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Translate the phrase into an algebraic expression: “the quotient of $3x$ and $8.\u201d$
The keyword is quotient ; it tells us that the operation is division. Look for the words of and and to find the numbers to divide.
This tells us that we need to divide $3x$ by $8.$ $\frac{3x}{8}$
Translate the phrase into an algebraic expression: the quotient of $9s$ and $14.$
$\frac{9s}{14}$
Translate the phrase into an algebraic expression: the quotient of $5y$ and $6.$
$\frac{5y}{6}$
Translate the phrase into an algebraic expression: the quotient of the difference of $m$ and $n,$ and $p.$
We are looking for the quotient of the difference of $m$ and , and $p.$ This means we want to divide the difference of $m$ and $n$ by $p.$
Translate the phrase into an algebraic expression: the quotient of the difference of $a$ and $b,$ and $cd.$
$\frac{a-b}{cd}$
Translate the phrase into an algebraic expression: the quotient of the sum of $p$ and $q,$ and $r.$
$\frac{p+q}{r}$
Our work with fractions so far has included proper fractions, improper fractions, and mixed numbers. Another kind of fraction is called complex fraction , which is a fraction in which the numerator or the denominator contains a fraction.
Some examples of complex fractions are:
To simplify a complex fraction, remember that the fraction bar means division. So the complex fraction $\frac{\phantom{\rule{0.2em}{0ex}}\frac{3}{4}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{5}{8}\phantom{\rule{0.2em}{0ex}}}$ can be written as $\frac{3}{4}\xf7\frac{5}{8}.$
Simplify: $\frac{\phantom{\rule{0.2em}{0ex}}\frac{3}{4}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{5}{8}\phantom{\rule{0.2em}{0ex}}}.$
$\frac{\phantom{\rule{0.2em}{0ex}}\frac{3}{4}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{5}{8}\phantom{\rule{0.2em}{0ex}}}$ | |
Rewrite as division. | $\frac{3}{4}\xf7\frac{5}{8}$ |
Multiply the first fraction by the reciprocal of the second. | $\frac{3}{4}\xb7\frac{8}{5}$ |
Multiply. | $\frac{3\xb78}{4\xb75}$ |
Look for common factors. | $\frac{3\xb7\mathrm{4\u0338}\xb72}{\mathrm{4\u0338}\xb75}$ |
Remove common factors and simplify. | $\frac{6}{5}$ |
Simplify: $\frac{\phantom{\rule{0.2em}{0ex}}\frac{2}{3}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{5}{6}\phantom{\rule{0.2em}{0ex}}}.$
$\frac{4}{5}$
Simplify: $\frac{\phantom{\rule{0.2em}{0ex}}\frac{3}{7}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{6}{11}\phantom{\rule{0.2em}{0ex}}}.$
$\frac{11}{14}$
Simplify: $\frac{-\frac{6}{7}}{3}.$
$\frac{-\frac{6}{7}}{3}$ | |
Rewrite as division. | $-\frac{6}{7}\xf73$ |
Multiply the first fraction by the reciprocal of the second. | $-\frac{6}{7}\xb7\frac{1}{3}$ |
Multiply; the product will be negative. | $-\frac{6\xb71}{7\xb73}$ |
Look for common factors. | $-\frac{\mathrm{3\u0338}\xb72\xb71}{7\xb7\mathrm{3\u0338}}$ |
Remove common factors and simplify. | $-\frac{2}{7}$ |
Simplify: $-\frac{\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{9}{10}\phantom{\rule{0.2em}{0ex}}}.$
$-\frac{10}{3}$
Simplify: $\frac{\phantom{\rule{0.2em}{0ex}}\frac{x}{2}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{xy}{6}\phantom{\rule{0.2em}{0ex}}}.$
$\frac{\phantom{\rule{0.2em}{0ex}}\frac{x}{2}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{xy}{6}\phantom{\rule{0.2em}{0ex}}}$ | |
Rewrite as division. | $\frac{x}{2}\xf7\frac{xy}{6}$ |
Multiply the first fraction by the reciprocal of the second. | $\frac{x}{2}\xb7\frac{6}{xy}$ |
Multiply. | $\frac{x\xb76}{2\xb7xy}$ |
Look for common factors. | $\frac{\mathrm{x\u0338}\xb73\xb7\mathrm{2\u0338}}{\mathrm{2\u0338}\xb7\mathrm{x\u0338}\xb7y}$ |
Remove common factors and simplify. | $\frac{3}{y}$ |
Simplify: $\frac{\phantom{\rule{0.2em}{0ex}}\frac{a}{8}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{ab}{6}\phantom{\rule{0.2em}{0ex}}}.$
$\frac{3}{4b}$
Simplify: $\frac{\phantom{\rule{0.2em}{0ex}}\frac{p}{2}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{pq}{8}\phantom{\rule{0.2em}{0ex}}}.$
$\frac{4}{q}$
Simplify: $\frac{2\frac{3}{4}}{\frac{1}{8}}.$
$\frac{2\frac{3}{4}}{\frac{1}{8}}$ | |
Rewrite as division. | $2\frac{3}{4}\xf7\frac{1}{8}$ |
Change the mixed number to an improper fraction. | $\frac{11}{4}\xf7\frac{1}{8}$ |
Multiply the first fraction by the reciprocal of the second. | $\frac{11}{4}\xb7\frac{8}{1}$ |
Multiply. | $\frac{11\xb78}{4\xb71}$ |
Look for common factors. | $\frac{11\xb7\mathrm{4\u0338}\xb72}{\mathrm{4\u0338}\xb71}$ |
Remove common factors and simplify. | $22$ |
Simplify: $\frac{\frac{5}{7}}{1\frac{2}{5}}.$
$\frac{25}{49}.$
Where does the negative sign go in a fraction? Usually, the negative sign is placed in front of the fraction, but you will sometimes see a fraction with a negative numerator or denominator. Remember that fractions represent division. The fraction $-\frac{1}{3}$ could be the result of dividing $\frac{\mathrm{-1}}{3},$ a negative by a positive, or of dividing $\frac{1}{\mathrm{-3}},$ a positive by a negative. When the numerator and denominator have different signs, the quotient is negative.
If both the numerator and denominator are negative, then the fraction itself is positive because we are dividing a negative by a negative.
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