Fractions

Partition Division

Example 1: 1 1/2 ÷ 3

What does 1 1/2 ÷ 3 mean? We can read this example as 1 1/2 divided by 3 and we can think of it as 1 1/2 shared into 3 equal parts. For example, we could be talking about 1 1/2 cakes shared equally among 3 people.

In this example, to find each person's share we had to find 1/3 of 1 1/2. Our division problem became a multiplication problem:

1 1/2 ÷ 3 =

1/3 of 1 1/2 =

1/3 x 1 1/2 =

1 1/2 x 1/3

So, we have seen that dividing by 3 is the same as finding 1/3 of something, which is multiplying by 1/3.

Dividing by a number (3) is multiplying by its reciprocal (1/3).

Multiplying by a number (1/3) is dividing by its reciprocal (3).

In partition division problems we think of sharing between. The divisor is a whole number which represents the number of parts to be shared between.

Example 2: 7/10 ÷ 5 = 7/50

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Quotition Division

Example 3: 6 ÷ 1/3 = 18

What does 6 ÷ 1/3 mean? It does not make much sense to say 6 shared between 1/3. To picture a quotition division problem we ask 'how many?'

Question	Thinking
How many thirds in 6?	There are 3 thirds in 1, so there must be 6 x 3 thirds in 6. Therefore there are 18 thirds in 6.

We can see that while thinking through this problem we turned a division problem into a multiplication problem. 6 divided by 1/3 became 6 multiplied by 3.

6 ÷ 1/3 = 6 x 3 = 18

So we can see that dividing by 1/3 is the same as multiplying by 3.

Dividing by a number (1/3) is multiplying by its reciprocal (3).

Multiplying by a number (3) is dividing by its reciprocal (1/3).

Example 4: 1/2 ÷ 1/4 = 2

There are a number of ways we can describe this expression. We can say, 'a half divided by a quarter' or, 'how many quarters in a half?'

Would we say, 1/2 shared between 1/4 (of a person)? No! This description does not make any sense to us and certainly does not help us picture the problem.

This is best understood as a quotition division problem. Asking 'how many quarters in a half?' helps us to picture this problem. From the square above we can see that the answer is 2. There are 2 quarters in a half. 1/2 ÷ 1/4 = 2

Let's compare to multiplication.

If we multiply 1/2 x 4 the answer is 2. 1/2 ÷ 1/4 = 1/2 x 4 = 2

So we can see that dividing by 1/4 is the same as multiplying by 4.

The relationship between division and multiplication can be shown as,

b ÷ 1/a = b x a (where a does not equal zero)

Example 5: 3/8 ÷ 1/6 = 2 1/4

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Example 6: 4/3 ÷ 5/7 = 1 13/15

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Example 7: 2 2/3 ÷ 1 3/5 = 1 2/3

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Steps for the division of fractions:

To divide fractions:

Step 1: Change the division into multiplication by inverting the divisor. This works because dividing by a number is the same as multiplying by its reciprocal.

Step 2: Carry out the multiplication

Step 3: Simplify the answer if required.

Using multiplication to check your answer

We have seen how multiplication and division are related.

We can use the fact that multiplication and division are the inverse of one another to help us check our answers to division problems.

Example 8: 4/3 ÷ 5/7 = ? Using multiplication to check our answer

Question

Setting Out

Thinking

4/3 ÷ 5/7

= 4/3 x 7/5

= 28/15

= 1 13/15

I know that dividing by a number is the same as multiplying by the inverse of that number, therefore instead of dividing by 5/7, I can multiply by 7/5.

I can check this answer with the answer obtained in the video of example 6.

General rule for the division of fractions

There are a number of ways of explaining why we can turn division problems into multiplication problems. The explanation below relies on our knowing that we can multiply the numerator and denominator of a fraction by the same number without changing its value.

We chose to multiply by d/c because we wanted the denominator to be 1.

We know that a number divided by 1 is the same number.

Thus, the general rule for division of fractions is:

Quick quiz

Find the answers to the following examples:

1.	a)	b)	c)
2.	a)	b)	c)

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