Analysis. Since 1 ton costs 21 dollars, of a ton will cost as much. Now, 1 fourth of 21 is 21; and 2 of 21 is 3 times as much; but 21 21X3 63 = X3=4 or 15 dollars. 4' 23. Multiply 136 by . 24. Multiply 432 by 1. 25. Multiply 635 by . Ans. 45. Operation. 51 3 Ans. 154 dolls 26. Multiply 360 by 3. QUEST. 215. How is a whole number multiplied by a fraction? 216. How find a frac tional part of a number? 28. Multiply 672 by . 29. Multiply 710 by 7. 30. Multiply 765 by 11. 31. Multiply 660 by . 216. Since multiplying by a fraction is taking a certain portion of the multiplicand as many times, as there are like portions of a unit in the multiplier, it is plain, that the process of finding a fractional part of a number, is simply multiplying the number by the given fraction, and is therefore performed by the same rule. Thus, of 12 dollars is the same as the product of 12 dollars multiplied by ; and 12×=8 dollars. OBS. The process of finding a fractional part of a number, is often a source of confusion and perplexity to the learner. The difficulty arises from the erroneous impression that finding a fractional part, implies that the given number must be divided by the fraction, instead of being multiplied by it. 34. What is of 457 ? 35. What is 16 of 16245? 36. What is 4 of 25000? 37. What is 12% of 4261 ? 41. Multiply 64 by 51. Operation. 2)64 51 320 32 Ans. 352. We first multiply 64 by 5, then by, and the sum of the products is 352. But multiplying by is taking one half of the multiplicand once. (Arts. 82, 214.) Hence, 217. To multiply a whole by a mixed number. Multiply first by the integer, then by the fraction, and add the QUEST.-217. How is a whole number multiplied by a mixed number? 51. Multiply 125 by 10. 52. Multiply 108 by 20. 53. Multiply 256 by 17. 54. Multiply 196 by 411. 55. Multiply 341 by 30. 56. Multiply 457 by 1273. 57. Multiply 107 by 4731. 58. Multiply 510 by 8513. 59. Multiply 834 by 8991. 60. Multiply 963 by 9581. CASE II. 218. To multiply a fraction by a fraction. Ex 1. A man bought of a bushel of wheat, at 7 of a dollar per bushel how much did he pay for it? Analysis. Since 1 bushel costs of a dollar, of a bushel must cost of 3, which is of a dollar; for, multiplying the denominator, divides the value of the fraction. (Art. 188.) Now, Or, we may reason thus: since 1 bushel costs of a dollar, of a bushel must cost of 7 of a dollar. 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. 198.) Solution.-x=28, or 7 dollars, Ans. Hence, 219. To multiply a fraction by a fraction. Multiply the numerators together for a new numerator, and the denominators together for a new denominator. OBS. 1. 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. (Art. 198.) 2. The reason of this rule may be thus explained. Multiplying by a fraction is taking a certain part of the multiplicand as many times, as there are like parts of a unit in the multiplier. (Art. 210.) Now multiplying the denominator of the multiplicand by the denominator of the multiplier, gives the value of only one of the parts denoted by the given multiplier; (Art. 188;) we there fore multiply this new product by the numerator of the multiplier, to find the number of parts denoted by the given multiplier. (Art. 186.) QUEST.-219. How is a fraction multiplied by a fraction? Obs. To what is the process of multiplying one fraction by another similar? 10. What cost 63 yards of cloth, at 44 dollars per yard? Analysis.-4 dollars=2, and 63 yards=2o. (Art. 197.) Now X20-180, or 30. (Art. 196.) Ans. 30 dollars. Hence, 220. 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. OBS. Mixed numbers may also be multiplied together, without reducing them to improper fractions. Take, for instance, the last example. We first multiply by 4, the whole number. Thus, 4 times are equal to 2 and 3; set down the , and carry the 2. Next, 4 times 6 are 24, and 2 to carry are 26. We then multiply by, the fractional part. Thus, of 6 is 3; and of 2 thirds is . The sum of the two partial products is 30 dollars, the same as before. 11. Multiply 63 by 214. 12. Multiply 8 by 64. Operation. 61 41 261 31 30 dolls. by . of 7. 23. Multiply 246 24. Multiply 640 by 25. Multiply 1475 by 3 of 21. 26. Multiply 34 by of 68. 27. Multiply 800 by of 1000. 28. Multiply of 75 by of 28. 29. Multiply 24 by 3 of 4 of 85. 30. Multiply of 2 by of 61. 31. Multiply of 103 by 3 of 84. 32. Multiply of 164 by 4 of 91. 33. Multiply of 1 of 20 by 25. 34. Multiply 23 of 65 by 461. bbls. of flour, at 72 dollars per barrel? acres of land, at 254 dollars per acre? 37. If a man travels 403 miles per day, how far will he travel in 135 days? σ QUEST.-220. When the multiplier and multiplicand are mixed numbers, how proceed? CONTRACTIONS IN MULTIPLICATION OF FRACTIONS. Ex. 1. Multiply by and and and . Operation. -= 2 22 Since the factors 3, 5 and 8 are common to the numerators and denominators, we may cancel them; (Art. 191;) and then multiply the remain ing factors together, as in reduction of compound fractions to simple ones. (Art. 199.) Hence, 221. To multiply fractions by CANCELATION. Cancel all the factors common both to the numerators and denominators; then multiply together the factors remaining in the numerators for a new numerator, and those remaining in the denominators for a new denominator, as in reduction of compound fractions. (Art. 199.) OBS. 1. The reason of this process may be seen from the fact that the product of the numerators is divided by the same numbers as that of the denominators, and therefore the value of the answer is not altered. (Art. 191.) 2. Care must be taken that the factors canceled in the numerators are exactly equal to those canceled in the denominators. 13. Multiply 12. Multiply by and and and 25. by 4 and 74 and ♬ and 1. into into 3. 16. Multiply into into into 17. What must a man pay for 34 barrels of flour, when flour is worth 6 dollars a barrel? QUEST.-221. How are fractions multiplied by cancelation? Obs. How does it appeal that this process will give the true answer? What is necessary to be observed with regard to canceling factors? |