MULTIPLICATION AND DIVISION OF FRACTIONS

62. In multiplication and division of fractions in algebra we have the same rules of operation as in arithmetic; namely,

1. To find the product of two fractions, multiply the numerators together for a new numerator, and the denominators together for a new denominator.

2. To find the quotient, invert the divisor and then proceed as in the multiplication of a fraction by a fraction.

It is easy to establish the truth of these rules by means of the equation.

For example, the second rule may be proved as follows:

This proves the rule for the division of one fraction by another.

Proceeding in a similar manner, prove the rule for the multiplication of fractions.

64. A complex fraction is one which contains one or more fractions in its numerator, or denominator, or both.

Complex fractions may be simplified in two ways:

1. Simplify the numerator and denominator, then divide the numerator by the denominator.

2. Multiply both numerator and denominator by the l. c. d. of their fractional terms.

The pupil is probably familiar with the first method from his study of arithmetic. In simplifying complex fractions arising in algebra, the second method is usually the easier. It should be thoroughly mastered.

Multiplying the numerator of the complex fraction by 36 gives 18 - 12. Multiplying the denominator of the complex fraction by 36 gives 9 – 4.

HIGHEST COMMON FACTOR AND LEAST COMMON MULTIPLE 66. Only expressions which contain no fractions or radicals. will be considered here.

The highest common factor (h. c. f.) of two or more expressions is the product of the prime factors that occur in each expression, every factor being taken the least number of times it occurs in any one expression.

This definition indicates the process of finding the h. c. f.

The lowest common multiple (1. c. m.) of two or more expressions is the product of all the prime factors that occur in the expressions, every factor being taken the greatest number of times it occurs in any one expression.

Observe

This definition indicates the process of finding the 1. c. m. that the 1. c. m. is the least expression which is exactly divisible by each of the given expressions, while the h. c. f. is the expression of highest degree by which each of the given expressions is exactly divisible.