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Division of Mixed Numbers by Fractions

Understand how to take a mixed number and divide it by a fraction.

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Division of Mixed Numbers by Fractions
Source: https://pixabay.com/en/food-bowl-fressnapf-dog-food-281980/
License: CC BY-NC 3.0

Audrey is checking to see how much dog food she has left. There is \begin{align*} 3 \frac{3}{4} \end{align*} pounds of dog food left in the bag. Her dog eats about a half pound of food per day. How many days worth of dog food does Audrey have left before she runs out?

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

Dividing Mixed Numbers by Fractions

You can divide a number by a fraction by multiplying the number by the reciprocal of the divisor. A mixed number consists of a whole number and a fraction, or "a whole and a part." When dividing a mixed number by a fraction, the quotient is the number of times the fraction divides into the mixed number. 

Here is a division problem.

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

In this problem, you are trying to figure out how many sets or groups of one-third can be made from one and one-half. 

Here is a drawing of one and one-half.

To figure out how many groups of one–third can be made from this quantity, you would have to divide these boxes up again into thirds. Dividing by a fraction using diagrams does not always provide clear answers. 

Instead, use these rules for dividing mixed numbers and fractions.

  1. Convert the mixed number to an improper fraction. 
  2. Change the division to its inverse, multiplication, and change the divisor to its reciprocal.
  3. Multiply and simplify to find the quotient.

Let's apply this information to the division problem.

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

First, convert the mixed number to an improper fraction.

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

Next, rewrite the problem.

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

Then, change the division to multiplication and change the divisor to its reciprocal. Flip the numerator and the denominator of a fraction to find the reciprocal. The reciprocal of \begin{align*}\frac {1}{3} \end{align*} is \begin{align*}\frac {3}{1}\end{align*}

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

Next, multiply and simplify the fractions.

\begin{align*}\frac{3 \times 3}{2 \times 1} = \frac{9}{2} = 4 \frac{1}{2}\end{align*}

The quotient is \begin{align*}4 \frac{1}{2}\end{align*}.

Examples

Example 1

Earlier, you were given a problem about Audrey and her dog.

Audrey's dog eats about a half pound of food per day and she has \begin{align*}3 \frac{3}{4}\end{align*} pounds of dog food left. Divide \begin{align*}3 \frac{3}{4}\end{align*} by \begin{align*}\frac{1}{2}\end{align*} to find out how many days worth of food she has.

First, write a division problem for the problem. 

\begin{align*}3 \frac{3}{4} \div \frac {1}{2}\end{align*}

Next, convert the mixed number to an improper fraction. 

\begin{align*}3 \frac{3}{4} = \frac {15}{4} \\ \frac {15}{4} \div \frac {1}{2}\end{align*}

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

\begin{align*}\frac {15}{4} \div \frac {1}{2} =\frac {15}{4} \times \frac {2}{1}\end{align*}

Next, multiply the fractions

\begin{align*}\frac {15}{\cancel{4}^2} \times \frac {\cancel{2}^1}{1}= \frac {15}{2}\end{align*}

Finally, convert the improper fraction to a mixed number.

\begin{align*}\frac {15}{2}=7\frac {1}{2}\end{align*}

The answer is \begin{align*}7 \frac{1}{2}\end{align*}.

Audrey will run out of dog food after 7 days. 

Example 2

Divide the mixed number: \begin{align*}6 \frac{2}{3} \div \frac{3}{4}\end{align*}. Answer in simplest form.

First, convert the mixed number to an improper fraction.

\begin{align*}6 \frac{2}{3} = \frac{20}{3}\end{align*}

Next, rewrite the problem.

\begin{align*}\frac {20}{3} \div \frac {3}{4}\end{align*}

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

\begin{align*} \frac{20}{3} \div \frac{3}{4} = \frac{20}{3} \times \frac{4}{3} \end{align*}

Next, multiply and simply the fractions.

\begin{align*} \frac{20 \times 4}{3 \times 3} = \frac{80}{9} = 8 \frac{8}{9}\end{align*}

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

Example 3

Divide the mixed numbers: \begin{align*}2 \frac{1}{3} \div \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, convert the mixed number to an improper fraction.

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

\begin{align*}2 \frac{1}{3} \div \frac{1}{4} =\frac{7}{3} \div \frac {1}{4}\\\end{align*}

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

\begin{align*}\frac{7}{3} \div \frac {1}{4} = \frac{7}{3} \times \frac {4}{1}\end{align*}

Next, multiply and simplify the fractions.

\begin{align*} \frac{7 \times 4}{3 \times 1} = \frac {28}{3} = 9 \frac {1}{3}\end{align*}

The quotient is \begin{align*}9 \frac{1}{3}\end{align*}.

Example 4

Divide the mixed numbers: \begin{align*}4 \frac{1}{2} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, convert the mixed number to an improper fraction.

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

\begin{align*}4 \frac{1}{2} \div \frac {1}{3} = \frac{9}{2} \div \frac {1}{3} \end{align*}

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

 \begin{align*}\frac{9}{2}\div \frac{1}{3} = \frac{9}{2}\times \frac{3}{1}\end{align*}

Next, multiply and simplify the fractions.

\begin{align*}\frac{9 \times 3}{2 \times 1}\times = \frac{27}{2} = 13 \frac{1}{2}\end{align*}

The quotient is \begin{align*}13 \frac{1}{2}\end{align*}.

Example 5

Divide the mixed numbers: \begin{align*}5 \frac{2}{3} \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, convert the mixed number to an improper fraction.

\begin{align*}5 \frac{2}{3} = \frac {17}{3}\\ \end{align*}

\begin{align*}5 \frac{2}{3} \div \frac{1}{2} = \frac {17}{3} \div \frac{1}{2}\\\end{align*}

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

\begin{align*}\frac {17}{3} \div \frac{1}{2} = \frac {17}{3} \times \frac{2}{1}\end{align*}

Next, multiply and simplify the fractions.

\begin{align*}\frac {17 \times 2}{3 \times 1} = \frac{34}{3} = 11 \frac {1}{3}\end{align*}

The quotient is \begin{align*}11 \frac{1}{3}\end{align*}.

Review

Divide the mixed number. Answer in simplest form.

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

Review (Answers)

To see the Review answers, open this PDF file and look for section 7.11.

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Vocabulary

Quotient

The quotient is the result after two amounts have been divided.

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