1.5 Visualize fractions  (Page 4/12)

There are several ways to remember which steps to take to multiply or divide fractions. One way is to repeat the call outs to yourself. If you do this each time you do an exercise, you will have the steps memorized.

• “To multiply fractions, multiply the numerators and multiply the denominators.”
• “To divide fractions, multiply the first fraction by the reciprocal of the second.”

Another way is to keep two examples in mind:

The numerators or denominators of some fractions contain fractions themselves. A fraction in which the numerator or the denominator is a fraction is called a complex fraction .

Some examples of complex fractions are:

To simplify a complex fraction, we remember that the fraction bar means division . For example, the complex fraction $\frac{\frac{3}{4}}{\frac{5}{8}}$ means $\frac{3}{4}÷\frac{5}{8}.$

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

Solution

	$\frac{\frac{3}{4}}{\frac{5}{8}}$
Rewrite as division.	$\frac{3}{4}÷\frac{5}{8}$
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.
Divide out common factors and simplify.	$\frac{6}{5}$

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

Solution

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

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Simplify expressions with a fraction bar

The line that separates the numerator from the denominator in a fraction is called a fraction bar. A fraction bar acts as grouping symbol. The order of operations then tells us to simplify the numerator and then the denominator. Then we divide.

To simplify the expression $\frac{5-3}{7+1},$ we first simplify the numerator and the denominator separately. Then we divide.

Where does the negative sign go in a fraction? Usually the negative sign is in front of the fraction, but you will sometimes see a fraction with a negative numerator, or sometimes with a negative denominator. Remember that fractions represent division. When the numerator and denominator have different signs, the quotient is negative.

Translate phrases to expressions with fractions

Now that we have done some work with fractions, we are ready to translate phrases that would result in expressions with fractions.