Percentages appear everywhere: discounts in shops, exam marks, price increases, survey results and progress towards a target. Most percentage questions are straightforward once you know which number represents the whole and which represents the part.
The word percent means 'per hundred'. A percentage is simply a proportion expressed out of 100. The practical challenge is identifying the starting amount, especially when calculating a percentage change or working backwards from a final price.
Find what percentage one number is of another
Use the formula: percentage = (part ÷ whole) × 100. For example, if 18 of 24 people answer a survey, calculate 18 ÷ 24 = 0.75, then multiply by 100 to get 75%.
The number representing the whole goes underneath in the division. If you reverse the two, you will get a very different answer that may not fit the question.
A quick sense-check helps. If the part is less than the whole and both are positive, the percentage must be below 100%.
Find a percentage of an amount
To find 20% of £80, convert 20% into a decimal by dividing by 100: 20 ÷ 100 = 0.20. Multiply £80 by 0.20 to get £16.
This works for any ordinary percentage. To find 7.5% of 240, calculate 0.075 × 240 = 18.
You can also work mentally from 10%, 5% and 1%. Ten percent of £80 is £8, so 20% is £16. Both routes give the same answer.
Calculate a percentage discount
A coat costs £120 and has 25% off. Find the discount first: 120 × 0.25 = £30. Subtract that from the original price, leaving £90.
An alternative single step is £120 × 0.75 = £90 because paying after a 25% discount leaves 75% of the original.
Take care with multiple discounts. Twenty percent off followed by another ten percent off is not generally a thirty percent discount on the original price.
Work out a percentage increase
Suppose a monthly subscription rises from £40 to £46. Find the change, which is £6. Divide the change by the original £40 and multiply by 100: (6 ÷ 40) × 100 = 15%.
The original amount is the base. Do not divide by the new £46 unless the question explicitly asks something different.
The amount increased by 15%, but the £46 total is 115% of the original £40. Keeping those two statements separate prevents confusion.
Calculate a percentage decrease
If a product price falls from £50 to £42, the decrease is £8. Divide by the original £50 and multiply by 100 to get 16%.
A decrease should be expressed as a positive percentage with the word 'decrease', or as a negative signed percentage change if using that convention.
For a general signed change, calculate (new - old) ÷ old × 100, provided the old value is nonzero. A negative result represents a decline relative to that base.
Work backwards from a discounted price
Suppose an item costs £72 after a 20% reduction. That £72 is 80% of the original price. Divide 72 by 0.8 to get £90.
Do not add 20% of £72 to the discounted price. That gives £86.40, not the original £90, because the two percentages would use different bases.
Backward percentage calculations are common in VAT, discounts and price-change questions. Identify the percentage that the known amount represents before dividing.
How to calculate VAT in a simple example
If a price is £100 before a hypothetical 20% tax, the tax is £20 and the total is £120. To remove a 20% tax from an inclusive £120 price, divide by 1.20, giving £100.
The tax percentage in the example is illustrative; actual tax treatment depends on jurisdiction, product and current rules. Do not assume every purchase is taxed at that rate.
The arithmetic idea is the same as other reverse percentages. A total that includes a 20% addition represents 120% of the starting amount, not 100%.
Understand percentage points
If a survey's approval rate rises from 40% to 45%, the increase is five percentage points. The relative percentage increase is (45 - 40) ÷ 40 × 100 = 12.5%.
Those are not interchangeable descriptions. Saying the result increased by five percent would mean something different unless you specifically mean a relative increase.
This distinction matters in news reports, interest rates, election figures and statistics. Always specify whether you are comparing percentages themselves or relative changes.
Handle percentages over 100%
A percentage can exceed 100 when the part is larger than the reference amount. If a new figure is 150 against an original 100, it is 150% of the original.
That is different from saying it increased by 150%. The increase is 50 compared with the base of 100, so the percentage increase is 50%.
You can also have percentages below zero in signed-change contexts. Make sure the phrasing makes clear whether you mean a proportion or a change.
Find an exam percentage
A student scores 42 marks out of 60. Divide 42 by 60 and multiply by 100: 42 ÷ 60 × 100 = 70%.
If some questions are weighted differently, do not assume raw marks can be treated equally. Follow the scoring scheme when calculating the final grade.
An exam percentage and a grade boundary are separate matters. Seventy percent is a numerical score; what letter grade it earns depends on the assessment system.
Percentage change when the original is zero
The familiar percentage-change formula divides by the original value. If the original is zero, that division is undefined.
A change from zero customers to ten customers is an increase of ten customers, but it cannot be described using the ordinary finite percentage increase formula relative to zero.
In reports, state the absolute change and explain the limitation rather than presenting an invented enormous percentage.
Use percentages to compare things fairly
Suppose one shop reduces a £40 item by £8 and another reduces a £60 item by £9. The first gives 20% off, while the second gives 15% off.
A larger cash discount does not necessarily mean a larger percentage discount. Convert both to proportions of their original prices before comparing.
Likewise, when measuring completion against a target, divide the achieved amount by the full target rather than by what remains.
Check your answer
If you calculated a 10% discount on £100 and got £1000, a decimal point has gone astray. Estimate first: 10% is one-tenth, so the result must be around £10.
Write a sentence that identifies the base: '£8 is 20% of the original £40 price.' This makes it harder to reverse the division accidentally.
With awkward values, keep adequate precision until the final step and round according to the currency or measurement requirements.
The three formulas worth keeping
Percentage of a whole is (part ÷ whole) × 100. A percentage of an amount is amount × (percentage ÷ 100). Percentage change is (new - old) ÷ old × 100 when the starting value is nonzero.
The calculations are short. Choosing the right starting amount is where accuracy comes from.