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108 Rh But b parts in (1) = mb parts in (2) ; 1 part in (1) = m parts in (2) ; a parts in (1) = ma parts in (2)

that is, a b ma mb

Conversely, ma mb a b

Hence, The value of a fraction is not altered if we multiply or divide the numerator and denominator by the same quantity.

137. Reduction of a Fraction to its Lowest Terms. As shown in the preceding article an algebraic fraction may be changed into an equivalent fraction by dividing numera- tor and denominator by any common factor ; if this factor be the highest common factor, the resulting fraction is said to be reduced to its lowest terms.

Ex. 1. Reduce to lowest terms 24a3c2x3

24 a3c^2x^3 36 a5x2 2^3 X 3 a^3c^2x^3 2^2 X 3^2 a^5x^2 2c^2x 3 a2

Ex. 2. Reduce to lowest terms

24 a3c2x2 18 a^3x2 - 12 a2x3 24 a3c2x2  6 a2x2(3 a-2x) 4ac2 3a -2x

Ex. 3. Reduce to lowest terms ^^^~^^-^ > 9xy -12 y^ 6 x2 — 8 xy _ 2x(3a; — 4?/) _ 2x 9xy - y^~ Sy(Bx - 4y) ~ Sy

Note. The beginner should be careful not to begin cancelling until he has expressed both numerator and denominator in the most convenient form, by resolution into factors where necessary.

EXAMPLES XIII. a.

Reduce to lowest terms :

. 12^mn^^ g ax g abx + bx"^ 15 mhip^ ' a^x^ — ax ' arx + cx^ 2 4Gx^y^z 4 3 a2 _ 6 ab g _ l^a^-b'^c 69 x'^y^z* 2 a-^b - 4 ab'^ 100 (a^ - a:^b)