Numbers, units and arithmetic

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# 3.13 More division with fractions

How about other fractions? What is 6 ÷ ? This means how many are there in 6? Consider six squares, divided into quarters, and count how many groups of three quarters there are.

## Activity 51

Three quarters is 3 × ? So dividing by is the same as dividing by 3 and by . Dividing by is multiplying by 4. So dividing by is the same as dividing by 3 and multiplying by 4, which is multiplying by . So 6 ÷ = 6 × = = 8.

The general rule for dividing by a fraction can be stated as follows.

To divide by a fraction, turn the fraction upside down and multiply.

Does this also work for dividing by ? Yes, because multiplying by n is the same as multiplying by . It also works for division of whole numbers: dividing by 4 (= ) is the same as multiplying by .

## Example 23

Evaluate:

• Using the rule above,

•
• This is

• (divide top and bottom by 3).

• Alternatively divide top and bottom by 3 by cancelling by 3 to give

•
• So .

Turning a fraction upside down is called finding its reciprocal.

The reciprocal of is . The reciprocal of is , i.e. 6.

So an alternative formulation of the rule for dividing by a number can be stated as follows.

Division by a number is the same as multiplication by its reciprocal.

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