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20. Divide of by of 1 23. Divide 4 of 74 by 4 of $. 21. Divide of 24 by 64. 24. Divide of # of by 4. 22. Divide 154 by of . 25. Divide 2 of 7 by of of 42. 0,6 Divide hea of 15 of 1 of 1 by o of 3 of 4 of 5.

CASE III.

233. Dividing a whole number by a fraction.

27. How many pounds of tea, at of a dollar a pound, can be bought for 15 dollars ?

Analysis.-Since of a dollar will buy 1 pound, 15 dollars will buy as many pounds as f is contained times in 15. Reducing the dividend 15, to the form of a fraction, it becomes 16; (Art. 197. Obs. 1 ;) then inverting the divisor and proceeding as before, we have 4X4=6,0, or 20. Ans. 20 pounds.

Or, we may reason thus : is contained in 15, as many times as there are fourths in 15, viz: 60 times. But 3 fourths will be contained in 15, only a third as many times as 1 fourth, and 60-3=20, the same result as before. Hence,

234. To divide a whole number by a fraction.

Reduce the whole number to the form of a fraction, (Art. 197. Obs. 1,) and then proceed according to the rule for dividing a fraction by a fraction. (Art. 229.)

Or, multiply the whole number by the denominator, and divide the product by the numerator.

Obs. 1. When the divisor is a mixed number, it must be reduced to an improper fraction; then proceed as above.

Or, reducing the dividend to a fraction having the same denominator, (Art. 197. Obs. 2,) we may divide one numerator by the other. (Art. 229. I.)

2. If the divisor is a unit or 1, the quotient is equal to the dividend ; if the divisor is greater than a unit, the quotient is less than the dividend; and if the divisor is less than a unit, the quotient is greater than the dividend.

28. How much cloth, at 31 dollars per yard, can you buy for 28 dollars ?

Obs. How by a mixed

QUEST.-234. How is a whole number divided by a fraction ? number?

Operation.

Since the divisor is a mixed number, 31)28

we reduce it to halves ; we also reduce 2 2

the dividend to the same denominator; 7) 56 halves. (Art. 197. Obs. 2 ;) then divide one nu. Ans. 8 yards. merator by the other. (Art. 229. I.) 29. Divide 75 by .

32. Divide 145 by 12%. 30. Divide 96 by 4.

33. Divide 237 by 254. 31. Divide 120 by 10%. 34. Divide 425 by 31%.

CONTRACTIONS IN DIVISION OF FRACTIONS.

235. When the divisor is 3}, 331, 333}, &c.

Multiply the dividend by 3, divide the product by 10, 100, or 1000, as the case may be, and the result will be the true quotient. (Art. 131.)

OBs. The reason of this contraction will be understood from the principle, that if the divisor and dividend are both multiplied by the same number, the quotient will not be altered. (Art. 146.) Thus 34X3=10; 331 X3=100; 333X3=1000, &c.

35. At 3} dollars per yard, how many yards of cloth can be bought for 561 dollars ? Operation.

We first multiply the dividend by 3, dolls. 561

then divide the product by 10; for, mul3

tiplying the divisor 3} by 3, it becomes 10. 110)168|3 (Art. 146.) Ans. 168 yds. 36. Divide 687 by 337.

Ans. 2017 37. Divide 453 by 33}, 38. Divide 2783 by 3333.

236. When the divisor is 1ḥ, 163, 166, &c.

Multiply the dividend by 6, and divide the product by 10, 100, or 1000, as the case may be.

Oes. This contraction also depends upon the principle, that if the divisor and dividend are both multiplied by the same number, the quotient will not be altered. (Art. 146.) Thus, 1* X6=10; 163 X6=100; 1663 X6=1000, &c.

39. What is the quotient of 725 divided by 16+ ?
Solution.--725 X 6=4350; and 4350-100=431 Ans.
40. Divide 367 by 13. 42. Divide 849 by 163.
41. Divide 507 by 163. 43. Divide 1124 by 1663.

237. When the divisor is 13, 115, 1111, &c.

Multiply the dividend by 9, and divide the product by 10, 100, or 1000, as the case may be.

Oes. This contraction depends upon the same principle as the preceding. Thus, 14 X9=10; 114X9=100; 1114X9=1000, &c.

44. Divide 587 by 113.
Solution.—587X9=5283, and 5283-100=5280 Ans.
15. Divide 861 by 15.

Ans. 7741%. 46. Divide 4263 by 115. 47. Divide 6037 by 1114. Note.--Other methods of contraction might be added, but they will naturally suggest themselves to the student, as he becomes familiar with the principles of fractions.

238. From the definition of complex fractions, and the manner of expressing them, it will be seen that they arise from di

41 vision of fractions. (Art. 183.) Thus, the complex fraction is

17 the same as 1:1; for, the numerator, 41=), and the denominator 14=; but the numerator of a fraction is a dividend, and the denominator a divisor. (Art. 184.) Now, 1==18. which is a simple fraction. Hence,

239. To reduce a complex fraction to a simple one.

Consider the denominator as a divisor, and proceed as in division of fractions. (Arts. 229, 232.)

OBS. The reason of this rule is evident from the fact that the denominator of a fraction denotes a divisor, and the numerator, a dividend ; (Art. 184;) hence the process required, is simply performing the division which is expressed by the given fraction.

GUEST.--238. From what do complex fractions arise ? 239. How reduce them to sim. ple fractions ?

443 48. Reduce

to a simple fraction.

77
Solution.—43=14, and 71-2. (Art. 197.)
Now 14-4-4x, or $4 Ans.
Reduce the following complex fractions to simple ones:

8
49. Reduce

53. Reduce 31

51 50. Reduce

54. Reduce 7.

154 2

28 동
51. Reduce

55. Reduce
33
61

7 좋을
52. Reduce

56. Reduce

35

by

240. To multiply complex fractions together.

First reduce the complex fractions to simple ones ; (Art. 239 ;) then arrange the terms, and cancel the common factors, as in multiplication of simple fractions. (Art. 219.)

Obs. The terms of the complex fractions may be arranged for reducing them to simple ones, and for multiplication at the same time.

31 15 57. Multiply

23

44 Operation. The numerator 3 =1. (Art. 197.) Place 217

the 7 on the right hand and 2 on the left of 1215

the perpendicular line. The denominator 23 712

=1,1, which must be inverted; (Art. 239 ;) 912

i. e. place the 12 on the left and the 5 on 9/5=. Ans. the right of the line, 14=12, and 41=1, both

of which must be arranged in the same manner as the terms of the multiplicand. Now, canceling the common factors, we divide the product of those remaining on the right of the line by the product of those on the left, and the answer is . (Art. 219.)

QUEST.-240. How are complex fractions multiplied together? 241. How is one com plex fraction divided by another?

21 42 58. Multiply by

21

23

4}
59. Multiply

by
83 177

مرد | داحم

or 60. Multiply by into

믛 24 23 1} 61. Multiply

into 13 16

by

241. To divide one complex fraction by another.

Reduce the complex fractions to simple ones, then proceed as in division of simple fractions. (Arts. 229, 239.)

41 3 62. Divide

by 21 1

3X721

Now,

18

41 9

4 36 7 - 을 1 4 4 Solution.

X Х

and
212

(Art. 239.)
9 18'
14

21
36 4 36 21 756

X
18. 21

=101. Ans.
4 72
9X4

1X4 Or, since the given dividend= and the divisors

2 X9

3x7 9 X4 3x7 then

x =the answer. (Art. 231.) 2x9°1X4

9X4, 3X7 9 X4X3X7 21 But, (Art. 232,) 2X9^1X42X9XIXA 7, or 104 Ans.

12 2} 63. Divide by

64. Divide

by
41 27

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APPLICATION OF FRACTIONS. 242. Ex. 1. A merchant bought 157 yards of domestic flannel of one customer, 194 of another, 12% of another, and 41% of another: how many yards did he buy of all ?

2. A grocer sold 161 lbs. of sugar to one customer, 112; to another, and 33} to another : how many pounds did he sell ?

3. A clerk spent 263 dollars for a coat, 9. dollars for pants, 61 dollars for a vest, 57 dollars for a hat, and 61 dollars for a pair of boots : how much did his suit cost him ?

4. A man having bought a bill of goods amounting to 85% dollars, handed the clerk a bank note of 100 dollars : how much change ought he to receive back ?

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