To reduce a mixed quantity to a fraction, multiply the integral part by the denominator, to the product annex the numerator, and under the result write the denominator.

136. The sign before the fraction shows that the number of things of the group b indicated by the numerator must be added or subtracted according as the sign is or from the number of things in the integral part of the kind in the b group, i. e., from Ab. If the sign precedes the fraction, when the numerator is annexed, the sign of every term in the numerator must be changed. x y Ab (xy) Ab x + y.

Thus:

A

137. Reduction of Fractions to a Lowest Common Denominator. Some propositions concerning fractions in Arithmetic will now be recalled, and be proved to hold universally in Algebra. In the following paragraphs the letters represent positive integers, unless it is otherwise stated.

138. 1. Rule for multiplying a fraction by an integer. Either multiply the numerator by that integer, or divide the denominator by it.

Or, in the fractions

b

the unit is divided into b equal parts,

ac

ас

and c times as many parts are taken in as in; hence is c times

b

a This proves the first part of the rule.

b

Again,

Or, in each of the fractions

b

a X c bex1

be

a

in each case, but each part in

b

is c times as large as the parts in

and the same number of parts is taken

b

a

be

9

[131]

2. Rule for dividing a fraction by an integer. Either multiply the denominator by that integer, or divide the numerator by it.

For,

Let be any fraction, and e any integer; then will

Therefore, is
is c times , that is,

first part of the theorem.

be

Let now be any fraction and c any integer; then prove that

3. We have from 131 a third rule of frequent use in reducing fractions to common denominators.

139. Rule for Reducing Fractions to a Common Denominator. Multiply the numerator of each fraction by all the denominators except its own for a new numerator of that fraction, and multiply all the denominators together for the common denominator.

Thus, letand

a adf

be the given fractions; then, by 138, 3,
с cbf e ebd

=
b bdf d a=df f= fbd

are fractions which have respectively the

ebd

bdfa
i d'

с

and and have the common denominator

same values as

bdf; and, further, each numerator is found according to the rule above.

140. If the denominators have one or more common factors, the rule for reducing them to equivalent fractions, with their lowest. common denominator, will be:

Find the L. C. M. of their denominators; then for a new numerator corresponding to each of the given fractions divide the L. C. M. by the denominator of that fraction and multiply its numerator by the quotient.

1. Suppose, for example, that the given fractions are The L. C. M. of the denominators is 10 abc.

a

b с

5 bc 10 ab 5b

Hence,

a × 2a = 2 a2, the new numerator of first fraction;
bx с =
bc, the new numerator of second fraction;

c × 2 ac = 2 ac2, the new numerator of third fraction.

The second members are the equivalent fractions of lowest common denominator, which are respectively equivalent to the given fractions.

and

The L. C. M. of the denominators x 1, x+1, and 1

(− 1) (x + 1) (x − 1) = − (x2 — 1) = 1 − x2;

x2 is

Hence, xx[− (x+1)]=-x(x+1), new num. of first fraction; 1 × [ — (x − 1)] = — (x − 1), new num. of second fraction; 1, new num. of third fraction.

1 x 1

=

Reduce to equivalent fractions with the lowest common denominator:

141. The sum or the difference of two fractions having a common denominator is a fraction whose numerator is the sum or the difference of the numerators of the given fraction, and whose denominator is the common denominator.

It follows from equation 2, 67, that,