Percent Of A Percent Calculator

What this calculator does

So, you want to take a percentage of another percentage. This tool works it out. Enter two percentages and it returns the first one taken as a percentage of the second, as a single percentage figure.

Two boxes, one for each percentage, and a button. It answers questions of the form "what is 15 percent of 40 percent?"

How to use it

1. Enter the first percentage.
2. Enter the second percentage.
3. Press Calculate.

How to take a percent of a percent

The word "of" in percentages means multiply, and that is the whole method. To find a percent of a percent, you turn each into its decimal form by dividing by 100, multiply them together, and turn the result back into a percentage by multiplying by 100. So for 15 percent of 40 percent, the tool computes 0.15 times 0.40, which is 0.06, and reports it as 6 percent. The two percentages are simply two fractions of 100 multiplied together, which is why the calculation is so short.

Why the answer is smaller than both

Taking a percent of a percent always gives something smaller than either one, and it helps to see why. A percentage less than 100 is a fraction of a whole, and taking a fraction of a fraction shrinks it further. Half of a half is a quarter, not a half; 40 percent of 40 percent is 16 percent, not 40. So whenever both percentages are under 100, the result drops below both. It only grows if one of the percentages is over 100, which represents more than a whole rather than a part of one.

Where this comes up

This calculation appears whenever percentages stack on top of each other. A discount on top of a discount is the classic case: an extra 10 percent off an item already reduced by 20 percent is not 30 percent off, because the second cut applies to the already-lowered price. Commission on a percentage, a tax on a portion, or a share of a share all work the same way. Any time one percentage acts on another rather than on the original whole, this is the calculation that gives the honest combined figure.

A worked example

Enter 15 percent and 40 percent. The tool converts them to 0.15 and 0.40, multiplies to get 0.06, and reports 6 percent. So 15 percent of 40 percent is 6 percent. Try 50 percent of 50 percent, and it returns 25 percent, since half of a half is a quarter.

Questions people ask

How do I take a percent of a percent?

Turn each into a decimal by dividing by 100, multiply them, then multiply by 100. So 15 percent of 40 percent is 6 percent.

Why is the result smaller than both percentages?

Because taking a fraction of a fraction shrinks it. When both are under 100 percent, the answer falls below both.

Is a discount on a discount just the two added?

No. The second discount applies to the already-reduced amount, so 20 percent then 10 percent is not 30 percent off. This tool gives the true combined effect.

What if a percentage is over 100?

Then it represents more than a whole, and the result can be larger. The shrinking only happens when both are under 100 percent.

What about growth over several years?

That is a different calculation. The percent growth rate calculator handles change over a number of years.

References

On percentages of percentages. A percent of a percent is found by multiplying the two as fractions of 100.

1. Eric W. Weisstein, "Percent," from MathWorld, a Wolfram resource, on percentages as ratios expressed relative to 100.
2. "Percentage," Wikipedia, on the meaning of per cent and calculations combining percentages.