If the numerator and denominator of a fraction be divided by the same number, the value of the fraction is not altered (Art. 66); and the greatest number which will divide the numerator and denominator is their greatest common measure.

Note. Sometimes it is unnecessary to find the greatest common measure, as it is easier to bring the fraction to its lowest terms by successive divisions of the numerator and denominator by common factors, which are easily determined by inspection.

Ex. 2. Reduce 540 to its lowest terms,

540

448=44, dividing numerator and denominator by 10,

=18, dividing numerator and denominator by 3.

Ex. XVI.

Reduce each of the following fractions to its lowest terms:

74. To reduce fractions to equivalent ones with a common denominator.

RULE. Find the least common multiple of the denominators: this will be the common denominator. Then divide the common multiple so found by the denominator of each fraction, and multiply each quotient so found into the numerator of the fraction which belongs to it for the new numerator of that fraction.

Note 1. If the given fractions be in their lowest terms, the above rule will reduce them to others having the least common denominator; if the least common denominator be required, the given fractions should be reduced to their lowest terms before the rule be applied.

9

Ex. Reduce, 18, 4, 33, into equivalent fractions with a common denominator.

Proceeding by the Rule given above,

therefore least common multiple = 2×2×2×3×2×11

The least common multiple of the denominators of the given fractions will evidently contain the denominator of any one of the fractions an exact number of times. If both the numerator and denominator of that fraction be multiplied by that number, the value of the fraction will not be altered (Art. 66); and the denominator will then be equal to the least common multiple of all the denominators. If this be done with all the fractions, they will evidently be, in like manner, reduced to others of the same value, and having the least common multiple of all the denominators for the denominator of each fraction.

Note 2. If the denominators have no common measure, we must then multiply each numerator into all the denominators, except its own, for a new numerator for each fraction, and all the denominators together for the common denominator.

Ex. Reduce,,, to equivalent fractions with a common denominator.

The least common multiple of the denominators

Reduce the fractions in each of the following sets to equivalent fractions, having the least common denominator:

(1),, and .

(3),, and .

(2)

%, and J.

(4) %, and 7.

Note 3. Whenever a comparison has to be made between fractions, in respect of their magnitudes, they must be reduced to equivalent ones with a common denominator; because then we shall have the unit divided, in the case of each fraction so obtained, into the same number of equal parts; and the respective numerators will shew us how many of such parts are taken in each case; or which is the greatest fraction, which the next, and so on.

Ex. Compare the values of,,,, and .

First, to find the least common multiple of the denominators;

227, 24, 6, 15, 5

+

therefore is the greatest, the next, 11 the next, the next, and

7

(5), 1, 2, 1, and .

(6) of of 4, of 3 of 5, of of 42, and 14.

75. RULE. Reduce the fractions to equivalent ones with their least common denominator; add all the new numerators together, and under their sum write the common denominator.

Proceeding by the Rule given above,

First, find the least common multiple of the denominators;