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It was shown in Arithmetic, Art. XXII, that a common denominator may frequently be found much smaller than that produced by the above rule. This is much more easily done in algebra than in arithmetic.

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Here the denominators will be alike, if each be multiplied by all the factors in the others not common to itself. If the first be multiplied by e g, the second by c2g, and the third by bce, each becomes b c2 e g. Then each numerator must be multiplied by the same quantity by which its denominator was multiplied, that the value of the fractions may not be altered. The fractions then become aeg

The answer is

c2 dg. and

beeg beeg

aeg+cdg+bcef
bceg

10. Add together

11. Add together

2 ac be

and

e c

3b f 2dg

5 am
2r' 36'

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eb cf

bceg

12. Add together

2m n

3 mp

13. Add together

14. Add together, and

15. Add together

16. Add together

2 a m

17. Add together

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and 13 c d.

and 2 a c-5 b.

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3 m2

2 ar

3mn3r

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2 ac 36

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But if they are reduced to a common denominator, the numerators may be subtracted.

3ac-2e

Ans.

2bc

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sign

- was changed to +. See Art. VI example 6th

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XIX. Division of whole numbers by Fractions, and Fractions by

Fractions.

How many times is contained in 7 ?

Ans.

is contained in 7, 35 times, and is contained as many times; that is, or 113 times.

2. How many times is contained in a?

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Ans. is contained in a, 8 a times, and is contained as many times; that is, .

84

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Ans.

is contained b c times in c, and is contained

1

α

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α

Ans. If c is the part of some number,

part of some number, — will be

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a

Arith. Art. XXIV.

Hence, to divide a whole number by a fraction, multiply it by the denominator of the fraction, and divide the product by the numera

tor.

How many times is contained in 7.

Solution. Reducing them to a common denominator, is , and is. 4 is contained in as many times as 24 is contained in 35; that is, or 111. Ans. 1.

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Hence, to divide a fraction by a fraction, multiply the nume rator of the dividend by the denominator of the divisor, and the denominator of the dividend by the numerator of the divisor.

Or more generally, when the divisor is a fraction, multiply the dividend (whether whole number or fraction) by the divisor inverted. Arith. Arts. XXIII. and XXIV.

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