2.02 Divide fractions and whole numbers

When we divide by a whole number, such as 12\div4, we ask the question "how many groups of 4 fit into 12?" It's just like thinking about "what number fills in the blank: 4\times⬚=12".

In this case, there are 3 whole groups of 4 in 12, so the result is 3.

We can think about dividing by a unit fraction in a similar way. The division 2\div\dfrac13 is equivalent to asking the question "how many parts of size \dfrac13 fit into 2 wholes?"

If we split two wholes up into thirds, we can see that there are 3 thirds in each whole, and so there are 2\times3=6 thirds in total.

The same thing happens for dividing by other unit fractions. If we calculated 3\div\dfrac15 this time, each of the three wholes will be divided into 5 fifths:

So 3\div\dfrac15 is the same as 3\times5=15.

Notice that this is just like thinking about "what number fills in the blank: \dfrac15\times⬚=3". We know that \dfrac15\times15=3, so it makes sense that 3\div\dfrac15=15.

Examples

Example 1

The number line below shows 4 wholes split into \dfrac13 sized parts.

a

If 4 is divided into parts that are \dfrac13 of a whole each, how many parts are there in total?

Worked Solution

Create a strategy

Divide the whole number by the unit fraction.

Apply the idea

\displaystyle \text{Number of parts}	\displaystyle =	\displaystyle 4\div\dfrac13	Divide the whole number by the unit fraction
	\displaystyle =	\displaystyle 4\times3	Multiply by the denominator
	\displaystyle =	\displaystyle 12	Evaluate

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b

How many parts would there be if we had 5 wholes?

Worked Solution

Create a strategy

Divide the whole number by the unit fraction.

Apply the idea

\displaystyle \text{Number of parts}	\displaystyle =	\displaystyle 5\div\dfrac13	Divide the whole number by the unit fraction
	\displaystyle =	\displaystyle 5\times3	Multiply by the denominator
	\displaystyle =	\displaystyle 15	Evaluate

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c

How many parts would there be if we split up 10 wholes?

Worked Solution

Create a strategy

Divide the whole number by the unit fraction.

Apply the idea

\displaystyle \text{Number of parts}	\displaystyle =	\displaystyle 10\div\dfrac13	Divide the whole number by the unit fraction
	\displaystyle =	\displaystyle 10\times3	Multiply by the denominator
	\displaystyle =	\displaystyle 30	Evaluate

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When we divide a whole number by a unit fraction, such as 2\div \dfrac{1}{3}, we ask the question "how many parts of size \dfrac{1}{3} fit into 2 wholes?"

For this example, the answer is 6, and we can get this result by multiplying the whole 2 by the denominator 3.

Dividing a unit fraction by a whole number is the reverse of this. Let's look at \dfrac{1}{3}\div 2 as an example:

We start with \dfrac13 of a whole, shown as the shaded area in the image above. We then divide each of these thirds into 2 parts:

How big is the remaining shaded area? Well, there are now 6 parts of equal area and 1 of them is shaded, so this is equal to \dfrac16 of the whole.

We can think about this using multiplication, in a similar way to dividing by a unit fraction, but this time the multiplication happens in the denominator:

\begin{aligned} \dfrac13\div2&=\dfrac{1}{2\times3}\\ &=\dfrac16 \end{aligned}

Examples