# Word Problems Involving Percent

Three Kinds of Percent Problems

Consider the statement " $x$ percent of $y$ is $z$ ."

If any two of the variables are given, you can use algebra to find out the missing one. This results in three different kinds of problems. In each one, the unknown is in a different position.

  problems where $x$ is the unknown (e.g. "What percent of $44$ is $11$ ?")

  problems where $y$ is the unknown (e.g. " $58$ is $25%$ of what number?")

  problems where $z$ is the unknown (e.g. "What is $88%$ of $5000$ ?")

When given a word problem involving percents, then, the first job is to find out what the unknown is.

Example 1:

A concert hall has $400$ seats, of which $325$ are occupied. Express the attendance at a percent of capacity.

The question here can be stated as, "What percent of $400$ is $325$ ?" So, the unknown here is $x$ .

Write a proportion.

$\frac{325}{400}=\frac{x}{100}$

You could cross multiply. Or, you could notice that the denominator on the left is $4$ times the denominator on the right.

So, to find $x$ , just divide the numerator on the left by $4$ .

$x=\frac{325}{4}=81.25$

So, the hall is filled to $81.25%$ capacity.

Example 2:

$71%$ of the earth's surface--that's about $362,100,000$ square kilometers--is covered by water. Use this information to find the total surface area of the earth.

The question here can be stated as, " $71%$ percent of what number is $362,100,000$ ?" So, the unknown here is $y$ .

Write a proportion.

$\frac{362,100,000}{y}=\frac{71}{100}$

Cross multiply.

$71y=36,210,000,000$

Divide both sides by $71$ .

$y=510,000,000$

So, the earth's surface area is about $510,000,000$ square kilometers.

Example 3:

The residents of a state were surveyed about their opinion of the public education system. $12%$ said "Good", $54%$ said "Bad", and the rest said "No opinion".

If $650$ people were surveyed, how many said "No opinion"?

The total of the three percentages should be $100$ . Find the percentage of people who said "No opinion."

$\begin{array}{l}12+54+c=100\\ 66+c=100\\ c=34\end{array}$

The question here can be stated as, "What is $34%$ of $650$ ?" So, the unknown here is $z$ .

Write a proportion.

$\frac{z}{650}=\frac{34}{100}$

Cross multiply.

$\begin{array}{l}100z=22100\\ z=221\end{array}$

So, $221$ people said "No opinion".