that the value of a fraction is not altered by dividing both the numerator and the denominator by the same quantity. Hence a fraction may be simplified by the rejection of any factor which is common to its numerator and denominator. For example, the fraction a2y/x'y takes the simpler form a2/x2, when the factor y, which is common to its numerator and denominator, is rejected. When the numerator and denominator of a fraction have no common factors, the fraction is said to be in its lowest terms. 160. Reduction of Fractions to their Lowest Terms. To reduce a fraction to its lowest terms, we must divide its numerator and denominator by their H. C. F.; for we thus obtain an equivalent fraction whose numerator and denominator have no common factors. The H. C. F. of the numerator and denominator is 3 axy; and a2b3 ÷ a2b3 = a5b+ ÷ a2b3 The H. C. F. of the numerator and denominator is a2b3; and a2b3 1 a3b 2 a3b2xy1 Ex. 3. Reduce to its lowest terms. 3 ab3x3y2 The H. C. F. of the numerator and denominator is ab2xy2; and 161. Instead of reducing a fraction to its lowest terms by dividing the numerator and denominator by their H. C. F., we may divide by any common factor, and repeat the process until the fraction is reduced to its lowest terms. The above process may be written down more compactly as follows: a2 162. When the numerator and denominator of a fraction are multinomial expressions whose factors can be seen by inspection, write the numerator and denominator as the product of factors of the lowest possible dimensions; the factors which are common to the numerator and denominator will then be obvious, and can be Now x-a = a2 x2 - (x-5)(x − 2) ах x2 a = (a− x) (a + x) (a − x); hence, dividing the numerator and x, we have the equivalent fraction, denominator by a - (α + x) usually left. a + x ; and this latter is the form in which the result is NOTE. - It should be remarked that the value of a fraction is not altered by changing the signs of all the terms in the numerator and also of all the terms in the denominator; for this is equivalent to multiplying both numerator and denominator by - 1. 163. When the factors of the numerator and denominator of a fraction cannot be found by inspection, their H. C. F. can be found by the rule given in Chapter XI.; and the fraction will be reduced to its simplest form by dividing the numerator and denominator by their H. C. F. The H. C. F. will be found to be x2-5x+2, and x3-23x + 10 = (x2. 5x + 2)(2 +5), 5 x3- 23x2 + 4 = (x2 − 5 x + 2) (5 x + 2). Hence the given fraction is equal to 164. Reduction of Fractions to a Common Denominator. Since the value of a fraction is unaltered by multiplying its numerator and denominator by the same quantity [Art. 158], any number of fractions can be reduced to equivalent fractions, all of which have the same denominator. The process is precisely the same as in arithmetic, and is as follows. First, find the L. C. M. of all the denominators; then divide the L. C. M. by the denominator of one of the N |