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9. Required the greatest common measure of the fraction х4 -- Захз 8a-x2 + 18ax ~~ 8a4

Ans. 2? + 2ux 2a. 23 - axa 8a-x +6Q? 10. Required the greatest common measure of the fraction 545 +10a4b + 5a/2

Ans. a+b. 236 + 2a2b2 + 2ab3 + 74

11. Required the greatest common measure of the fraction 605 +15a46 - 4Q?2 10a2bc2

Ans. 3a - 20%.

. 9a'b --- 27a^bc - 6abca + 18bc3?

CASE II. To reduce fractions to their lowest or most simple terns.

RULE.—Divide the terms of the fraction by any number, or quantity, that will divide each of them without leaving a remainder; or find their greatest common measure, as in the last rule, by which divide both the numerator and denomina tor, and it will give the fraction required.

EXAMPLES.

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abc

oc? 1. Reduce and

to their lowest terms. 50262

ax + 2012 aabc

22 Here Ans. And

Ans. 5a72 56

ax + x2

at 30

cx + 22 2. It is required to reduce

to its lowest terms.

dc + arx Here cx + 22 ac tax ctx aạc + aʻx (aa

ac tarx

or

*

Whence c + x is the greatest commion measure ; and c + ac)

the fraction required, a-c + a-x an

cx + x2

х

to a single dimension only, I divide the same into the parts x2 + 2ax, and-bx— Zab; which, by inspection, appear to be equal to (+-+-2aXat

, and (x + 2a) b. Therefore 2+2a is a divisor to both the parts, and likewise to the whole, expressed by. (-+-2a) X(2-6); so that one of these two factors, if the fraction given can he reduced to lower terms, must also measure the numerator; but the former will be found to su.cceed, the quotient coming out 2c2 -- ax+bx— ab, exactly; whence the fraction itself is reduced to axt-bu-ab

which is not reducible farther by 2-6, since the di-O vision does not terminate without a remainder, as upon trial will be found.

This rule is sometimes of great utility, because it spares great labour, and is very expeditious in reducing several fractions.-Ed.

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3 C

72x 3. It is required to reduce

to its lowest terms.

202 +2bic +62 2? + 2x + 5)23 - 12x (a

203 +2522 + 6 x

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Whence & to b is the greatest common measure; and a +b) 2003 - 62x 22 bac

the fraction required. 22 + 2x + 32

And the same answer would have been found, if 203 — bax had been made the divisor instead of 202 + 25x + 62.

204 a4 4. It is required to reduce

to its lowest terms. ac5 a-23

ox2 + a2 Ans.

103

Ans.

Ans. %

6a2 + 7ax

3x 5. It is required to reduce

to its lowest 6a + 11ax + 3x~

За ° terms.

3a +

2 203 - 162 - 6 6. It is required to reduce

to its lowest 3.2?

248 9 terms.

9x5 + 2003 + 4x2

X: +1 7. It is required to reduce

to 1584 2x3 + 10x9

XC + 2

3x3 + 2a + 1 its lowest terms.

Ans.

5x2 + 0 +2

a da cada a2c2 + A 8. It is required to reduce

to its 4a'd - 4acd

4acd - 2aca +- 2c3 lowest terms.

ada t- cd ac2

03 Ans.

4ad 2c2

2

CASE III.

To reduce a mixed quantity to an improper fraction. RULE.-Multiply the integral part by the denominator of the fraction, and to the product add the numerator, when it is affirmative, or subtract it when negative; then the result, placed over the denominator, will give the improper fraction required.

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Here 33

ac

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3 x 5 + 2 15 +2 17

Ans.
5

5 5
B
axc-b

b

Ans.
C
a

a? X2 2. Reduce a +

to improper fractions.

And a

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and a

C

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a tx?

a

5. Let x

be reduced to an improper fraction.

2ax

22 Ans.

2a 2x 77 명 6. Let 5+

be reduced to an improper fraction. 3x

17x by Ans.

-a-1 7. Let 1 be reduced to an improper fraction.

2a

+1 Ans.

X

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22

.

* XXX = X. In adding the numerator a2-x2, the sign -affixed to the fraction

denotes that the whole of that fraction is to be

> subtracted, and consequently that the signs of each term of the numerator must be changed when it is combined with 22; hence the impro

X2--02-22 2.22 per fraction is

ED,

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CASE IV. To reduce an improper fraction to a whole or mixed quantity.

RULE.—Divide the numerator by the denominator, for the integral part, and place the remainder, if any, over the denominator, for the fractional part; then the two, joined together, with the proper sign between them, will give the mixed quantity required.

EXAMPLES. 27

ax + a2 1. Reduce and

to mixed quantities.
5

27
Here 27 = 5= 53. Ans.

5
dix + a?

al
And

=+a

(ax t-ao) = x = a + Ans.

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to a

20

Ans. a

- 2c 2. It is required to reduce the fraction whole quantity.

.xx2.

ab - 2012 3. It is required to reduce the fraction

to a mixed

ab quantity.

2a Ans. 1

b

a2 + x2 4. It is required to reduce the fraction

to a mixed

a

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5. It is required to reduce the fraction

to a whole

Y quantity

Ans. 22 + xy + yo 10x2 – 5x + 3

to a 6. It is required to reduce the fraction

5x

3 mixed quantity

Ans, 2x - 1+

50

CASE V.

To reduce fractions to other equivalent ones, that shall have a

common denominator. RULE. -- Multiply each of the numerators, separately, into all the denominators, except its own, for the new numerators, and all the denominators together for a common denominator.*

EXAMPLES.

a

and to fractions that shall have a common b

с denominator.

Here a Xcr ac the new numerators.

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ac

bcs

a

ac

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Xc=bc the common denominator.
a
b

72 Whence and

and the fractions required. b.

bc

2x b 2. Reduce and to equivalent fractions having a com

C mon denominator.

2cm

ab Ans.

and

ac a

a+b 3. Reduce and to equivalent fractions having a

7 common denominator.

ac ab +33 Ans.

and bc

be 3x 26 4. Reduce and d, to equivalent fractions having a

2a' 30' common denominator.

9ca. 4ab 6acd Ans.

and

6ac' bac 6ac 3 28

4x 5. Reduce

to fractions having a com4' 3'

5' mon denominator.

45 40x 60a t-48x Ans.

and 60' 60

60 atx 6. Reduce

and to fractions having a common 2' ing?

x denominator.

7a2 7ах бах 622 14a + 14* Ans.

and 14a -- 14x140 14X 14a 14x

and at

а Зr

* It may here be remarked, that if the numerator and denominator of a fraction be either both multiplied, or both divided, by the same number or quantity, its value will not be altered; thus 2 2 X 3 6 3 3 : 3 1

ab and

; or

and
3
3 X3
12 12:34 b

bc which method is often of great use in reducing fractions more readily to a common denominator.

a

ac

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9

bci

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