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28. Multiply 672 by .. 29. Multiply 710 by . 30. Multiply 765 by 14.

31. Multiply 660 by
32. Multiply 840 by 46.
33. Multiply 975 by 145.

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 x š=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 I of 457 ?

Ans. 266,7. 35. What is as of 16245 ? 38. What is 144 of 5268 ? 36. What is of 25000 ? 39. What is the one of 45260 ? 37. What is the of 4261 ? 40. What is best of 452120 ? 41. Multiply 64 by 57. Operation. 2)64 We first multiply 64 by 5, then by }, and the

52 sum of the products is 352. But multiplying by 320 * is taking one half of the multiplicand once.

32 (Arts. 82, 214.) Hence, Ans. 352.

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 7.

Ans. 597. 43. Multiply 45 by 83. 47. Multiply 225 by 30. 44. Multiply 72 by 104. 48. Multiply 342 by 20%. 45. Multiply 93 by 123. 49. Multiply 432 by 35%. 46. Multiply 184 by 183. 50. Multiply 685 by 42.

QUEST.-217. How is a whole number multiplied by a mixed number 3

51. Multiply 125 by 104. 52. Multiply 108 by 2012. 53. Multiply 256 by 177. 54. Multiply 196 by 4135. 55. Multiply 341 by 30-73.

56. Multiply 457 by 124.
57. Multiply 107 by 4731.
58. Multiply 510 by 857.
59. Multiply 834 by 899.
60. Multiply 963 by 9545.

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 i bushel costs of a dollar, † of a bushel must cost of , which is no of a dollar ; for, multiplying the denominator, divides the value of the fraction. (Art. 188.) Now, if 3 of a bushel costs Lc of a dollar, $ of a bushel will cost 4 times as much and 4 times to are 4, or to dolls. (Art. 195.)

Ans. of a dollar. Or, we may reason thus: since i 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.-7x=18, or to 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 ? of multiplying one fraction by another similar ?

Obs. To what is the prores, 43 263

2. Multiply s by 4.

Ans. 192=1. 3. Multiply by to.

6. Multiply by 46. 4. Multiply 11 by 25. 7. Multiply 3? by H. 5. Multiply by 44. 8. Multiply $8 by 150. 9. What is the product of $ into into 11 into + into 1? 10. What cost 64 yards of cloth, at 41 dollars per yard ?

Analysis.--41 dollars=, and 6 yards=20. (Art. 197.) Now #X20-19, 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, Operation the whole number. Thus, 4 times ģ are $; equal to 2 and 3 ; 63 set down the , and carry the 2. Next, 4 times 6 are 24, and 2 to carry are 26. We then multiply by 1, the fractional part. Thus, t of 6 is 3; and 1 of 2 thirds is 1. The sum of the two partial products is 30 dollars, the same as before.

30 dolls, 11. Multiply 6 by 214. 23. Multiply 246-175 by 53. 12. Multiply 86 by 64. 24. Multiply 6401 by of . 13. Multiply 13} by 17%. 25. Multiply 1475 by of 21. 14. Multiply 15 by 203. 26. Multiply 343 by } of 68. 15. Multiply 30 by 44%. 27. Multiply 800 by of 1000. 16. Multiply 634} by 50ž. 28. Multiply of 75 by j of 28. 17. Multiply 17 by 2511. 29. Multiply 2 by i of of 85. 18. Multiply 4731 by 1713. 30. Multiply of 2 by i of 61. 19. Multiply 6115 by 3237. 31. Multiply of 104 by i of 81. 20. Multiply 7134 by 45.75. 32. Multiply cf 161 by 2 of 9 21. Multiply 83" by 6135. 33. Multiply 1 of Ps of 20 by 25+ 22. Multiply 9645 by 7234. 34. Multiply ** of 65} by 465. 35. What cost 1251 bbls. of flour, at the dollars per

barrel ? 36. What cost 250 acres of land, at 251 dollars per acre ?

37. If a man travels 40 miles per day, how far will he travel in 1351 days?

31

QUEST.-220. When the multiplier and multiplicand are mixed numbers, how proceed ! CONTRACTIONS IN MULTIPLICATION OF FRACTIONS.

Ex. 1. Multiply by # and i and 1 and 1.
Operation.

Since the factors 3, 5 and 8 are 1.7 7 common to the numerators and denomXX

22 inators, we may cancel them; (Art.

191 ;) and then multiply the remaining 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 numero ors 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.

2. Multiply í by and . Ans. ģ. 3. Multiply by into 7. Multiply of by to. 4. Multiply by to into . 8. Multiply by 19 of 13. 5. Multiply by } into 1. 9. Multiply 1 of 4 by to 6. Multiply 1 by of 10. Multiply 34 by 13 of 8. 11. Multiply by } and and; and 12. Multiply $ by $ and 1 and 1 and 46. 13. Multiply #% by $ and 1 and and fu 14. Multiply it into * ; into into into ao into 4. 15. Multiply is into into $1 into 7 into into H. 16. Multiply 4 into & into / into into 4 into g.

17. What must a man pay for 3} barrels of flour, when flour is worth 6 dollars a barrel ?

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 re gard to canceling factors ?

Analysis.--37 bbls. is * of 10

Operation. bbls.; now since i bbl. costs 6 dolls. 6 price of 1 bbl. dollars, 10 bols. will cost 10 times

10 as much, or 60 dollars. But we 3)60 of 10 bbls. wished to find the cost of only 3} dolls. 20

of 3} bbls. barrels, which is } of 10 bbls. Therefore if we take } of the cost of 10 bbls., it will of course be the price of 3 bbls.

PROOF.—6 dolls. X3}=20 dolls., the same as before.

Note.-In like manner, when the multiplier is 331, 333$, &c., if we multiply by 100, 1000, &c., of the product will be the answer. Hence,

222. To multiply a whole number by 31, 333, 333}, &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 34 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 3}, 33} &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 333.

Ans. 52663. 19. Multiply 148 by 31. 22. Multiply 297 by 3334. 20. Multiply 256 by 331. 23. Multiply 5611 by 31. 21. Multiply 1728 by 333. 24. Multiply 426} by 333. 223. To multiply a whole number by 63, 663, 6663, &c.

Annex as many ciphers to the multiplicand as there are 6s in the integral part of the multiplier ; then take s of the number thus produced, and the result will be the answer required.

OBs. The rearon of this contraction is manifest from the fact that 6} is of 10; 663 is į of 100, &c.

25. What will 6-3 tons of iron cost, at 75 dollars per ton ?

QCEST.-222. How may a whole number be multiplied by 31, 33}, &c. ? 223. How may a whole number be inultiplied by 67, SF3, &c.

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