88. To reduce a Fraction to a Mixed Quantity.

When any term of the numerator is divisible by any term of the denominator, the transformation can be effected by division. Hence the rule: :

Perform the indicated division, continuing the operation as far as possible; then write the remainder over the denominator, and annex the result to the quotient found.

89. To reduce a Mixed Quantity to a Fractional Form.

This transformation is the converse of the preceding, and may be effected by the following rule:

Multiply the entire part by the denominator of the fraction, and add to the product the numerator. Write the result over the denominator of the fraction.

90. To reduce Fractions having Different Denominators to Equivalent Fractions having the Least Common Denominator.

This transformation is effected by finding the L. C. M. of the denominators.

Reduce

1 3 3' 4'

and

5 12

to their least common denominator.

The L. C. M. of the denominators is 12, which is also the least common denominator of the required fractions. If each fraction be multiplied by 12, and the result divided by 12, the values of the fractions will not be changed.

5 12

9

Hence

and are the new equivalent fractions. 12' 12'

From the preceding example we deduce the following rule:

Find the least common multiple of the denominators.

Multiply each fraction by it, and cancel the denominator. Write each product over the common multiple, and the results will be the required fractions.

Or, in general,

Multiply each numerator by all the denominators except its own, for the new numerators; and all the denominators together, for a common denominator.

The L. C. M. of the denominators is (a + b)(a — b).

x (a+b)(a - b) = c(a - b).

ADDITION OF FRACTIONS.

91. Fractions can be added only when they have a common unit; that is, when they have a common denominator. In that case, the sum of the numerators will indicate how many times that unit is taken in the entire collection. Hence the rule:

Reduce the fractions to be added, to a common denominator. Add the numerators together, for a new numerator, and write the sum over the common denominator.

By reducing to a common denominator, we have

6 × 3 × 590, 1st numerator.

4× 2 × 540, 2d numerator.

2 × 3 × 2 = 12, 3d numerator.

2× 3 × 530, the denominator.

Hence the expression for the sum of the fractions becomes

which, being reduced to the simplest form, gives 411.