129. We have seen (115) that a fraction is equal to the reciprocal of its denominator multiplied by the numerator. Hence, if two or more fractions have a common denominator, they will have a common fractional unit, which may be made the unit of addition. Thus,

The intermediate steps may be omitted; hence the following

RULE. I. Reduce the fractions to their least common denomi

nator.

II. Add the numerators, and write the result over the common denominator.

NOTES. 1. If there are mixed quantities, we may add the entire and fractional parts separately.

2. Any fractional result should be reduced to its lowest terms.

SUBTRACTION.

130. If two fractions have a common denominator, they will have the same fractional unit; and the one may be subtracted from the other, by taking the difference of the numerators.

Thus,

RULE. I. Reduce the fractions to their least common denominator II. Subtract the numerator of the subtrahend from the numerator of the minuend, and write the result over the common denominator.

131. Any fraction may be multiplied by an entire quantity in

two ways:

1st. By multiplying its numerator; or

2d. By dividing its denominator; (119, I and II).

132. A general rule for the multiplication of fractions is furnished by the following example:

By observing the result, we find that the new numerator is the product of the given numerators, and the new denominator is the product of the given denominators. Hence the following

RULE. I. Reduce entire and mixed quantities to fractional forms. II. Multiply the numerators together for a new numerator, and the denominators for a new denominator, canceling all factors common to the numerator and denominator of the indicated product.