# Products of Three Fractions

## Multiply a series of fractions

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Products of Three Fractions
Credit: trenttsd
Source: https://www.flickr.com/photos/84335369@N00/6614318221/
License: CC BY-NC 3.0

Roberto is working on his math homework. He needs to find the volume of a rectangular prism. The formula is for the volume of a rectangular prism is:

\begin{align*}V = \frac{3}{5} \times \frac{1}{4} \times \frac{5}{6}\end{align*}

The given dimensions are length = \begin{align*}\frac{3}{5}\end{align*}, width = \begin{align*}\frac{1}{4}\end{align*}, and height \begin{align*}\frac{5}{6}\end{align*}. The units are in inches. What is the volume of the  prism?

In this concept, you will learn how to multiply three fractions.

### Multiplying Three Fractions

Multiplying three fractions only sounds more complicated than multiplying two fractions. The procedure is the same. Multiply the numerators and the denominators to find the product. But before multiplying, simplify the fractions if you can. This way, you won’t end up with a large fraction that is challenging to simplify at the end.

There are two ways to simplify before multiplying:

1. Simplify any whole fraction.

Example: \begin{align*}\frac{3}{6}=\frac{1}{2}\\ \frac{3}{6}\times \frac{1}{9}& =\frac{1}{2}\times \frac{1}{9}\end{align*}

1. Cross simplify using the greatest common factor (GCF).

Example: The GCF of 3 and 9 is 3. \begin{align*}\frac{\cancel{3}}{6}\times \frac{1}{\cancel{9}}=\frac{1}{6}\times \frac{1}{3}\end{align*}

Let's look at an example.

Multiply the following fractions.

\begin{align*}\frac{1}{4} \times \frac{2}{6} \times \frac{4}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}

First, simplify the fractions. Check to see if any of the individual fractions can be simplified. The fraction \begin{align*}\frac{2}{6}\end{align*} can be simplified.

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

\begin{align*}\frac{1}{4}\times \frac{1}{3} \times \frac{4}{5}\end{align*}

Also check the diagonals. The two 4s can be cross simplified with the GCF of 4. Each one simplifies to 1.

\begin{align*}\frac{1}{\cancel{4}^1}\times \frac{1}{3} \times \frac{\cancel{4}^1}{5} = \frac{1}{1}\times \frac{1}{3} \times \frac{1}{5}\end{align*}

Then, multiply the fractions. Find the product of the numerators over the product of the denominators.

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

The final answer is \begin{align*}\frac{1}{15}\end{align*}.

Here is another example.

\begin{align*}\frac{5}{9} \times \frac{7}{14} \times \frac{3}{5} = \underline{\;\;\;\;\;\;}\end{align*}

First, simplify the fraction. The 5s simplify with the GCF of 5. The 3 and 9 also simplify with the GCF of 3.

\begin{align*}\frac{^1 \cancel{5}}{^3 \cancel{9}} \times \frac{7}{14} \times \frac{^1 \cancel{3}}{^1 \cancel{5}} = \frac{1}{3} \times \frac{7}{14} \times \frac{1}{1}\end{align*}

The fraction \begin{align*}\frac{7}{14}\end{align*} can also be simplified.

\begin{align*}\frac{1}{3} \times \frac{\cancel{7}^1}{\cancel{14}^2} \times \frac{1}{1} & = \frac{1}{3} \times \frac{1}{2} \times \frac{1}{1}\end{align*}

Then, multiply the fractions. Find the product of the numerators over the product of the denominators.

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

The final answer is \begin{align*}\frac{1}{6}\end{align*}.

### Examples

#### Example 1

Earlier, you were given a problem about Roberto's math question.

Roberto needs to find the volume of a rectangular prism with the dimensions of length = \begin{align*} \frac{3}{5} \end{align*}, width = \begin{align*}\frac{1}{4} \end{align*}, and height = \begin{align*}\frac{5}{6}\end{align*}. Use the formula for the volume of a rectangular prism to find the volume.

\begin{align*}V=l \times w \times h\end{align*}

First, substitute the known values.

\begin{align*}V= \frac{3}{5} \times \frac{1}{4} \times \frac{5}{6}\end{align*}

Then, simplify the fractions. \begin{align*}V= \frac{\cancel{3}^1}{\cancel{5}^1} \times \frac{1}{4} \times \frac{\cancel{5}^1}{\cancel{6}^2}= \frac{1}{1} \times \frac{1}{4} \times \frac{1}{2}\end{align*}

Next, multiply the fractions. Remember to include the units.

\begin{align*}V= \frac{1 \times 1 \times 1}{1 \times 4 \times 2} = \frac{1}{8} \text{inches}^3\end{align*}

The volume of the rectangular prism is \begin{align*}\frac{1}{8} \text{inches}^3\end{align*}.

#### Example 2

Multiply the following fractions. Answer in simplest form.

\begin{align*} \frac{5}{6} \times \frac{2}{4} \times \frac{6}{9}\end{align*}

First, simplify the fractions. Simplify the second and third fractions before multiplying.

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

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

Next, rewrite the problem.

\begin{align*} \frac{5}{6} \times \frac{1}{2} \times \frac{2}{3}\end{align*}

Also, simplify the twos.

\begin{align*} \frac{5}{6} \times \frac{1}{\cancel{2}^1} \times \frac{\cancel{2}^1}{3} = \frac{5}{6} \times \frac{1}{1} \times \frac{1}{3}\end{align*}

Then, multiply the fractions.

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

The product is \begin{align*} \frac{5}{18}\end{align*}.

#### Example 3

Multiply the following fractions: \begin{align*}\frac{1}{5} \times \frac{5}{6} \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, simplify the fractions.

\begin{align*}\frac{1}{\cancel{5}^1} \times \frac{\cancel{5}^1}{6} \times \frac{1}{2} = \frac{1}{1} \times \frac{1}{6} \times \frac{1}{2} \end{align*}

Then, multiply.

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

The product is \begin{align*} \frac{1}{12}\end{align*}.

#### Example 4

Multiply the following fractions: \begin{align*}\frac{8}{9} \times \frac{3}{4} \times \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, simplify the fractions.

\begin{align*}\frac{\cancel{8}^2}{9} \times \frac{\cancel{3}^1}{\cancel{4}^1} \times \frac{1}{\cancel{3}^1} = \frac{2}{9} \times \frac{1}{1} \times \frac{1}{1}\end{align*}

Then, multiply. Note that \begin{align*}\frac{1}{1}\end{align*} is also equal to 1.

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

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

#### Example 5

Multiply the following fractions: \begin{align*}\frac{6}{7} \times \frac{7}{14} \times \frac{2}{10} = \underline{\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, simplify the fractions.

\begin{align*}\frac{\cancel{6}^3}{\cancel{7}^1} \times \frac{\cancel{7}^1}{\cancel{14}^7} \times \frac{\cancel{2}^1}{\cancel{10}^5} = \frac{3}{1} \times \frac{1}{7} \times \frac{1}{5}\end{align*}

Then, multiply.

\begin{align*} \frac{3 \times 1 \times 1}{1\times7\times 5}=\frac {3}{35}\end{align*}

The product is \begin{align*} \frac{3}{35}\end{align*}.

### Review

Multiply the following fractions. Answer in simplest form.

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

### Review (Answers)

To see the Review answers, open this PDF file and look for section 7.4.

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### Vocabulary Language: English

TermDefinition
Product The product is the result after two amounts have been multiplied.

### Image Attributions

1. [1]^ Credit: trenttsd; Source: https://www.flickr.com/photos/84335369@N00/6614318221/; License: CC BY-NC 3.0

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