166. To reduce a mixed number to a fraction:

Multiply the integral part by the denominator, add the numerator, and place the sum over the denominator.

x

=

NOTE. When the sign of a fraction is minus, it produces a change

of sign in each term of the numerator.

167. To reduce fractions to equivalent ones having a common denominator.

As the value of a fraction is not changed by multiplying both terms by the same quantity, it is evident that any number of fractions can be reduced to fractions which are equivalent and have a common denominator.

The least common denominator of several fractions is the least common multiple of their denominators.

Hence, in reducing several fractions to their least common denominator, divide the denominator of each into the least common multiple of their denominators, multiply the corresponding

numerator by the quotient thus obtained, and place this product over the least common denominator.

Each fraction, before its reduction to the least common denominator, should be in its lowest terms.

L.C.D. stands for "Least Common Denominator."

Multiplying the corresponding numerators by the results, we have the required fractions,

Reduce the following to fractions with L. C. D. :

CHAPTER X.

ADDITION AND SUBTRACTION OF FRACTIONS.

168. Addition of Fractions: Reduce the given fractions to fractions having a common denominator, add the numerators, and place the sum over the common denominator.

169. Subtraction of Fractions: Reduce the given fractions to fractions having a common denominator, subtract the numera tor of the subtrahend from the numerator of the minuend, and place the remainder over the common denominator.

(1) Add the following fractions:

The least common denominator of a2 a2 and x2 - 2 ax + a2 is (x + a)(x − a)(x − a); dividing the given denominators into the L. C. D. and multiplying the corresponding numerators by the quotient thus obtained, we have,

Adding these numerators and placing the result over the L. C. D., we have,