GCSE Reverse percentages Practice Questions

Decide what the given amount stands for

The $20\%$ was taken off the original, not taken off $\pounds 48$. So $\pounds 48$ is $80\%$ of the original — this is a reverse percentage problem, and the original is the number we have to find.

Divide by the multiplier

$80\%$ as a decimal multiplier is $0.8$. The original was multiplied by $0.8$ to get $\pounds 48$, so dividing by $0.8$ undoes it and gives $\pounds 60$.

Check by working forwards

The round-trip check: $\pounds 60$ with the change applied gives $\pounds 48$, exactly as the question says. (Doing $\pounds 48 \times 1.2 = \pounds 57.60$ instead would be wrong — the $20\%$ was a percentage of the original, not of $\pounds 48$.)

Decide what the given amount stands for

The $35\%$ was taken off the original, not taken off $\pounds 52$. So $\pounds 52$ is $65\%$ of the original — this is a reverse percentage problem, and the original is the number we have to find.

Divide by the multiplier

$65\%$ as a decimal multiplier is $0.65$. The original was multiplied by $0.65$ to get $\pounds 52$, so dividing by $0.65$ undoes it and gives $\pounds 80$.

Check by working forwards

The round-trip check: $\pounds 80$ with the change applied gives $\pounds 52$, exactly as the question says. (Doing $\pounds 52 \times 1.35 = \pounds 70.20$ instead would be wrong — the $35\%$ was a percentage of the original, not of $\pounds 52$.)

See what the student has done

The student has added $25\%$ of $\pounds 90$ (which is $\pounds 22.50$) on to $\pounds 90$. That treats the discount as $25\%$ of the SALE price.

See what the discount really was a percentage of

The shop took $25\%$ off the ORIGINAL price. The original is bigger than $\pounds 90$, so $25\%$ of it is bigger than $\pounds 22.50$. The student used the wrong base.

Write the correct relationship

After a $25\%$ discount you pay $75\%$ of the original, so $\pounds 90$ is $75\%$ of the original.

Divide to find the original

The original price was $\pounds 120$.

Round-trip check the correct answer

$25\%$ of $\pounds 120$ is $\pounds 30$, and $\pounds 120 - \pounds 30 = \pounds 90$ — the sale price in the question. The answer checks out.

Round-trip check the student's answer to show it fails

Taking $25\%$ off the student's $\pounds 112.50$ gives $\pounds 84.38$ (to the nearest penny), not $\pounds 90$. The student's price fails the check, so it cannot be the original.

Recognise a reverse percentage problem

The $\pounds 84$ is the price AFTER the discount, and the price before the sale is unknown. The $30\%$ was $30\%$ of that unknown original, so we must work backwards.

Name the multiplier

A $30\%$ discount multiplies the original price by $0.7$.

Divide to find the price before the sale

The dress cost $\pounds 120$ before the sale.

Round-trip check the original price

$30\%$ of $\pounds 120$ is $\pounds 36$, and $\pounds 120 - \pounds 36 = \pounds 84$ — the sale price in the question. Correct.

Now go forwards for the second part

The shop increases the SALE price by $30\%$. This part is a forward calculation: the starting amount $(\pounds 84)$ is known, so multiply by $1.3$. Notice how the same question contains one reverse step and one forward step — deciding which is which is the skill being tested.

Work out the new price

After the sale ends, the dress costs $\pounds 109.20$.

Compare with the price before the sale

The dress does NOT return to $\pounds 120$. A $30\%$ rise on the reduced price adds less money than the $30\%$ cut took away, because it is $30\%$ of a smaller amount.

Show the two changes in money

The sale took off $\pounds 36$ but the increase only put back $\pounds 25.20$. The difference between those two amounts is exactly what is missing.

Answer the question that was asked

The dress now costs $\pounds 10.80$ less than it did before the sale.

Check with the combined multiplier

The two changes together multiply the original by $0.91$ — a $9\%$ fall overall, not a return to the start. And $9\%$ of $\pounds 120$ is exactly $\pounds 10.80$, confirming the answer.

Read what the $\pounds 96$ is

$\pounds 96$ is the price after BOTH discounts. The original price is unknown, and neither discount was a percentage of $\pounds 96$, so this is a reverse percentage problem.

Name the first multiplier

The first discount multiplies the original price by $0.75$.

Name the second multiplier

The further $20\%$ is taken off the already-reduced price, so it multiplies that price by $0.8$.

Combine the multipliers

Applying one multiplier and then the other is the same as multiplying by $0.6$.

Say what the combined multiplier means

The customer pays $60\%$ of the original price, so the saving is $40\%$ — not the $45\%$ the customer claims, and not the $25 + 20 = 45\%$ the advert seems to promise.

Write the relationship

The original price, multiplied by $0.6$, gives the $\pounds 96$ paid.

Divide to find the original price

The coat was $\pounds 160$ before any discount.

Round-trip check the first discount

Taking $25\%$ off $\pounds 160$ gives $\pounds 120$.

Round-trip check the second discount

Taking a further $20\%$ off $\pounds 120$ gives $\pounds 96$, exactly what the customer paid. The original of $\pounds 160$ is confirmed.

Find the saving in money

The customer saved $\pounds 64$ altogether.

Work out the actual percentage saved

The saving is $40\%$ of the original price. The customer's claim of $45\%$ is wrong.

Explain why the customer is wrong

The first discount saved $\pounds 40$, but the second saved only $\pounds 24$, because it was $20\%$ of the reduced $\pounds 120$ — not $20\%$ of the original $\pounds 160$ (which would be $\pounds 32$). Adding the percentages, $25 + 20 = 45$, assumes both were taken from $\pounds 160$, and they were not.

Check the two savings add up

The two separate savings total $\pounds 64$, which matches the overall saving. Everything is consistent.

Check the claimed $45\%$ fails

A genuine $45\%$ saving on $\pounds 160$ would have left the customer paying $\pounds 88$, not $\pounds 96$. The customer paid $\pounds 8$ more than a $45\%$ saving would give.

State the answer

The actual saving is $40\%$ of the original price of $\pounds 160$, not $45\%$.