Hard GCSE Percentages as operators Questions

Plan the two stages

Two percentage changes happen one after the other. Do them in order, using a multiplier for each stage.

Stage $1$ multiplier (increase $40\%$)

A $40\%$ increase means multiplying by $1.4$.

Apply stage $1$

After stage $1$ the amount is £350.

Check stage $1$ the long way

$40\%$ of £250 is £100, the size of the first change.

Combine stage $1$

This confirms the amount after stage $1$ is £350.

Stage $2$ multiplier (decrease $15\%$)

A $15\%$ decrease means multiplying by $0.85$. Important: this acts on the new amount, not the original.

Apply stage $2$

After stage $2$ the amount is £297.50.

Check stage $2$ the long way

$15\%$ of £350 is £52.50, the size of the second change.

Combine stage $2$

This confirms the final amount is £297.50.

State the final amount

So the final amount is £297.50.

Combine into one multiplier

The two changes together are the same as multiplying by this single combined multiplier.

Check with the combined multiplier

Multiplying the original by the combined multiplier gives the same final amount, a strong check.

Reflect on the result

The combined multiplier is $1.19$, so the selling price is still $19\%$ above the £250 cost, because the $15\%$ discount acted on the marked-up £350.

Sense check

£297.50 is above £250, which fits a net rise of $19\%$ after markup and discount.

State the answer

The final amount is £297.50.

Set up with a friendly number

Try both orders on £100 and compare the final prices.

Route $1$: increase by $10\%$ first

A $10\%$ rise gives £110.

Route $1$: then increase by $20\%$

A $20\%$ rise on £110 gives £132.

Route $1$ final price

Doing $+10\%$ then $+20\%$ ends at £132.

Route $2$: increase by $20\%$ first

A $20\%$ rise gives £120.

Route $2$: then increase by $10\%$

A $10\%$ rise on £120 gives £132.

Route $2$ final price

Doing $+20\%$ then $+10\%$ also ends at £132.

Both routes agree

Both orders give exactly the same final price.

Explain why

Multiplication can be done in any order, so the order of the two rises does not matter.

The combined multiplier

Either way, the overall multiplier is $1.32$.

It is not a $30\%$ rise

Two $10\% + 20\%$ rises give a $32\%$ increase, not $30\%$, because the second rise acts on a larger amount.

Works for any price

For any starting price $P$, both orders give $1.32P$.

Reflect

Because you just multiply by $1.1$ and $1.2$ in some order, and multiplication order does not affect the answer.

Sense check

£132 is a little more than a straight $30\%$ rise (£130), as expected.

Conclude

The order makes no difference: both give £132.

Recognise a reverse problem

We are told the amount AFTER the change and must find the ORIGINAL. Work backwards.

Name the multiplier

A $20\%$ decrease multiplies the original by $0.8$.

Write the relationship

The original amount, times $0.8$, gives £480.

Rearrange to find the original

To undo a multiplication, divide. So divide the final amount by the multiplier.

Do the division

Dividing gives an original of £600.

State the original amount

So the original amount was £600.

Check by working forwards

Multiplying the original by $0.8$ returns £480, which matches, so it is correct.

Check the change the long way

$20\%$ of £600 is £120.

Confirm

Applying the change to the original gives £480, confirming the answer.

Spot the common mistake

A common error is to apply $20\%$ to the final amount instead. That uses the wrong starting number.

Show why it is wrong

That shortcut gives £576, which is not the true original of £600.

Remember the safe method

The reliable rule for any reverse percentage is: divide the final amount by the multiplier.

Reflect

For an decrease, the original is larger than the final amount, which matches our answer.

Sense check

£600 times $0.8$ is £480, the sale price, so £600 is correct.

State the answer

The original amount was £600.

Recognise a reverse problem

We are told the amount AFTER the change and must find the ORIGINAL. Work backwards.

Name the multiplier

A $20\%$ increase multiplies the original by $1.2$.

Write the relationship

The original amount, times $1.2$, gives £30.

Rearrange to find the original

To undo a multiplication, divide. So divide the final amount by the multiplier.

Do the division

Dividing gives an original of £25.

State the original amount

So the original amount was £25.

Check by working forwards

Multiplying the original by $1.2$ returns £30, which matches, so it is correct.

Check the change the long way

$20\%$ of £25 is £5.

Confirm

Applying the change to the original gives £30, confirming the answer.

Spot the common mistake

A common error is to apply $20\%$ to the final amount instead. That uses the wrong starting number.

Show why it is wrong

That shortcut gives £24, which is not the true original of £25.

Remember the safe method

The reliable rule for any reverse percentage is: divide the final amount by the multiplier.

Reflect

For an increase, the original is smaller than the final amount, which matches our answer.

Sense check

£25 times $1.2$ is £30, the wage after the rise, so £25 is correct.

State the answer

The original amount was £25.

Set up with a friendly number

Using £80 makes the arithmetic clean. The conclusion will hold for any starting price.

Increase by $25\%$

A $25\%$ rise multiplies by $1.25$, giving £100.

Size of the rise

The price went up by £20.

Now decrease by $20\%$

A $20\%$ fall multiplies by $0.8$, giving £80.

Size of the fall

The price came down by £20, exactly what was added.

Final amount

The price ends back at £80, the original.

Combine into one multiplier

The two changes together multiply by exactly $1$.

Read the combined multiplier

Multiplying by $1$ leaves any amount unchanged.

Show it with fractions

A $25\%$ rise is multiplying by $\frac{5}{4}$, and a $20\%$ fall is multiplying by $\frac{4}{5}$.

The fractions cancel

$\frac{5}{4}$ times $\frac{4}{5}$ cancels to $1$, so the price returns exactly.

It works for any price

For any starting price $P$, the result is $P$, the original.

The statement is true

So increasing by $25\%$ then decreasing by $20\%$ always returns to the start.

Reflect

These two multipliers undo each other, which is why the price comes back exactly.

Sense check

The final price equals the original, as claimed.

Conclude

The combined multiplier is $1$, so the price returns to the original.