21. Divide 18 a m3-54 a m2 + 42 a3 m*

4

by 30 a' m3 d-12 a' cm3.

22. Divide (a+b) (13 ac+be) by (m2 —c) (a + b).

23. Divide 3 c2 (a-2c)3 by 2 b c3 (a — 2 c)3.

24. Divide 36 b3 c2 (2a + d)2 (7 b—d)3

5

This addition may be expressed by writing the fractions one

after the other with the sign of addition between them; thus

N. B. When fractions are connected by the signs + and -, the sign should stand directly in a line with the line of the fraction.

It is frequently necessary to add the numerators together, in which case, the fractions, if they are not of the same denomination, must first be reduced to a common denominator, as in Arithmetic, Art. XIX.

These must be reduced to a common denominator. It has been shown above that if both numerator and denominator be multiplied by the same number, the value of the fraction will not be altered. If both the numerator and denominator of the first fraction be multiplied by 7, and those of the second by 5, the fractions become and . They are now both of the same denomination, and their numerators may be added. answer is 3.

The

Multiply both terms of the first by d, and of the second by b, they become and

ad b c

b d b d

The denominators are now alike

In all cases the denominators will be alike if both terms of each fraction be multiplied by the denominators of all the others. then they will all consist of the same factors.

Applying this rule to the above example, the fractions be

f

and

bdfg

bd fh

The answer is a dƒh + b c f h + b d e h + b d f g

bdfh

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.

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 + c d g + b c e f

10. Add together

11. Add together

bceg

ebcf

cb2 eg

2 ac be

5 am

ec

2 r

36

anden.

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

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

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

Ans. is contained in 7, 35 times, and is contained as many times; that is, 35 or 113 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, 8. a.