Hard GCSE Reverse percentages Questions

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 $45\%$.

Read what the $0.36\,\mathrm{kg}$ is

$0.36\,\mathrm{kg}$ is the amount AFTER the reduction. The $40\%$ was $40\%$ of the ORIGINAL amount of flour, so this is a reverse percentage problem.

Decide: forward or reverse?

We are given the final amount and asked for the original, so we must divide by the multiplier, not multiply by it.

Name the multiplier

A $40\%$ reduction leaves $60\%$ of the flour, so the original is multiplied by $0.6$.

Divide to find the original amount

The original recipe used $0.6\,\mathrm{kg}$ of flour.

Round-trip check

Reducing $0.6\,\mathrm{kg}$ by $40\%$ gives $0.36\,\mathrm{kg}$, exactly as the question says. Correct. (Notice that $0.6 \times 0.6 = 0.36$ is a coincidence of these numbers, not a rule.)

Guard against the classic error

Adding $40\%$ back on to $0.36\,\mathrm{kg}$ gives $0.504\,\mathrm{kg}$, which fails the check: reducing $0.504\,\mathrm{kg}$ by $40\%$ gives $0.3024\,\mathrm{kg}$, not $0.36\,\mathrm{kg}$. The $40\%$ removed was $40\%$ of the larger original.

Recall the unit conversion

The answer must be in grams, so a conversion is needed.

Convert the original amount to grams

The original recipe used $600\,\mathrm{g}$ of flour.

Check the round trip in grams as well

$0.36\,\mathrm{kg}$ is $360\,\mathrm{g}$, and $60\%$ of $600\,\mathrm{g}$ is $360\,\mathrm{g}$. The check works in either unit, which confirms the conversion was done correctly.

Find the amount of flour removed

The scaling down removed $240\,\mathrm{g}$ of flour.

Check that removal as a percentage

$240\,\mathrm{g}$ out of the original $600\,\mathrm{g}$ is $40\%$, matching the reduction in the question.

Look at the wrong base once more

Comparing the $240\,\mathrm{g}$ removed with the NEW amount of $360\,\mathrm{g}$ gives $66.7\%$, which is not the reduction. Percentage change is always measured against the original.

Sense check the size of the answer

The original recipe must use more flour than the scaled-down one, and it does.

Note a quick check with fractions

A $40\%$ reduction leaves three fifths, so dividing by three fifths (multiplying by five thirds) gives the original — the same answer by an exact fraction method.

State the answer

The original recipe used $600\,\mathrm{g}$ of flour.

Work out what percentage of the donations was passed on

If $15\%$ is kept, then $85\%$ is passed on. So the $\pounds 2550$ is $85\%$ of the total donated — not $100\%$ of it.

Recognise a reverse percentage problem

We know the amount AFTER the $15\%$ was taken out, and we want the total before. That is a reverse percentage.

Name the multiplier

Keeping $15\%$ and passing on the rest multiplies the total donations by $0.85$.

Write the relationship

The total donated, multiplied by $0.85$, gives the $\pounds 2550$ passed on.

Divide to find the total donated

The total donated that week was $\pounds 3000$.

Round-trip check the total

$85\%$ of $\pounds 3000$ is $\pounds 2550$, exactly the amount passed on. Correct.

Answer the question that was asked

The charity kept $\pounds 450$ for its costs.

Check the amount kept as a percentage

$15\%$ of the $\pounds 3000$ total is $\pounds 450$, which agrees. The costs are a percentage of the TOTAL donated.

Guard against the classic error

Taking $15\%$ of the $\pounds 2550$ passed on gives $\pounds 382.50$ — the wrong answer, because the $15\%$ was $15\%$ of the total donations, not $15\%$ of the money passed on.

Show why that error fails the check

If the charity had kept only $\pounds 382.50$, the total would have been $\pounds 2932.50$, and $15\%$ of that is $\pounds 439.88$ — not $\pounds 382.50$. The figures do not fit together, so the method is wrong.

Check the split adds up

The money kept plus the money passed on equals the total donated, as it must.

Check the ratio of the split

The kept and passed-on amounts are in the ratio $15$ : $85$, which simplifies to $3$ : $17$ — exactly the split the charity promised.

Sense check the size of the answer

The charity keeps much less than it passes on, which fits keeping only $15\%$.

Note the shortcut for next time

Because kept : passed on is $15$ : $85$, the amount kept is $15/85$ of $\pounds 2550$, which is $\pounds 450$ again — a quicker route, but only once the reverse step is understood.

State the answer

The charity kept $\pounds 450$ for its costs, out of $\pounds 3000$ donated.

Read the chain of changes

$\pounds 27300$ is the amount after every change has happened. The original is unknown, and each percentage was a percentage of the amount at the time — never a percentage of $\pounds 27300$. So we must work backwards.

Decide: forward or reverse?

A forward problem gives you the original and asks for the final amount, so you multiply. Here we are given the final amount and asked for the original, so we must divide. Getting this decision right is the whole skill.

Name the first multiplier

A $5\%$ increase multiplies the original by $1.05$.

Name the second multiplier

A $4\%$ increase multiplies the original by $1.04$.

Combine the multipliers

One multiplier after another is the same as multiplying by their product.

Say what the overall multiplier means

The overall multiplier is $1.092$, so Meera now earns $109.2\%$ of her old salary — a $9.2\%$ rise overall, not the $9\%$ you get by adding $5\%$ and $4\%$. The extra $0.2\%$ is the $4\%$ rise being applied to the money the $5\%$ rise had already added.

Write the relationship

The original, multiplied by $1.092$, gives $\pounds 27300$.

Divide to find the original

Dividing by the overall multiplier undoes every change at once. The original was $\pounds 25000$.

Round-trip check, change $1$

Applying the $5\%$ increase to $\pounds 25000$ gives $\pounds 26250$.

Round-trip check, change $2$

Applying the $4\%$ increase to $\pounds 26250$ gives $\pounds 27300$. This is the amount given in the question, so the original of $\pounds 25000$ is confirmed.

Guard against the classic error

Adding the percentages gives $9\%$ and a salary of about $\pounds 25045.87$, which fails the round-trip check: $\pounds 25045.87 \times 1.092$ is about $\pounds 27350.09$, not $\pounds 27300$. The second rise was $4\%$ of the already-increased salary.

State the overall percentage rise

The two rises together are worth $9.2\%$, not $9\%$. The extra $0.2\%$ is the $4\%$ rise being paid on the money the $5\%$ rise had already added.

Check the rise in money

Meera is $\pounds 2300$ a year better off. As a percentage of her old salary that is $2300/25000 = 9.2\%$, which matches the combined multiplier exactly.

Check each rise separately

The first rise added $\pounds 1250$ and the second added $\pounds 1050$ — a total of $\pounds 2300$, as above. The second rise is worth more than $4\%$ of $\pounds 25000$ $(\pounds 1000)$ because it was calculated on the higher salary.

State the answer

Meera earned $\pounds 25000$ before the two rises.

Find the true original price

The $\pounds 120$ is the price AFTER the $20\%$ increase, so this is a reverse percentage problem.

Divide by the multiplier

The price before the increase was $\pounds 100$.

Round-trip check the original

$20\%$ of $\pounds 100$ is $\pounds 20$, and $\pounds 100 + \pounds 20 = \pounds 120$. So $\pounds 100$ is definitely the original.

Now test the student's method

Decreasing $\pounds 120$ by $20\%$ gives $\pounds 96$, not $\pounds 100$. The student's method has lost $\pounds 4$.

Say why the student's method fails

The increase added $20\%$ of $\pounds 100$, which is $\pounds 20$. The student then removes $20\%$ of $\pounds 120$, which is $\pounds 24$ — a bigger amount, because it is a percentage of a bigger number. Taking away more than was added lands below the start.

Look at the multipliers

To undo a change, the two multipliers must multiply to $1$. Here they give $0.96$, so the price ends at $96\%$ of the original — a $4\%$ fall overall, not back to the start.

Work out the multiplier that DOES undo the increase

The undoing multiplier is the reciprocal of $1.2$, which is five sixths.

Turn that multiplier into a percentage decrease

A multiplier of five sixths means removing one sixth.

Write the percentage decrease

A decrease of one sixth, about $16.7\%$, is what really undoes a $20\%$ increase.

Check the correct decrease

Five sixths of $\pounds 120$ is exactly $\pounds 100$, the true original. The round-trip check passes.

Check with the multipliers

The two multipliers now multiply to exactly $1$, which is what "undoing" means.

See the general rule

You undo a percentage change by DIVIDING by its multiplier, never by applying the opposite percentage. The opposite percentage is measured from the wrong base.

Note the size of the error

The student's method is $\pounds 4$ out on this price — and the error grows with the size of the percentage.

Note when the student's idea nearly works

For a $1\%$ increase, undoing with a $1\%$ decrease is out by only $0.01\%$. That is why the mistake feels reasonable — but it is still wrong.

Choose the correct statement

The student is wrong: $\pounds 120$ decreased by $20\%$ is $\pounds 96$, not $\pounds 100$. The decrease that undoes a $20\%$ increase is one sixth, about $16.7\%$.