Percentages: Percentage of Amounts, Percentage Change — AQA GCSE Maths Revision Notes
What you'll learn
This topic covers percentages in all the forms the specification asks for. By the end of this guide you should be able to find a percentage of an amount, both mentally and with a multiplier; increase or decrease a quantity by a percentage; express one quantity as a percentage of another; and calculate percentage change, including percentage profit and loss.
You should also be able to work backwards to find an original amount after a percentage change — the "reverse percentage" — and to handle repeated percentage change, including compound interest and depreciation.
The single most useful idea in the topic is the multiplier. Nearly every percentage question becomes a one-step multiplication once the right multiplier has been found, and reverse percentage questions become a division by that same multiplier. Learning to write the multiplier down first will make this topic far quicker and more reliable than working with separate steps.
Key terms and definitions
Percentage — a number expressed as a fraction of 100. The symbol % means "out of 100", so 35% is 35/100 or 0.35.
Multiplier — the decimal you multiply by to carry out a percentage operation in one step. For 15% of an amount the multiplier is 0.15; for a 15% increase it is 1.15; for a 15% decrease it is 0.85.
Percentage increase — a rise expressed as a percentage of the original amount.
Percentage decrease — a fall expressed as a percentage of the original amount.
Percentage change — the general term for either, calculated as the change divided by the original, then multiplied by 100.
Reverse percentage — finding the original amount when only the amount after a percentage change is known.
Compound interest — interest calculated each year on the running total, including interest already earned, rather than on the starting amount alone.
Depreciation — a repeated percentage decrease in value over time, typical of cars and electronic goods.
Core concepts
Percentages, decimals and fractions
The three are different notations for the same thing, and moving between them freely saves a great deal of time.
To convert a percentage to a decimal, divide by 100: 35% = 0.35. To convert a decimal to a percentage, multiply by 100: 0.6 = 60%. A percentage is already a fraction out of 100, so 35% = 35/100, which cancels to 7/20.
A few conversions are worth knowing without calculation: 50% = 0.5 = ½, 25% = 0.25 = ¼, 10% = 0.1 = 1/10, 75% = 0.75 = ¾, and 1% = 0.01 = 1/100.
Finding a percentage of an amount
The reliable method is the multiplier. Convert the percentage to a decimal and multiply.
To find 35% of 240: the multiplier is 0.35, so 0.35 × 240 = 84.
Where a calculator is not allowed, build the answer from parts you can find mentally. 10% is found by dividing by 10, 1% by dividing by 100, and 50% by halving. So 35% of 240 can be assembled as 10% (24) three times, giving 72, plus 5% — half of 10%, so 12 — giving 84 again. The two routes must agree, and checking one against the other is a good use of thirty seconds.
Increasing and decreasing by a percentage
Here the multiplier saves the most time, because it does the whole operation at once.
For an increase of p%, the multiplier is 1 + p/100. A 15% increase uses 1.15, a 7% increase uses 1.07, and a 120% increase uses 2.2.
For a decrease of p%, the multiplier is 1 − p/100. A 15% decrease uses 0.85, a 7% decrease uses 0.93, and a 40% decrease uses 0.6.
So increasing 320 by 15% is simply 320 × 1.15 = 368, and decreasing 320 by 15% is 320 × 0.85 = 272. The alternative — find 15%, then add or subtract it — gives the same answer but takes two steps and offers two chances to slip.
Expressing one quantity as a percentage of another
Divide the part by the whole, then multiply by 100.
If 18 out of 40 students walk to school, the proportion is 18 ÷ 40 = 0.45, which is 45%.
The order matters: the quantity that comes after the word "of" goes on the bottom. "18 as a percentage of 40" puts 40 on the bottom; "40 as a percentage of 18" would put 18 there and give a figure above 100%.
Percentage change
Percentage change compares the size of a change with the original amount:
percentage change = (change ÷ original) × 100
The word original is doing important work. The denominator is always the starting value, never the finishing value and never the change itself. A price rising from 40 to 50 has changed by 10, and 10 ÷ 40 = 0.25, so the increase is 25%. Using 50 on the bottom would give 20%, which is the wrong answer to a different question.
Percentage profit and percentage loss are the same calculation, with the change being the profit or loss and the original being the cost price.
Reverse percentages
This is the type of question students most often get wrong, and the error is always the same: adjusting the final amount by the percentage instead of dividing by the multiplier.
If a price after a 20% increase is £60, then £60 represents 120% of the original, so the original is 60 ÷ 1.2 = £50. Reducing £60 by 20% would give £48, which is not the answer — 20% of 50 and 20% of 60 are different amounts.
The rule is simple: to undo a percentage change, divide by the multiplier you would have multiplied by.
A question is a reverse percentage whenever the amount you are given is described as being after the change: "after a 20% discount", "including 20% VAT", "after depreciation of 15%".
Repeated percentage change
When the same percentage change happens several times, apply the multiplier once for each period. Do not multiply the percentage by the number of years.
For compound interest at 5% a year over 3 years, the multiplier is 1.05 applied three times, so the final amount is the starting amount × 1.05³. Depreciation works identically with a decrease multiplier: a car losing 20% of its value each year for 3 years is worth its starting value × 0.8³.
The reason compound interest exceeds simple interest is that each year's interest is calculated on a larger running total. Over three years at 5%, the total growth is 1.05³ ≈ 1.158, or about 15.8%, rather than the 15% that three separate 5% additions on the original would give.
Worked examples
Example 1: Percentage change
A shop buys a jacket for £40 and sells it for £50. Calculate the percentage profit.
The change is 50 − 40 = £10, and the original is the cost price, £40.
Percentage change = (10 ÷ 40) × 100 = 0.25 × 100 = 25%.
Check the denominator before writing anything down. Dividing by the selling price of £50 would give 20%, and that is the single commonest error in this question type.
Example 2: Reverse percentage
The price of a coat is £60 in a sale offering 25% off. What was the price before the sale?
The sale price is 75% of the original, so the multiplier for the change was 0.75.
Original = 60 ÷ 0.75 = £80.
Check it forwards: 80 × 0.75 = 60. ✓
Adding 25% to £60 would give £75, which is wrong, because that adds 25% of the sale price rather than removing 25% of the original price.
Example 3: Compound interest
£2000 is invested at 3% compound interest per year. Find the value after 4 years, to the nearest penny.
The multiplier for a 3% increase is 1.03, applied four times.
Value = 2000 × 1.03⁴ = 2000 × 1.12550881 = £2251.02 to the nearest penny.
Simple interest would have given 4 × 3% = 12% of £2000, which is £240, for a total of £2240. The extra £11.02 is the interest earned on interest, and questions often ask for exactly that difference.
Common mistakes and how to avoid them
Dividing by the new amount in percentage change. The denominator is always the original. Write "original =" before you write the number.
Treating a reverse percentage as an ordinary one. If the question says "after" or "including", divide by the multiplier rather than adjusting the given figure.
Multiplying the percentage by the number of years. Compound change uses the multiplier raised to a power; 5% for 3 years is 1.05³, not 15%.
Using the wrong multiplier direction. A 15% increase is 1.15 and a 15% decrease is 0.85. Writing the multiplier down before calculating makes this visible.
Forgetting that a percentage increase followed by the same percentage decrease does not return to the start. Increasing 100 by 10% gives 110; decreasing 110 by 10% gives 99, not 100, because the second percentage is taken of a larger amount.
Rounding too early. Keep the full figure in the calculator and round only the final answer, particularly in compound interest questions.
Exam technique for "Percentages"
Write the multiplier down as a separate step before doing any arithmetic. It makes the method visible to the examiner and makes the direction of the change obvious to you.
Look for the word after. It is the clearest signal that a question is a reverse percentage, and those questions carry more marks than an ordinary percentage of an amount.
Show the division in percentage change questions as a fraction, so the denominator is visible. An examiner can award method marks for the correct fraction even when the arithmetic that follows is wrong.
Check a reverse percentage by working forwards. It takes one multiplication and confirms the answer completely.
Give money answers to two decimal places and include the units. A correct value written without the pound sign, or as £2251.0157, can lose the final accuracy mark.
Quick revision summary
A percentage is a fraction out of 100. Convert to a decimal by dividing by 100.
The multiplier turns nearly every percentage question into one multiplication. For an increase of p% use 1 + p/100; for a decrease use 1 − p/100.
Percentage change = (change ÷ original) × 100. The denominator is always the starting value.
For a reverse percentage, divide by the multiplier. If a price after 20% off is £60, the original is 60 ÷ 0.8 = £75; if a price after a 20% increase is £60, the original is 60 ÷ 1.2 = £50.
For repeated change, raise the multiplier to the power of the number of periods: 1.05³ for three years at 5% compound interest, 0.8³ for three years of 20% depreciation.
A percentage rise followed by the same percentage fall does not return to the original, because each is taken of a different amount.