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# Products of Mixed Numbers

## Multiply fractions > 1.

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Products of Mixed Numbers

Cynthia just got the recipe for her grandmother’s famous Italian bread. The recipe makes 3 loaves of bread, which is more than Cynthia wants right now. Cynthia only wants to make 2 loaves of bread so she is going to need \begin{align*}\frac{2}{3} \end{align*} of all of the ingredients listed in the recipe. The recipe calls for \begin{align*}4 \frac{1}{2}\end{align*} tablespoons of olive oil along with 7 other ingredients. How can Cynthia figure out how much olive oil she should use to make only 2 loaves of bread?

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

### Multiplying Fractions and Mixed Numbers

Recall that in a fraction, the number written above the bar is the numerator and the number written below the bar is the denominator. When multiplying fractions, you will multiply the numerators and multiply the denominators.

To multiply fractions:

Here is an example.

Find the product of \begin{align*}\frac{1}{2} \cdot \frac{4}{5}\end{align*}.

First, multiply the numerators and multiply the denominators. Note that you do not need to find a common denominator first!

\begin{align*}\frac{1}{2} \cdot \frac{4}{5}=\frac{1 \cdot 4}{2 \cdot 5} =\frac{4}{10}\end{align*}

Next, simplify your answer. The greatest common factor of 4 and 10 is 2, so divide by 4 and 10 by 2 to simplify the fraction.

\begin{align*}\frac{4}{10} = \frac{4 \div 2}{10 \div 2}=\frac{2}{5}\end{align*}

The answer is \begin{align*}\frac{1}{2} \cdot \frac{4}{5}=\frac{2}{5}\end{align*}.

You can also multiply fractions and whole numbers. You will first need to turn the whole number into a fraction by writing it over 1.

Here is an example.

Find the product of \begin{align*}5 \cdot \frac{1}{2}\end{align*}.

First, rewrite 5 as \begin{align*}\frac{5}{1}\end{align*}.

\begin{align*}\frac{5}{1} \cdot \frac{1}{2}\end{align*}

Next, multiply the numerators and multiply the denominators.

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

Now, simplify your answer. You can rewrite \begin{align*}\frac{5}{2}\end{align*} as a mixed number.

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

The answer is \begin{align*}5 \cdot \frac{1}{2}=2 \frac{1}{2}\end{align*}.

You can also multiply mixed numbers. Because mixed numbers involve both wholes and parts, multiplying mixed numbers requires an extra step. Before multiplying mixed numbers, you will convert them to improper fractions.

To multiply mixed numbers:

1. Convert the mixed numbers to improper fractions.

Here is an example.

Find the product of \begin{align*}3 \frac{1}{2} \cdot 2 \frac{1}{3}\end{align*}.

First, rewrite both mixed numbers as improper fractions.

\begin{align*}3 \frac{1}{2} = \frac{(2 \cdot 3 )+1}{2} = \frac{7}{2} \\ 2 \frac{1}{3} = \frac{(3 \cdot 2 )+1}{3} = \frac{7}{3} \end{align*}

Next, multiply the numerators and multiply the denominators.

\begin{align*}\frac{7}{2} \cdot \frac{7}{3}=\frac {49}{6}\end{align*}

Now, simplify your answer. You can rewrite \begin{align*}\frac{49}{6}\end{align*} as a mixed number.

\begin{align*}\frac{49}{6} = 8 \frac {1}{6}\end{align*}

The answer is \begin{align*}3 \frac{1}{2} \cdot 2 \frac{1}{3}=8 \frac {1}{6}\end{align*}.

Sometimes when you multiply fractions or mixed numbers you will end up with very large numbers. To avoid having to deal with such big numbers, you can simplify before multiplying. To do this, look for common factors and divide them out.

Here is an example.

Find the product of \begin{align*}\frac{2}{9} \cdot \frac{18}{30}\end{align*}.

First, write the numerator and the denominator as a product.

\begin{align*}\frac{2}{9} \cdot \frac{18}{30} = \frac{2 \cdot 18}{9 \cdot 30}\end{align*}

Next, look for common factors along the diagonals in the numerator and the denominator. First look at 2 and 30. Both 2 and 30 have a factor of 2. Divide both 2 and 30 by 2.

\begin{align*}\frac{2 \cdot 18}{9 \cdot 30} = \frac{1 \cdot 18}{9 \cdot 15}\end{align*}

Now, look at 9 and 18. Both 9 and 18 have a factor of 9. Divide both 9 and 18 by 9.

\begin{align*}\frac{1 \cdot 18}{9 \cdot 15} = \frac{1 \cdot 2}{1 \cdot 15}\end{align*}

Now, multiply the numbers in the numerator and the numbers in the denominator.

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

The answer is \begin{align*}\frac{2}{9} \cdot \frac{18}{30}=\frac{2}{15}\end{align*}.

### Examples

#### Example 1

Earlier, you were given a problem about Cynthia and her Italian bread.

Her recipe is for 3 loaves but she only wants to make 2 loaves. Her plan is to use \begin{align*}\frac{2}{3}\end{align*} of all the ingredients listed in the recipe. The recipe calls for \begin{align*}4 \frac{1}{2}\end{align*} tablespoons of olive oil. Cynthia wants to figure out how much olive oil she should use to make only 2 loaves of bread.

Cynthia needs to find \begin{align*}\frac{2}{3}\end{align*} of \begin{align*}4 \frac{1}{2}\end{align*}. She will need to multiply.

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

First, Cynthia should rewrite the mixed number as an improper fraction.

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

Next, she should multiply the numerators and multiply the denominators.

\begin{align*}\frac{2}{3} \cdot \frac{9}{2}=\frac{18}{6}\end{align*}

Now, she should simplify her answer.

\begin{align*}\frac{18}{6}=3\end{align*}

The answer is Cynthia should use 3 tablespoons of olive oil to make 2 loaves of bread.

#### Example 2

Deirdre claims that it takes her \begin{align*}6 \frac{3}{4}\end{align*} hours to complete her homework every night. Carlos thinks he can finish his homework in \begin{align*}\frac{2}{3}\end{align*} that time. How long does Carlos think it will take him to complete his homework?

To figure out how long Carlos thinks it will take him to complete his homework, you need to find \begin{align*}\frac{2}{3}\end{align*} of \begin{align*}6 \frac{3}{4}\end{align*}. The word “of” indicates multiplication, so you will need to multiply.

\begin{align*}\frac{2}{3} \cdot 6 \frac{3}{4}\end{align*}

First, rewrite the mixed number as an improper fraction.

\begin{align*}6 \frac{3}{4} = \frac{27}{4}\end{align*}

Next, multiply the numerators and multiply the denominators.

\begin{align*}\frac{2}{3} \cdot \frac{27}{4} = \frac{54}{12}\end{align*}

\begin{align*}\frac{54}{12}= 4 \frac{6}{12}=4 \frac{1}{2}\end{align*}

The answer is Carlos thinks he can complete his homework in \begin{align*}4 \frac{1}{2}\end{align*} hours.

#### Example 3

Find the product of \begin{align*}\frac{1}{3} \cdot \frac{5}{6}\end{align*}.

First, multiply the numerators and multiply the denominators.

\begin{align*}\frac{1}{3} \cdot \frac{5}{6} = \frac{1 \cdot 5}{3 \cdot 6}=\frac{5}{18}\end{align*}

Next, check to see if you need to simplify your answer. In this case, \begin{align*}\frac{5}{18}\end{align*} is in simplest form.

The answer is \begin{align*}\frac{1}{3} \cdot \frac{5}{6}=\frac{5}{18}\end{align*}.

#### Example 4

Find the product of \begin{align*}\frac{18}{20} \cdot \frac{4}{9}\end{align*}.

First, write the numerator and the denominator as a product.

\begin{align*}\frac{18}{20} \cdot \frac{4}{9} =\frac{18 \cdot 4}{20 \cdot 9}\end{align*}

Next, look for common factors along the diagonals in the numerator and the denominator. First look at 18 and 9. Both 18 and 9 have a factor of 9. Divide both 18 and 9 by 9.

\begin{align*}\frac{18 \cdot 4}{20 \cdot 9} = \frac{2 \cdot 4}{20 \cdot 1}\end{align*}

Now, look at 4 and 20. Both 4 and 20 have a factor of 4. Divide both 4 and 20 by 4.

\begin{align*}\frac{2 \cdot 4}{20 \cdot 1} = \frac{2 \cdot 1}{5 \cdot 1}\end{align*}

Now, multiply the numbers in the numerator and the numbers in the denominator.

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

The answer is \begin{align*}\frac{18}{20} \cdot \frac{4}{9}= \frac{2}{5}\end{align*}.

#### Example 5

Find the product of \begin{align*}2 \frac{1}{5} \cdot 3 \frac{1}{2}\end{align*}.

First, rewrite both mixed numbers as improper fractions.

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

Next, multiply the numerators and multiply the denominators.

\begin{align*}\frac{11}{5} \cdot \frac{7}{2} =\frac{77}{10}\end{align*}

Now, simplify your answer. You can rewrite \begin{align*}\frac{77}{10}\end{align*} as a mixed number.

\begin{align*}\frac{77}{10} = 7 \frac{7}{10}\end{align*}

The answer is \begin{align*}2 \frac{1}{5} \cdot 3 \frac{1}{2}=7 \frac{7}{10}\end{align*}.

### Review

Multiply.

1. \begin{align*}\frac{1}{4} \cdot \frac{3}{7}\end{align*}
2. \begin{align*}\frac{5}{6} \cdot \frac{2}{3}\end{align*}
3. \begin{align*}\frac{3}{10} \cdot \frac{10}{12}\end{align*}
4. \begin{align*}\frac{4}{7} \cdot \frac{2}{3}\end{align*}
5. \begin{align*}\frac{1}{3} \cdot 2\frac{2}{3}\end{align*}
6. \begin{align*}2\frac{5}{7} \cdot 1\frac{1}{5}\end{align*}
7. \begin{align*}2\frac{3}{10} \cdot 2\frac{1}{4}\end{align*}
8. \begin{align*}7\frac{1}{5} \cdot \frac{1}{11}\end{align*}
9. \begin{align*}4\frac{5}{8} \cdot 2\end{align*}
10. \begin{align*}\frac{1}{7} \cdot \frac{1}{6}\end{align*}
11. \begin{align*}3\frac{5}{6} \cdot 1\frac{2}{3}\end{align*}
12. \begin{align*}\frac{1}{5} \cdot \frac{7}{12}\end{align*}
13. \begin{align*}\frac{2}{3} \cdot \frac{9}{12} \cdot \frac{6}{7}\end{align*}
14. \begin{align*}\frac{1}{3} \cdot 1\frac{4}{5} \cdot \frac{3}{4}\end{align*}
15. \begin{align*}\frac{4}{9} \cdot \frac{5}{8} \cdot \frac{3}{7}\end{align*}
16. \begin{align*}\frac{10}{12} \cdot \bigg(3\frac{1}{5} \cdot \frac{7}{10}\bigg)\end{align*}

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### Vocabulary Language: English

TermDefinition
fraction A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a rational number.
Greatest Common Factor The greatest common factor of two numbers is the greatest number that both of the original numbers can be divided by evenly.
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 $4\frac{3}{5}$.