2

Here, the least common multiple of 4cx2, 2x, and Sacs ; (Art. 147), is Saco x2 ; then, Sac x 2

X Sa'b=2ac + 3a2b=Ca'bc 4cxa Bacara

new numeraXy=4aco Xy=4acy 2x Sac? x2

X 5x2=X 5x2=524 8ac2

6a3bc Hence

and

are the 8ac2x2' Sacox?? 8acx2 fractions required.

tors;

4acʻy

5x

1 Ex. 6. Reduce

and to a comat 3

2.0 mon denominator. 30x2 2a2x - 2x3

3a + 3x Ans.

and Gax+6x2 Gar+6x? Gax +6x2

х

2x+3 5.0 +2 Ex. 7. Reduce

and

to a com

3ab mon denominator.

6abx +9ab 5x2 + 2x Ans.

and 3abx

3ab.

Ex. 8. Reduce

3' mon denominator.

4x2 +42 Ans.

12r+12

3x2+6x +3

a

2c2

За — х2
Ex. 9. Reduce

and +

b' d' common denominator.

adx 2bc x Ans.

and bdx' bdx

3abd bdx

2025

40-15 Ex. 10. Reduce

and 7+ 54' 3x

2 to a common denominator.

6x9 30axy-10~2 y + 50y Ans.

and

30xy 60xy-15xy

30xy

a

36

5& Ex, 11. Reduce

and
a
4a-4x

ata
to other equivalent fractions having the least com-
mon denominator.
4a 3ab +3630 20ax20x2

and 4a2-4x3' 4a2 - 4x2?

4a --4x2

§ Ans.

a

-x2

1

1 Ex. 12. Reduce

and a2 + 2ax + x2 5y

to the least common denominator. -24

a3 +ara + ax + x3 as-ax4 ta' x a' - ax4 t-a4 x 75

5ay +5xy
and
25

-a x4 tra4x

-25

§ V. ADDITION AND SUBTRACTION OF ALGEBRAIC

FRACTIONS.

To add fractional quantities together.

RULE,

154. Reduce the fractions, if necessary, to a common denominator, by the rules in the last case, then add all the numerators together, and under their sum put the common denominator; bring the resulting fraction to its lowest terms, and it will be the sum required.

C

20 5x Ex. 1. Add

and together.

3' 7 ? 9 2x X7X9=126x

126x +135x + 21x_282x 5x X 3X9=13500

189

139 X7 X3= 21x

932

is the sum required.

189 3X7X9= 189

+

a

2a

56 Ex. 2. Add

and together. Õ 36 4a

12ao b+8ab-+-1563 a X 36 X 4a=12026

12ab2 2a X6 X 4a= 8a2b

20a2b-+1563

-(dividing by 6) 56 x 36 xb= 1563 12ab2

20a2 +1562

is the sum b X 36 X 4a=12ab2 12ab

quired.

Or, the least common multiple of the denominators may be found, and then proceed, as in (Art.

153),

It is generally understood that mixed quantities are reduced to improper, fractions, before we perform any of the operations of Addition and Subtraction. But it is best to bring the fractional parts only to a common denominator, and to affix their sum or difference to the sum or difference of the integral part interposing the proper sign.

30* Ex. 3. It is required to find the sum of a

b 2ar and bt

с

3.22
ab- 3x

2ax bc + 2ax Here, a

and 6+ 6

6 Then, (ab — 3x2) xc=abc - 3cx 2

numerators. (bc+ 2ax) xbbc-2abx

с

-

bXc=bc= denominator.
abc-3cx° +6°C-+-2abr abc+c

+ bc

& bc 2abx - 3cx?

2abx - 3era zatot bc

bc is the sum required.

Or, bringing the fractional parts only to a common denominator, Thus, 302 Xc=3cx2

numerators, 2ax Xb=2abx And bxc=bc common denominator. 3cx2 2abx

2abx - 3cx Whence a

+6+ =a+b+1 bc

be the sum.

bc

Ex. 4. It is required to find the sum of 5:0+ -2

2x - 3 and 4x 3

5x Here, ( *—2) X 5x=5x2 -- 10x

numerators. (x-3) X3 =6x - 9

=

And 3 X 5x=15x common denominator. 5x2-10x

6x -9 Whence 5x --

+4x

=9x + 150

15x 5x +-10%, 9-6.0 5x2-16x+9 + =9x+ to

the sum 150 153

15x required. 6x -9

9-62 Here, is evidently

(Art. 128); 153

15.30 but we might change the fractions into other equivalent forms before we begin to add or subtract ; thus, the fractional part of the proposed quantity

2x - 3 4x

may be transformed by changing the

5x signs of the numerator, (Art. 128), and the quantity

3-260 itself can be written thus, 4x+ : It is well to

5x keep this transformation in mind, as it is often necessary to make use of it in performing several algebraical operations.

3a2 2a b Ex. 5. Add and together. 26' 5

7.

105a +-28ab +1002 Ans.

706