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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.、

α d

9. Add together be

and £

cg

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 cg, and the third by bce, each becomes b c 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. cdg, and The fractions then become aeg

The answer is

beeg beeg

a e g + c2 dg + b c e f

10. Add together

11. Add together

12. Add together

13. Add together

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14. Add together 3 m

15. Add together

16. Add together

2 ac 36

2 am

17. Add together

and 13 cd.

and 2 a c-5 b.

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

3αc-2e

Ans.

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- 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, 35 or 11 times.

2. How many times is contained in a?

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

84

3. How many times is

contained in c?

Ans.

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is contained b c times in c, and is contained

be

α

Arith. Art. XXIII.

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α

Ans. If c is the part of some number,

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will be

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 7 is . is contained in as many times as 24 is contained in 35; that is, or 11. Ans. 14.

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Hence, to divide a fraction by a fraction, multiply the numerator 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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