Grade 6

2.03 Divide fractions

Lesson

Practice

Introduction

When we divide whole numbers we split the whole number into equally sized parts and find the size of one part.

For example, consider 6 \div 3.

Since there are 3 equal groups of 2 in 6 we can say that 6 \div 3 = 2.

Now consider multiplication of fractions from the previous section. When we multiply by a fraction we are finding what that fraction of the quantity represents.

For example, consider 6 \times \dfrac{1}{3}. Here we want to find how much is one third of six.

To do this, we split 6 into 3 equal groups and find the size of one group.

And we can see that 6 \times \dfrac{1}{3} = 2.

Notice that we can take the same approach with 6 \div 3 as with 6 \times \dfrac{1}{3}. This means that 6 \div 3 = 6 \times \dfrac{1}{3}.

This isn't a coincidence. Dividing by any number is the same as multiplying by the fraction one over that number.

We call the fraction made by taking 1 over a whole number a reciprocal.

Ideas

Dividing fractions

Dividing fractions

If multiplying by a fraction is like dividing by a whole number, what is dividing by a fraction like?

Consider the question: Evaluate \dfrac{8}{9} \div \dfrac{2}{9}.

There are a few approaches we can take with this question. We could rephrase it as, "How many groups of \dfrac{2}{9} make up \dfrac{8}{9}?" We could use the same approach as we used in multiplication, and divide the numerators and denominators separately. Or we could try to apply what we found with dividing whole numbers to fractions.

This image shows a circle split into 9 parts. 8 sectors are shaded

Let's start by asking, "How many groups of \dfrac{2}{9} make up \dfrac{8}{9} .

First we can draw a diagram of \dfrac{8}{9}.

This image shows a circle split into 9 parts. 8 sectors are shaded. Ask your teacher for more information.

Then we can split it into groups of \dfrac {2}{9}.

We can see that there are 4 equal groups of \dfrac {2}{9} in \dfrac {8}{9}. So \dfrac {8}{9} \div \dfrac {2}{9} = 4.

We can try another method which starts with the question: "What fraction do we multiply by which gives the same result as dividing by \dfrac{8}{9}?"

When we take the reciprocal of a whole number, we take the fraction of one over that whole number. However, we can also think of the whole number as a fraction over one. That is, 3 = \dfrac{3}{1}. So to find the reciprocal we can swap the numerator and the denominator of a fraction.

We can do this with any fraction. So to find the reciprocal of \dfrac{2}{9} we swap the numerator, 2 and the denominator 9 to get \dfrac{9}{2}.

Now we can say that:

\displaystyle \dfrac {8}{9} \div \dfrac {2}{9}	\displaystyle =	\displaystyle \dfrac {8}{9} \times \dfrac {9}{2}	Multiply \dfrac{8}{9} by the reciprocal of \dfrac{2}{9}
	\displaystyle =	\displaystyle \dfrac {8\times 9}{9\times 2}	Multiply the numerators together and denominators together
	\displaystyle =	\displaystyle \dfrac {8}{2}	Evaluate and simplify
	\displaystyle =	\displaystyle 4	Further simplify

Both methods give us the same result. It might seem like this third method is more complicated than the other two. However, in most situations it is actually the best one to use.

Examples

Example 1

Evaluate \dfrac18\div\dfrac15.

Worked Solution

Create a strategy

Divide fractions by multiplying the 1st fraction by the reciprocal of the 2nd.

Apply the idea

\displaystyle \dfrac18\div\dfrac15	\displaystyle =	\displaystyle \dfrac{1}{8} \times \dfrac{5}{1}	Multiply by the reciprocal
	\displaystyle =	\displaystyle \dfrac{1\times5}{8\times1}	Multiply numerators and denominators
	\displaystyle =	\displaystyle \dfrac58	Evaluate

Example 2

Evaluate \dfrac45\div\dfrac{36}{35}. Give your answer as a fully simplified fraction.

Worked Solution

Create a strategy

Divide fractions by multiplying the 1st fraction by the reciprocal of the 2nd.

Apply the idea

\displaystyle \dfrac45\div\dfrac{36}{35}	\displaystyle =	\displaystyle \dfrac{4}{5} \times \dfrac{35}{36}	Multiply by the reciprocal
	\displaystyle =	\displaystyle \dfrac{4\times35}{5\times36}	Multiply numerators and denominators
	\displaystyle =	\displaystyle \dfrac{140}{180}	Evaluate
	\displaystyle =	\displaystyle \dfrac79	Simplify

Idea summary

Fractions are a way of writing division of one whole number by another.

The reciprocal of a number 1 divided by the number.

The reciprocal of a whole number is a fraction of 1 over the whole number
The reciprocal of a fraction can be found by swapping the numerator and denominator

To divide one fraction by another, multiply the first fraction by the reciprocal of the second.

Outcomes

6.NS.A.1

Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g. By using visual fraction models and equations to represent the problem.