Multiplying Fractions

Learning Outcomes

  • Multiply two or more fractions
  • Multiply a fraction by a whole number

Multiply Fractions

Just as you add, subtract, multiply, and divide when working with whole numbers, you also use these operations when working with fractions.   There are many times when it is necessary to multiply fractions. A model may help you understand multiplication of fractions.

When you multiply a fraction by a fraction, you are finding a “fraction of a fraction.” Suppose you have [latex]\dfrac{3}{4}[/latex] of a candy bar and you want to find [latex]\dfrac{1}{2}[/latex] of the [latex]\dfrac{3}{4}[/latex]:

3 out of four boxes are shaded. This is 3/4.

By dividing each fourth in half, you can divide the candy bar into eighths.

Six of 8 boxes are shaded. This is 6/8.

Then, choose half of those to get [latex]\dfrac{3}{8}[/latex].

Six of 8 boxes are shaded, and of those six, three of them are shaded purple. The 3 purple boxes represent 3/8.

In both of the above cases, to find the answer, you can multiply the numerators together and the denominators together.

Multiplying Two Fractions

[latex]\dfrac{a}{b}\cdot\dfrac{c}{d}=\dfrac{a\cdot c}{b\cdot d}=\dfrac{\text{product of the numerators}}{\text{product of the denominators}}[/latex]

Multiplying More Than Two Fractions

[latex]\dfrac{a}{b}\cdot\dfrac{c}{d}\cdot\dfrac{e}{f}=\dfrac{a\cdot c\cdot e}{b\cdot d\cdot f}[/latex]

Example

Multiply [latex]\dfrac{2}{3}\cdot\dfrac{4}{5}[/latex]

To review: if a fraction has common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing the common factors.

For example,

  • Given [latex]\dfrac{8}{15}[/latex], the factors of [latex]8[/latex] are: [latex]1, 2, 4, 8[/latex] and the factors of [latex]15[/latex] are: [latex]1, 3, 5, 15[/latex].  [latex]\dfrac{8}{15}[/latex] is simplified because there are no common factors of [latex]8[/latex] and [latex]15[/latex].
  • Given [latex]\dfrac{10}{15}[/latex], the factors of [latex]10[/latex] are: [latex]1, 2, 5, 10[/latex] and the factors of [latex]15[/latex] are: [latex]1, 3, 5, 15[/latex]. [latex]\dfrac{10}{15}[/latex] is not simplified because [latex]5[/latex] is a common factor of [latex]10[/latex] and [latex]15[/latex].

Note that you can simplify the fractions first, before you multiplying them, to make your work easier. This allows you to work with smaller numbers when you multiply.

In the following video you will see an example of how to multiply two fractions, then simplify the answer.

Think About It

Multiply [latex]\dfrac{2}{3}\cdot\dfrac{1}{4}\cdot\dfrac{3}{5}[/latex]. Simplify the answer.

What makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would multiply three fractions together.