# 7.5: Products of Three Fractions

**At Grade**Created by: CK-12

**Practice**Products of Three Fractions

Have you ever become so interested in something that you forgot everything else? Well, this has happened to Julie.

Julie has become so interested in the rainforest that she hasn't been keeping up on her math homework. Her mom found out, and as a result, Julie has to stay in until her work is all completed. Julie was moving right along until she got to a section on multiplying several fractions at one time. Then she got stuck.

Here is the problem Julie was working on.

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

Julie isn't sure what to do from here.

**This Concept is all about multiplying three fractions. By the end of it, you will know just how to help Julie.**

### Guidance

**How do we multiply three fractions?**

Multiplying three fractions is just a bit more complicated than multiplying two fractions. The procedure is the same, you multiply the numerators and the denominators and up with a new fraction.

**The key to multiplying three fractions is to simplify first, like we learned in the last Concept.**

This way, you won’t end up with a fraction that is too large when multiplying or is challenging to simplify at the end.

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

To start, let’s only look at the first two fractions.

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

We start by simplifying. We can simplify these two fractions in two different ways. We can either cross simplify the two and the four with the GCF of 2, or we can simplify two-sixths to one-third.

Let’s simplify two-sixths to one-third. Now rewrite the problem with all three fractions.

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

Next, we can multiply and then simplify, or we can look and see if there is anything else to simplify. One-fourth and one-third are in simplest form, four-fifths is in simplest form. **Our final check is to check the diagonals.**

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

The two fours can be simplified with the greatest common factor of 4. Each one simplifies to one.

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

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

**Let's look at another one.**

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

**To start simplifying, look at the fractions themselves and the diagonals.** **You can see right away that seven-fourteenths can be simplified to one-half.** **Also, the fives simplify with the GCF of 5.** **Finally, the 3 and 9 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}} & = \underline{\;\;\;\;\;\;}\\ \frac{1}{3} \times \frac{1}{2} \times \frac{1}{1} & = \frac{1}{6}\end{align*}

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

Practice finding these products. Be sure to simplify.

#### Example A

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

**Solution: \begin{align*} \frac{1}{12}\end{align*}**

#### Example B

\begin{align*}\frac{8}{9} \times \frac{3}{4} \times \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

**Solution: \begin{align*} \frac{2}{9}\end{align*}**

#### Example C

\begin{align*}\frac{6}{7} \times \frac{7}{14} \times \frac{2}{10} = \underline{\;\;\;\;\;\;\;}\end{align*}

**Solution: \begin{align*} \frac{3}{35}\end{align*}**

Now back to Julie and her homework.

Julie has become so interested in the rainforest that she hasn't been keeping up on her math homework. Her mom found out, and as a result, Julie has to stay in until her work is all completed. Julie was moving right along until she got to a section on multiplying several fractions at one time. Then she got stuck.

Here is the problem Julie was working on.

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

Julie isn't sure what to do from here.

First, we can simplify the fives and then look at what we have.

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

Next, we multiply across.

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

**This is our answer.**

### Vocabulary

Here are the vocabulary words in this Concept.

- Product
- the answer to a multiplication problem.

### Guided Practice

Here is one for you to do on your own.

Multiply the following fractions. Be sure your answer is in simplest form.

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

**Answer**

It makes the most sense to simplify the second two fractions before multiplying.

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

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

Now we can rewrite the problem.

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

Next, simplify the twos.

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

**Our answer is \begin{align*} \frac{5}{18}\end{align*}.**

### Video Review

Here is a video for review.

James Sousa: Ex: Multiplying Three Fractions

### Practice

Directions: Multiply the following fractions. Be sure that your answer is 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*}

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### Image Attributions

Here you'll learn to multiply three fractions.