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 is contained times in 15. Reducing the dividend 15, to the form of a fraction, it becomes ; (Art. 197. Obs. 1;) then inverting the divisor and proceeding as before, we have X, 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 32 dollars per yard, can you buy for 28 dollars?

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

235. When the divisor is 31, 331, 3331, &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 3X3=10; 33×3=100; 333X3=1000, &c.

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

Operation. dolls. 561

We first multiply the dividend by 3, then divide the product by 10; for, multiplying the divisor 34 by 3, it becomes 10. (Art. 146.)

3

10)1683

Ans. 168 yds.

36. Divide 687 by 331.

37. Divide 453 by 331,

Ans. 20.

38. Divide 2783 by 3331.

236. When the divisor is 13, 16, 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, 1×6=10; 163×6=100; 1663×6=1000, &c.

39. What is the quotient of 725 divided by 16a? Solution.-725×6=4350; and 4350÷100=43

40. Divide 367 by 1.

41. Divide 507 by 16%.

Ans.

42. Divide 849 by 16.

43. Divide 1124 by 166.

237. When the divisor is 11, 111, 111, &c.

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

OBS. This contraction depends upon the same principle as the preceding. Thus, 19=10; 111×9=100; 111+9=1000, &c.

44. Divide 587 by 11.

Solution.-587×9=5283, and 5283÷100=5283 Ans.

45. Divide 861 by 14.

46. Divide 4263 by 111.

Ans. 7741.

47. Divide 6037 by 1111.

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 di41 vision of fractions. (Art. 183.) Thus, the complex fraction is the same as÷4; for, the numerator, 4, and the denominator 14-4; but the numerator of a fraction is a dividend, and the denominator a divisor. (Art. 184.) Now, ÷ 16. 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 ?

Solution.-4-14, and 74-29. (Art. 197.)

Now 1422-4X4, or 5 Ans.

Reduce the following complex fractions to simple ones:

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.

Operation.

27

125

712

92

The numerator 3. (Art. 197.) Place the 7 on the right hand and 2 on the left of the perpendicular line. The denominator 2 =12, which must be inverted; (Art. 239;) i. e. place the 12 on the left and the 5 on 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.)

915=5. Ans.

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

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.)

X =the answer. (Art. 231.)

9X4 3X7 9X4X3X7 21

But, (Art. 232,) 2X9 1X4 2X9X1X4 2 or 10 Ans.

,

APPLICATION OF FRACTIONS.

242. Ex. 1. A merchant bought 157 yards of domestic flannel of one customer, 19 of another, 12 of another, and 41 of another: how many yards did he buy of all?

2. A grocer sold 164 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, 92 dollars for pants, 63 dollars for a vest, 5 dollars for a hat, and 64 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?