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Multiplying Fractions
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Introduction

In this lesson, you will learn how to multiply fractions and integers.


This video illustrates the lesson material below. Watching the video is optional.


Multiplying by Fractions

Now that you've learned how to add and subtract fractions, you will learn how to multiply fractions. Multiplying fractions is a lot simpler than adding or subtracting fractions because you don’t need to find a common denominator. The rule for multiplying fractions is simply to multiply across the numerator and multiply across the denominator.

Example 1
\(\frac{2}{3}\times\frac{3}{5}\)

2/3 times 3/5 is 6/15

Figure 1

The numerator will be \(2\times3=6\), and the denominator is \(3\times5=15\). If you reduce \(\frac{6}{15}\), it is equal to \(\frac{2}{5}\).

Example 2
\(\frac{5}{8}\times\frac{3}{2}\)

5/8 times 3/2 is 15/16

Figure 2

Multiply straight across. When you multiply the fractions across, you get \(\frac{15}{16}\).

Example 3
\(\frac{1}{2}\times\frac{1}{2}\)

According to the rules of multiplying fractions, this becomes \(\frac{1}{4}\):

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

Why does this work? Remember, a fraction actually indicates division. So \(\frac{1}{2}\times\frac{1}{2}\) is really the same as \(\frac{1}{2}\times1\div2\).

Example 4
\(\frac{1}{4}\times\frac{1}{2}\)

According to the algorithm, this should equal \(\frac{1}{8}\):

1/4 times 1/2 is 1/8

Figure 3

Consider this equation visually. You started out with one fourth, as shown in the rectangle below divided into four pieces. One of the pieces is shaded.

rectangle divided into four pieces. One is shaded.

Figure 4

The fourths were then cut in half until the rectangle was divided into eight pieces. The one-eighth is represented by the shaded area. So \(\frac{1}{4}\times\frac{1}{2}=\frac{1}{8}\).

rectangle divided into 8 pieces and one is shaded

Figure 5

Multiplying Fractions with Integers

You may have already caught on to this, but another important aspect of multiplying fractions is when you’re multiplying a fraction by an integer. Remember, any whole number can be represented as a fraction by putting it over 1.

Example 5
\(3=\frac{3}{1}\)

Using what you know about how to represent 3 as a fraction, complete this equation:

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

Rewrite 3 as \(\frac{3}{1}\) and then multiply the fractions across to get the answer.

3/1 times 3/4 is 9/4

Figure 6

The answer is \(\frac{9}{4}\).

Example 6
The same is true going the other direction, as in this equation:
\begin{align*} \frac{2}{3}\times 5 \end{align*}

Two-thirds times five is the same as \(\frac{2}{3}\times \frac{5}{1}\).

3/1 times 3/4 is 9/4

Figure 7

All you have to do is take the \(2\times 5\) on the top, which equals 10, and \(3\times 1\) on the bottom, which equals 3, and that gives you the answer: \(\frac{10}{3}\).


Things to Remember


  • When multiplying fractions, make sure to multiply across. The numerators are multiplied together and the denominators are multiplied together.
  • When multiplying fractions with an integer, the integer will be divided by one, making it a fraction so you can multiply across.

Practice Problems

Multiply the following fractions:
  1. \(\displaystyle \frac{1}{4}\cdot \frac{1}{3}=\) (
    Solution
    x
    Solution: \(\displaystyle \frac{1}{4}\cdot \frac{1}{3}=\frac{1}{12}\)
    Details:
    The problem 1/4 times 1/3 is listed.  1/4 is shown in green and 1/3 in blue. To the right of the problem there’s an equal sign and the problem is rewritten as 1 times 1 on top of a fraction line, and 4 times 3 written below the line in the denominator of the fraction.  To the right of this is another equal sign and 1/12 is written in red.
    When you multiply two fractions together you multiply straight across.
    You start with the following:
    \(\displaystyle \frac{1}{4}\times\frac{1}{3}\)

    Which can be written:
    \(\displaystyle \frac{1\times1}{4\times3}\)

    Which equals:
    \(\dfrac{1}{12}\)
    )
  2. \(\displaystyle \frac{1}{4}\cdot \frac{5}{8}=\) (
    Solution
    x
    Solution: \(\displaystyle \frac{1}{4}\times \frac{5}{8}=\frac{5}{32}\)
    Details:
    The problem 1/4 times 5/8 is listed. 1/4 is shown in green and 5/8 in blue. To the right of the problem, there’s an equal sign. Right of that, the problem is rewritten as 1 times 5 on the top of a fraction line in the numerator. Below that line, in the denominator, is written 4 times 8. To the right of this is another equal sign and 5/32 is written in red.
    When you multiply two fractions together you multiply straight across.
    You start with the following:
    \(\displaystyle \frac{1}{4}\times\frac{5}{8}\)

    Which can be written:
    \(\displaystyle \frac{1\times5}{4\times8}\)

    Which equals:
    \(\dfrac{5}{32}\)
    )
  3. \(\displaystyle \frac{3}{7}\cdot \frac{2}{5}=\) (
    Solution
    x
    Solution:
    \(\displaystyle \frac{3}{7}\cdot \frac{2}{5}=\frac{6}{35}\)
    )
  4. \(\displaystyle \frac{3}{4}\cdot \frac{2}{9}=\) (
    Video Solution
    x
    Solution: \(\displaystyle \frac{3}{4}\cdot \frac{2}{9}=\frac{1}{6}\)
    Details:

    (Multiplying Fractions #4 (01:09 mins) | Transcript)
    )
  5. \(\displaystyle \frac{3}{4}\cdot 10=\) (
    Solution
    x
    Solution:
    \(\displaystyle \frac{3}{4}\cdot10=\frac{3}{4}\cdot\frac{10}{1}=\frac{30}{4}=\frac{15}{2}\)
    )
  6. \(\displaystyle 6\cdot \frac{2}{3}=\) (
    Video Solution
    x
    Solution: \(\displaystyle 6 \cdot \frac{2}{3}=4\)
    Details:

    (Multiplying Fractions #6 (01:14 mins) | Transcript)
    )

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