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# Products of Two Fractions

## Multiply a fraction by a fraction

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Products of Two Fractions

Have you ever wondered about how much water there is in the world? How about in the rainforest?

Julie is amazed by all of the things that she is learning about the rainforest. One of the most interesting things that she has learned is that two-thirds of all of the fresh water on the planet is found in the Amazon River. Well, it isn’t exactly found in the Amazon, but in its basin, rivers, streams and tributaries. Julie is working on a drawing to show this. She draws the earth in one corner of the page and the Amazon River in the other corner of the page. As she reads on in her book on the Rainforest, she learns a new detail about the water of the Amazon. One-fifth of the water found in the Amazon River is found in its basin. Julie draws this on the page. She has the fraction two-thirds written near the top of the Amazon River and one-fifth written near its basin. “I wonder how much this actually is?” Julie thinks to herself. “How much is one-fifth of two-thirds?”

She leans over to her friend Alex in the next desk and asks him how to find one-fifth of two-thirds. Alex smiles and takes out a piece of paper and a pencil. Before Alex shows Julie, you need to learn this information. This Concept will teach you all about multiplying fractions. Then you can see how Alex applies this information when helping Julie.

### Guidance

Multiplying fractions can be a little tricky to understand. When we were adding fractions, we were finding the sum, when we subtracted fractions we were finding the difference, when we multiplied a fraction by a whole number we were looking for the sum of a repeated fraction or a repeated group.

What does it mean to multiply to fractions?

When we multiply two fractions it means that we want a part of a part.

This means that we want one-half of three-fourths. Here is a diagram.

Here are three-fourths shaded. We want one-half of the three-fourths. If we divide the three fourths in half, we will have a new section of the rectangle.

The black part of this rectangle shows \begin{align*}\frac{1}{2} \end{align*} of \begin{align*}\frac{3}{4} = \frac{3}{8}\end{align*}.

Now we can’t always draw pictures to figure out a problem, so we can multiply fractions using a few simple steps.

How do we multiply fractions?

We multiply fractions by multiplying the numerator by the numerator and the denominator by the denominator. Then we simplify.

Numerator \begin{align*} \times\end{align*} numerator \begin{align*}=\end{align*} 1 \begin{align*}\times\end{align*} 3 \begin{align*}=\end{align*} 3

Denominator \begin{align*}\times\end{align*} denominator \begin{align*}=\end{align*} 2 \begin{align*}\times\end{align*} 4 \begin{align*}=\end{align*} 8

Our final answer is \begin{align*}\frac{3}{8}\end{align*}. We have the same answer as the one that we found earlier.

To find this product we can do the same thing. We multiply across.

Next, we simplify the fraction \begin{align*}\frac{3}{54}\end{align*} by dividing by the GCF of 3.

Our answer is \begin{align*}\frac{1}{18}\end{align*}.

To solve this problem, we multiplied and then simplified. Sometimes, we can simplify BEFORE we do any multiplying.

There are two ways that we can simplify first when looking at a problem.

1. Simplify any fractions that can be simplified.

Here three-sixths could be simplified to one-half. Our new problem would have been \begin{align*}\frac{1}{2} \times \frac{1}{9} = \frac{1}{18}\end{align*}.

2. We could also CROSS-SIMPLIFY. How do we do this?

To cross-simplify, we simplify on the diagonals by using greatest common factors to simplify a numerator and a denominator.

We look at the numbers on the diagonals and simplify any that we can. 1 and 6 can’t be simplified, but 3 and 9 have the GCF of 3. We can simplify both of these by 3.

Now we insert the new numbers in for the old ones.

Notice that you can simplify in three different ways, but you will always end up with the same answer.

Try a few of these on your own. Be sure that your answer is in simplest form.

#### Example A

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

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

#### Example B

\begin{align*}\frac{6}{9} \times \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

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

#### Example C

\begin{align*}\frac{5}{6} \times \frac{2}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

Solution: \begin{align*} \frac{5}{9}\end{align*}

Now let's go back to Julie's rainforest dilemma.

Julie leans over to her friend Alex in the next desk and asks him how to find one-fifth of two-thirds. Alex smiles and takes out a piece of paper and a pencil.

Now here is Alex’s explanation.

“We want to find one-fifth of two-thirds. To do this, we can multiply,” Alex explains.

“This is the same as one-fifth of two-thirds. The word “of” means multiply. Now we can multiply across.”

“This amount is two-fifteenths of the water. This means one-fifth of the two-thirds would be the same as two-fifteenths of the water in the basin,” Alex says as Julie takes some notes.

### Vocabulary

Product
the answer to a multiplication problem.

### Guided Practice

Here is one for you to try on your own.

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

To solve this problem, we multiply numerator times numerator and denominator times denominator. Then we simplify.

Our answer is \begin{align*} \frac{2}{7}\end{align*}.

### Practice

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

1. \begin{align*}\frac{1}{6} \times \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

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

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

4. \begin{align*}\frac{6}{7} \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}

5. \begin{align*}\frac{1}{8} \times \frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}

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

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

8. \begin{align*}\frac{2}{5} \times \frac{3}{6} = \underline{\;\;\;\;\;\;\;}\end{align*}

9. \begin{align*}\frac{7}{9} \times \frac{2}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

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

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

12. \begin{align*}\frac{4}{7} \times \frac{2}{14} = \underline{\;\;\;\;\;\;\;}\end{align*}

13. \begin{align*}\frac{6}{7} \times \frac{2}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}

14. \begin{align*}\frac{4}{9} \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}

15. \begin{align*}\frac{8}{9} \times \frac{2}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}

16. \begin{align*}\frac{3}{8} \times \frac{2}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

### Vocabulary Language: English

Product

Product

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