4.3 Multiply and divide mixed numbers and complex fractions (Page 2/4)

Prealgebra Page 2 / 4

Translate the phrase into an algebraic expression: “the quotient of $3 x$ and $8 .”$

Solution

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.

The quotient of 3 x and 8 .

This tells us that we need to divide $3 x$ by $8 .$ $\frac{3 x}{8}$

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Translate the phrase into an algebraic expression: the quotient of $9 s$ and $14 .$

$\frac{9 s}{14}$

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Translate the phrase into an algebraic expression: the quotient of $5 y$ and $6 .$

$\frac{5 y}{6}$

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Translate the phrase into an algebraic expression: the quotient of the difference of $m$ and $n,$ and $p .$

Solution

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 $p .$

\frac{m - n}{p}

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Translate the phrase into an algebraic expression: the quotient of the difference of $a$ and $b,$ and $c d .$

$\frac{a - b}{c d}$

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Translate the phrase into an algebraic expression: the quotient of the sum of $p$ and $q,$ and $r .$

$\frac{p + q}{r}$

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Simplify complex fractions

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:

\frac{\frac{6}{7}}{3} \frac{\frac{3}{4}}{\frac{5}{8}} \frac{\frac{x}{2}}{\frac{5}{6}}

To simplify a complex fraction, remember that the fraction bar means division. So the complex fraction $\frac{\frac{3}{4}}{\frac{5}{8}}$ can be written as $\frac{3}{4} \div \frac{5}{8} .$

Simplify: $\frac{\frac{3}{4}}{\frac{5}{8}} .$

Solution

	$\frac{\frac{3}{4}}{\frac{5}{8}}$
Rewrite as division.	$\frac{3}{4} \div \frac{5}{8}$
Multiply the first fraction by the reciprocal of the second.	$\frac{3}{4} \cdot \frac{8}{5}$
Multiply.	$\frac{3 \cdot 8}{4 \cdot 5}$
Look for common factors.	$\frac{3 \cdot 4̸ \cdot 2}{4̸ \cdot 5}$
Remove common factors and simplify.	$\frac{6}{5}$

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Simplify: $\frac{\frac{2}{3}}{\frac{5}{6}} .$

$\frac{4}{5}$

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Simplify: $\frac{\frac{3}{7}}{\frac{6}{11}} .$

$\frac{11}{14}$

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Simplify a complex fraction.

Rewrite the complex fraction as a division problem.
Follow the rules for dividing fractions.
Simplify if possible.

Simplify: $\frac{- \frac{6}{7}}{3} .$

Solution

	$\frac{- \frac{6}{7}}{3}$
Rewrite as division.	$- \frac{6}{7} \div 3$
Multiply the first fraction by the reciprocal of the second.	$- \frac{6}{7} \cdot \frac{1}{3}$
Multiply; the product will be negative.	$- \frac{6 \cdot 1}{7 \cdot 3}$
Look for common factors.	$- \frac{3̸ \cdot 2 \cdot 1}{7 \cdot 3̸}$
Remove common factors and simplify.	$- \frac{2}{7}$

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Simplify: $\frac{- \frac{8}{7}}{4} .$

$- \frac{2}{7}$

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Simplify: $- \frac{3}{\frac{9}{10}} .$

$- \frac{10}{3}$

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Simplify: $\frac{\frac{x}{2}}{\frac{x y}{6}} .$

Solution

	$\frac{\frac{x}{2}}{\frac{x y}{6}}$
Rewrite as division.	$\frac{x}{2} \div \frac{x y}{6}$
Multiply the first fraction by the reciprocal of the second.	$\frac{x}{2} \cdot \frac{6}{x y}$
Multiply.	$\frac{x \cdot 6}{2 \cdot x y}$
Look for common factors.	$\frac{x̸ \cdot 3 \cdot 2̸}{2̸ \cdot x̸ \cdot y}$
Remove common factors and simplify.	$\frac{3}{y}$

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Simplify: $\frac{\frac{a}{8}}{\frac{a b}{6}} .$

$\frac{3}{4 b}$

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Simplify: $\frac{\frac{p}{2}}{\frac{p q}{8}} .$

$\frac{4}{q}$

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Simplify: $\frac{2 \frac{3}{4}}{\frac{1}{8}} .$

Solution

	$\frac{2 \frac{3}{4}}{\frac{1}{8}}$
Rewrite as division.	$2 \frac{3}{4} \div \frac{1}{8}$
Change the mixed number to an improper fraction.	$\frac{11}{4} \div \frac{1}{8}$
Multiply the first fraction by the reciprocal of the second.	$\frac{11}{4} \cdot \frac{8}{1}$
Multiply.	$\frac{11 \cdot 8}{4 \cdot 1}$
Look for common factors.	$\frac{11 \cdot 4̸ \cdot 2}{4̸ \cdot 1}$
Remove common factors and simplify.	$22$

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Simplify: $\frac{\frac{5}{7}}{1 \frac{2}{5}} .$

$\frac{25}{49} .$

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Simplify: $\frac{\frac{8}{5}}{3 \frac{1}{5}} .$

$\frac{1}{2}$

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Simplify expressions with a fraction bar

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 $\frac{−1}{3},$ a negative by a positive, or of dividing $\frac{1}{−3},$ a positive by a negative. When the numerator and denominator have different signs, the quotient is negative.

Negative 1 over positive 3 is equal to negative one third. Negative over positive equals negative. Positive 1 over negative 3 is equal to negative one third. Positive over negative equals 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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Source: OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9

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