How to calculate percentages, percentage change and reverse percentages the right way

Almost every percentage question you meet in daily life is one of three: what is X% of Y, what percent is X of Y, and by what percent did something change from A to B. Each has a one-line formula, and the same handful of traps catch people every time: mixing up the base, adding percentages that should be multiplied, and confusing percentage points with percent. This guide walks through the formulas, then the traps, with numbers you can check by hand.

The three everyday questions

What is X% of Y? Convert the percentage to a decimal and multiply. 15% of 200 is 0.15 × 200 = 30. The word “of” always means multiply, and the percentage is the multiplier.

X is what percent of Y? Divide the part by the whole and multiply by 100. 50 is what percent of 400? 50 ÷ 400 = 0.125, so 12.5%. The whole is the denominator, and the order matters: 400 is 800% of 50, not 12.5%.

What is the percent change from A to B? Subtract the old value from the new value, divide by the old value, multiply by 100. Going from 80 to 100 is (100 − 80) ÷ 80 × 100 = +25%. Going back from 100 to 80 is (80 − 100) ÷ 100 × 100 = −20%. The old value is always the base, which is why the same two numbers give different percentages in each direction.

The percentage calculator has one row for each of these questions and updates as you type, so it is a quick way to check your hand arithmetic while you are learning the patterns.

Percentage points are not percent

When a rate moves, you have two ways to describe the change and they can differ enormously. If a mortgage rate rises from 10% to 12%, it has risen by 2 percentage points, because 12 − 10 = 2. It has also risen by 20%, because 2 ÷ 10 × 100 = 20. Both statements are true; they answer different questions.

Percentage points are the plain difference between two percentages. Percent describes the change relative to the starting rate. A headline that says a tax “went up 20%” when the rate moved from 10% to 12% is technically accurate but makes the change sound ten times larger than the “2 points” phrasing. Whenever you read about rates, ask which one is meant. When you write about them, say “points” if you mean the difference.

Why a 50% drop needs a 100% rise

Percentages compound on whichever number they are applied to, so a fall and a rise of the same percentage never cancel out. If a share price drops 50% from 100 to 50, getting back to 100 means gaining 50 on a base of 50, which is a 100% rise. The smaller the base becomes, the bigger the percentage needed to climb back.

The formula for the required recovery is 1 ÷ (1 − drop) − 1, expressed as a percentage.

Drop	Value left from 100	Rise needed to recover
10%	90	11.1%
20%	80	25%
25%	75	33.3%
33.3%	66.7	50%
50%	50	100%
75%	25	300%
90%	10	900%

The same asymmetry works the other way. Something that rises 20% and then falls 20% does not return to where it started: 100 becomes 120, and 20% off 120 is 24, leaving 96. You end up 4% down. If you ever need to undo a percentage change, divide by the multiplier rather than applying the opposite percentage.

Stacked discounts are multiplied, not added

A sign that reads “20% off, plus an extra 10% off at the till” is not a 30% discount. The second discount is taken from the already reduced price, so the two multipliers are chained.

Here is the full working for an item priced at 80.00.

1. First discount: 20% of 80.00 is 16.00, leaving 80.00 × 0.80 = 64.00.
2. Second discount: 10% of 64.00 is 6.40, leaving 64.00 × 0.90 = 57.60.
3. Total saving: 80.00 − 57.60 = 22.40.
4. Combined discount: 22.40 ÷ 80.00 × 100 = 28%.

If the discounts were additive, 30% off 80.00 would be 56.00, which is 1.60 less than you actually pay. The shortcut is to multiply the remaining fractions: 0.80 × 0.90 = 0.72, so you pay 72% and save 28%. The order of the discounts makes no difference to the final price, since multiplication is commutative, but which one a store applies first can change how a fixed-amount coupon interacts with a percentage. The discount calculator handles a second stacked discount and sales tax, and it reports the true combined percentage off so you are never misled by the sign.

Reverse percentages: finding the original figure

A reverse percentage question gives you the final number and the rate and asks for the starting number. The classic example is a price after tax or after an increase. Suppose a subscription now costs 138 after a 15% increase. The temptation is to take 15% off 138, which gives 138 × 0.85 = 117.30. That is wrong, because the 15% was charged on the smaller original price, not on 138.

The correct method is to divide by the multiplier. An increase of 15% means the final price is 115% of the original, so the original is 138 ÷ 1.15 = 120. Check it: 120 × 1.15 = 138. For a decrease, divide by (1 − rate): a sale price of 60 after 25% off came from 60 ÷ 0.75 = 80. The “is what percent of what number” row on the percentage calculator does this reversal, and the GST and VAT calculator does the same thing for tax-inclusive prices where you need the pre-tax amount for an expense claim or an invoice.

Common mistakes

• Using the new value as the base for a change. Percent change always divides by the old value. From 80 to 100 is 25%, not 20%.
• Adding stacked percentages. 20% then 10% is 28% off, not 30%. Multiply the remaining fractions instead.
• Subtracting a percentage to reverse an increase. Divide by 1 plus the rate. A 15% increase is undone by dividing by 1.15, not by taking 15% off.
• Reporting points as percent, or the reverse. A move from 10% to 12% is 2 points and 20%. Say which you mean.
• Dividing by zero. “What percent of 0” and “change from 0” have no answer, because there is no base. If your starting figure is zero, describe the change in absolute terms.
• Rounding too early. Keep full precision through the steps and round only the final answer, or the pennies will drift on longer chains of discounts and tax.