(84.) To subtract one fractional quantity from an

other.

RULE.

Reduce the fractions to a common denominator; subtract one numerator from the other, and place their difference over the common denominator.

QUEST.-How do we subtract one fraction from another?

(85.) To multiply fractional quantities together.

QUEST.-How do we multiply fractional quantities together?

RULE.

Multiply all the numerators together for a new numerator, and all the denominators together for a new denominator.

Before demonstrating this rule, we must establish. the two following principles :

1. In order to multiply a fraction by any number, we must multiply its numerator or divide its denominator by that number.

ab

Thus, the value of the fraction is b. If we mul

α

a2b

tiply the numerator by a, we obtain or ab; and

a

if we divide the denominator of the same fraction by a, we also obtain ab; that is, the original value of the fraction has been multiplied by a.

2. In order to divide a fraction by any number, we must divide its numerator or multiply its denominator by that number.

Thus, the value of the fraction

a2b

is ab. If we di.

a

ab

or b; and if

α

vide the numerator by a, we obtain

we multiply the denominator of the same fraction by a, we obtain

a2b

or b; that is, the original value of

the fraction ab has been divided by a.

Let it now be required to multiply

α

b

by

by

QUEST.-Upon what principles does this rule depend? How do we multiply a fraction by any number? How do we divide a fraction by any number?

First, let us multiply by c. According to the first

of the preceding principles, the product must be

с

ac

b

But the proposed multiplier was; that is, we have

used a multiplier d times too great. We must there

ac

fore divide the result by d; and according to the

b

second of the preceding principles, we obtain

ac

bd'

which result is the same as would have been obtained by multiplying together the numerators of the two fractions for a new numerator, and their denominators for a new denominator.

If the quantities to be multiplied are mixed, they must first be reduced to fractional forms.

(86.) To divide one fractional quantity by another.

RULE.

Invert the divisor, and proceed as in multiplication. If the two fractions have the same denominator, then the quotient of the fractions will be the same as the quotient of their numerators.

QUEST.-How do we divide one fraction by another? Explain the reason of the rule.