MULTIPLICATION OF 15. Multiply 45 by 10. 16. Multiply 814 by 9. Ans. 19. Multip 17. Multiply 31 18 by 20. 20. Multip 18. Multiply 14810 by 25 21. Multip 214. Multiplying by a fraction, we hav certain portion of the multiplicand as many like portions of a unit in the multiplier. Hend To multiply by: Divide the multiplicand l To multiply by : Divide the multiplicand l To multiply by : Divide the multiplicand b To multiply by : Divide by 3, and multiply To multiply by : Divide by 4, and multiply 215. Hence, to multiply a whole number b Divide the multiplicand by the denominator, quotient by the numerator. Or, multiply the given number by the numerato product by the denominator. Obs. 1. When the given number cannot be divided by without a remainder, the latter method is generally preferre 2. Since the product of any two numbers is the same, for the multiplier, (Art 83,) the fraction may be taken fo and the whole number for the multiplier, when it is more co 22. If 1 ton of hay costs 21 dollars, how much 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 23. Multiply 136 by . 24. Multiply 432 by 1. 26. Multiply 360 25. Multiply 635 by . 27. Multiply 580 QUEST.-215. How is a whole number multiplied by a fraction? 216. tional part of a number? 28. Multiply 672 by §. 29. Multiply 710 by 7. 30. Multiply 765 by 12. 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 num. ber must be divided by the fraction, instead of being multiplied by it. 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. 56. Multiply 457 by 123. 57. Multiply 107 by 47%. 58. Multiply 510 by 8513. 59. Multiply 834 by 89. 60. Multiply 963 by 95. CASE II. 218. To multiply a fraction by a fraction. Ex 1. A man bought of a bushel of wheat, at 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, which is of a dollar; for, multiplying the denominator, divides the value of the fraction. (Art. 188.) if of a bushel costs Now, of a dollar, 4 of a bushel will cost 4 times as much; and 4 times are 28, or 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 3 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? Obe. To what is the process of multiplying one fraction by another similar ↑ 9. What is the product of into into 1 into into? 10. What cost 63 yards of cloth, at 4 dollars per yard? Analysis.--4 dollars=2, and 63 yards=2o. (Art. 197.) Now 2018, 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. Operation. 61 41 261 31 30 dolls. of 1000. 11. Multiply 63 by 214. 23. Multiply 246 by 12. Multiply 8 by 6%. 24. Multiply 9133 by 3 of 7. 13. Multiply 13 by 173. 25. Multiply 1475 by 7 of 21. 14. Multiply 15 by 20%. 26. Multiply 34 by of 68. 15. Multiply 305 by 44. 27. Multiply 800 by 16. Multiply 6321 by 50%. 28. Multiply of 75 by of 28. 17. Multiply 171% by 2517. 29. Multiply 24 by of 2 of 85. 18. Multiply 4739 by 1713. 30. Multiply of 24 by 2 of 61. 19. Multiply 617 by 3224. 31. Multiply 20. Multiply 7134 by 45. 32. Multiply of 164 by 9 of 93. 21. Multiply 83 by 6135. 33. Multiply of of 20 by 251. 22. Multiply 964 by 7234. 34. Multiply 22 of 651 ły 461. 35. What cost 125 bbls. of flour, at 73 dollars per barrel? 36. What cost 250 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? of 103 by of 84. 1 QUEST.-220. When the multiplier and multiplicand are mixed numbers, how proceed ing factors together, as in reduction of compound ple ones. (Art. 199.) Hence, 221. To multiply fractions by CANCELATION. Cancel all the factors common both to the num nominators; then multiply together the factors r numerators for a new numerator, and those rema nominators for a new denominator, as in reductio fractions. (Art. 199.) OBS. 1. The reason of this process may be seen from the fa of the numerators is divided by the same numbers as that of and therefore the value of the answer is not altered. (Art. 19 2. Care must be taken that the factors canceled in the n actly equal to those canceled in the denominators. into 24 into 16. Multiply into into into 17. What must a man pay for 33 barrels of flour, w worth 6 dollars a barrel? QUEST.-221. How are fractions multiplied by cancelation? Obs. How that this process will give the true answer? What is necessary to be ob gard to canceling factors? |