28. Multiply 672 by . 29. Multiply 710 by 7. 30. Multiply 765 by 1. 31. Multiply 660 by 2. 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 number must be divided by the fraction, instead of being multiplied by it. 34. What is of 457 ? 35. What is 8 of 16245? 36. What is 44 of 25000 ? 37. What is of 4261? 41. Multiply 64 by 51. Operation. 2)64 51 320 32 Ans. 352. Ans. 266. 38. What is 44 of 5268? 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 products together. (Art. 214.) 42. Multiply 83 by 74. Ans. 5973. 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 307. 56. Multiply 457 by 12. 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 7, which is of a dollar; for, multiplying the denominator, divides the value of the fraction. (Art. 188.) Now, of a dollar, 4 of a bushel will cost 4 times are 28, or if of a bushel costs as much and 4 times dolls. (Art. 195.) Ans. 7 of a dollar. Or, we may reason thus: since 1 bushel costs of a dollar, of a bushel must cost of 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 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 process of multiplying one fraction by another similar? 5. Multiply by 44. Ans. 1. 6. Multiply 35 by 48. 8. Multiply by 188. 9. What is the product of into into into into ? 10. What cost 63 yards of cloth, at 42 dollars per yard ? Analysis.-42 dollars=, and 6 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; 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 62. 13. Multiply 13 by 177. 14. Multiply 153 by 20§. 15. Multiply 305 by 44. 16. Multiply 632 by 50%. 17. Multiply 17 by 2514. 18. Multiply 473 by 1712. 19. Multiply 617 by 3234. 20. Multiply 7134 by 45. 21. Multiply 83% by 6135. 22. Multiply 964 by 7284. 35. What cost 125 36. What cost 250 Operation 63 41 26 31 30 dolls. 23. Multiply 246 by $1. of 1000. by 3 of 28. of 2 of 85. of 24 by 3 of 61. 31. Multiply of 10 by 3 of 84. 32. Multiply of 161 by § of 93. 33. Multiply of of 20 by 254, 34. Multiply 23 of 651 by 461. bbls. of flour, at 73 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? 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. X X = 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. QUEST.-221. How are fractions multiplied by cancelation? Obs. How does it appear that this process will give the true answer? What is necessary to be observed with regard to canceling factors? PROOF.-6 dolls. ×3=20 dolls., the same as before. * Note.-In like manner, when the multiplier is 331, 3331, &c., if we multiply by 100, 1000, &c., of the product will be the answer. Hence, 222. To multiply a whole number by 31, 33, 3331, &c. Annex as many ciphers to the multiplicand as there are 3s in the integral part of the multiplier; then take of the number thus produced, and the result will be the answer required. OBS. 1. The reason of this contraction is evident from the principle that annexing a cipher to a number multiplies it by 10, annexing two ciphers multiplies it by 100, &c. (Art. 98.) But 3 is of 10; 33 is of 100, &c.; therefore annexing as many ciphers to the multiplicand, as there are 3s in the integral part of the multiplier, gives a product 3 times too large; consequently of this product must be the true answer. 2. When the multiplicand is a mixed number, and the multiplier is 31, 331 &c., it is evident we may multiply by 10, 100, &c., as the case may be, and § of the number thus produced will be the answer required. 18. Multiply 158 by 331. Ans. 5266. 22. Multiply 297 by 3331. 23. Multiply 5611 by 31. 24. Multiply 4263 by 331. 223. To multiply a whole number by 63, 66, 666, &c. Annex as many ciphers to the multiplicand as there are 6s in the integral part of the multiplier; then take of the number thus produced, and the result will be the answer required. OBS. The rea on of this contraction is manifest from the fact that 63 is of 10; 663 is of 100, &c. 25. What will 6 tons of iron cost, at 75 dollars per ton? QUEST.-222. How may a whole number be multiplied by 34, 33, &c.? 223. How may a whole number be multiplied by 63, 6fg, &c. |