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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/

Maria can run 713\begin{align*}7\frac{1}{3}\end{align*} miles an hour. She decides to see how far she can run in 212\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.

6×114=\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 114\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 114\begin{align*}1\frac{1}{4}\end{align*} in terms of parts. Convert 114\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.

114=(4×1)+14=54\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.

6×114=6×54\begin{align*}6 \times 1 \frac{1}{4} = 6 \times \frac{5}{4}\end{align*}

Convert 6 into a fraction over one.

61×54\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.

61×54=31×52=152\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.

152=712\begin{align*} \frac {15}{2}= 7\frac{1}{2}\end{align*}

The product is 712\begin{align*}7 \frac{1}{2}\end{align*}.

Here is another mixed number multiplication problem.

12×214=\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  214\begin{align*}2 \frac{1}{4}\end{align*} .

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

214=94\begin{align*}2 \frac{1}{4} = \frac{9}{4}\end{align*}

12×94\begin{align*}\frac{1}{2} \times \frac{9}{4}\end{align*}

The fractions are in simplest form.

Then, multiply the fractions.

12×94=98\begin{align*} \frac{1}{2} \times \frac {9}{4} = \frac{9}{8} \end{align*}

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

98=118\begin{align*} \frac{9}{8} = 1\frac{1}{8}\end{align*}

The product is 118\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

214×112=\begin{align*}2\frac{1}{4} \times 1\frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}

First, convert each mixed number to an improper fraction.

214112=94=32\begin{align*}2\frac{1}{4} & = \frac{9}{4}\\ 1\frac{1}{2} & = \frac{3}{2}\end{align*}

Rewrite the problem.

94×32=\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.

94×32=278\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.

278=338\begin{align*}\frac{27}{8} = 3\frac{3}{8}\end{align*}

The product is 338\begin{align*}3\frac{3}{8}\end{align*}.

### Examples

#### Example 1

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

Maria can run 713\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*}

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

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Color Highlighted Text Notes

### Vocabulary Language: English

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 $4\frac{3}{5}$.