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2.05 Multiplying fractions

Lesson

Multiply fractions

We've seen how to multiply whole numbers by fractions. Can we use the same techniques to multiply fractions by fractions?

Evaluate \dfrac{2}{3} \times \dfrac{4}{5}.

This image shows a circle split into 5 sectors, 4 are shaded and 1 is not.

Finding \dfrac{2}{3} \times \dfrac{4}{5} is the same as finding \dfrac{2}{3} of \dfrac{4}{5}.

We can find this by starting with a diagram of \dfrac{4}{5}.

A circle split into 15 sectors, 12 sectors are shaded and 3 are not.

Then we can split each of these fifths into thirds.

This image shows a circle split into 15 parts. 12 sectors are shaded and 3 are not.

Notice that the circle is now divided into fifteenths (that is, 3 \times 5) and twelve parts have been shaded (3 \times 4).

Now we can shade in two thirds of each of the original pieces.

And we finish with eight fifteenths. So \dfrac{2}{3} \times \dfrac{4}{5} = \dfrac{8}{15}.

Each of these steps we've done before. We can think of \dfrac{2}{3} of \dfrac{4}{5} as \dfrac{2}{3} of \dfrac{12}{15} (since \dfrac{4}{5} and \dfrac{12}{15} are

equivalent fractions). Since \dfrac{12}{15} is 12 fifteenths we then want to find \dfrac{2}{3} of 12, and this is the number of fifteenths we are left with.

This suggests another method for multiplying fractions. By equivalent fractions, \dfrac{2}{3} \times \dfrac{4}{5} = \dfrac{2}{3} \times \dfrac{4 \times 3}{5 \times 3}.

Since this is \dfrac {2}{3} \times 4 \times 3 fifteenths, we are multiplying a fraction by a whole number, so we can write

\dfrac {2}{3} \times 4 \times 3 = \dfrac{2 \times 4 \times 3}{3}.

If we cancel the common factor of 3, we get 2 \times 4 fifteenths which is \dfrac{8}{15}.

So \dfrac{2}{3} \times \dfrac{4}{5} = \dfrac{2 \times 4}{3 \times 5}.

We can generalize this method to any fractions. So whenever we want to multiply two fractions, we can multiply the numerators and the denominators separately. Sometimes we might have to simplify the resulting fraction afterwards.

Let's use this method from now on.

Exploration

The following applet explores multiplication of fractions with area models. The intersection of the two area models represents the product.

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We can use area models to show a visual representation of the product of two fractions. In area models, the total number of equally partitioned pieces represents the denominator of the product.

Examples

Example 1

Evaluate \dfrac35\times\dfrac47.

Worked Solution
Create a strategy

Multiply numerators and denominators separately.

Apply the idea
\displaystyle \dfrac35\times\dfrac47\displaystyle =\displaystyle \dfrac{3\times4}{5\times7}Multiply numerators and denominators
\displaystyle =\displaystyle \dfrac{12}{35}Evaluate

Example 2

Evaluate \dfrac53\times\dfrac{21}{2}.

Worked Solution
Create a strategy

Multiply numerators and denominators separately.

Apply the idea
\displaystyle \dfrac53\times\dfrac{21}{2}\displaystyle =\displaystyle \dfrac{5\times21}{3\times2}Multiply numerators and denominators
\displaystyle =\displaystyle \dfrac{105}{6}Evaluate
\displaystyle =\displaystyle \dfrac{35}{2}Simplify
Idea summary

To multiply two fractions, multiply the numerators and the denominators separately.

Outcomes

MA4-5NA

operates with fractions, decimals and percentages

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