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Find the quotient: 7 27 ÷ ( 35 36 ) .

4 15

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Find the quotient: 5 14 ÷ ( 15 28 ) .

2 3

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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:

This is an image with two columns. The first column reads “One fourth of two pizzas is one half of a pizza. Below this are two pizzas side-by-side with a line down the center of each one representing one half. The halves are labeled “one half”. Under this is the equation “2 times 1 fourth”. Under this is another equation “two over 1 times 1 fourth.” Under this is the fraction two fourths and under this is the fraction one half. The next column reads “there are eight quarters in two dollars.” Under this are eight quarters in two rows of four. Under this is the fraction equation 2 divided by one fourth. Under this is the equation “two over one divided by one fourth.” Under this is two over one times four over one. Under this is the answer “8”.

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 .

Complex fraction

A complex fraction    is a fraction in which the numerator or the denominator contains a fraction.

Some examples of complex fractions are:

6 7 3 3 4 5 8 x 2 5 6

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

Simplify: 3 4 5 8 .

Solution

3 4 5 8
Rewrite as division. 3 4 ÷ 5 8
Multiply the first fraction by the reciprocal of the second. 3 4 8 5
Multiply. 3 8 4 5
Look for common factors. .
Divide out common factors and simplify. 6 5
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Simplify: 3 7 6 11 .

11 14

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Simplify: x 2 x y 6 .

Solution

x 2 x y 6
Rewrite as division. x 2 ÷ x y 6
Multiply the first fraction by the reciprocal of the second. x 2 6 x y
Multiply. x 6 2 x y
Look for common factors. .
Divide out common factors and simplify. 3 y
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Simplify: a 8 a b 6 .

3 4 b

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Simplify: p 2 p q 8 .

4 2 q

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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 5 3 7 + 1 , we first simplify the numerator and the denominator separately. Then we divide.

5 3 7 + 1
2 8
1 4

Simplify an expression with a fraction bar.

  1. Simplify the expression in the numerator. Simplify the expression in the denominator.
  2. Simplify the fraction.

Simplify: 4 2 ( 3 ) 2 2 + 2 .

Solution

4 2 ( 3 ) 2 2 + 2 Use the order of operations to simplify the numerator and the denominator. 4 6 4 + 2 Simplify the numerator and the denominator. −2 6 Simplify. A negative divided by a positive is negative. 1 3

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Simplify: 6 3 ( 5 ) 3 2 + 3 .

3 4

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Simplify: 4 4 ( 6 ) 3 2 + 3 .

2 3

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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.

−1 3 = 1 3 negative positive = negative 1 −3 = 1 3 positive negative = negative

Placement of negative sign in a fraction

For any positive numbers a and b ,

a b = a b = a b

Simplify: 4 ( −3 ) + 6 ( −2 ) −3 ( 2 ) 2 .

Solution

The fraction bar acts like a grouping symbol. So completely simplify the numerator and the denominator separately.

4 ( −3 ) + 6 ( −2 ) −3 ( 2 ) 2 Multiply. −12 + ( −12 ) −6 2 Simplify. −24 −8 Divide. 3

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Simplify: 8 ( −2 ) + 4 ( −3 ) −5 ( 2 ) + 3 .

4

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Simplify: 7 ( −1 ) + 9 ( −3 ) −5 ( 3 ) 2 .

2

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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.

Practice Key Terms 7

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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