all the denominators, except its own, for the new numerators, and all the denominators together for a common denominator.*

It may here be remarked, that if the numerator and denominator of a fraction be either both multiplied, or both divided, by the same number or quantity, its value will not be altered; thus

2 2X3 6 3 3÷3 1

and

3

3 X 3 9

=

a ; or

ac

and 12 12÷3 4 b bc'

ab a

bc

which method is often of great use in reducing fractions more readily to a common denominator.

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.

cxbxf=cbf the numerators.
exbxd=ebd

And bx dxf = bdf the common denominator.
adf, cbf ebd adf+cbf+ebd

Whence + +
bdf bdf bdf

bdf

3. It is required to find the sum of a

the sum.

3x2

2ax

and b +

b

C

Here, taking only the fractional parts,
3x2 Xc3cx2

we shall have { 2ax × b= 2abx 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.

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.*

1

EXAMPLES.

1. It is required to find the product of and

2x

9'

2. It is required to find the continued product of

x 4x

2' 5

3. It is required to find

the product of

a + x

* 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.