Percentage Increase and Decrease
In Percentages of an Amount, you learnt how to find a share of a whole. Here, you will use that share to make an amount larger or smaller.
Start with the two-stage methods. We will then explain multipliers, find percentage changes and finish with reverse-percentage examples for both tiers.
What changes, and what is the final amount?
The original amount is the starting amount. It represents 100%. The new amount is what you have after the change.
Increase means make larger. Decrease means make smaller.
Finding 20% of £50 gives £10. In a 20% increase or decrease, that £10 is the change amount, not the final answer.
| Instruction | Answer |
|---|---|
| Find 20% of £50 | £10 — the share only. |
| Increase £50 by 20% | £50 + £10 = £60 — add the change. |
| Decrease £50 by 20% | £50 − £10 = £40 — subtract the change. |
For either change, work out the percentage of the original £50.
Example Question:
A price of £50 increases by 20%. Find the new price.
First find the extra amount. Then decide what to do with it. Example 1 shows both stages.
Exam Tip
Read what the question asks for: the change amount, the final amount, the percentage change or the original amount.
For a final price, do not stop after finding the percentage share.
Keep the original units. In money questions, keep full values during working and round only the final answer to the nearest penny when needed.
Method — Find the Change, Then Use It
1. Identify the original amount. This is the amount the percentage applies to.
2. Find the percentage of that amount. Use familiar shares, find 1%, or use a calculator as in the previous lesson.
3. Increase? Add the change to the original.
4. Decrease? Subtract the change from the original.
For a positive starting amount, an increase gives a larger answer. A decrease between 0% and 100% gives a smaller, still-positive answer.
Worked Examples
Example 1 — Increase using a mental method
Increase £50 by 20%.
First find 10%, then double it:
10% of £50 = £5
20% of £50 = 2 × £5 = £10
The increase is £10. Add it to the original price:
£50 + £10 = £60
The new price is £60.
Example 2 — Decrease using the same idea
Decrease £50 by 20%.
The change amount is still 20% of the original £50:
20% of £50 = £10
This time, subtract it:
£50 − £10 = £40
The new amount is £40.
Example 3 — A shop discount
A bag costs £80. It has a 15% discount. Find the sale price.
A discount is a reduction in the price. Find 10% + 5% of £80:
10% of £80 = £8
5% of £80 = £4
£8 + £4 = £12
The discount is £12, but the question asks for the price you pay:
£80 − £12 = £68
The sale price is £68.
Why decimal multipliers work
A multiplier is the number you multiply the original amount by. It represents the whole new amount as a share of the original, not just the change.
For a 20% increase, keep the original 100% and add 20%:
100% + 20% = 120%
120% = 1.20
So multiply the original amount by 1.20. For Example 1:
£50 × 1.20 = £60
For a 20% decrease, remove 20% from the original 100%:
100% − 20% = 80%
80% = 0.80
So multiply the original amount by 0.80. For Example 2:
£50 × 0.80 = £40
These are faster versions of the same add-or-subtract methods.
A positive percentage increase uses a multiplier greater than 1. A decrease between 0% and 100% uses a multiplier between 0 and 1.
Example 4 — An increase multiplier
A weekly payment of £240 increases by 5%. Find the new weekly payment.
The new payment is 105% of the original:
100% + 5% = 105%
105% = 1.05
£240 × 1.05 = £252
The new weekly payment is £252.
Compare with the two-stage method:
5% of £240 = £12
£240 + £12 = £252
Both methods agree.
Example 5 — A decrease multiplier
A sample has mass 500 g. Its mass decreases by 12%. Find its new mass. The unit g means grams.
The amount left is 88% of the original:
100% − 12% = 88%
88% = 0.88
500 g × 0.88 = 440 g
The new mass is 440 g.
A check using the change amount:
12% of 500 g = 60 g
500 g − 60 g = 440 g
Finding the percentage change
Sometimes you know both the original and new amounts, but not the percentage change.
Find the difference between them. Then compare that difference with the original amount.
Percentage change = (change ÷ original) × 100%. Use a positive original amount. This calculation cannot give a percentage change from zero, because division by zero is undefined.
Here, use the positive size of the change and state whether it is an increase or a decrease. We divide by the original because it is the starting 100%.
Example 6 — Find a percentage increase
A price rises from £40 to £50. Find the percentage increase.
Find the increase:
£50 − £40 = £10
Compare £10 with the original £40:
10 ÷ 40 = 0.25
Convert that share to a percentage:
0.25 × 100 = 25
The percentage increase is 25%.
The £10 tells us the change in money. The 25% tells us how large that change is compared with the starting price.
Example 7 — Find a percentage decrease
A club’s membership falls from 80 to 60. Find the percentage decrease.
Find how many members were lost:
80 − 60 = 20
Compare the decrease with the original 80 members:
20 ÷ 80 = 0.25
0.25 × 100 = 25
The percentage decrease is 25%.
Do not divide by 60. That is the new membership, not the starting 100%.
Working backwards — both tiers
In a reverse-percentage question, the new amount is known but the original amount is unknown.
First find the multiplier for the stated change, just as before.
Going forwards multiplies by that number. Going backwards divides by the same number. Division undoes multiplication.
Example 8 — Reverse percentages: reverse a decrease
A jacket costs £80 after a 20% reduction. Find its original price.
After the reduction, 80% of the original price remains:
100% − 20% = 80%
80% = 0.80
The £80 sale price represents 80%, not 100%.
Write the forward calculation in words:
Original price × 0.80 = £80
Divide by 0.80 to undo that multiplication:
£80 ÷ 0.80 = £100
The original price was £100.
Check by applying the reduction again:
£100 × 0.80 = £80
The check gives the stated sale price.
Example 9 — Reverse percentages: reverse an increase
A price is £66 after a 10% increase. Find the original price.
After the increase, the new price is 110% of the original:
100% + 10% = 110%
110% = 1.10
Original price × 1.10 = £66
Divide by 1.10 to work backwards:
£66 ÷ 1.10 = £60
The original price was £60.
Check:
£60 × 1.10 = £66
Do not subtract 10% of £66. That would take a percentage of the new price, not undo the original increase.
Keep the question types separate
| Question asks for… | Your route |
|---|---|
| Change amount | Find the stated percentage of the original. |
| New amount | Find the change and add/subtract, or multiply by the final-amount multiplier. |
| Percentage change | Change ÷ original, then convert to a percentage. |
| Original amount — both tiers | Divide the new amount by the forward multiplier. |
Check that your answer makes sense
For a positive amount, an increase should give more than you started with. A decrease below 100% should leave a smaller positive amount.
A 50% decrease should halve the original. A 10% increase should give only slightly more than the original.
Check that the final answer has the right unit. A percentage change is given in %, while a new price is given in pounds.
For reverse percentages, apply the original change to your answer. You should recover the given new amount.
Think Like a Mathematician
Does increasing by 20% and then decreasing by 20% get you back to the start?
Start with £100. First increase it by 20%:
20% of £100 = £20
£100 + £20 = £120
Now consider a 20% decrease from £120:
20% of £120 = £24
£120 − £24 = £96
You finish at £96, not £100.
The two changes use different starting amounts. The increase adds £20, but the decrease removes £24.
That is why subtracting the same percentage does not reverse an increase. To reverse the first change, divide £120 by its multiplier:
£120 ÷ 1.20 = £100
This example explains the reverse-percentage trap; methods for repeated growth and decay belong in a later lesson.
Common mistakes to avoid
Stopping at the change amount. For an increase or decrease, add or subtract that amount to find the final value.
Using 0.20 for a 20% increase. That finds only the 20% share. The final amount is 120%, so use 1.20.
Using an increase multiplier for a decrease. A 20% decrease leaves 80%, so use 0.80, not 1.20.
Dividing by the new amount. When finding percentage change, compare the difference with the original amount, which represents 100%.
Reversing by applying the same percentage. The new amount is a different base. Divide by the forward multiplier instead.
Rounding too early. Keep the full calculator value until the final money answer. Then round to the nearest penny if needed.
Dropping the unit. Keep £ for prices and the appropriate unit for quantities. Use % for a percentage change.
Try it yourself
Show your working and read carefully which amount the question asks for. Questions 9–10 are reverse-percentage questions for both tiers.
1. Increase 80 by 10%.
Find the change:
10% of 80 = 8
Add it to the original:
80 + 8 = 88
The new amount is 88.
2. Decrease 60 by 25%.
25% is one quarter:
60 ÷ 4 = 15
Subtract the change:
60 − 15 = 45
The new amount is 45.
3. A £48 bag has a 15% discount. Find both the discount and the sale price.
Find 10% and 5% of the original price:
10% of £48 = £4.80
5% of £48 = £2.40
£4.80 + £2.40 = £7.20
The discount is £7.20.
Subtract it from £48:
£48 − £7.20 = £40.80
The sale price is £40.80.
4. Use a multiplier to increase £250 by 8%.
The final amount is 108% of the original:
100% + 8% = 108%
108% = 1.08
£250 × 1.08 = £270
The new amount is £270.
5. Use a multiplier to decrease 450 g by 12%.
88% remains:
100% − 12% = 88%
88% = 0.88
450 g × 0.88 = 396 g
The new mass is 396 g.
6. A price rises from £75 to £90. Find the percentage increase.
Find the increase:
£90 − £75 = £15
Divide by the original price:
15 ÷ 75 = 0.20
0.20 × 100 = 20
The percentage increase is 20%.
7. A club’s membership falls from 120 to 90. Find the percentage decrease.
Find the decrease:
120 − 90 = 30
Divide by the original membership:
30 ÷ 120 = 0.25
0.25 × 100 = 25
The percentage decrease is 25%.
8. A coat costs £39.99 before a 15% discount. Find the sale price to the nearest penny.
The sale price is 85% of the original:
100% − 15% = 85%
85% = 0.85
39.99 × 0.85 = 33.9915
Keep this full result until the final step. The third decimal digit is 1, so round down to two decimal places.
The sale price is £33.99.
9. Reverse percentages: a jacket costs £72 after a 20% reduction. Find its original price.
80% remains, so the forward multiplier is 0.80.
Original price × 0.80 = £72
Divide to undo the multiplication:
£72 ÷ 0.80 = £90
The original price was £90.
Check by applying the reduction:
£90 × 0.80 = £72
10. Reverse percentages: a weekly payment becomes £132 after a 10% increase. Find the original weekly payment.
The new payment is 110% of the original, so use 1.10.
Original payment × 1.10 = £132
Divide to work backwards:
£132 ÷ 1.10 = £120
The original weekly payment was £120.
Check by applying the increase:
£120 × 1.10 = £132
Frequently Asked Questions
What is the difference between “20% of” and “increase by 20%”?
“20% of” finds the share only. “Increase by 20%” means add that share to the starting amount. For £50, the answers are £10 and £60 respectively.
Why does a 20% increase use 1.20?
You keep 100% of the original and add 20%. The final amount is 120% of the original, which is 1.20 as a decimal.
Must I use a multiplier?
No. For a forward change, finding the percentage amount and then adding or subtracting is equally valid. Choose the clearest method for the numbers given.
Why divide by the original when finding percentage change?
The question measures the change relative to where you started. The original is the reference amount, or 100%.
Reverse percentages: how do I recognise a reverse-percentage question?
You are given an amount after a percentage change and asked what it was before. Divide by the multiplier for the forward change.
What happens with a 100% decrease?
Nothing remains: the multiplier is 0. You cannot recover an original amount from that result, because every original amount would become 0.
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