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GCSE Maths · EdexcelChapter 5 · Percentages
GCSE Maths · Percentages · Higher · Lesson 5.6

GCSE Maths: Reverse percentages: finding the original amount (Higher)

Part of the GCSE Maths percentages chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)

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5.6

Reverse percentages: finding the original amount

Edexcel 1MA1: R9
Grade 4-7

In a sale a coat is reduced by 30% and now costs £63. What did it cost before the sale? This is a reverse percentage (or original value) problem. The amount you are given is the new amount, after the change, and you must work backwards to find the original.

The tempting mistake is to work out 30% of £63 and add it on. That is wrong, because the 30% was taken off the original price, which was bigger than £63, not off £63. Instead use the idea that the original is always 100%.

Bar model (not to scale of the numbers, to scale of the percentages)Original price = 100%New = 70% = £63
A 30% reduction leaves 70% of the original, so £63 is 70% of the original price.

After a 30% reduction the coat costs 100% − 30% = 70% of the original, and we know that 70% is £63.

Method 1: find 1%, then 100%

If 70% is £63, then 1% is £63 ÷ 70 = £0.90, and 100% is 100 × £0.90 = £90. This is the same “unitary” idea you use for ratio: go to one, then scale up.

Method 2: divide by the multiplier

The multiplier for a 30% decrease is 0.7. We know original × 0.7 = £63. To undo a multiplication we divide: original = 63 ÷ 0.7 = £90. For an increase of, say, 20%, the multiplier is 1.2 and the original is the new amount ÷ 1.2.

It is just the same when a percentage of a whole is given. If 15% of the pupils in a school is 21 pupils, then 1% is 21 ÷ 15 = 1.4 pupils, and the whole school (100%) is 140 pupils.

Higher tier. You can also set this up as an equation. Let the original be x. After an increase of 25% the new amount is 1.25x. If the new amount is 75 then 1.25x = 75, so x = 75 ÷ 1.25 = 60. When the division does not come out exactly, keep all the digits on your calculator and round only at the end (to the nearest penny for money).

Key ideas

  • The original amount is 100%.
  • After an increase of p%, the new amount is (100 + p)% of the original. After a decrease of p%, it is (100 − p)%.
  • Method 1: new amount = (100 ± p)%, so divide by (100 ± p) to get 1%, then multiply by 100.
  • Method 2: original = new amount ÷ multiplier.
  • Check by going forwards: original × multiplier should give the new amount.
  • Never find p% of the new amount and add or subtract it.
Worked example 5.6.1
In a sale all prices are reduced by 30%. A coat costs £63 in the sale. Work out its price before the sale.

Key words. Per cent means “out of 100”. A percentage is a fraction whose denominator is 100, written with the % sign. A multiplier is the number you multiply by to get the new amount in one step.

Method. In a reverse percentage question the amount we are given is NOT the original. It is a percentage of the original. Work backwards to find 100%.

  1. Find the percentage that the new amount represents. The original amount is 100%. After a decrease of 30% the new amount is 100% − 30% of the original.
    100% − 30% = 70%
  2. Write down what we know. The new amount is a known percentage of the unknown original.
    70% = £63
  3. Find 1%. Divide by 70 to go from 70% down to 1%.
    £63 ÷ 70 = £0.90
  4. Find 100%. The original is 100%, so multiply 1% by 100.
    £0.90 × 100 = £90

Check. Check by going forwards: £90 × 0.7 = £63, which matches the new amount. ✓

Watch out. Do not work out 30% of the new amount and add it. The percentage was taken from the original, not from the new amount.

Worked example 5.6.2
A bill including VAT at 20% is £144. Work out the bill before VAT was added.

Method. In a reverse percentage question the amount we are given is NOT the original. It is a percentage of the original. Work backwards to find 100%.

  1. Find the percentage that the new amount represents. The original amount is 100%. After an increase of 20% the new amount is 100% + 20% of the original.
    100% + 20% = 120%
  2. Write down what we know. The new amount is a known percentage of the unknown original.
    120% = £144
  3. Find 1%. Divide by 120 to go from 120% down to 1%.
    £144 ÷ 120 = £1.20
  4. Find 100%. The original is 100%, so multiply 1% by 100.
    £1.20 × 100 = £120

Check. Check by going forwards: £120 × 1.2 = £144, which matches the new amount. ✓

Watch out. Do not work out 20% of the new amount and subtract it. The percentage was taken from the original, not from the new amount.

Worked example 5.6.3
Fatima’s salary was increased by 4%. It is now £27,248. Work out her salary before the increase.

Method. In a reverse percentage question the amount we are given is NOT the original. It is a percentage of the original. Work backwards to find 100%.

  1. Find the multiplier. The original is 100%. After an increase of 4% the new amount is 104% of the original, which is 1.04 as a decimal.
    104% = 104 ÷ 100 = 1.04
  2. Write what the multiplier does. original × multiplier = new amount.
    original × 1.04 = £27,248
  3. Go backwards: divide by the multiplier. To undo a multiplication, divide.
    £27,248 ÷ 1.04 = £26,200

Check. Check by going forwards: £26,200 × 1.04 = £27,248, which matches the new amount. ✓

Watch out. Do not work out 4% of the new amount and subtract it. The percentage was taken from the original, not from the new amount.

Worked example 5.6.4
In a school, 21 pupils walk to school. This is 15% of the pupils in the school. How many pupils are in the school?

Method. If you know what a percentage of the whole is, find 1% first, then multiply by 100 to find the whole (100%).

  1. Write down what we know. Part of the whole has been given as a percentage.
    15% of the whole = 21 pupils
  2. Find 1%. Divide by 15. That takes us from 15% to 1%.
    21 pupils ÷ 15 = 1.4 pupils
  3. Find 100%. The whole is 100%, so multiply 1% by 100.
    1.4 pupils × 100 = 140 pupils

Check. 15% of 140 pupils = 0.15 × 140 pupils = 21 pupils. ✓

Watch out. The whole is 100%. Do not stop at 1%.

Worked example 5.6.5
After a 12% price increase, a laptop bag costs £250. Work out the original price. Give your answer to the nearest penny.

Method. In a reverse percentage question the amount we are given is NOT the original. It is a percentage of the original. Work backwards to find 100%.

  1. Find the multiplier. The original is 100%. After an increase of 12% the new amount is 112% of the original, which is 1.12 as a decimal.
    112% = 112 ÷ 100 = 1.12
  2. Write what the multiplier does. original × multiplier = new amount.
    original × 1.12 = £250
  3. Go backwards: divide by the multiplier. To undo a multiplication, divide.
    £250 ÷ 1.12 = £223.2143…
  4. Round the answer. Round to 2 decimal places.
    £223.2143… = £223.21

Check. Check by going forwards: £223.21 × 1.12 = £249.9952, which matches the new amount. ✓

Watch out. Do not work out 12% of the new amount and subtract it. The percentage was taken from the original, not from the new amount.

Worked example 5.6.6
After an increase of 25% a number is 75. Use an equation to find the original number.

Method. Call the original number x, write the increase as a multiplier, and solve the equation.

  1. Write the multiplier. An increase of 25% means the new number is 125% of the original, which is 1.25.
    100% + 25% = 125% = 1.25
  2. Form an equation. Original × multiplier = new number.
    1.25x = 75
  3. Solve it: divide both sides by 1.25. Dividing both sides by the same number keeps the equation balanced and leaves x on its own.
    x = 75 ÷ 1.25 = 60
  4. Check. Increase 60 by 25%: 25% of 60 = 15 and 60 + 15 = 75.
    60 × 1.25 = 75

Check. 60 + 25% of 60 = 60 + 15 = 75. ✓

Watch out. Make sure the equation uses the multiplier, 1.25, not 0.25.

Watch out

Do not take 30% of the new price and add it on. £63 + 30% of £63 = £81.90, which is wrong. The 30% was 30% of the original. Going forwards: £90 reduced by 30% is £63. A good habit is to check by applying the percentage change to your answer.

Exercise 5J  Reverse percentages (non-calculator)

Non-calculator
1
FFind the original amount in each case.
(a)After a 10% decrease: £45
(b)After a 20% increase: £60
(c)After a 25% decrease: £30
(d)After a 50% increase: £45
(e)After a 30% decrease: £35
(f)After a 5% increase: 21 kg
2
FWork out the whole amount in each case.
(a)10% is 8
(b)25% is 15
(c)20% is 30
(d)5% is 6
(e)40% is 18
(f)15% is 21
3
FAfter a 25% increase, a number is 50. What was the number before the increase?
4
PSIn a sale, trainers are reduced by 40%. The sale price is £54. What was the price before the sale?
5
PSA garage bill is £180 including VAT at 20%. How much was the bill before VAT?
6
PSThe number of students in a school increased by 5% this year. There are now 1,260 students. How many were there last year?
7
PSAisha’s hourly pay was increased by 5%. She now earns £12.60 an hour. What did she earn an hour before the rise?
8
PSA club has some members. 36 of them are girls, which is 60% of the members.
(a)How many members does the club have?
(b)How many boys are in the club?
9
PSTwo shops sell the same bike. Shop A reduces its price by 20% and sells the bike for £360. Shop B reduces its price by 10% and sells it for £396. Which shop had the higher price before its reduction? Show your working.
10
PSThe price, p pounds, of a camera was increased by 12%. The new price is £84. Form an equation and solve it to find p.
11
FThe 45 students in the school choir are 25% of all the students in the school. How many students are in the school?
12
R20% of a number is 12. Work out the number, and then work out 35% of the number.
13
RThe price of a coat is reduced by 30% in a sale, and it now costs £63. Priya says, “The original price was £63 + 30% of £63 = £81.90.” Explain her mistake and find the original price.

Exercise 5K  Reverse percentages (calculator)

Calculator allowed
1
FFind the original amount in each case. Give money answers to the nearest penny.
(a)After a 12% increase: £250
(b)After a 20% decrease: £399
(c)After a 17.5% increase: £84.60
(d)After a 13% decrease: £13,050
(e)After a 3.5% increase: 12,420
(f)After a 2% decrease: £7.84
2
PSA car loses 18% of its value in its first year. After one year it is worth £10,578. What was it worth when it was new?
3
PSA house price rose by 6% in a year. It is now £236,000. What was its price a year ago, to the nearest pound?
4
PSA dress is reduced by 28% in a sale. The sale price is £37.80. Work out the original price.
5
PSJo’s yearly pay is increased by 3.2% to £28,200. Work out her yearly pay before the rise. Give your answer to the nearest penny.
6
FWork out the original amount in each case.
(a)A population falls by 4.5% to 71,625
(b)A mass falls by 12% to 5.28 kg
7
PSA shop takes 20% off the price of a coat. VAT of 20% is then added to the reduced price, and the customer pays £72. Work out the price of the coat before either change.
8
RA phone costs £96 after a 20% increase. Hamza says the price before the increase was £76.80, because 96 × 0.8 = 76.80. Is he right? Show how you know.

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