Percentages of percentages turn up whenever one share is taken out of another: a portion of a portion, a discount on a sale price, a tax on a fee, a growth rate on top of a growth rate. The rule is always to multiply, never to add — and this calculator makes that concrete for three common questions.
How to use the percent of a percent calculator
- Choose what you want to find:
- P% of Q% — a share of a share, optionally applied to an amount;
- R% is what percent of Q%? — comparing two percentages;
- Overall effect of successive percentage changes — several increases and decreases in a row.
- Enter the percentages (and the starting amount, if you have one). For a chain of changes, list them in order with signs, such as
+20, -10, +5. - Read the result on the tape; the steps show the decimal multipliers behind it.
Formulas
A percent of a percent is the product of the two decimals:
Comparing two percentages is a plain division — the percent signs cancel:
Successive changes multiply their growth factors:
Worked examples
A share of a share. In a town of 1,200 households, 35% have a garden and 20% of those grow tomatoes. 0.20 × 0.35 = 0.07, so 7% of households grow tomatoes — that is 0.07 × 1,200 = 84 households.
Comparing percentages. 7% of the town grows tomatoes and 35% has a garden. 7 ÷ 35 = 0.20, so tomato growers are 20% of gardeners.
Successive changes. A price rises 20%, falls 10%, then rises 5%. Factors: 1.20 × 0.90 × 1.05 = 1.134, an overall rise of 13.4% — not the 15% you get by adding. Starting from $1,200, the final price is $1,360.80.
Percentage points versus percent
When both numbers are already percentages, there are two different ways to describe the gap between them:
- Percentage points subtract: going from 35% to 42% is a rise of 7 percentage points.
- Percent change divides: the same move is a 20% increase, because 7 ÷ 35 = 0.20.
Interest rates, polling numbers and test pass rates are usually discussed in percentage points, while growth of a share relative to itself is a percent change. Mixing the two is one of the most common errors in news reports.
Why successive percentages don’t add
Each percentage applies to the amount as it stands at that moment. After a 50% increase, 100 becomes 150; a 50% decrease then removes half of 150, leaving 75. The changes are equal in size but act on different bases, so the overall result is a 25% loss. The table in the chain mode shows this base shifting row by row, along with the cumulative change after each step.
The same logic explains why “an extra 20% off an already-reduced price” is less generous than it sounds, and why an investment that loses 50% needs a 100% gain to recover.
Related calculators
For a single percentage of a number, use the percentage calculator. The percentage change calculator measures one change between two values, and the percentage difference calculator compares two values symmetrically. To combine percentages from groups of different sizes, use the average percentage calculator.
Frequently asked questions
How do I find a percent of a percent?
Convert both to decimals and multiply. 20% of 35% is 0.20 × 0.35 = 0.07, which is 7%. If 35% of a class plays sports and 20% of those play soccer, 7% of the whole class plays soccer.
Do successive percentage changes add up?
No. A 20% rise followed by a 10% fall is not a 10% rise. Multiply the growth factors: 1.20 × 0.90 = 1.08, an overall rise of 8%. Each change applies to a different base.
Are two discounts of 20% the same as 40% off?
No. Two 20% discounts multiply the price by 0.80 × 0.80 = 0.64, which is 36% off, not 40%.
How do I undo a percentage change?
Divide by the growth factor instead of reversing the percentage. Undoing a 25% increase (factor 1.25) needs a 20% decrease, because 1/1.25 = 0.80. The calculator shows the change needed to undo any chain.