Percentage of a Percentage

A percentage of a percentage is found by multiplying the two percentages in decimal form, then writing the result as a percentage again. This works because a percentage is already a fraction of a whole, so taking a percentage of it means taking a fraction of a fraction. You do not add the percentages together; you multiply them because the second percentage is being applied to only part of the original whole.

How the method works

Start by treating each percentage as a share of a whole. Convert both percentages into decimal form, multiply those decimals, and then express the product as a percentage.

In symbols, if the first percentage is written as a% and the second as b%, the calculation is:

decimal form of a% × decimal form of b%

The answer is the share of the original whole after both percentage ideas have been applied. If you want the final answer as a percentage, convert that product back into percentage form.

The key point is that “of” means multiplication in this setting. A percentage of a price, a percentage of a group, a percentage of a tax base, and a percentage of a probability all use the same idea.

Why you multiply, not add

Adding percentages would describe two separate shares of the same whole. A percentage of a percentage is different. It narrows the base before the second percentage is applied.

Think of it as layers. The first percentage selects part of the whole. The second percentage selects part of that selected part. Because each layer reduces the base, multiplication gives the correct result.

This is the same reason successive discounts do not combine by simple addition. If a discount is applied after an earlier discount, the later discount applies to the reduced price, not to the starting price. The calculation is chained, so multiplication is the right operation.

Where this shows up

You see percentage-of-a-percentage calculations anywhere a share is taken from another share.

In probability, it appears when two conditions both need to happen and the second condition applies inside the first. The combined probability comes from multiplying the relevant probabilities, not from adding them.

In tax calculations, it can appear when a rate applies only to a portion of something already defined by another percentage. The exact rules depend on the tax system, but the arithmetic idea is the same: apply the second percentage to the first percentage’s base.

In successive discounts, it explains why the final reduction is not usually the simple sum of the displayed discounts. Each new discount acts on what remains after the earlier one.

A quick way to check your answer

Ask what the second percentage is being taken from. If it is being taken from the original whole, you may be comparing separate percentages. If it is being taken from another percentage, multiply.

The final percentage should describe the share of the original whole after both percentage steps have been applied. If your method treats both percentages as if they applied to the full original amount, it is probably addition in disguise, not a percentage of a percentage.