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GCSE Maths · EdexcelChapter 5 · Percentages
GCSE Maths · Percentages · Higher · Lesson 5.5

GCSE Maths: Percentage change, profit and loss (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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5.5

Percentage change, profit and loss

Edexcel 1MA1: R9
Grade 4-6

“The price of a phone went from £400 to £500.” That is an increase of £100, but is it a big increase? £100 matters more on a £400 phone than on a £40,000 car. To judge a change fairly we write it as a percentage of the original amount. This is the percentage change.

Old£400New£400+£100The change is compared with the ORIGINAL (old) amount.
A price rises from £400 to £500. The change is £100, which is 25% of the original £400.

The rule is easy to remember:

percentage change = changeoriginal × 100%

where the change is the bigger number take away the smaller number. If the new amount is bigger it is a percentage increase; if it is smaller it is a percentage decrease.

Why divide by the original? Because the percentage tells us what share of the starting amount has been added or lost. The starting amount is the 100%. This is the most common mistake: dividing by the new amount.

Profit and loss

A shop buys goods at a cost price and sells them at a selling price. If the selling price is higher there is a profit; if it is lower there is a loss. The profit or loss is a change, so the same rule works: percentage profit = profitcost price × 100%. The cost price is the original amount.

Percentages bigger than 100%

If a quantity more than doubles, the percentage increase is bigger than 100%. A plant that grows from 12 cm to 30 cm has grown by 18 cm, and 18 ÷ 12 × 100 = 150%. A decrease, however, can never be more than 100%.

Higher tier. Percentage points. If an interest rate goes from 4% to 6%, it has gone up by 2 percentage points (6 − 4 = 2). But as a percentage change it has gone up by 24 × 100% = 50%. Always check whether a question asks for percentage points or a percentage change. Also notice that a rise and the fall that brings you back are different percentages: a price that rises from £50 to £60 (up 20%) must fall by only 16.7% to return to £50, because the fall is measured against the bigger number £60.

Key ideas

  • Percentage change = change ÷ original × 100%.
  • Change = bigger − smaller. Say increase or decrease.
  • Always divide by the original (the starting amount).
  • Profit = selling price − cost price. Percentage profit = profit ÷ cost price × 100%. Loss works the same way.
  • An increase can be over 100%. A decrease cannot.
Worked example 5.5.1
The price of a ticket rises from £40 to £50. Work out the percentage increase.

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. Percentage change = change ÷ original × 100.

  1. Work out the change. Subtract the smaller value from the bigger one. The change is the difference.
    £50 − £40 = £10 (increase)
  2. Write the change as a fraction of the ORIGINAL. Percentage change always compares the change with the starting (original) value.
    1040
  3. Divide the top by the bottom. This changes the fraction into a decimal.
    10 ÷ 40 = 0.25
  4. Multiply by 100. To change a decimal into a percentage multiply by 100, because a percentage is a number out of 100.
    0.25 × 100 = 25%
  5. Say whether it is an increase or a decrease.
    25% increase

Check. Check: 25% of £40 = £10, and £40 + £10 = £50. ✓

Watch out. Divide by the original value, not the new one.

Worked example 5.5.2
The number of students at a school falls from 250 to 215. Work out the percentage decrease.

Method. Percentage change = change ÷ original × 100.

  1. Work out the change. Subtract the smaller value from the bigger one. The change is the difference.
    250 − 215 = 35 (decrease)
  2. Write the change as a fraction of the ORIGINAL. Percentage change always compares the change with the starting (original) value.
    35250
  3. Divide the top by the bottom. This changes the fraction into a decimal.
    35 ÷ 250 = 0.14
  4. Multiply by 100. To change a decimal into a percentage multiply by 100, because a percentage is a number out of 100.
    0.14 × 100 = 14%
  5. Say whether it is an increase or a decrease.
    14% decrease

Check. Check: 14% of 250 = 35, and 250 − 35 = 215. ✓

Watch out. Divide by the original value, not the new one.

Worked example 5.5.3
A shop buys a chair for £60 and sells it for £84. Work out the percentage profit.

Method. Percentage change = change ÷ original × 100.

  1. Work out the change. Subtract the smaller value from the bigger one. The change is the difference.
    £84 − £60 = £24 (profit)
  2. Write the change as a fraction of the ORIGINAL. Percentage change always compares the change with the starting (original) value.
    2460
  3. Divide the top by the bottom. This changes the fraction into a decimal.
    24 ÷ 60 = 0.4
  4. Multiply by 100. To change a decimal into a percentage multiply by 100, because a percentage is a number out of 100.
    0.4 × 100 = 40%
  5. Say whether it is an increase or a decrease.
    40% profit

Check. Check: 40% of £60 = £24, and £60 + £24 = £84. ✓

Watch out. Divide by the original value, not the new one.

Worked example 5.5.4
A dealer buys a motorbike for £850 and sells it for £680. Work out the percentage loss.

Method. Percentage change = change ÷ original × 100.

  1. Work out the change. Subtract the smaller value from the bigger one. The change is the difference.
    £850 − £680 = £170 (loss)
  2. Write the change as a fraction of the ORIGINAL. Percentage change always compares the change with the starting (original) value.
    170850
  3. Divide the top by the bottom. This changes the fraction into a decimal.
    170 ÷ 850 = 0.2
  4. Multiply by 100. To change a decimal into a percentage multiply by 100, because a percentage is a number out of 100.
    0.2 × 100 = 20%
  5. Say whether it is an increase or a decrease.
    20% loss

Check. Check: 20% of £850 = £170, and £850 − £170 = £680. ✓

Watch out. Divide by the original value, not the new one.

Worked example 5.5.5
A plant is 12 cm tall. A month later it is 30 cm tall. Work out the percentage increase in its height.

Method. Percentage change = change ÷ original × 100.

  1. Work out the change. Subtract the smaller value from the bigger one. The change is the difference.
    30 cm − 12 cm = 18 cm (increase)
  2. Write the change as a fraction of the ORIGINAL. Percentage change always compares the change with the starting (original) value.
    1812
  3. Divide the top by the bottom. This changes the fraction into a decimal.
    18 ÷ 12 = 1.5
  4. Multiply by 100. To change a decimal into a percentage multiply by 100, because a percentage is a number out of 100.
    1.5 × 100 = 150%
  5. Say whether it is an increase or a decrease.
    150% increase

Check. Check: 150% of 12 cm = 18 cm, and 12 cm + 18 cm = 30 cm. ✓

Watch out. Divide by the original value, not the new one.

Worked example 5.5.6
A house was worth £245,000. It is now worth £261,500. Work out the percentage increase. Give your answer to 1 decimal place.

Method. Percentage change = change ÷ original × 100.

  1. Work out the change. Subtract the smaller value from the bigger one. The change is the difference.
    £261,500 − £245,000 = £16,500 (increase)
  2. Write the change as a fraction of the ORIGINAL. Percentage change always compares the change with the starting (original) value.
    16,500245,000
  3. Divide the top by the bottom. This changes the fraction into a decimal.
    16,500 ÷ 245,000 = 0.067347…
  4. Multiply by 100. To change a decimal into a percentage multiply by 100, because a percentage is a number out of 100.
    0.067347… × 100 = 6.734694…%
  5. Round the answer. Round to 1 decimal place as the question asks.
    6.734694…% = 6.7% (to 1 d.p.)
  6. Say whether it is an increase or a decrease.
    6.7% increase

Check. Check: 6.7% of £245,000 is about 16,415.00, which is very close to the actual change of £16,500. ✓

Watch out. Divide by the original value, not the new one.

Worked example 5.5.7
A ticket price falls from £80 to £60. Later it rises from £60 back to £80. Work out the percentage change each time. Give the second answer to 1 decimal place.

Method. Use percentage change = change ÷ original × 100% twice. The original is different each time.

  1. Fall from £80 to £60: the change. Bigger take away smaller.
    £80 − £60 = £20
  2. Divide by the original, £80, and multiply by 100. The original is where the first change started.
    £20 ÷ £80 × 100 = 25%
  3. Rise from £60 to £80: the change. The change is again £20.
    £80 − £60 = £20
  4. Divide by the new original, £60, and multiply by 100. This time the original is £60.
    £20 ÷ £60 × 100 = 33.333…% = 33.3%
  5. Say what happened. The same £20 is a different percentage because the original amount is different.
    A 25% decrease, then a 33.3% increase.

Check. Check: 25% of 80 = 20 and 80 − 20 = 60. 33.3% of 60 = 20 and 60 + 20 = 80. ✓

Watch out. Divide by the original each time. The original changes when the direction changes.

Watch out

Divide by the original amount, not the new amount. A price that goes from £40 to £50 rises by 25% (10 ÷ 40), not 20% (10 ÷ 50).

Exercise 5H  Percentage change, profit and loss (non-calculator)

Non-calculator
1
FWork out the percentage change in each case. Say whether it is an increase or a decrease.
(a)£40 to £50
(b)80 to 60
(c)£200 to £250
(d)50 to 65
(e)400 to 300
(f)£15 to £18
2
FWork out the percentage profit or loss in each case.
(a)Bought for £60, sold for £84
(b)Bought for £500, sold for £450
(c)Bought for £25, sold for £30
(d)Bought for £80, sold for £60
3
PSA gym has 400 members in January and 460 members in June. Work out the percentage increase in the number of members.
4
PSThe price of a bus pass goes up from £80 to £88. Work out the percentage increase in the price.
5
PSA shop buys a pair of trainers for £50 and sells them for £65. Work out the percentage profit.
6
PSThe population of a village falls from 800 to 680. Work out the percentage decrease.
7
PSA phone costs £500. After a year its price falls to £375. Work out the percentage decrease in the price.
8
PSA market trader buys scarves for £8 each and sells them for £14 each. Work out the trader’s percentage profit.
9
PSA sunflower is 12 cm tall in week 1 and 30 cm tall in week 2. Work out the percentage increase in height.
10
PSShop A raises the price of a kettle from £120 to £150. Shop B raises the price of a toaster from £80 to £104. Which shop has the bigger percentage increase?
11
RA price rises from £50 to £60, which is a 20% increase. Lee says, “So it needs to fall by 20% to get back to £50.” Is he right? Show how you know.
12
R(a) The unemployment rate fell from 8% to 6%. By how many percentage points did it fall?  (b) Work out the percentage decrease in the rate.
(a)Change in percentage points
(b)Percentage decrease
13
RA shopkeeper buys a jacket for £200 and marks it up by 40% to get the price on the label. In a sale the jacket is sold for 25% off the label price. Does the shopkeeper make a profit or a loss on the jacket? Work out the percentage.

Exercise 5I  Percentage change (calculator)

Calculator allowed
1
FWork out the percentage change in each case. Give your answers correct to 1 decimal place.
(a)36 to 41
(b)8.5 to 7.2
(c)1,250 to 980
(d)64 to 100
(e)15.5 to 21.7
2
PSA shop buys a lamp for £18.50 and sells it for £27.95. Work out the percentage profit. Give your answer to 1 decimal place.
3
PSNaz buys a car for £12,400 and sells it two years later for £9,650. Work out his percentage loss. Give your answer to 1 decimal place.
4
PSThe rainfall in a town was 84 mm in March and 99 mm in April. Work out the percentage increase. Give your answer to 1 decimal place.
5
PSThe price of a share falls from £4.20 to £3.57. Work out the percentage decrease.
6
PSA town’s population was 24,800 in 2015, 27,280 in 2020 and 29,100 in 2025. In which five-year period was the percentage increase bigger? Show your working.
7
FWrite down the percentage change when a number goes from 80 to 45. Give your answer to the nearest whole number.
8
RSales were 200 in 2023, 300 in 2024 and 450 in 2025. Ava says, “Sales rose by 50% each year, so over the two years they rose by 100%.” Is she right? Show how you know.

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