Percentages
Percentage of a Percentage
To find a percentage of a percentage, write both as decimals and multiply. 20% of 50% is 0.2 × 0.5 = 0.1, which is 10%. This guide explains why that works, then applies it to the places it matters most: stacked discounts, repeated increases and decreases, and the difference between a change in percent and a change in percentage points.
How to find a percentage of a percentage
Convert each percentage to a decimal by dividing by 100, multiply the decimals, then multiply the result by 100 to turn it back into a percentage. That is all there is to it.
The shortcut in the formula box skips the conversions: multiply the two percentage numbers and divide by 100 once. 20 × 50 = 1,000, and 1,000 ÷ 100 = 10, so the answer is 10%. Both routes always agree.
The answer is always smaller than both starting percentages, as long as both are under 100%. Taking part of a part leaves less than either part on its own. If your answer comes out larger, a conversion went wrong somewhere. The percent to decimal calculator and the decimal to percent calculator handle the conversions if a value has awkward decimals.
Why multiplying percentages works
The word “of” means multiply. 20% of 80 is 0.2 × 80 = 16, and that rule does not care whether the second number is 80 or 50%. The general method for any “X% of Y” question is set out in how to calculate percentage; this is the same method with a percentage in the Y slot.
It also helps to picture a real group. Start with 100 people. Half of them, 50 people, are in a club. Twenty percent of the club members hold an office, which is 10 people. Those 10 office holders are 10 out of the original 100, or 10% of everyone.
Written as fractions, the same calculation is 20/100 × 50/100 = 1,000/10,000 = 10/100. Multiplying two “per hundred” values gives “per ten thousand,” and dividing by 100 brings it back to “per hundred.” That is the divide-by-100 step in the shortcut. The rules for multiplying fractions like these are in how to multiply fractions.
Worked examples
Example: 30% of 60%
Check with the shortcut: 30 × 60 ÷ 100 = 18.
Example: 12% of 35%
Small percentages multiply to very small results. 4.2% is about one person in 24.
Example: a share of a share
A store finds that 60% of its customers used a coupon last month, and 25% of the coupon users shopped online. What share of all customers used a coupon online?
It is not 25%. The 25% describes only the coupon users, so it has to be scaled down by the 60% they make up.
Example: a percentage of a percentage of a number
A budget of $1,200 puts 40% toward marketing, and 30% of the marketing money goes to online ads. How much is that?
You can do it in two steps instead: 40% of $1,200 is $480, and 30% of $480 is $144. The percentage calculator handles either step on its own.
Stacked discounts: 20% off then 10% off
The most common place people meet a percentage of a percentage is a sale with an extra discount on top. “20% off, plus an extra 10% at checkout” sounds like 30% off. It is not.
The second discount applies to the already-reduced price, so it takes 10% of $120, not 10% of $150. Each discount leaves a fraction of the price: 20% off leaves 80%, and 10% off leaves 90%. What is left after both is 90% of 80%, a percentage of a percentage:
0.8 × 0.9 = 0.72, so you pay 72% of the original price and save 28%. The order does not matter, since multiplication can be done in any order: 10% off first and then 20% off still leaves 72%.
With three discounts the gap grows. 25%, then 20%, then 10% off a $200 item leaves 0.75 × 0.8 × 0.9 = 0.54, or $108. That is a 46% saving, not 55%. The percent off calculator stacks discounts this way, and the guide to calculating a discount in your head shows the mental shortcuts.
Repeated increases and decreases
The same multiplication explains why percentage changes do not add up. A 10% increase multiplies by 1.1, so two of them multiply by 1.1 × 1.1 = 1.21: a 21% increase, not 20%.
| Changes | Multipliers | Combined | Overall change |
|---|---|---|---|
| +10%, then +10% | 1.1 × 1.1 | 1.21 | +21% |
| +50%, then −50% | 1.5 × 0.5 | 0.75 | −25% |
| −20%, then +25% | 0.8 × 1.25 | 1 | no change |
| −10%, then −15% | 0.9 × 0.85 | 0.765 | −23.5% |
| +5% three times | 1.05 × 1.05 × 1.05 | ≈ 1.1576 | ≈ +15.76% |
The second row surprises people most. A 50% rise followed by a 50% fall does not get you back where you started, because the fall is 50% of a larger number. Going up and down by the same percentage always leaves you below the start. The third row shows the fix: to undo a 20% drop you need a 25% rise. The percentage increase formula guide covers this in detail, and the reverse percentage calculator finds the original value before any change.
This is compounding. Each change applies to the result of the one before, which is the difference between compound growth and the flat, non-compounding kind used in the simple interest calculator.
Percentage points vs percent
When the thing that changes is itself a percentage, there are two ways to describe the change, and they give different numbers.
Both statements are true. “Up 1 percentage point” describes the gap; “up 25%” describes how big the gap is relative to where it started. The percentage change calculator gives the second figure if you enter 4 and 5.
The ambiguity bites when someone says “increase 20% by 10%.” If they mean 10% of the 20%, a percentage of a percentage, the result is 20 × 1.1 = 22%. If they mean 10 percentage points, the result is 30%. When a rate of 3.5% rises by 20%, the new rate is 3.5 × 1.2 = 4.2%, not 23.5%. Saying “points” for an absolute change and “percent” for a relative one removes the doubt.
A related mix-up, comparing two values when neither one is the starting point, is covered in percent difference vs percent change.
Percentage of a percentage table
Each cell is the row percentage of the column percentage. Read across from 20% and down from 50% to get 10%.
| of → | 10% | 20% | 25% | 50% | 75% |
|---|---|---|---|---|---|
| 10% | 1% | 2% | 2.5% | 5% | 7.5% |
| 20% | 2% | 4% | 5% | 10% | 15% |
| 25% | 2.5% | 5% | 6.25% | 12.5% | 18.75% |
| 50% | 5% | 10% | 12.5% | 25% | 37.5% |
| 75% | 7.5% | 15% | 18.75% | 37.5% | 56.25% |
The table is symmetric: 20% of 50% and 50% of 20% are both 10%. Multiplication does not care about order, so neither does a percentage of a percentage. The diagonal gives squares: 50% of 50% is 25%, and 10% of 10% is just 1%.
Common mistakes
- Multiplying without dividing by 100.
20 × 50 = 1,000, and 1,000% is clearly wrong for “part of a part.” Divide by 100 once to get 10%. - Adding instead of multiplying. 20% of 50% is not 70%, and 20% off then 10% off is not 30% off.
- Subtracting the percentages. 50% minus 20% is 30 percentage points, which answers a different question.
- Dividing by 100 twice. If you convert both to decimals and multiply, you get 0.1. Multiply by 100 to read it as 10%, rather than writing 0.1%.
- Assuming equal ups and downs cancel. +50% then −50% is a 25% loss.
- Mixing up points and percent. A rate going from 4% to 5% is up 1 point and up 25%. Say which you mean.
Frequently asked questions
What is 20% of 50%?
20% of 50% is 10%. Convert both to decimals and multiply: 0.2 × 0.5 = 0.1, which is 10%. The shortcut is to multiply the two numbers and divide by 100 once: 20 × 50 ÷ 100 = 10. In a group of 100 people, half is 50, and a fifth of those 50 is 10 people.
What is 40% of 25%?
40% of 25% is 10%. As decimals, 0.4 × 0.25 = 0.1. Notice that it matches 20% of 50%; different pairs can give the same product. The order also does not matter, so 25% of 40% is 10% too, because multiplication gives the same result either way round.
How do you calculate a percentage of a percentage?
Divide each percentage by 100 to make it a decimal, multiply the decimals, then multiply by 100. For 30% of 60%, 0.3 × 0.6 = 0.18, so the answer is 18%. As a check, the answer should always be smaller than both percentages when both are below 100%.
Is 20% off and then 10% off the same as 30% off?
No, it is 28% off. The second discount applies to the reduced price. On $150, 20% off gives $120, and 10% off that gives $108, a total saving of $42, or 28%. In multiplier form, 0.8 × 0.9 = 0.72, so you pay 72% of the original price.
What is 50% of 50%?
50% of 50% is 25%, because 0.5 × 0.5 = 0.25. Half of a half is a quarter. Taking 50% a third time gives 12.5%, since 0.5 × 0.5 × 0.5 = 0.125. Each repeat halves what is left rather than taking away another fixed 50 points.
What is the difference between a percentage point and a percent?
A percentage point is the plain difference between two percentages; a percent change compares that difference with the starting value. A rate rising from 4% to 5% is up 1 percentage point, and also up 25%, since 1 ÷ 4 = 0.25. Use points for the gap and percent for the relative change.
How do you find 25% of 40% of 300?
Multiply all three: 0.25 × 0.4 × 300 = 30. You can also go one step at a time: 40% of 300 is 120, and 25% of 120 is 30. Combining the percentages first, 25% of 40% is 10%, and 10% of 300 is 30. Every route gives the same answer.
Calculators for this
Every one of these shows the working, so you can check the method as well as the answer.
More guides
https://fourfunctioncalculator.com/percentage-of-a-percentage/