GCSE Percentages as operators Practice Questions

Understand what $10\%$ means

$10\%$ means $10$ out of every $100$, which is exactly one tenth. To find one tenth of something you just divide by $10$.

Divide the amount by $10$

Dividing £$80$ by $10$ makes every digit one place smaller. That gives £$8$.

State the answer

So $10\%$ of £$80$ is £$8$. This is an exact value, no rounding needed.

Find $10\%$ first

Start with the easy building block: $10\%$ of £$20$ is £$2$.

Multiply by $3$ to make $30\%$

$30\%$ is three lots of $10\%$, so multiply the $10\%$ amount by $3$.

State the answer

So $30\%$ of £$20$ is £$6$.

Read the claim carefully

Kim says $25\%$ of £$80$ is £$25$. We need to work out the real answer and spot her mistake.

Recall what $25\%$ means

$25\%$ is one quarter, so we must divide £$80$ by $4$ - not just copy the number $25$.

Work out a quarter of £$80$

A quarter of £$80$ is £$20$, so the correct answer is £$20$.

Compare with Kim's answer

Kim wrote £$25$, but the correct value is £$20$, so she is wrong.

Name the mistake

Kim treated the $25$ in '$25\%$' as £$25$. A percentage is a fraction of the amount, not a fixed number of pounds.

State the correct reasoning

The right method is: $25\%$ is a quarter, and a quarter of £$80$ is £$20$.

State what must be shown

We must show the left side and the right side give the same amount of money.

Turn the left percentage into a decimal

$12\%$ divided by $100$ is $0.12$.

Work out the left side

$0.12$ times $250$ is $30$, so the left side is £$30$.

Check the left side with blocks

Building $12\%$ from $10\%$ and two lots of $1\%$ also gives £$30$.

Left side equals £$30$

Both methods agree the left side is £$30$.

Turn the right percentage into a decimal

$15\%$ divided by $100$ is $0.15$.

Work out the right side

$0.15$ times $200$ is $30$, so the right side is £$30$.

Check the right side with blocks

Building $15\%$ from $10\%$ and $5\%$ also gives £$30$.

Right side equals £$30$

Both methods agree the right side is £$30$.

Conclude

Both sides equal £$30$, so $12\%$ of £$250$ does equal $15\%$ of £$200$.

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 £$297.50$.