“Percentage of a Percentage” Problems

How to Solve Two-Step Percentage Problems

How to Solve "Percentage of Percentage" Problems

Understanding the Problem Type

"Percentage of a percentage" problems have two steps. These problems typically follow this pattern:

A certain percentage of a group has one characteristic
Of that subgroup, another percentage has a second characteristic
You need to find what percentage of the entire original group has both characteristics

The key insight is that you multiply the two percentages together (after converting them to decimals) to find your answer.

The Strategy

Follow these three simple steps:

Step 1: Convert both percentages to decimals by dividing by 100 (or moving the decimal point two places to the left).

Step 2: Multiply the two decimals together.

Step 3: Convert your answer back to a percentage by multiplying by 100 (or moving the decimal point two places to the right).

Quick Formula:

Final percentage = (First percentage) × (Second percentage)

In decimal form: Answer = (First %) ÷ 100 × (Second %) ÷ 100 × 100

Or more simply: (First %) × (Second %) ÷ 100 = Answer

Common Mistakes to Avoid

❌ DON'T add the two percentages together. If 40% of items are red and 30% of red items have stripes, the answer is NOT 70%.
❌ DON'T subtract one percentage from the other. The answer is NOT 40% − 30% = 10%.
❌ DON'T forget that you're finding a percentage of a percentage. You must multiply, not perform other operations.

Example 1: Students and Sports

Problem: At a high school, 60% of the students participate in extracurricular activities. Of the students who participate in extracurricular activities, 45% play a sport. What percentage of all students at the high school play a sport?

Solution:

Let's use the variable x to represent the total number of students.

First percentage: 60% of students participate in extracurricular activities = 0.60x students
Second percentage: 45% of those students play a sport = 0.45 × (0.60x)

Now multiply:

0.45 × 0.60x = 0.27x

Since 0.27x represents 27% of x, and x is the total number of students, 27% of all students at the high school play a sport.

Quick calculation method:

Convert to decimals: 60% = 0.60 and 45% = 0.45
Multiply: 0.60 × 0.45 = 0.27
Convert to percentage: 0.27 = 27%

Answer: 27%

Example 2: Books in a Library

Problem: In a library collection, 75% of the books are nonfiction. Of all the nonfiction books, 40% are biographies. What percentage of all the books in the library are nonfiction biographies?

Solution:

Let x represent the total number of books in the library.

First percentage: 75% of books are nonfiction = 0.75x books
Second percentage: 40% of nonfiction books are biographies = 0.40 × (0.75x)

Now multiply:

0.40 × 0.75x = 0.30x

Since 0.30x represents 30% of x, and x is the total number of books, 30% of all the books in the library are nonfiction biographies.

Quick calculation method:

Convert to decimals: 75% = 0.75 and 40% = 0.40
Multiply: 0.75 × 0.40 = 0.30
Convert to percentage: 0.30 = 30%

Answer: 30%

Practice Tip

When you see a problem that says "Of all the [first group], [some percentage] are [second characteristic]," this is your signal that you need to multiply percentages. The word "of" in mathematics often indicates multiplication!

Why This Works

Think about it logically: if 75% of books are nonfiction, that's three-quarters of all books. If 40% of those are biographies, you're taking 40% of that three-quarters. Since 40% means "40 out of 100," you're really taking 40/100 of 75/100 of all books:

(40/100) × (75/100) = 3,000/10,000 = 30/100 = 30%

The multiplication makes sense because you're finding a fraction of a fraction, which always results in a smaller number than either of the original percentages.

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