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CASE. VI. 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 fractions required.*

EXAMPLES.

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1. It is required to find the sum of - and

2 3
X X 3

Зх
Here

the numerators.
X X 2= 2x
And 2 X3=6 the common denominator.

Зах 28 50
Whence

the sum required. 6 6 6'

+

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e

2. It is required to find the sum of and

d f Here a xdxf=adf

cxbxf=cbf the numerators.

e xbxd=ebd
And b xdxf=bdf the common denominator.

adf , cbf", ebd adf + cbf + ebd Whence

the sum. bdfbdfbdf bdf

3.22

2 ax 3. It is required to find the sum of a

and b +

b Here, taking only the fractional parts,

= 3cx

=2abx
And 6 x c= bc the common denominators.
3cx2
2aba

2aba

Зcar2 Whence a +6+ =a+b+

the sum. be bic

be

20 4. It is required to find the sum of and

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we shall have {

5X

5

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39x Ans.

35

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

and 2a 5.

15x + 2ax Ans.

10a * In the adding or subtracting of mixed quantities, it is best to bring the fractional parts only to a common denominator, and then to affix their sum or difference to the sum or difference of the integral parts, interposing the proper sign.

C C

a 6. It is required to find the sum of and

2' 3' 4

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X

4x

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

7

5

272 - 14 Ans.

35

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

5:

9

22x Ans. 6a

45

2x

and a

a

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Зх 9. Required the sum of 2a +5a

and

3a-x - 3ax: + 5* Ans. 2a +2+

5a? 5ax

a

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2

2x -3 10. Required the sum of 5x + and 4x

3

5x

522 - 16x + 9 Ans. 9x +

153

11. It is required to find the sum of 5%, and

3х2)

4x

8a + 3ax + 62: Ans. 5x +

12.x2

a + 2x

CASE VII.

To subtract one fractional quantity from another. RULE.-Reduce the fractions to a common denominator, if necessary, as in addition; then subtract the less numerator from the greater, and under the difference write the common denominator, and it will give the difference of the fractions required.

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And 3 x 5= 15 the common denominator.

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And 26 x 3c = 6be the common denominator.

3cx Зас 4ab 8bx 3cx - 3ac-4ab-t 8bx Whence

the 6bc 6bc

6bc difference required.

123 3x

42 3. Required the difference of and

Ans. Cat 7 5

35

1 + 2y 4. Required the difference of 15y and

8

118y - 1 Ans.

8

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a

and a

a.

and a

atx 7. Required the difference of a +

ata

ac 2a + 2x2 Ans.

a2 2x2 2x + 7

5x 6 8. Required the difference of ax to

8

21

86x 99 Ans. as

168

3х 5 9 Required the difference of 2 +

and 3x +

7 118 - 10

320 + 5

Ans. x +

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a to 20 10. Required the difference of a +

and a(a + 2) a(a - 2)

4x Ans. a

a2 а CASE VIII.

To multiply fractional quantities together. Rule.-Multiply the numerators together for a new nunerator and the denominators for a new denominator; and the former of these being placed over the latter, will give the product of the fractions, as required.*

EXAMPLES.

2x 1. It is required to find the product of a and

6

9
X X 2x 2x2 x2
Here

the product required.
6 X 9

54 27

4x

2' 59

a

a

ox2 + ax

2

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2. It is required to find the continued product of

10x and 21.

x 4x x 10x 4023 423 Here

the product. 2 X 5 X 21 210 21

atX 3. It is required to find the product of and.

* X (a + x) Here

the product. a X (a - x)

3x 52 4. It is required to find the product of

and
2 37

5x2 Ans.

26

2x 3х2 5. It is required to find the product of and

5
2a .

323 Ans.

5a. * When the numerator of one of the fractions to be multiplied, and the denominator of the other, can be divided by some quantity, which is common to each of them, the quotients may be used instead of the fractions themselves.

Also, when a fraction is to be multiplied by an integer, it is the same thing whether the numerator be multiplied by it, or the denominator divided by it. Or if an integer is to be multiplied by a fraction, or a fraction by an integer, the integer may be considered as having unity for its denominator, and the two be then multiplied together as usual.

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2x 402 6. It is required to find the continued product of

3373

8αα3 ? and

Ans. a ta

218 +21x

22. 3ab 7. It is required to find the continued product of

a 5ac and

Ans. 15ax. 26.

bx

b 8. It is required to find the product of 2a + and 3a

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с

a

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26 Ans. 60% + 3bx

a? 9. It is required to find the continued product of 3x, X + 1 XC 1

3203 and

Ans. 2a at6

202 + 2ab

a" x2 10. It is required to find the continued product of ad - 72

a25 and at

Ans.

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3 a

ax tona

CASE IX.

To divide one fractional quantity by another. RULE.-Multiply the denominator of the divisor by the numerator of the dividend, for the numerator; and the numerator of the divisor by the denominator of the dividend, for the denominator. Or, which is more convenient in practice, multiply the dividend by the reciprocal of the divisor, and the product will be the quotient required.*

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* When a fraction is to be divided by an integer, it is the same thing whether the numerator be divided by it, or the denominator multiplied

Also, when the two numerators, or the two denominators, can be divided by some common quantity, that quantity may be thrown out of each, and the quotients used instead of the fractions first proposed.

by it.

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