Multiplication of Fractions

151. By the definition of multiplication, multiplying by a fraction is taking a part of the multiplicand, as many times as there are like parts of an unit in the multiplier. (Art. 88.) Now the denominator of a fraction shows into what parts the integral unit is supposed to be divided ; and the numerator showahow many of those parts belong to the given fraction. In multiplying by a fraction, therefore, the multiplicand is to be divided into such parts, as are denoted by the denominator ; and then one of these parts is to be repeated, as many times, as is required by the numerator.

Suppose $a$ is to be multiplied by $\frac{3}{4}$.

A fourth part of $a$ is $\frac{a}{4}$.

This taken $3$ times is $\frac{a}{4} + \frac{a}{4} + \frac{a}{4} = \frac{3a}{4}$.(Art. 145 )

Again, suppose $\frac{a}{b}$ is to be multiplied by $\frac{3}{4}$.

One fourth of $\frac{a}{b}$ is $\frac{a}{4b}$. (Art. 135.)

This taken $3$ times is $\frac{a}{4b}+\frac{a}{4b}+\frac{a}{4b}=\frac{3a}{4b}$, the product required.

In a similar manner, any fractional multiplicand may be divided into parts, by multiplying the denominator; and one of the parts may be repeated, by multiplying the numerator. We have then the following rule:

152. TO MULTIPLY FRACTIONS, MULTIPLY THE NUMERATORS TOGETHER, FOR A NEW NUMERATOR, AND THE DENOMINATORS TOGETHER, FOR A NEW DENOMINATOR.

Ex. 1. Multiply $\frac{3b}{c}$ into $\frac{d}{2m}$. Product $\frac{3bd}{2cm}$.

2. Multiply $\frac{a + d}{y}$ into $\frac{4h}{m - 2}$. Product $\frac{(4ah+4dh)}{(my - 2y)}$.

153. The method of multiplying is the same, when there are more than two fractions to be multiplied together.

Multiply together $\frac{a}{b}, \frac{c}{d}$, and $\frac{m}{y}$. Product $\frac{acm}{bdy}$.

For ix $\frac{a}{b}\cdot\frac{c}{d}$ is, by the last article $\frac{ac}{bd}$, and this into $\frac{m}{y}$ is $\frac{acm}{bdy}$.

2. Mult. $\frac{3 + b}{n}, \frac{1}{h}$ and $\frac{d}{r + 2}$.

154. The multiplication may sometimes be shortened, by rejecting equal factors, from the numerators and denominators.

1. Multiply $\frac{a}{r}$ into $\frac{h}{a}$ and $\frac{d}{y}$. Product $\frac{dh}{ry}$.

Here a, being in one of the numerators, and in one of the denominators, may be omitted. If it be retained, the product will be $\frac{adh}{ary}$. But this reduced to lower terms, by Art. 142, will become $\frac{dh}{ry}$ as before.

It is necessary that the factors rejected from the numerators be exactly equal to those which are rejected from the denominators. In the last example, a being in two of the numerators, and in only one of the denominators, must be retained in one of the numerators.

2. Multiply $\frac{a + d}{y}$ into $\frac{my}{ah}$. Product $\frac{am + dm}{ah}$.

Here, though the same letter a is in one of the numerators, and in one of the denominators, yet as it is not in every term of the numerator, it must not be cancelled.

3. Multiply $\frac{am + d}{h}$ into $\frac{h}{m}$ and $\frac{3r}{5a}$.

If any difficulty is found, in making these contractions, it mill be better to perform the multiplication, without omitting any of the factors; and to reduce the product to lower terms afterwards.

155. When a fraction and an integer are multiplied together, the numerator of the fraction is multiplied into the integer. The denominator is not altered; except in cases where division of the denominator is substituted for multiplication of the numerator, according to Art. 136.

Thus $a\cdot\frac{m}{y} = \frac{am}{y}$. For $a = \frac{a}{1}$; and $\frac{a}{1}\cdot\frac{m}{y} = \frac{am}{y}$.

So $r\cdot\frac{x}{d}\cdot\frac{h + 1}{3} = \frac{hrx + rx}{3d}$. And $a\cdot\frac{1}{b} = \frac{a}{b}$. Hence,

156. A FRACTION IS MULTIPLIED INTO A QUANTITY EQUAL TO ITS DENOMINATOR, BT CANCELLING THE DENOMINATOR.

Thus $\frac{a}{b}\cdot b = a$. For $\frac{a}{b}\cdot b = \frac{ab}{b}$. But the letter $b$, being in both the numerator and denominator, may be set aside, (Art. 142.)

So $\frac{3m}{a - y}\cdot(a - y) = 3m$.

On the same principle, a fraction is multiplied into any factor in its denominator, by cancelling that factor.

Thus $\frac{a}{by}\cdot y = \frac{ay}{by}= \frac{a}{b}$. And $\frac{h}{24}\cdot 6 = \frac{h}{4}$.

157. From the definition of multiplication by a fraction, it follows that what is commonly called a compound fraction, is the product of two or more fractions. Thus $\frac{3}{4}$ of $\frac{a}{b}$ is $\frac{3}{4}\cdot\frac{a}{b}$. For $\frac{3}{4}$ of $\frac{a}{b}$ is $\frac{1}{4}$ of $\frac{a}{b}$ taken three times, that is, $\frac{a}{4b}+\frac{a}{4b}+\frac{a}{4b}$. But this is the same as $\frac{a}{b}$ multiplied by $\frac{3}{4}$.(Art. 151.)

Hence, reducing a compound fraction into a simple one, is the same as multiplying fractions into each other.

Ex. 1. Reduce $\frac{2}{7}$ of $\frac{a}{b+2}$. Ans. $\frac{2a}{7b+14}$.

2. Reduce $\frac{2}{3}$ of $\frac{4}{5}$ of $\frac{b + h}{2a - m}$. Ans. $\frac{8b + 8h}{30a - 15m}$.

3. Reduce $\frac{1}{7}$ of $\frac{1}{3}$ of $\frac{1}{8 - d}$. Ans. $\frac{1}{168 - 21d}$.

158. The expression $\frac{2}{3}a, \frac{1}{5}b, \frac{4}{7}y, \& c$. are equivalent to $\frac{2a}{3}, \frac{b}{5}, \frac{4y}{7}$. For $\frac{2}{3}a$ is $\frac{2}{3}$ of a, which is equal to $\frac{2}{3}\cdot a=\frac{2a}{3}$.(Art. 155.)

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