Teaching mathematics
Teaching mathematics

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Teaching mathematics

3.3 Finding the original amount when given the result of a percentage change

It is necessary to understand what percentage change is before this type of problem can be understood.

Finding the original amount from the result of percentage change requires some understanding of what happens when algebra is manipulated. However, first consider two easier examples where intuitive thinking can be sufficient.

Activity 11 25% more sunscreen

Timing: Allow 10 minutes

A ‘special offer’ bottle of sunscreen promises 25% more than usual and holds 500 ml. How much does the bottle of sunscreen normally hold?

Discussion

Here is one way to arrive at the solution:

25% more means one-quarter more than the usual amount in the bottle of sunscreen. This means that the special offer holds:

125 percent or postfix times five divided by four postfix times of the usual amount

You need to find out how much 100% or 1 whole of 500 ml.

divide postfix times 500 postfix times m times l times s postfix times by postfix times five postfix times to find postfix times 25 percent open or times one divided by four close postfix times of the amount in the normal size bottle

This gives 100 ml. Then multiply by 4 to get the whole of the amount in the normal size bottle, i.e. 400 ml.

You can see that the solution was found by dividing by 5 and multiplying by 4. Remembering that a fraction can also denote numerator divided by denominator, this is equivalent to multiplying by the fraction

four divided by five
four divided by five postfix times is the inverse of the fraction times five divided by four postfix times which is the multiplying factor for an increase in 25 percent

This means that one is the inverted (upside-down) version of the other and, if they are multiplied together, you get 1.

It also works by using the equivalent decimals 0.8 and 1.25, which can be checked on a calculator.

Activity 12 A coat is reduced in a sale

Timing: Allow 10 minutes

A coat is reduced in a sale by 10% and the sale price is £81. What was the original price of the coat?

Have a go at this problem yourself.

Discussion

The sale price is 90% of the original price and you need to find 100% of the original price.

90% of the original price = £81

10% of the original price = £9 (dividing by 9)

100% of the original price is £90 (multiplying by 10)

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