15. Multiply 451 by 10. Ans. 451%. 19. Multiply 1273 by 35. 20. Multiply 48 by 47. 21. Multiply 25012 by 50. 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, 3 of 3 a ton will cost as much. Now, 1 fourth of 21 4 4' or 153 dollars. 23. Multiply 136 by 1. 24. Multiply 432 by 1. 25. Multiply 635 by . Ans. 45. 26. Multiply 360 by . QUEST.-215. How is a whole number multiplied l y a fraction? 216. How find a fracHional part of a number? 28. Multiply 672 by §. 29. Multiply 710 by 7. 30. Multiply 765 by H. 31. Multiply 660 by 2. 46 216. 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 12X-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 num. ber must be divided by the fraction, instead of being multiplied by it. 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 ? 50. Multiply 685 by 42. 26 51 Multiply 125 by 10. 52. Multiply 26 by 101? 53. Multiply 256 by 17%. 54. Multiply 196 by 4111. 55. Multiply 341 by 301. 56. Multiply 457 by 1231. 57. Multiply 107 by 47. 58. Multiply 510 by 8513. 59. Multiply 834 by 89. 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 3 of a dollar per bushel: how much did he pay for it? Analysis. Since 1 bushel costs 0 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, if of a bushel costs of a dollar, 4 of a bushel will cost 4 times as much; and 4 times 40 0 are 28, or dolls. (Art. 195.) Ans. 7 of a dollar. since 1 bushel costs of a dollar. Now 0 of a dollar, of is a com 8 Or, we may reason thus: 4 of a bushel must cost of pound fraction, whose value is found by multiplying the numera tors together for a new numerator, and the denominators for a new denominator. (Art. 198.) Solution.—3×4=28, or 1% 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 therefore 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 procese of multiplying one fraction by another similar? 9. What is the product of into § into 1 into into? 10. What cost 6 yards of cloth, at 42 dollars per yard? Analysis.--4 dollars, and 6 yards=20. (Art. 197.) Now & X=18, or 30. (Art. 196.) Ans. 30 dollars. Hence, 6 220. 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 ; 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 81 by 64. 13. Multiply 13% by 177. 1.5 28 21 bbls. of flour, at 74 dollars per barrel? acres of land, at 254 dollars per acre? 37. If a man travels 40 miles per day, how far will he travel in 135 days? QUEST.-20. When the multiplier and multiplicand are mixed numbers, how proceed CONTRACTIONS IN MULTIPLICATION OF FRACTIONS. Ex. 1. Multiply by and rand and . Operation. N 7 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 de nominators; 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 ex actly equal to those canceled in the denominators. 16 5 13. Multiply 1 by 4 and 74 and 1 and . 6 12 10 14. Multiply into 4 into into into & into 4. 14 6 3 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 appea that this process will give the true answer? What is necessary to be observed with ro gard to canceling factors? |