2r

8. It is required to find the sum of 3a +

and

5

8r

α

9

a

9. It is required to find the sum of a,, and

4x

x-2
3

10. It is required to find the sum of a + and

2x-3
5.x

-

11. It is required to find the sun of

a + . ს

c-d

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.

(x-a) x 3c=3cx-3ac

(2a-4x) × 2b=4ab - 8bx

}

the numerators.

3cx-3ac

Whence

2b x 3c6bc the common denominator. 4ab-8bx 3cx-3ac-4ab + 8bx

the

difference required.

3. It is required to find the difference of

and 3+5

7

4. It is required to find the difference of 5y and 34

ax

3y

8

5. It is required to find the difference of ax and

b-c

+ C

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 (q).

(9) When the numerator of one of the fractions to be multi

x 4x

2. It is required to find the product of 2' 5'

It is required to find the product of 1⁄2 and

x × (a + x)

ax (a-x) a2 ax

a

5x

36

x2 + ax

the product.

3r

4. It is required to find the product of

and

2

2x

5.

It is required to find the product of

and

5

plied, 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: thus,

7. It is required to find the continued product of

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 (r).

(~) 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.

(G) INVOLUTION is the raising of powers from any proposed root; or the method of finding the square, cube, biquadrate, &c, of any given quantity (s).

(s) In the denominations of powers, considerations peculiar to Geometry have led us to call the second and third powers, the