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126. To reduce a mixed quantity to the form of a fraction.

This case is the converse of the last, and may be explained by it. Hence the following

RULE. Multiply the entire part by the denominator of the fraction; add the numerator if the sign of the fraction be plus, but subtract it if the sign be minus, and write the result over the denominator.

EXAMPLES FOR PRACTICE.

Reduce the following mixed quantities to fractions :

bt ' 1. 1+at

6 32

26–3xta 2. 26

Ans.

Ans. 6+abta

-a

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ax

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127. To transfer a factor from the denominator to the numerator, or the reverse.

Let us take any fraction, as and multiply both numerator and denominator by y-n, observing that any factor having zero for its exponent is equal to unity, (88,1), and may therefore be omitted. We shall have

ax" ax"y-ax"y" ax"y-*
by" by"-"

by" 6 If we multiply both numerator and denominator of the same frao tion by a-m, we shall have

axo by* by"am byx-m byxm In like manner we may transfer any factor having a negative exponent. For example, let us take the fraction, -, and multiply

axm

azm-m

a

ах

ax

ax-s+8

both numerator and denominator by aco; we shall have

axo 6 bx® bx* - bx* By the same principle also, any fraction may be reduced to the form of an entire quantity; thus,

xm xmyn xmyn ocny
э yn yr-n

1

=x"ye In all operations of this kind, the intermediate steps may be omitted, and the results obtained by the following

RULE. I. To transfer any factor from the denominator to the numerator, or the reverse : -Change the sign of its exponent. II. To reduce

any

fraction to the form of an entire quantity :-Transfer all the factors of the denominator to the numerator, observing to change the signs of the exponents of the factors transferred.

EXAMPLES FOR PRACTICE.

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In each of the following fractions, transfer the unknown factors, or factors containing unknown quantities, to the numerator. 1.

Ans.

axy c'yo

ca 3a"

3aa2.

Ans. 5moco

5m 1

Ans.

x-ly3.

axy%

a

Ans. C(2-y)

a

5.

Ans. 2a’xy

7.

4.

ay)
2aa'yo
5aRxy'

5a"
4x®z

4x®z

Ans.
6.
3ax-8

За 36'(a)

36'(a)'

Ans. 5cm(a-x)

5cm 3c(1-3)*(0-4)

3c*(1-x)(3-y)

Ans. 4m(c−)*(1−c)

4m In each of the following fractions, transfer the known factors to the depominator, and the unknown factors to the numerator. a’bcoach

Ans. 9. 5xy*%

5a6-4c-8 (ab)(x—a)'

(-a)

Ans. 10. (z–a)-(-6)

(a-6) 5a'la'yo

Ans. 11. cb-sayt

5-12-06-26

14.

In the following fractions, transfer the factors having negative exponents.

3a-xz 12.

3xzo

Ans. 5mz

5aRm 13. 5x(x-1)

5x*(x-1)'

Ans. 3ax*(x*—-1)

За 5ab-cd

5a'c

Ans. 12a--8-

126's Reduce each of the following fractions to the form of an entire quantity

5a’b 15.

Ans. 5aobx. zcz 16. 7xy"

Ans. 7a-may. aʼmo

a'l 17.

Ans. 4-'a'bay. 4xy

4ab 18.

Ans. 4ab'(a-*-*

OASE V.

128. To reduce one or more fractions to a common denominator.

We have seen (124) that a fraction may be reduced to lower terms by division. Conversely, a fraction must be reduced to higher terms by multiplication, and each of the higher denominators it may have, must be some multiple of its lowest denominator. Hence,

1.—A common denominator to which two or more fractions may be reduced, must be a common multiple of their lowest denominators; and

2.-The least common denominator of two or more fractions, must be the least common multiple of their denominators.

d 1. Reduce and to their least common denominator.

a’l

с

at?'

We find by inspection that the least common multiple of the given denominators is aobe. And

a’79:- a’l=3

aʼl;- abs=a If, therefore, we multiply both numerator and denominator of the first fraction by b', and of the second by a, we shall reduce the two fractions to their least common denominator, a’l'. Thus,

cXb=l'c, new numerator of first fraction;
dxa=ad, new numerator of second fraction.

d

62c ad Hence

Ans. a'bab

°7a'zo' From these principles and illustrations we deduce the following

RULE. I. Find the least common multiple of all the denominators, for the least common denominator.

II. Divide this common denominator by each of the given denominators, and multiply each numerator by the corresponding quotient. The products will be the new numerators.

NOTE.—Mixed numbers should first be reduced to fractions, and all fractions to their lowest terms.

с

EXAMPLES FOR PRACTICE.

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5a

In each of the following examples, reduce the fractions and mixed quantities to their least common denominator : 2a 36

4ac 3bx 1. and

Ans. 2c

2c« 2cx 2a 3a+26

4ac3ab+269 2. and

Ans. 2c

2bc

2bc 36

10ac 9bx 24cdx 3. and 4d.

Ans.
3.c 2c

6cx 6cx' 6cx
x + 1
and

y
4.
'

x + a acx + a'c

(bx + b) (x + a) bcy Ans.

bcx + abc bcx + abc a°у?

cy 5. a't and

Ans.

ay-y aya - y

a

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с

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bcx + abc'

a

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a

ay-1

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