Percentages of Percentages

How to apply the percentage concept to itself

Example problem:
In a class, 40% of the students have blue eyes and 20% of those students with blue eyes have brown hair. What percentage of the whole class has blue eyes and brown hair?

This problem does not say how many students are there in the class. So, we can pick 100 to simplify the first step. If there are 100 students in the class, then 40 of them have blue eyes because 40% of 100 is 40.
Now, 20% of those 40 have brown hair. Because 20% of 40 is (20/100)(40) = (2/10)(40) = (1/5)(40) = 40/5 = 8, there are 8 students in the class with blue eyes and brown hair.
So the answer to this problem is 8%.


Another way to state this result is by saying that the 20% of 40% is 8% because 20% of 40 is 8.

A very important key to solve this kind of problem is to pay close attention to the words following "percentage of." The phrase after "of" completely determines the situation.

If the problem had said instead:
40% of the class has blue eyes, 20% of the class has brown hair, and all the students with brown hair have blue eyes. What percentage of the students with blue eyes has brown hair?
This would be a completely different problem.

Here the answer would be 50% because 20% of the class is exactly one half of 40% of the same class.
We could express this by saying that the 50% of 40% is 20%.

The difference between the two problems above is that 20% of the class is some number, but 20% of the students with blue eyes is another number. These numbers are different unless we knew all the students in the class had blue eyes. Usually though, the numbers are going to be quite different and the key to figure them out is to pay attention to the words that come after the word "of" in "percentage of."

By the way, when translating these phrases into equations, the word "of" is translated as multiplication, the word "is" is translated as the equal sign, and both the word "percentage" as well as the sign "%" are translated as "division by 100".
For example, the phrases:

"The number B is x% of the number A"
"x% of A is B"
"What percentage of A is represented by B?"
"B is what percentage of A?"

are all equivalent and they can be translated into an equation as follows:

(x/100)(A) = B

Remember a percentage is a fraction with a denominator equal to 100.

This other post is related to percentages, as well:
Successive Changes Given as Percentages.
It explains how to calculate the final percent change after two different percent changes have been applied one after another.