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Ans.

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6. a 16 and

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a(x"+x+x+1), B(x* +1),

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8.

2 + x + 1

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and
2 - 6x + 6x-5 x - 4x - 4x – 5
Ans.
(2*+x+1)

(x*—*+1)
ac -5x4+x*—5xo +2–5' 2–5x*+x—5x® +3—5*

ADDITION

129. We have seen (115) that a fraction is equal to the reciprocal of its denominator multiplied by the numerator. Hence, if two or more fractions have a common denominator, they will have a common fractional unit, which may be made the unit of addition. Thus, 7 1

1
1

a +6
+ X at

X (a+b)

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The intermediate steps may be omitted; hence the following

RULE. I. Reduce the fractions to their least common denominator.

II. Add the numerators, and write the result over the common (lenominator.

NotEs. 1. If there are mixed quantities, we may add the entire and fractional parts separately.

2. Any fractional result should be reduced to its lowest terms.

EXAMPLES FOR PRACTICE.

5' 7 , and

Add Q and

с

3x 2x X

63x+30x+35x 128x 1. Add

Ans. 3

105

105 a+b

actab+6 2.

Ans.

bc a'+

a'+ax+3a'+32" 3. Add and

Ans.
3 ata

3(a+x)
a+b
a-6

2a'+269 4. Add and

Ans.
ab
a+6

a' _69
a+3
2a-5

14a-13 and 4a+

Ans. bat
5
4

20
2x-3

5x'_4x49 6. Add 5x+ and 4x+

Ans. 9x+
3
5x

15x

5. Add 2a+a73

a

с

Ans. ate

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Ans. 2+2y

2c 7. Add

and atc are

atc 8. Ada x*y—37' 3x+3yo 5x 52°y

10y

10 a +6

b + c 9. Add

,and

ct a (6 —c) (c-a)' (c-a) (a - b)' (a-6) (6-0)

Ans. 0. a2 - 6 10. Add

ta

1+ ab

and a — '

(1- a) (1+6)

Ans. 0.

ab 11. Add

and (a-6) (a -c)' (6c) (6—a)' (c—a) (c-6)

Ans. 1 - 3 - 2

-1 12. Add

and
x*— 3x + 2 ** — 4x +3 2*— 5x + 6

3x*-12x +14
Ans.

x'- 6x + 11x - 6

-2 13. Add

and
2 + + 8x + 2 2° + 6xo + 11x + 6

3.x(+ 1) · Ans.

2 + 5x + 6

bc

ac

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2C

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SUBTRACTION.

a

a

с

с

130. If two fractions have a common denominator, they will have the same fractional unit; and the may be subtracted from the other, by taking the difference of the numerators. Thus, b 1

1
1

-b
X Xb= -x (a - b)
Or thus,

B

ac--—bc-'=(ab)cHence the following RULE. I. Reduce the fractions to their least common denominator

II. Subtract the numerator of the subtrahend from the numerator of the minuend, and write the result over the common denominator.

a

a

с

с

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3. From

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1

1
subtract
y

a+y
lla . 10

3a_5 4. From 3a +

take 2a + 15

7

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3a + 6 9. From

a + 76

take
a + 3ab + 20 a' + 5ab + 660

2(a - b)
Ans.

a' + 4ab + 30° 4a-36

8a 10. From

take Tab(a-1)-2(a – 69) 3al(a+b)-2(a+)

2 Ans.

a' - 20

MULTIPLICATION.

131. Any fraction may be multiplied by an entire quantity in

two ways:

1st. By multiplying its numerator; or
2d. By dividing its denominator ; (119, I and II).

132. A general rule for the multiplication of fractions is fur. nished by the following example:

1. Multiply s by a

OPERATION.

ac
o Xx=ab-
=al-Xcd-=acb-d-=

bd By observing the result, we find that the new numerator is the product of the given numerators, and the new denominator is the product of the given denominators. Hence the following

RULE. I. Reduce entire and mixed quantities to fractional forms. II. Multiply the numerators together for a new numerator,

and the denominators for a new denominator, canceling all factors common to the numerator and denominator of the indicated product.

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8. Multiply a + by a—

7a

2x+3y
2a

2x+By 3. Multiply

Ans. 5x

5x a 2c, 2a

(a — wa 4. Multiply by

Ans. 2y ata

y 4yo 5. Multiply

15y – 30 by

Ans. by. 57 - 10

2y ab a

'+ 6. Multiply by

Ans. atb abba

b aRx-2

6a 7. Multiply by

Ans. 3(a+x). 2ax-2x y

a'b'+abu-aby- y

Ans. 7

39 3x'_53

3ax-5a 9. Multiply by

Ans.
14
2._-3x

4x*__6 cy 10. Multiply together

and

Ans. a. atý 4a_1669

56 11. Multiply -26 8a'+32ab +326

56 Ans.

2a+46 +1 2-1

and 3a.
2a
ato

3(3-1) Ans.

2(a+b) a2-x2 a2b2 13. Multiply together a +6 acta

a'lab) Ans.

a

by

a

12. Multiply together to

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