What Is the Percentage Formula? The 3 You Actually Need

Quick answer: Three formulas cover every percentage question. Percentage of an amount = (percent / 100) x amount. Percentage change = (change / original) x 100. Reverse percentage = final value / multiplier. Learn the multiplier method and all three become the same move: multiply going forwards, divide going backwards.

I dropped four marks on a mock last year on percentages, and not because I couldn't do them. I could find 15 percent of anything in my head. What I couldn't do was notice that the question had handed me the answer and was asking for the start. That is percentages: everyone assumes they finished the topic in Year 8, then quietly leaks marks on it through IGCSE, A-Level, CBSE and WASSCE papers. So here are the three formulas properly, with the one idea that ties them together.

Formula 1: percentage of an amount

Divide the percent by 100, then multiply by the amount. So 18 percent of 250 = (18 / 100) x 250 = 0.18 x 250 = 45.

On a non-calculator paper, build it from chunks instead. Ten percent is the number divided by 10. One percent is the number divided by 100. Five percent is half of ten percent. For 18 percent of 250, the fastest route is 20 percent minus 2 percent: 20 percent = 50, 2 percent = 5, so the answer is 45. Same result, no calculator, about six seconds.

One shortcut worth knowing: "of" always means multiply, so 15 percent of 80 and 80 percent of 15 give the same number (12). Flip the question whenever the other version is easier.

Formula 2: percentage change (increase and decrease)

Percentage change = (new value - original value) / original value, then x 100. The number you divide by is always the one you started with. That single sentence is worth more marks than any other line in this article.

Worked example: a test score goes from 40 to 46. Change = 6. 6 / 40 = 0.15. Multiply by 100 = 15 percent increase.

Decrease example: a mass falls from 80 g to 62 g. Change = 18. 18 / 80 = 0.225 = 22.5 percent decrease.

Mark schemes for GCSE and Cambridge papers usually split this into two marks: M1 for a correct 6/40 (method), A1 for 15 (answer). That means if you write 6/40 and then fumble the arithmetic, you still bank the method mark. Write the fraction down before you touch the calculator. I once did it all in my head, wrote one wrong number, and earned zero.

Formula 3: reverse percentages, where most marks disappear

When the question gives you the value after the change, divide by the multiplier instead of multiplying. Original = final / multiplier.

Worked example: a laptop costs 540 pounds after a 10 percent discount. What was the original price? A 10 percent discount means you pay 90 percent, so the multiplier is 0.9. Original = 540 / 0.9 = 600.

The tempting wrong method is to find 10 percent of 540 (which is 54) and add it on to get 594. Check it: 594 reduced by 10 percent is 594 x 0.9 = 534.60, not 540. So 594 is wrong. Ten percent of the original and ten percent of the reduced price are different amounts, and that is the whole trap.

To spot one in the wording, look for "after", "including", "the sale price", or "this was a 20 percent increase on last year". If the number you were given is the end of the story, divide.

The multiplier method: one idea instead of three

Turn every percentage change into a single decimal you multiply by. An increase of p percent has multiplier 1 + p/100. A decrease of p percent has multiplier 1 - p/100.

• Increase 15 percent, multiply by 1.15
• Decrease 15 percent, multiply by 0.85
• Increase 7.5 percent, multiply by 1.075
• Decrease 2.5 percent, multiply by 0.975
• Decrease 0.5 percent, multiply by 0.995 (not 0.95 — write the subtraction out)

Once you think in multipliers, repeated change is easy. Three years of 5 percent growth is 1.05^3 = 1.157625, a 15.76 percent total increase, not 15 percent. Depreciation of 12 percent a year for 4 years is 0.88^4 = 0.5997, so the item keeps about 60 percent of its value. GCSE, AP, Cambridge and CBSE all reuse this idea in compound interest and exponential growth, so it is not a one-topic trick.

And to reverse any of it, you divide by the same multiplier. That is the entire method: forwards multiply, backwards divide.

A full exam-style worked example

In 2024 a school had 640 students. This was a 28 percent increase on its 2019 number. Between 2024 and 2026 the number fell by 12.5 percent. Find the overall percentage change from 2019 to 2026.

1. Find 2019. The 640 is the value after the increase, so this part is reverse. The multiplier for a 28 percent increase is 1.28, so the 2019 number = 640 / 1.28 = 500.
2. Find 2026. This part runs forwards. The multiplier for a 12.5 percent decrease is 0.875, so the 2026 number = 640 x 0.875 = 560.
3. Overall change. The original is the 2019 figure, so change = 560 - 500 = 60, and 60 / 500 = 0.12 = 12 percent increase.

Notice you cannot get 12 percent by combining 28 and 12.5 in your head, because they are percentages of different totals. Working straight through the multipliers also gets there: 1.28 x 0.875 = 1.12, a 12 percent increase in one line. If a question like this comes out ugly, put your working through the math solver and compare it line by line with yours rather than only checking the final number.

Mistakes that cost the most marks

• Dividing by the new value instead of the original in percentage change.
• Treating a reverse percentage as a normal one (the 594 mistake above).
• Assuming 20 percent up then 20 percent down returns you to the start. It doesn't: 1.2 x 0.8 = 0.96, a 4 percent loss overall.
• Confusing percentage points with percent. A rise from 30 percent to 36 percent is 6 percentage points, but a 20 percent increase.
• Rounding halfway through. Keep full accuracy in the calculator and round only the final answer, to whatever the question asks for.
• Adding percentages taken from different totals, which is only valid when the totals are the same.

Test yourself

1. A jacket costs 68 pounds after a 15 percent discount. What was the original price?
2. A population grows from 2,400 to 2,880. What is the percentage increase?
3. A machine worth 9,000 depreciates by 20 percent per year. What is it worth after 3 years?

Answers: 80 pounds (68 / 0.85); 20 percent (480 / 2400); 4,608 (9000 x 0.8^3). If you got the first one wrong, that is exactly the reverse-percentage gap, and it is worth drilling with a short mixed set on quiz until spotting it becomes automatic.

Work through profit and loss word problems solved in exam layout.

FAQ

Why doesn't 20 percent up then 20 percent down get me back to the start?

Because the two 20 percents are taken from different numbers. The increase is 20 percent of the original; the decrease is 20 percent of the bigger new value, so it removes more than was added. In multipliers: 1.2 x 0.8 = 0.96, a 4 percent overall fall.

What multiplier do I use for a 7.5 percent decrease?

0.925. Take 7.5 / 100 = 0.075 and subtract it from 1. The common slip is writing 0.93 or 0.95 because of the decimal, so write the subtraction out rather than guessing it.

Do I always divide by the original value for percentage change?

Yes. For percentage change and percentage error, the denominator is the starting or true value. The only time you divide by something else is percentage of a total, like percentage yield or a share of a whole, where the denominator is the total.

How do I answer these on a non-calculator paper?

Build from 10 percent, 5 percent and 1 percent, and use fraction equivalents: 25 percent = 1/4, 12.5 percent = 1/8, 20 percent = 1/5. For reverse questions, set it up as a fraction and simplify instead of dividing by a decimal. If a method still doesn't click, ask for it in different words on explain, or work back through the surrounding number topics on the maths hub.

In short: percentage of an amount, percentage change and reverse percentages are three faces of one skill. Convert the change into a multiplier, multiply when you're going forwards, divide when the question has handed you the end of the story, and always divide by the original when you're asked how much something changed. Write the fraction down before you calculate, and the method marks stay yours even on a bad day.