1. Operations with Fractions

Fractions describe parts of a whole, but they can also describe parts of a quantity.

Find \dfrac{1}{12} of 36.

We can also work this out using arithmetic. We know that \dfrac{1}{12} of 36 can be written using multiplication, \dfrac{1}{12} \cdot 36.

This is the same as \dfrac{1}{12} \cdot \dfrac{36}{1} because the whole number 36 can be written as a fraction over 1.

First, if we evaluate the multiplication of the numerators we get 36. And if we evaluate the multiplication in of the denominators we get 12.

Next we can simplify the fraction by factoring out the greatest common factor, which is 12. This gives us \dfrac{3}{1} which is the same as 3.

We can check this answer by multiplying back. 12 \cdot 3 = 36, so we know that 3 is \dfrac{1}{12} of 36.

Use the dropdown menus to choose a whole number and fraction to be multiplied.

Drag the slider to combine the models.

Check the 'Arrange' box to reorganize the model.

Use the applet to model a multiplication problem then answer the following questions:

- What is happening when you slide the slider? How does this represent the multiplication?
- What does checking the 'Arrange' checkbox do? How might this help with understanding the result?
- Use the applet with a few more multiplication problems. What patterns do you notice between the numbers being multiplied and the result?
- What can you say about multiplying a whole number by a fraction between 0 and 1?

When a whole number is multiplied by a fraction between 0 and 1, the result is smaller than the original whole number.

Each rectangle represents 1 whole.

a

Write the product that is represented by the model.

Worked Solution

b

Evaluate the product using the model.

Worked Solution

Evaluate \dfrac{2}{5}\cdot 35

Worked Solution

Idea summary

Finding a fraction of a quantity is the same as multiplying a fraction by a whole number.

To multiply a fraction by a whole number, multiply the numerator by the whole number.

When multiplying any whole number by a fraction between 0 and 1 the result is smaller than the original whole number.

To multiply two fractions together, we'll start by thinking of the fractions as **multiples** of **unit fractions,** and work towards a more efficient strategy.

Let's take an example of \,\dfrac{2}{3} \cdot \dfrac{4}{5} . We can rewrite these fractions as

\dfrac{2}{3} = 2 \cdot \dfrac{1}{3} \quad\text{and}\quad \dfrac{4}{5} = 4 \cdot \dfrac{1}{5}

We can then multiply the whole parts together:

\begin{aligned} \dfrac{2}{3} \cdot \dfrac{4}{5} &=2 \cdot \dfrac{1}{3} \cdot 4 \cdot \dfrac{1}{5}\\\\ &=8 \cdot \dfrac{1}{3} \cdot \dfrac{1}{5} \end{aligned}

What can we do with the product of the unit fractions \dfrac{1}{3} and \dfrac{1}{5}?

We can now finish our multiplication:

\begin{aligned} \dfrac{2}{3} \cdot \dfrac{4}{5} &=8 \cdot \dfrac{1}{3} \cdot \dfrac{1}{5}\\\\ &=8 \cdot \dfrac{1}{15}\\\\ &=\dfrac{8}{15} \end{aligned}

Do you notice the pattern that has happened here?

In a fraction, the denominator tells us the size of the pieces, and the numerator tells us how many pieces there are. When we multiply two fractions, the denominators multiply together to tell us the new size of the pieces, and the numerators also multiply together to tell us how many of the new pieces there are.

That is:

\begin{aligned} \dfrac{2}{3} \cdot \dfrac{4}{5} &=\dfrac{2\cdot4}{3\cdot5}\\\\ &=\dfrac{8}{15} \end{aligned}

We can multiply with mixed numbers as well, because they're really just fractions. We just have the added step of converting the mixed number to a fraction first.

Use the dropdown menus to choose the fractions to be multiplied.

Drag the slider to combine the models.

Use the applet to model a multiplication problem then answer the following questions:

- What is happening when you slide the slider? How does this represent the multiplication?
- Use the applet with a few more multiplication problems. What patterns do you notice between the numbers being multiplied and the result?
- What can you say about multiplying a fraction by a fraction between 0 and 1?

When a fraction is multiplied by a fraction between 0 and 1, the result is smaller than the original fraction.

Demonstrate how to multiply 1\dfrac{1}{4} and \dfrac{2}{3} using a number line. Then find the product.

Worked Solution

Find the value of the following:

a

\dfrac{1}{3}\cdot\dfrac{7}{10}

Worked Solution

b

\dfrac{5}{3}\cdot\dfrac{21}{2}

Worked Solution

c

5 \dfrac{4}{7} \cdot 4 \dfrac{2}{3}

Worked Solution

Danielle takes 4 \dfrac {7}{9} minutes to drive from her home to the local shopping center. She spends \dfrac {1}{9} of this time waiting at traffic lights.

Find the number of minutes she spends waiting.

Worked Solution

Idea summary

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

To multiply mixed numbers by a fraction or by another mixed number, convert the mixed number to an improper fraction first. Then, multiply the numerators and denominators separately.

When multiplying any number by a fraction between 0 and 1 the result is smaller than the original number.