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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 \begin{align*}\frac{3}{4}\end{align*} cups of flour, but Corey can only find the \begin{align*}\frac{1}{3}\end{align*}-cup measuring cup. How many \begin{align*}\frac{1}{3}\end{align*}-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.

\begin{align*}\frac{1}{2} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

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.

\begin{align*}\frac{1}{2} \div \frac{1}{3} = \frac{1}{2} \times \frac{3}{1}\end{align*}

Then, multiply the fractions.

\begin{align*}\frac{1}{2} \times \frac{3}{1} = \frac{3}{2}\end{align*}

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

\begin{align*}\frac{3}{2} = 1\frac{1}{2}\end{align*}

The quotient is \begin{align*}1\frac{1}{2}\end{align*}.

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

\begin{align*}\frac{8}{9} \div \frac{1}{3}\end{align*}

First, change the operation and change \begin{align*}\frac{1}{3}\end{align*} to its reciprocal.

\begin{align*}\frac{8}{9} \div \frac{1}{3} = \frac{8}{9} \times \frac{3}{1} \end{align*}

Then, multiply the fractions.

\begin{align*} \frac{8}{9} \times \frac{3}{1} = \frac{24}{9} \end{align*}

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

\begin{align*}\frac{24}{9} = 2\frac{5}{9}\end{align*}

The quotient is \begin{align*}2\frac{5}{9}\end{align*}.

Examples

Example 1

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

Corey needs to measure out \begin{align*}\frac{3}{4}\end{align*} cups of flour for his waffles, but can only find a \begin{align*}\frac{1}{3}\end{align*} measuring cup. Divide \begin{align*}\frac{3}{4}\end{align*} by \begin{align*}\frac{1}{3}\end{align*} to find how many \begin{align*}\frac{1}{3}\end{align*} cups Corey should use. 

First, write an expression.

\begin{align*} \frac{3}{4} \div \frac{1}{3}\end{align*}

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

\begin{align*} \frac{3}{4} \div \frac{1}{3} = \frac{3}{4} \times \frac{3}{1}\end{align*}

Next, multiply the fractions. 

\begin{align*} \frac{3}{4} \times \frac{3}{1} = \frac {9}{4}\end{align*}

Finally, convert the improper fraction to a mixed number. 

\begin{align*}\frac {9}{4} = 2\frac {1}{4}\end{align*}

Corey can use a little more than 2 of the \begin{align*}\frac{1}{3}\end{align*} measuring cup to make his waffles.  

Example 2

Divide the fractions: \begin{align*} \frac{4}{9} \div \frac{1}{2}\end{align*}= _____. Answer in simplest form.

First, change the operation to multiplication and \begin{align*}\frac{1}{2}\end{align*} to its reciprocal.

\begin{align*} \frac{3}{4} \div \frac{1}{3}\end{align*}

Next, multiply the fractions. 

\begin{align*}\frac{4}{9} \times \frac{2}{1}= \frac {8}{9}\end{align*}

The quotient is \begin{align*} \frac{8}{9}\end{align*}.

Example 3

Divide the fractions: \begin{align*}\frac{1}{4} \div \frac{3}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

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

\begin{align*}\frac{1}{4} \div \frac{3}{4} = \frac {1}{4} \times \frac {4}{3}\end{align*}

Then, multiply.

\begin{align*} \frac {1}{\cancel{4}^1} \times \frac {\cancel{4}^1}{3} = \frac {1}{3} \end{align*}

The quotient is \begin{align*} \frac{1}{3}\end{align*}.

Example 4

Divide the fractions: \begin{align*}\frac{7}{8} \div \frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

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

\begin{align*}\frac{7}{8} \div \frac{1}{4} = \frac{7}{8} \times \frac{4}{1} \end{align*}

Then, multiply. 

\begin{align*} \frac{7}{\cancel{8}^2} \times \frac{\cancel{4}^1}{1} = \frac {7}{2}\end{align*}

Next, convert the improper fraction to a mixed number. 

\begin{align*}\frac{7}{2} = 3 \frac {1}{2}\end{align*}

The quotient is \begin{align*}3 \frac{1}{2}\end{align*}

Example 5

Divide the fractions: \begin{align*}\frac{1}{4} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

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

\begin{align*}\frac{1}{4} \div \frac{1}{3} = \frac {1}{4} \times \frac{3}{1}\end{align*}

Then, multiply. 

\begin{align*}\frac {1}{4} \times \frac{3}{1} = \frac {3}{4}\end{align*}

The quotient is \begin{align*} \frac{3}{4}\end{align*}.

Review

Divide the fractions. Answer in simplest form.

  1. \begin{align*}\frac{1}{2} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
  2. \begin{align*}\frac{1}{4} \div \frac{1}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}
  3. \begin{align*}\frac{2}{5} \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
  4. \begin{align*}\frac{4}{7} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
  5. \begin{align*}\frac{6}{8} \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
  6. \begin{align*}\frac{4}{9} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
  7. \begin{align*}\frac{5}{6} \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
  8. \begin{align*}\frac{6}{10} \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
  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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