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Quotients of Mixed Numbers

Understand how to find a quotient between two mixed numbers.

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Quotients of Mixed Numbers
Credit: Dick Culbert
Source: https://www.flickr.com/photos/92252798@N07/15439579910/
License: CC BY-NC 3.0

Keith loves snakes. Last summer, he found a \begin{align*}2 \frac{1}{4}\end{align*} foot garter snake in his backyard. Keith read that the average anaconda is between 12 and 18 feet long. In the book, there was a picture of a \begin{align*}13 \frac{1}{2}\end{align*} feet long anaconda. How many garter snakes would it take to equal that anaconda?

In this concept, you will learn how to divide mixed numbers. 

Dividing Mixed Numbers

You can divide a mixed number by another mixed number. This means that you are looking for how many groups and parts of groups can be made from another whole and part. This seems complicated, but if you follow a few simple steps, you can figure it out. 

These are the rules for dividing a mixed number by a fraction.

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

Here is a division problem.

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

The first step to dividing a mixed number by another mixed number is to convert both mixed numbers to improper fractions.

\begin{align*}3 \frac{1}{2} & = \frac{7}{2} \\ 1 \frac{1}{4} & = \frac{5}{4}\end{align*}

Next, rewrite the problem with the improper fractions.

\begin{align*}\frac{7}{2} \div \frac{5}{4} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

Then, change the operation to multiplication and multiply by the reciprocal.

\begin{align*}\frac{7}{2} \div \frac{5}{4} = \frac{7}{2} \times \frac{4}{5} \end{align*}

Next, multiply the fractions and simplify the answer.

 \begin{align*}\frac{7 \times 4}{2 \times 5} = \frac{28}{10} = 2 \frac{ \cancel{8}^4}{\cancel{10}^5} = 2 \frac{4}{5}\end{align*}

The quotient is \begin{align*}2 \frac{4}{5}\end{align*}. Therefore, \begin{align*}1 \frac{1}{4}\end{align*} can divide into \begin{align*}3\frac{1}{2}\end{align*} a total of \begin{align*}2\frac{4}{5}\end{align*} times. 

Examples

Example 1

Earlier, you were given a problem about Keith and the garter snake.

Keith wrote a division problem to find out how many \begin{align*}2\frac{1}{4}\end{align*} feet long garter snakes would equal the length of a \begin{align*}13\frac{1}{2}\end{align*} feet long anaconda. Divide the length of the anaconda by the length of the garter snake. 

\begin{align*}13 \frac{1}{2} \div 2 \frac{1}{4} & = \underline{\;\;\;\;\;\;\;\;} \\ \end{align*}

First, convert both mixed numbers to improper fractions.

\begin{align*}13 \frac {1}{2} \div 2 \frac {1}{4} = \frac{27}{2} \div \frac{9}{4}\end{align*}

Then, change the operation to multiplication and multiply by th reciprocal.

\begin{align*}\frac{27}{2} \div \frac{9}{4} = \frac{27}{2} \times \frac{4}{9}\end{align*}

Next, multiply the fractions and simplify the answer. You can cross simplify before multiplying the fractions.

\begin{align*}\frac{\cancel{27}^3}{\cancel{2}^1} \times \frac{\cancel{4}^2}{\cancel{9}^1} = \frac{3\times2}{1\times1} = \frac {6}{1} = 6\end{align*}

It would take 6 garter snakes to equal the length of the one anaconda in Keith's book.

Example 2

Divide the mixed number: \begin{align*}12 \frac{1}{2} \div 2 \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, convert both mixed numbers to improper fractions.

\begin{align*}12\frac{1}{2} = \frac {25}{2}\\ 2\frac{1}{3} = \frac {7}{3}\\\end{align*}

Rewrite the problem with the improper fractions.

\begin{align*}\frac{25}{2} \div \frac{7}{3} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

Then, change the operation to multiplication and multiply by the reciprocal.

\begin{align*} \frac{25}{2} \div \frac{7}{3} = \frac{25}{2} \times \frac{3}{7} \end{align*}

Next, multiply the fractions and simplify the answer. 

\begin{align*} \frac{25 \times 3}{2 \times 7} = \frac{75}{14} = 5 \frac {5}{14}\end{align*}

The quotient is \begin{align*}5 \frac{5}{14}\end{align*}.

Example 3

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

First, convert both mixed numbers to improper fractions.

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

Then, change the operation to multiplication and multiply by the reciprocal.

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

Next, multiply the fractions and simplify the answer. 

\begin{align*} \frac {9\times2}{4\times3} = \frac {\cancel{18}^3}{\cancel {12}^2} = 1\frac{1}{2}\end{align*}

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

Example 4

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

First, convert both mixed numbers to improper fractions.

 \begin{align*}3 \frac{1}{3} \div 1 \frac{1}{4} = \frac {10}{3} \div \frac {5}{4}\end{align*}

Then, change the operation to multiplication and multiply by the reciprocal.

\begin{align*}\frac {10}{3} \div \frac {5}{4} = \frac {10}{3} \times \frac {4}{5}\end{align*}

Next, multiply the fractions and simplify the answer. 

\begin{align*}\frac {10\times 4}{3\times 5} = \frac {\cancel {40}^8}{\cancel{15}^3} = 2 \frac{2}{3}\end{align*}

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

Example 5

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

First, convert both mixed numbers to improper fractions.

\begin{align*}2 \frac{1}{5} \div 1 \frac{1}{2} = \frac {11}{5} \div \frac{3}{2}\end{align*}

Then, change the operation to multiplication and multiply by the reciprocal.

\begin{align*}\frac {11}{5} \div \frac{3}{2} = \frac {11}{5} \times \frac{2}{3}\end{align*}

Next, multiply the fractions and simplify the answer. 

\begin{align*}\frac{11\times 2}{5\times 3}= \frac{22}{15}= 1\frac{7}{15}\end{align*}

The quotient is \begin{align*}1 \frac{7}{15}\end{align*}.

Review

Divide the mixed number. Answer in simplest form.

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

Review (Answers)

To see the Review answers, open this PDF file and look for section 7.12. 

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Vocabulary

Quotient

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

Image Attributions

  1. [1]^ Credit: Dick Culbert; Source: https://www.flickr.com/photos/92252798@N07/15439579910/; License: CC BY-NC 3.0

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