Percentages, fractions and ratios are three ways of describing the same idea — a part of a whole. This guide walks through each with worked examples, including the two calculations people get wrong most often: percentage change and reverse percentages.
"Per cent" means "per hundred", so a percentage is just a fraction with 100 on the bottom. Almost every percentage problem is one of three questions.
What is X% of a number? Convert the percentage to a decimal and multiply. 15% of 80 is 0.15 × 80 = 12.
What percentage is one number of another? Divide the part by the whole, then multiply by 100. 25 out of 200 is 25 ÷ 200 × 100 = 12.5%.
What is the whole, given a part and its percentage? Divide the part by the decimal. If 20 is 40% of something, the whole is 20 ÷ 0.4 = 50.
Our percentage calculator handles all three, and a few more besides.
Percentage change trips people up because the answer depends on which number you start from.
Going from 40 to 50 is (50 − 40) ÷ 40 × 100 = +25%. But going back from 50 to 40 is (40 − 50) ÷ 50 × 100 = −20%. Same two numbers, different percentages, because the denominator — the starting point — changed.
This is the one that catches people out at the checkout and on tax. If a price already includes a discount or a tax, you cannot simply take the same percentage off to undo it.
Say an item costs $120 after a 20% discount, and you want the original price. The $120 represents 80% of the original (100% − 20%), so the original is 120 ÷ 0.8 = $150. Taking 20% off $120 would wrongly give $96.
The same logic removes GST. A price including 10% GST is 110% of the pre-tax price, so you divide by 1.1 — not multiply by 0.9.
A fraction is a part of a whole: the top (numerator) counts the parts, the bottom (denominator) says how many make a whole.
Adding requires a common denominator. To add 1/2 and 1/3, rewrite both over 6: 3/6 + 2/6 = 5/6. You cannot add the tops until the bottoms match.
Multiplying is easier — multiply tops and bottoms straight across. 2/3 × 3/4 = 6/12, which simplifies to 1/2.
Dividing means multiplying by the reciprocal — flip the second fraction and multiply. 1/2 ÷ 1/4 = 1/2 × 4/1 = 4/2 = 2. That is why "how many quarters fit in a half" is 2. Our fraction calculator shows each step.
A ratio compares quantities. The ratio 3:2 means three of one thing for every two of another — five parts in total.
To share by a ratio, add the parts and divide. Splitting $15 in the ratio 3:2 gives 5 parts, each worth 15 ÷ 5 = $3, so the shares are $9 and $6.
To simplify a ratio, divide both sides by their greatest common factor. 12:8 divides by 4 to become 3:2 — the same relationship in smaller numbers.
Ratios and fractions are close cousins: the ratio 3:2 is the same relationship as the fraction 3/5 of the total being the first share. Our ratio calculator simplifies, scales, and solves for a missing term.
This distinction appears constantly in news and finance, and confusing the two produces genuinely misleading statements.
Suppose an interest rate rises from 5% to 7%. That is a rise of 2 percentage points — but it is a 40% increase in the rate itself (2 ÷ 5 × 100). Both statements are true, and they describe the same change in very different-sounding ways.
When you see a percentage quoted about another percentage, pause and ask whether it means points or a proportional change. The gap between the two is often large enough to change the story entirely.
How do I calculate a percentage of a number?
Convert the percentage to a decimal by dividing by 100, then multiply. For example, 15 per cent of 80 is 0.15 × 80 = 12. To find what percentage one number is of another, divide the part by the whole and multiply by 100.
Why doesn't a percentage increase and decrease cancel out?
Because each percentage is taken from a different starting number. A 25 per cent rise from 100 gives 125, but a 25 per cent fall is then taken from 125, not 100, landing you at 93.75. The changing base is why they don't cancel.
How do I work out the original price before a discount?
Divide, don't subtract. If $120 is the price after a 20 per cent discount, it represents 80 per cent of the original, so the original is 120 ÷ 0.8 = $150. The same method removes GST: divide the tax-inclusive price by 1.1.
How do I add two fractions?
Give them a common denominator first, then add the numerators. To add 1/2 and 1/3, rewrite as 3/6 and 2/6, which add to 5/6. You can only add the tops once the bottoms match.
How do I divide by a fraction?
Multiply by its reciprocal — flip the second fraction upside down and multiply. So 1/2 ÷ 1/4 becomes 1/2 × 4/1 = 2. This is why dividing by a fraction less than one gives a larger answer.
How do I share an amount in a given ratio?
Add the parts of the ratio to get the total number of shares, divide the amount by that total, then multiply by each part. Sharing $15 in the ratio 3:2 gives 5 shares of $3, so $9 and $6.