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and the same number of parts is taken, but each part b

α

α in is c times as large as each part in because in b bc'

a the unit is divided into c times as many parts as in

bc

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133. Rule for dividing a fraction by an integer. Either multiply the denominator by that integer, or divide the numerator by that integer.

a

Let denote any fraction, and c any integer; then

a

a

willc. For is c times by Art. 132; and

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b

a

bc ,

This demonstrates the first form of the Rule.

ac

Again; let denote any fraction, and c any integer;

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and therefore

b

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ac

α

+c=
÷c= For is c times, by Art. 132;

b

a 1
is th of

b

ac

b'

This demonstrates the second form of the Rule.

134. If the numerator and denominator of any fraction be multiplied by the same integer, the value of the fraction is not altered.

For if the numerator of a fraction be multiplied by any integer, the fraction will be multiplied by that integer; and the result will be divided by that integer if its denominator be multiplied by that integer. But if we multiply

any number by an integer, and then divide the result by the same integer, the number is not altered.

The result may also be stated thus; if the numerator and denominator of any fraction be divided by the same integer, the value of the fraction is not altered.

Both these verbal statements are included in the alge

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This result is of very great importance: many of the operations in Fractions depend on it, as we shall see in the next two Chapters.

135. The demonstrations given in this Chapter are satisfactory only when every letter denotes some positive whole number; but the results are assumed to be true whatever the letters denote. For the grounds of this assumption the student may hereafter consult the larger Algebra. The result contained in Art. 134 is the most important; the student will therefore observe that henceforth we assume that it is always true in Algebra that a ac whatever a, b, and c may denote.

be'

b

a -a

For example, if we put -1 for c we have

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XV. Reduction of Fractions.

136. The result contained in Art. 134 will now be applied to two important operations, the reduction of a fraction to its lowest terms, and the reduction of fractions to a common denominator.

137. Rule for reducing a fraction to its lowest terms. Divide the numerator and denominator of the fraction by their greatest common measure.

For example; reduce

16a4b2c 20a3b3d

to its lowest terms.

The G. C. M. of the numerator and the denominator is 4a3b2; dividing both numerator and denominator by 4a3b2,

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equal to

16a4b2c
20a3b3d'

but it is expressed in a more simple

form; and it is said to be in the lowest terms, because it cannot be further simplified by the aid of Art. 134.

Again; reduce

x2-4x+3
4x3-9x2-15x+18

to its lowest terms.

The G. C. M. of the numerator and the denominator is x-3; dividing both numerator and denominator by x-3

we obtain for the required result

x-1
4x2+3x-6*

In some examples we may perceive that the numerator and denominator have a common factor, without using the rule for finding the G. C. M. Thus, for example,

(a - b)2 - c2 (a−b+c) (a−b−c) a-b+c

=

− ̄ ̄

a2 (b+c) (a+b+c) (a-b-c)

=

a+b+c°

138. Rule for reducing fractions to a common denominator. Multiply the numerator of each fraction by all the denominators except its own, for the numerator corresponding to that fraction; and multiply all the denominators together for the common denominator.

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The Rule given in this Article will always reduce fractions to a common denominator, but not always to the lowest common denominator; it is therefore often convenient to employ another Rule which we shall now give.

139. Rule for reducing fractions to their lowest common denominator. Find the least common multiple of the denominators, and take this for the common denominator; then for the new numerator corresponding to any of the proposed fractions, multiply the numerator of that fraction by the quotient which is obtained by dividing the least common multiple by the denominator of that fraction.

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mon denominator. The least common multiple of the denominators is xyz; and

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