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Products of Mixed Numbers

Multiply fractions > 1.

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Products of Mixed Numbers
Credit: Skeeze
Source: https://pixabay.com/en/runner-training-high-leg-jogging-811877/
License: CC BY-NC 3.0

Maria can run \begin{align*}7\frac{1}{3}\end{align*} miles an hour. She decides to see how far she can run in \begin{align*}2\frac{1}{2}\end{align*} hours. If she maintains her speed, how many miles did Maria run?

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

Multiplying Mixed Numbers

A mixed number consists of a whole number and a fraction.

Here is multiplication problem involving a whole number and a mixed number.

\begin{align*}6 \times 1 \frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}

Multiplication is a shortcut for repeated addition. The product of this expression is the total quantity of 6 groups of \begin{align*}1 \frac{1}{4}\end{align*}. You can multiply 6 wholes and 6 parts and then find the sum of both. Or you can think of \begin{align*}1\frac{1}{4}\end{align*} in terms of parts. Convert \begin{align*}1\frac{1}{4}\end{align*} to an improper fraction. Remember that to convert a mixed number to an improper fraction, multiply the denominator by the whole number. Then, add the numerator to the product and write that sum over the original denominator. 

\begin{align*}1 \frac{1}{4} = \frac{\left(4 \times 1\right)+1}{4} = \frac{5}{4}\end{align*}

Here is the problem again with the improper fraction.

\begin{align*}6 \times 1 \frac{1}{4} = 6 \times \frac{5}{4}\end{align*}

Convert 6 into a fraction over one. 

\begin{align*}\frac{6}{1} \times \frac{5}{4} \end{align*}

Then you can either multiply the fraction and simplify or simplify first and then multiply. Here you can cross simplify the fractions first.

 \begin{align*}\frac {6}{1} \times \frac {5}{4} =\frac {3}{1} \times \frac {5}{2} = \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 product is \begin{align*}7 \frac{1}{2}\end{align*}.

Here is another mixed number multiplication problem.

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

A fraction is a "part" and a mixed number consists of "wholes and a part." When multiplying a fraction and a mixed number, you are looking for "a part of a whole and a part." The product of this expression is half of  \begin{align*}2 \frac{1}{4}\end{align*} .

The first step is to convert the mixed number to an improper fraction.

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

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

The fractions are in simplest form.

Then, multiply the fractions.

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

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

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

The product is \begin{align*}1\frac{1}{8}\end{align*}.

Here is a multiplication problem with two mixed numbers.

The product of this expression is a whole and a part of another whole and a part. The key is to follow the same steps to find the solution.

  1. Convert the mixed numbers to improper fractions.
  2. Simplify if possible
  3. Multiply
  4. Check that your answer is in simplest form.

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

First, convert each mixed number to an improper fraction.

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

Rewrite the problem.

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

The fractions cannot be simplified so at this point. 

Then, multiply the fractions. 

\begin{align*}\frac{9}{4} \times \frac{3}{2} = \frac{27}{8} \end{align*}

Finally, check if the fraction is in simplest form. Convert the improper fraction to a mixed number.

\begin{align*}\frac{27}{8} = 3\frac{3}{8}\end{align*}

The product is \begin{align*}3\frac{3}{8}\end{align*}.

Examples

Example 1

Earlier, you were given a problem about Maria's run.

Maria can run \begin{align*}7\frac{1}{3}\end{align*} miles per hour and decides to run for \begin{align*}2\frac{1}{2}\end{align*} hours. Multiply her speed times the number of hours to find the total number of miles run.

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

First, convert the mixed numbers to improper fractions.

\begin{align*}7\frac{1}{3} &= \frac{22}{3} \\ 2\frac{1}{2} &= \frac{3}{2}\\ \frac{22}{3}&\times\frac{3}{2}\end{align*}

Then, simplify the fractions. You can simply 2 and 22 with the GCF of 2 and 3 and 3 with the GCF 3.

 \begin{align*}\frac{22}{3}&\times\frac{3}{2}=\frac{11}{1}\times\frac{1}{1}\end{align*}

Next, multiply.

 \begin{align*}11\times 1=11\end{align*}

Maria ran 11 miles.

Example 2

Find the product. Answer in simplest form.

\begin{align*}\frac{1}{3} \times 2\frac{1}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}

First, convert the mixed number to an improper fraction. 

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

The fractions are in simplest form.

Then, multiply the fractions. 

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

The product is \begin{align*} \frac{11}{15}\end{align*}.

Example 3

Find the product: \begin{align*}4 \times 2 \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, convert the whole number and mixed number to fractions. 

\begin{align*}\frac{4}{1}\times \frac{5}{2}\end{align*}

Then, cross simplify the fractions. 

\begin{align*}\frac{4}{1}\times \frac{5}{2}=\frac{2}{1}\times \frac{5}{1}\end{align*}

Next, multiply the fractions. 

\begin{align*}\frac{2}{1}\times \frac{5}{1}=\frac{10}{1}=10\end{align*}

Note that a fraction with a number over the denominator of 1 is a whole number.  

The product is 10.

Example 4

Find the product: \begin{align*}\frac {1}{6}\times 1 \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, convert the mixed number to an improper fraction. 

\begin{align*}\frac{1}{6}\times \frac{4}{3}\end{align*}

Then, cross simplify the fractions.

\begin{align*}\frac{1}{6}\times \frac{4}{3}=\frac{1}{3}\times \frac{2}{3}\end{align*}

Next, multiply the fractions. 

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

The fraction is in simplest form.

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

Example 5

Find the product: \begin{align*}4\frac{1}{3} \times 1 \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, convert the mixed numbers to improper fractions. 

\begin{align*}4\frac{1}{3}&=\frac{13}{3}\\ 1\frac{1}{2}& = \frac{3}{2}\end{align*}

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

Then, cross simplify the fractions.

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

Next, multiply the fractions.

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

Finally, convert the improper fraction to a mixed number.

 \begin{align*}\frac{13}{2}=7\frac{1}{2}\end{align*}

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

Review

Find the product in simplest form.

  1. \begin{align*}7 \times 1\frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
  2. \begin{align*}8 \times 2\frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
  3. \begin{align*}6 \times 3\frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
  4. \begin{align*}5 \times 3\frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
  5. \begin{align*}9 \times 2\frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
  6. \begin{align*}7 \times 4\frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
  7. \begin{align*}9 \times 2\frac{1}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}
  8. \begin{align*}6 \times 4\frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
  9. \begin{align*}8 \times 2\frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
  10. \begin{align*}6 \times 6\frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
  11. \begin{align*}\frac{1}{3} \times 2\frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
  12. \begin{align*}\frac{1}{2} \times 4\frac{2}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
  13. \begin{align*}\frac{1}{4} \times 6\frac{2}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
  14. \begin{align*}\frac{2}{3} \times 4\frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
  15. \begin{align*}\frac{1}{5} \times 5\frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
  16. \begin{align*}\frac{2}{3} \times 2\frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
  17. \begin{align*}\frac{4}{7} \times 2\frac{1}{7} = \underline{\;\;\;\;\;\;\;}\end{align*}
  18. \begin{align*}3\frac{1}{2} \times 2\frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
  19. \begin{align*}3\frac{1}{2} \times 2\frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
  20. \begin{align*}5\frac{1}{2} \times 3\frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
  21. \begin{align*}1\frac{4}{5} \times 3\frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
  22. \begin{align*}1\frac{1}{2} \times 2\frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
  23. \begin{align*}9\frac{1}{2} \times 9\frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
  24. \begin{align*}\frac{1}{8} \times 8\frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
  25. \begin{align*}\frac{4}{7} \times 2\frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 7.6. 

Resources

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Vocabulary

improper fraction

An improper fraction is a fraction in which the absolute value of the numerator is greater than the absolute value of the denominator.

Mixed Number

A mixed number is a number made up of a whole number and a fraction, such as 4\frac{3}{5}.

Image Attributions

  1. [1]^ Credit: Skeeze; Source: https://pixabay.com/en/runner-training-high-leg-jogging-811877/; License: CC BY-NC 3.0

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