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Quotients of Fractions

Understand the process of how to find a quotient between two fractions.

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Quotients of Fractions
Credit: Efraimstochter
Source: https://pixabay.com/en/waffle-waffle-irons-waffle-bake-203024/
License: CC BY-NC 3.0

Corey is planning on making waffles. The recipe says to use 34 cups of flour, but Corey can only find the 13-cup measuring cup. How many 13-cups does Corey need to make his waffles?

In this concept, you will learn how to divide a fraction by a fraction.

Dividing Fractions

When dividing whole numbers and fractions, you first change the operation to multiplication and then change the divisor to its reciprocal. The same rule applies to dividing a fraction by another fraction. Here is a division problem.

12÷13=

Start by applying the first part of the rule and change the sign to multiplication. Then apply the second part of the rule, the reciprocal of one-third is three over one.

12÷13=12×31

Then, multiply the fractions.

12×31=32

Next, simply the fraction. Convert the improper fraction a mixed number. 

32=112

The quotient is 112.

As long as you apply the rules, the problem is very straightforward and simple to figure out. Here is another one.

89÷13

First, change the operation and change 13 to its reciprocal.

89÷13=89×31

Then, multiply the fractions.

89×31=249

Next, simplify the fraction. Convert the improper fraction to a mixed number. 

249=259

The quotient is 259.

Examples

Example 1

Earlier, you were given a problem about Corey and his waffles.

Corey needs to measure out 34 cups of flour for his waffles, but can only find a 13 measuring cup. Divide 34 by 13 to find how many 13 cups Corey should use. 

First, write an expression.

34÷13

Then, change the operation to multiplication and change the divisor to its reciprocal. 

34÷13=34×31

Next, multiply the fractions. 

34×31=94

Finally, convert the improper fraction to a mixed number. 

94=214

Corey can use a little more than 2 of the 13 measuring cup to make his waffles.  

Example 2

Divide the fractions: 49÷12= _____. Answer in simplest form.

First, change the operation to multiplication and 12 to its reciprocal.

34÷13

Next, multiply the fractions. 

49×21=89

The quotient is 89.

Example 3

Divide the fractions: 14÷34=. Answer in simplest form.

First, change the expression. Multiply by the inverse of the divisor. 

14÷34=14×43

Then, multiply.

141×413=13

The quotient is 13.

Example 4

Divide the fractions: 78÷14=. Answer in simplest form.

First, change the expression. Multiply by the inverse of the divisor.

78÷14=78×41

Then, multiply. 

782×411=72

Next, convert the improper fraction to a mixed number. 

72=312

The quotient is 312

Example 5

Divide the fractions: 14÷13=. Answer in simplest form.

First, change the expression. Multiply by the inverse of the divisor. 

14÷13=14×31

Then, multiply. 

14×31=34

The quotient is 34.

Review

Divide the fractions. Answer in simplest form.

  1. 12÷13=
  2. 14÷15=
  3. 25÷12=
  4. 47÷13=
  5. 68÷12=
  6. 49÷13=
  7. 56÷12=
  8. 610÷12=
  9. \begin{align*}\frac{9}{18} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
  10. \begin{align*}\frac{8}{9} \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
  11. \begin{align*}\frac{15}{16} \div \frac{1}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}
  12. \begin{align*}\frac{8}{11} \div \frac{3}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
  13. \begin{align*}\frac{12}{16} \div \frac{3}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
  14. \begin{align*}\frac{20}{24} \div \frac{3}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
  15. \begin{align*}\frac{18}{20} \div \frac{4}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 7.10. 

Resources

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Vocabulary

Inverse Operation

Inverse operations are operations that "undo" each other. Multiplication is the inverse operation of division. Addition is the inverse operation of subtraction.

reciprocal

The reciprocal of a number is the number you can multiply it by to get one. The reciprocal of 2 is 1/2. It is also called the multiplicative inverse, or just inverse.

Image Attributions

  1. [1]^ Credit: Efraimstochter; Source: https://pixabay.com/en/waffle-waffle-irons-waffle-bake-203024/; License: CC BY-NC 3.0

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