We have seen that a fraction may be divided by multiplying its denominator, because the parts are made smaller; on the contrary, a fraction may be multiplied by dividing its denominator, because the parts are made larger. Arith. Art. XVIII. If the denominator be divided by 2, the unit is divided into only one half as many parts; consequently the parts must be twice as large as before. If the denominator be divided by 5, the unit is divided into only one fifth as many parts; hence the parts must be five times as large as before, and if the same number of parts be used as at first, the value of the fraction will be five times as great, and so on.

If we divide the denominator by b, the fraction becomes

the parts, and consequently the fraction is b times as large as

Dividing the denominator by 5 it becomes, or 3.

1

α

= 1, and being a times

as much as, must give a product a times as large, or ●

times 1, which is a.

Hence, if a fraction be multiplied by its denominator, the p duct will be the numerator.

Two ways have been shown to multiply fractions, and two

ways to divide them.

Reducing Fractions to Lower Terms.

XVII. If both numerator and denominator be multiplied by the same number, the value of the fraction will not be altered.

Arith. Art. XIX. For multiplying the numerator multiplies the fraction, and multiplying the denominator divides it; hence it will be multiplied and the product divided by the multiplier, which reproduces the multiplicand.

α

In other words, signifies that a contains & a certain num

b

ber of times, if a is as large or larger than b; or a part of ontime, if b is larger than a. Now it is evident that 2 a will contain 2 b just as often, since both numbers are twice as large as before.

So dividing both numerator and denominator, both divides and multiplies by the same number.

Hence, if a fraction contain the same factor both in the numerator and denominator, it may be rejected in both, that is, both may be divided by it. This is called reducing fractions to lower terms.

7. Reduce

to its lowest terms.

108 a x2+81x90 m2 x3

8. Divide 35 a bmx by 7 a3n m3 x.

Write the divisor under the dividend in the form of a frac

tion, and reduce it to its lowest terms.

5b m3 x2

Ans.

mʻ

2 a c-7 a b c + 15 ac d

21. Divide 18 a2 m2-54 a3 m2 + 42 a3 m1

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

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

23. Divide 3 c2 (a-2c)3 by 2bc3 (a — 2 c)3.

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

by 1265 (2a + d)2 (7 b — d)3 (a —d).

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.