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.

1. It is required to find the sum of and

2

8 im

6 the common denominator.

5x

Whence + 6

the sum required.

3x3

2ax

we shall have {

2ax Xb 2abx

3. It is required to find the sum of a

Here, taking only the fractional parts,
3x2 × c=3cx2

And b × c = bc the common denominators.

and b +

b

the numerators.

* 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.

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.

And 3 × 5 = 15 the common denominator.

And 2b × 3c = 6bc the common denominator.

3cx Зас 4ab 8bx

3cx-3ac-4ab+8bx

Whence

the

To multiply fractional quantities together.

RULE.-Multiply the numerators together for a new numerator 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 and and.

x X 2x 2002

Here

the product required.

6 X 9

54 27

2. It is required to find the continued product of

3x

5x

4. It is required to find the product of and

2

* 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.

6. It is required to find the continued product of

α

and

a + x

Ans.

2x 4x2

3' 7 8αx3 21a+21a 2x 3ab

7. It is required to find the continued product of

α C

and

5ac 26.

Ans. 15ax.

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

and 3a

Ans. 6a2+3bx

X

9. It is required to find the continued product of 3x,

a

10. It is required to find the continued product of

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.*

* 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 by it.

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.