Common Denominator.

For the value of each fraction is, from art. 55, not changed by this process; and as each of the denominators thus obtained is the product of all the denominators, the fractions are all reduced to the same denominator.

68. But fractions can be reduced to a common denominator which is smaller than their continued product, whenever their denominators have a common multiple less than this product. For, by art. 55,

Fractions may be reduced to a common denominator, which is a common multiple of their denominators, by multiplying both their terms by the quovely obtained from the division of the ninator by their denominators.

69. Corollary. An entire quantity may, by the preceding article, be reduced to an equivalent fractional expression having any required denominator, by regarding it as a fraction, the denominator of which is unity.

71. Problem. To find the least common multiple of given quantities.

Solution. When the given quantities are decomposed into their simplest factors, as is the case with monomials, their least common multiple is readily obtained; for it is obviously equal to the product of all the unlike factors, each factor being raised to a power equal to the highest power which it has in either of the given quantities.

But the common factors can always be obtained from the process of finding the greatest common divisor.

72. EXAMPLES.

1. Find the least common multiple of 2 a3 b2 c x, 3 a5 b c3 x2, 6 a cx=2.3 a cx, 9 c7 x10 32 c7 x10, 24 a8 = 23.3 a3. Ans. 23.32. a8 b2 c7 x10 = 72 a8 b2 c7 x10. 2. Find the least common multiple of 16 a x, 25 a7 b3 x2.

40 65 x,

Ans. 400 a b5 x2..

xn−2; Ans. x.

3. Find the least common multiple of x, x − 1, xn−3, x.

4. Find the least common multiple of 6 (a + b) xm, 54 (a—b)3, (a+b)7, 81 (a—b)3 xm+2, 8(a+b)5 xm—8.

Ans. 648 (a+b)7 (a — b)3 x2+2.

Sum and Difference of Fractions.

5. Find the least common multiple of a2+2 a b + b2, a2 + 4 ab + 4 b2, a2 — b2, a2 + 3 a b + 2 b2, a3 + a2 b Ans. (a+b)2 (a—b) (a +2 b)?.

a b2 — 3.

SECTION II.

Addition and Subtraction of Fractions.

73. Problem. To find the sum or difference of given fractions.

Solution. When the given fractions have the same denominator, their sum or difference is a fraction which has for its denominator the given common denominator, and for its numerator the sum or the difference of the given numerators.

When the given fractions have different denominators, they are to be reduced to a common denominator by arts. 67 and 68.

Sum and Difference of Fractions.

5. Reduce to one fraction the expression +c.

a

b

α

Ans.

8. Reduce to one fraction+z+a—ž

1

+

x2

a x2 — b x + 1

x3

75. Corollary. It follows, from examples 3 and 4,

that the sum of half the sum and half the difference

Product and Quotient of Fractions.

of two quantities is equal to the greater of the two quantities; and that the difference of half their sum and half their difference is equal to the smaller of them.

SECTION III.

Multiplication and Division of Fractions.

76. Problem.

several fractions.

To find the continued product of

Solution. The continued product of given fractions is a fraction the numerator of which is the continued product of the given numerators, and the denominator of which is the continued product of the given denominators.

77. Problem. To divide by a fraction.

Solution. Multiply by the divisor inverted.

The preceding rules for the addition, subtraction, multiplication, and division of fractions require no other demonstrations than those usually given in arithmetic.

78. When the quantities multiplied or divided contain fractional terms, it is generally advisable to reduce them to a single fraction by means of art. 73.