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We’ll use a model to show you how to multiply two fractions and to help you remember the procedure. Let’s start with $\frac{3}{4}.$
Now we’ll take $\frac{1}{2}$ of $\frac{3}{4}.$
Notice that now, the whole is divided into 8 equal parts. So $\frac{1}{2}\xb7\frac{3}{4}=\frac{3}{8}.$
To multiply fractions, we multiply the numerators and multiply the denominators.
If $a,b,c\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d$ are numbers where $b\ne 0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d\ne 0,$ then
To multiply fractions, multiply the numerators and multiply the denominators.
When multiplying fractions , the properties of positive and negative numbers still apply, of course. It is a good idea to determine the sign of the product as the first step. In [link] , we will multiply negative and a positive, so the product will be negative.
Multiply: $-\phantom{\rule{0.2em}{0ex}}\frac{11}{12}\xb7\frac{5}{7}.$
The first step is to find the sign of the product. Since the signs are the different, the product is negative.
$\begin{array}{cccccc}& & & & & -\phantom{\rule{0.2em}{0ex}}\frac{11}{12}\xb7\frac{5}{7}\hfill \\ \text{Determine the sign of the product; multiply.}\hfill & & & & & -\phantom{\rule{0.2em}{0ex}}\frac{11\xb75}{12\xb77}\hfill \\ \begin{array}{c}\text{Are there any common factors in the numerator}\hfill \\ \text{and the denominator? No}\hfill \end{array}\hfill & & & & & -\phantom{\rule{0.2em}{0ex}}\frac{55}{84}\hfill \end{array}$
Multiply: $-\phantom{\rule{0.2em}{0ex}}\frac{10}{28}\xb7\frac{8}{15}.$
$-\phantom{\rule{0.2em}{0ex}}\frac{4}{21}$
Multiply: $-\phantom{\rule{0.2em}{0ex}}\frac{9}{20}\xb7\frac{5}{12}.$
$-\phantom{\rule{0.2em}{0ex}}\frac{3}{16}$
When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, a , can be written as $\frac{a}{1}.$ So, for example, $3=\frac{3}{1}.$
Multiply: $-\phantom{\rule{0.2em}{0ex}}\frac{12}{5}\left(\mathrm{-20}x\right).$
Determine the sign of the product. The signs are the same, so the product is positive.
$-\phantom{\rule{0.2em}{0ex}}\frac{12}{5}(\mathrm{-20}x)$ | |
Write $20x$ as a fraction. | $\frac{12}{5}\left(\frac{20x}{1}\right)$ |
Multiply. | |
Rewrite 20 to show the common factor 5 and divide it out. | |
Simplify. | $48x$ |
Multiply: $\frac{11}{3}\left(\mathrm{-9}a\right).$
$\mathrm{-33}a$
Multiply: $\frac{13}{7}\left(\mathrm{-14}b\right).$
$\mathrm{-36}b$
Now that we know how to multiply fractions, we are almost ready to divide. Before we can do that, that we need some vocabulary.
The reciprocal of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator. The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}.$
Notice that $\frac{2}{3}\xb7\frac{3}{2}=1.$ A number and its reciprocal multiply to 1.
To get a product of positive 1 when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.
The reciprocal of $-\phantom{\rule{0.2em}{0ex}}\frac{10}{7}$ is $-\phantom{\rule{0.2em}{0ex}}\frac{7}{10},$ since $-\phantom{\rule{0.2em}{0ex}}\frac{10}{7}\left(-\phantom{\rule{0.2em}{0ex}}\frac{7}{10}\right)=1.$
The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}.$
A number and its reciprocal multiply to one $\frac{a}{b}\xb7\frac{b}{a}=1.$
To divide fractions, we multiply the first fraction by the reciprocal of the second.
If $a,b,c\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d$ are numbers where $b\ne 0,c\ne 0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d\ne 0,$ then
To divide fractions, we multiply the first fraction by the reciprocal of the second.
We need to say $b\ne 0,c\ne 0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d\ne 0$ to be sure we don’t divide by zero!
Divide: $-\phantom{\rule{0.2em}{0ex}}\frac{2}{3}\xf7\frac{n}{5}.$
$\begin{array}{cccccc}& & & & & -\phantom{\rule{0.2em}{0ex}}\frac{2}{3}\xf7\frac{n}{5}\hfill \\ \begin{array}{c}\text{To divide, multiply the first fraction by the}\hfill \\ \text{reciprocal of the second.}\hfill \end{array}\hfill & & & & & -\phantom{\rule{0.2em}{0ex}}\frac{2}{3}\xb7\frac{5}{n}\hfill \\ \text{Multiply.}\hfill & & & & & -\phantom{\rule{0.2em}{0ex}}\frac{10}{3n}\hfill \end{array}$
Divide: $-\phantom{\rule{0.2em}{0ex}}\frac{3}{5}\xf7\frac{p}{7}.$
$-\phantom{\rule{0.2em}{0ex}}\frac{21}{5p}$
Divide: $-\phantom{\rule{0.2em}{0ex}}\frac{5}{8}\xf7\frac{q}{3}.$
$-\phantom{\rule{0.2em}{0ex}}\frac{15}{8q}$
Find the quotient: $-\phantom{\rule{0.2em}{0ex}}\frac{7}{8}\xf7\left(-\phantom{\rule{0.2em}{0ex}}\frac{14}{27}\right).$
$-\phantom{\rule{0.2em}{0ex}}\frac{7}{18}\xf7\left(\mathrm{-}\phantom{\rule{0.2em}{0ex}}\frac{14}{27}\right)$ | |
To divide, multiply the first fraction by the reciprocal of the second. | $-\phantom{\rule{0.2em}{0ex}}\frac{7}{18}\cdot \mathrm{-}\phantom{\rule{0.2em}{0ex}}\frac{27}{14}$ |
Determine the sign of the product, and then multiply.. | $\frac{7\cdot 27}{18\cdot 14}$ |
Rewrite showing common factors. | |
Remove common factors. | $\frac{3}{2\cdot 2}$ |
Simplify. | $\frac{3}{4}$ |
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