An Introduction to Algebra

a X

3

a3 + x2 4. It is required to reduce the fraction to a mixed quantity.

5. It is required to reduce the fraction to a whole quantity.

10x2 - 5x+3 6. It is required to reduce the fraction

5x to a mixed quantity.

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 de nominator (h).

EXAMPLES.

a

1. Reduce

6 and - to fractions that shall have a com

b с mon denominator. Here a Xc=MC

the new numerators. 6 xb=62

}

+ Xc=bc the common denominator.

(h) It may here be remarked, that if the numerator and denominator of a fraction be either both multiplied, or bo: h divided, by the ame number, or quantity, its value wiil not be altered: thus 2 2x:3 6 33 1

ab
and
= ; or

and 33x9 12, 123 4 bbc bc which method is of:en of great use in reducing fractions more rea dily to a common denominator,

3

ас.

a

с

ac

с

(

с

a b

62 Whence, and

and

the fractions rebe

bc' quired.

2x 2. Re ice and to equivalent fractions having a common denominator.

a+b 3. Reduce and to equivalent fractions having

b a common denominator.

3x 26 4. Reduce and d, to equivalent fractions hav

2a' 3c ing a common denominator. 3 2x

4x 5. Reduce and a to to fractions having a 4' 3

5' common denominator.

3x 6. Reduce

and 2:7

to fractions having a common

denominator

a

a X

CASE VÍ.

To add fractional quantities together.

RULE.

Reduce the fractions, if necessary, to a common denominator ; then add all the numerators together, and under their sum put the common denominator, and it will give the sum of the fractions required (2).

(i) In the adding or subtracting of mixed quantities, it is best to bring the fractional parts only to a common denomaates, and then to affix their siim or difference to the sun or difference of the integral parts, interposing the proper sign.

EXAMPLES.

1. It is required to find the sum of Land

2
Here
X X3=32

the numerators.

а с

And 2X3=0 the common denominator.

3x 2x 5x Whence +

the sum required. 6 6 6

e 2. It is required to find the sum of and

od

f
Here a Xdxf=udf)

c Xbxf=cbf the numerators.
exbxdebu

+

And bxdxf=bdf the common denominator.

ebd adf _ cbj Whence

adf+cbf+ebd +

the sum. bdf' brif bdf bdf

3x2 3. It is required to find the sum of a

and b+

6 20x Here, taking only the fractional parts, 3x3 Xc=3cx"

the numerators.

We shall have {

And 6 Xc=bc the comrnon denominator.
3cx2
Qab.x

2ubx - 3cx3 Whence a - +6+ =a+b+

the bc

bc

sum.

2x

5% 4. It is required to find the sum of

and 5

7

3х
5. It is required to find the sum of and

enl8 al

.

43

6. It is required to find the sum of

X X

and
23
2' 3'

1

2
7. It is required to find the sum of

-2
and
7

5

2x
8. Required the sum of 2a, 3a +

.
5.

9
9. Required the sum of 2at:

and

8x

and a

3x

a

2x

5x

2 10. Required the sum of 5x+

and 4x 3

2a 11. It is required to find the sum of 5x,

302' a+2.x

4x

and

CASE VII.

To subtract one fractional quantity from another.

RULE.

శాతంగా ఉన్న

Reduce the fractions to a common denominator, if necessary, as in addition ; then subtrart the less numerator from the greater, and under the difference write the common denominator, and it will give the difference of the fractions required.

EXAMPLES

3x and

2x
1. It is required to find the difference of

3
Here
23 X5=10x

the numerators.
3x X3=93