Percentages: the one question that fixes most mistakes
A percentage is a fraction with 100 fixed as the denominator, which turns any part, change or rate onto a single scale you can compare at a glance. Say “20%” and you mean 20 out of every 100 — so 20% of 150 is 30, because you take 20 hundredths of 150. Percentages can also pass 100: 150% of 40 is 60. Almost every percentage mistake, from a mispriced discount to a misleading headline, comes from losing track of one thing — what the percentage is of. Fix the base and the answer falls out; lose it and the maths turns to nonsense.
The three questions every percentage reduces to
Nearly every percentage problem is one of three questions in disguise.
- What is X% of Y? You have a rate and a base, and you want the part. 20% of 150 is 30.
- What percent is A of B? You have two amounts and want the rate between them. 30 is 20% of 150.
- How much did something change? You compare a new figure to an old one, relative to the old one. Going from 80 to 100 is a 25% increase, because the 20-unit rise is measured against the starting 80.
The base is different in each case, and naming it before you calculate is the whole skill.
The traps worth knowing exist
Four quirks of percentages catch people out repeatedly. Each has its own guide; here is the shape of the danger.
Falls need bigger rises to undo. 100 reduced by 20% is 80, but getting from 80 back to 100 needs a 25% rise, not 20% — the second base is smaller. Recovering from a 50% fall needs a 100% rise.
Increase and decrease are not symmetric. From 80 to 100 is a 25% increase; from 100 to 80 is a 20% decrease. Same gap, different bases, different answers. (If you want one figure that treats both ends fairly, the symmetric percentage difference between 80 and 100 is 22.22%, measured against their average.)
Discounts don’t add. A 15% cut then another 15% cut is not 30% off, because the second cut applies to an already-reduced price.
“Percent” and “percentage points” are different words. A rate rising from 2% to 4% is up 2 percentage points, but up 100% — and confusing the two is how numbers mislead.
What you can do from here
The tools below let you run all three questions on your own figures, and the guides open up each trap in full — including compound interest, which is simply a percentage change applied again and again. Start with whichever number is bugging you.
Last reviewed 7 July 2026
Calculate it
Understand it
The three questions every percentage problem reduces to, and how to answer each one without a formula sheet.
Guide Percentage change, explainedHow percentage increase and decrease work, why the direction matters, and why a fall needs a bigger rise to undo it.
Guide Reverse percentages: finding the original numberHow to get back to the price before a percentage was added — and why subtracting the percentage gives the wrong answer.
Guide Percentage points vs percentA rate rising "by 2%" and "by 2 percentage points" are different claims — how to read each and why headlines blur them.
Guide Fractions, decimals and percentagesThe same number in three costumes: converting between fractions, decimals and percentages in both directions.
GuideCompare
See it in numbers
A worked example of percentage asymmetry: what falls by 20% must rise by 25% just to get back where it started.
Example Why 20% off then 10% off isn’t 30% offA worked example of stacked percentages: successive discounts multiply rather than add, and the gap is real money.
ExampleKey terms
Follow the rabbit hole
One idea leads to the next. Start anywhere.
- What is compound interest?A plain explanation of compound interest, why it accelerates, and why it matters for anyone saving or borrowing.
- How compound interest worksThe four moving parts — principal, rate, frequency and time — and how each one changes what you end up with.
- The compound interest formula, explainedWhat FV = P(1 + r/n)^(nt) actually says, term by term, with a worked calculation you can follow.
- The Rule of 72A mental shortcut for how long money takes to double — how it works, how accurate it is, and where it breaks down.
- Compound interest and inflation: real returnsWhy the growth you see is not the growth you get, and how to think about returns after inflation.
- Why compound interest makes pensions workHow decades of compounding do most of the work in a pension, and why starting earlier beats contributing more.
Questions people ask
Did Einstein really call compound interest the eighth wonder of the world?
There is no credible evidence Einstein said it. Researchers at Quote Investigator traced the "eighth wonder of the world" attribution and found it appearing decades after his death, with no source in his own writings. Treat it as apocryphal: the maths is impressive enough without the celebrity endorsement.
What is a realistic interest rate to assume?
It depends entirely on what the money is in: deposit accounts, bonds and shares behave differently, and Around doesn't predict markets. Our worked examples use a 3% to 7% range purely to show how the maths responds to different rates.
How do I work out a percentage of a number?
Divide by 100, then multiply by the rate: 20% of 150 is 150 divided by 100 (1.5) and scaled up by the rate to give 30. For mental arithmetic, build from 10% (move the decimal point one place) and 1% (two places).
How do I add a percentage to a number?
Multiply by one plus the rate: adding 15% to 200 gives 230. Removing a percentage later is not the mirror image — undoing an increase means dividing, not subtracting.
Can something increase by more than 100%?
An increase has no upper limit — something that triples has risen 200%. A decrease is capped, because losing everything means reaching zero and there is nothing left to lose.
How do I find the original price before a percentage was added?
Divide by one plus the rate that was added: a total of 123 after a 23% increase began as 123 divided by 1.23, which is 100. Subtracting 23% instead gives 94.71, which is wrong because the 23% was charged on the smaller original, not the final total.
What is a percentage point?
A percentage point is the unit for the gap between two percentages: a rate moving from 4% to 6% has risen two percentage points. The same move is also a 50 percent increase relative to where it started — two true statements that sound alike and mean different things.
How do I turn a percentage into a fraction or decimal?
Put it over 100 and simplify: 25% is 25/100, which is one quarter. For a decimal, divide by 100 instead: 25% is 0.25. The three forms are the same number written three ways.
What are basis points?
A basis point is one hundredth of a percentage point: 100 basis points make one percentage point, so a rate moving 25 basis points has moved 0.25 percentage points. Finance uses them because rate moves are small and 'percent' invites confusion with relative change.
Do two discounts add together?
No — they multiply. 20% off followed by 10% off leaves 72% of the original price, an effective discount of 28%, not 30%, because the second discount applies to the already-reduced price.