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Ans.
Ans. ing.
Ans. 137
Ans.
Ans. ds.
Ans. og
Ans. 53
Ans. 17g

3

EXAMPLES FOR PRACTICE. 3. Divide i by 3. 4. Divide is by 6. 5. Divide 11 by 12. 6. Divide it by 8. 7. Divide by 9. 8. Divide y by 15. 9. Divide 1: by 75.

563. 10. Divide 7 by 12.

11. John Jones owns of a share in a railroad valued at $ 117; this he bequeathes to his five children. What part of a share will each receive ?

Ans. 7. 12. Divide as by 15. 13. Divide ft by 28.

14. James Page's estate is valued at $10,000, and he has given of it to the Seamen's Society; } of the remainder he gave to his good minister; and the remainder he divided equally among his 4 sons and 3 daughters. What sum will each of his children receive?

Ans. $ 6804

Ans. Tis.
Ans. zig

OPERATION.

13 ;

Art. 160. To divide a whole number by a fraction.
Ex. 1. How many times will 13 contain 4.

Ans. 301.
For convenience, we invert the

terms of the divisor, and then 3 13 x 7 91

multiply the whole number by 7 3 3

30} the original denominator, and

divide the product by the numer

ator. The reason of this operation is evident, since 13 will contain as many times as there are sevenths in 13, equal 91 sevenths. Now if

13 contain 1 seventh 91 times, it will contain ; as many times as 91 -will contain 3, equal to 30%.

RULE. — Multiply the whole number by the denominator of the fraction, and divide the product by the numerator.

EXAMPLES FOR PRACTICE. 2. Divide 18 by }. 3. Divide 27 by 17. 4. Divide 23 by :

Ans. 204. Ans. 29-1:

Ans. 92.

QUESTIONS. — Art. 160. What is the rule for dividing a whole number by a fraction ? Give the reason for the rule.

OPERATION.

43

43

5. Divide 5 by $.

Ans. 25. 6. Divide 12 by .

Ans. 16. 7. Divide 16 by 4:

Ans. 32. 8. Divide 100 by 1%.

Ans. 1111. 9. I have 50 square yards of cloth; how many yards, { of a yard wide, will be sufficient to line it? Ans. 83) yards.

10. A. Poor can walk 311 miles in 60 minutes; Benjamin can walk iş as fast as Poor. How long will it take Benjamin to walk the same distance ?

Ans. 73} minutes. Art. 161. To divide a mixed number by a whole number. Ex. 1. Divide 173 by 6.

Ans. 245.

Having divided the whole number 6)173

as in simple division, we have a re

mainder of 5$, which we reduce to 2-53=4; 7 x6=45;

an improper fraction, and divide it by

the divisor as in Art. 159. Annexing 2+4= 24%.

this fraction to the quotient 2, we ob

tain 24. for the answer. Rule. Divide as in whole numbers, as far as the division can be carried, and, if the remainder is a mixed number, reduce it to an improper fraction, and then divide it by the divisor; but if the remainder

, is a simple fraction only, merely divide it by the divisor. (Art. 159.)

EXAMPLES FOR PRACTICE. 2. Divide 173 by 7.

Ans. 21% 3. Divide 18% by 8.

Ans. 231. 4. Divide 2713 by 9.

Ans. 31us. 5. Divide 314 by 11.

Ans. 2016. 6. Divide 784 by 12.

Ans. 617. 7. Divide 1894 by 4.

Ans. 4718 8. Divide 1071' by 3.

Ans. 351 9. Divide $14% among 7 men.

Ans. $ 245 10. Divide $1067 among 8 boys.

Ans. $ 133. 11. What is the value of 4 of a dollar.

Ans. $ 0.3413 12. Divide $ 107]among 4 boys and 3 girls, and give the girls twice as much as the boys.

Ans. Boy's share $ 104; Girl's share $ 21... 13. If $14 will purchase zz of a ton of copperas, what quantity will $ 1 purchase ?

Ans. lcwt. Oqr. 241b.

Questions. — Art. 161. What is the rule for dividing a mixed number by a whole number?

OPERATION.

43/25

Art. 162. To divide a whole number by a mixed number. Ex. 1. Divide 25 by 43.

Ans. 549. We first reduce the divisor and dividend to fifths,

and then divide as in whole numbers. 5 5

The reason why the answer is in whole numbers, 23)125(51%

and not in fifths, is because the divisor and dividend

were both multiplied by the same number, 5, and 115

therefore their relation to each other is the same as 10 before, and the quotient will not be altered. RULE. Reduce the divisor and dividend to the same parts as are denoted by the denominator of the fraction in the divisor, and then divide as in whole numbers,

Ans. 131.

EXAMPLES FOR PRACTICE. 2. Divide 36 by 97.

Ans. 355. 3. Divide 97 by 1311.

Ans. 6164. 4. Divide 113 by 217.

Ans. 514's. 5. Divide 342 by 1417

Ans. 23 31 6. There is a board 19 feet in length, which I wish to saw into pieces 24 feet long; what will be the number of pieces, and how many feet will remain ? Ans. 7 pieces, 2 feet.

Art. 163. To divide a fraction by a fraction.
Ex. 1. Divide by $.

In this operation, we invert the }==}x==131. terms of the divisor and then pro

ceed as in Art. 156. The reason of this process will be seen, when we consider that the divisor , is an expression denoting that 4 is to be divided by 9. Now, regarding 4 as a whole number, we divide the fraction } by it, by multiplying the denominator ; thus, x= Bat the divisor 4 is 9 times too great, since it was to be divided by 9, as seen in the original fraction; therefore the quotient, 32, is 9 times too small, and must be multiplied by 9; thus, 32 32

= 134. By this operation, we have multiplied the denominator of the dividend by the numerator of the divisor, and the numerator of the dividend by the denominator of the divisor. Hence the

OPERATION.

7 X 9

63

QUESTIONS. - Art. 162. What is the rule for dividing a whole by a mixed number?

How does it appear that this process does not alter the quotient? Art. 163. How do you divide a fraction by a fraction? Give the reason why this process divides the fraction of the dividend.

Rule. — Invert the divisor, and cancel all the factors common to the numerators and denominators, and then proceed as in multiplication of fractions.

Note. — When the divisor and dividend have a common denominator, their denominators cancel each other, and the division may be performed by simply dividing the numerator of the dividend by the numerator of the divisor.

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EXAMPLES FOR PRACTICE.

Ans. 136. 2. Divide ] by 4.

Ans. 33. 3. Divide ž by .

Ans. 4. Divide i} by 13.

Ans. 27. 5. Divide by i

Ans. 6% 6. Divide lo by 7.

Ans. 4. 7. Divide by it.

Ans. 6. 8. Divide it by 26.

Ans. 24. 9. Divide 13 by

Ans. 183. 10. Divide of 7 by 4 of f.

Ans. At 11. Divide of of 16 by of off.

Ans. 31 12. Divide i of of by f of of ig.

Art. 164. To divide a mixed number by a mixed number, it is only necessary to reduce them to improper fractions and proceed as in the foregoing rule. (Art. 163.)

Ans. 276. Ex. 1. Divide 74 by 37.

OPERATION.

7=*; 3=4

X =26=256.

EXAMPLES FOR PRACTICE. 2. Divide 7 by 44. 3. Divide 31 by 71. 4. Divide 117 by 58. 5. Divide 47 by 17. 6. Divide 116 by 144. 7. Divide 817 by 9}. 8. Divide of 54 of 7 by of 3%.

Ans. 138

Ans. 75
Ans. 255.
Ans. 27.

Ans. 8% 3.
Ans. 8131
Ans. 113

QUESTIONS. – What is the rule for dividing one fraction by another? How may fractions be divided when they have a common denominator ? Does this process differ in principle from the other ? - Art. 164. How do you divide a mixed number by a mixed number ?

COMPLEX FRACTIONS.
Art. 165. To reduce complex to simple fractions.

Ex. 1. Reduce

to a simple fraction.

Ans.

OPERATION.

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Ans. 17.

OPERATION.

Since the numerator of a fraction is š xf = at the dividend, and the denominator the di

visor (Art. 132), it will be seen by this operation, that we simply divide the numerator f by the denominator , as in division of fractions. (Art. 163.)

8 Ex: 2. Reduce

to a simple fraction. 41

We reduce the numerator, 8

= { x 4

= yo = 17 8, and the denominator, 41, to

improper fractions, and then

proceed as in Ex. 1. Ex. 3. Reduce to a simple fraction.

Ans. 14. of

We here reduce the = * X = H = 14

denominator, f of , to of

å simple fraction, and

then proceed as before. From the preceding illustrations we deduce the following

Rule. -- Reduce whole and mixed numbers to improper fractions, and compound fractions to simple ones, and then divide the numerator of the complex fraction by the denominator, according to the rule for the division of fractions.

EXAMPLES FOR PRACTICE.

12 4. Reduce to a whole number.

Ans. 28.

OPERATION

5. Reduce

to a simple fraction.

Ans.

14

.

47 6. Reduce

to a simple fraction. 9

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Ans. di

QUESTIONS. - Art. 165. What is the rule for reducing complex to simple fractions? How does this process differ from division of fractions ?

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