Lesson 2: Solving Percent Problems

Property

A percent proportion is an equation where a percent is equal to an equivalent ratio. The amount is to the base as the percent is to 100.
$\frac{\text{amount}}{\text{base}} = \frac{\text{percent}}{100}$
We can restate this as: The amount out of the base is the same as the percent out of one hundred.

Examples

• To solve "What number is 45% of 80?", set up the proportion $\frac{n}{80} = \frac{45}{100}$. Cross-multiply to get $100n = 3600$, so $n=36$.
• To solve "6.5% of what number is 1.56 dollars?", set up $\frac{1.56}{n} = \frac{6.5}{100}$. Cross-multiply to get $156 = 6.5n$, so $n=24$.
• To solve "What percent of 72 is 9?", set up $\frac{9}{72} = \frac{p}{100}$. Cross-multiply to get $900 = 72p$, so $p=12.5$. The answer is 12.5%.

Explanation

This special proportion is a powerful tool for any percent problem. It turns sentences like "What is 25% of 80?" into an equation you can easily solve by finding the missing piece. Just fill in what you know!

Property

To solve percent problems, we translate English sentences into algebraic equations and then solve them. We must be sure to change the given percent to a decimal when we put it in the equation. The three basic types of percent equations are:

1. Finding the amount: What number is 35% of 90? translates to $n = 0.35 \cdot 90$
2. Finding the base: 6.5% of what number is 1.17 dollars? translates to $0.065 \cdot n = 1.17$
3. Finding the percent: 144 is what percent of 96? translates to $144 = p \cdot 96$

Examples

• What number is 25% of 160?

Translate the sentence into an equation: $n = 0.25 \cdot 160$. Solving this gives $n = 40$. So, 40 is 25% of 160.

• 4.5% of what number is 18 dollars?

Translate this as $0.045 \cdot n = 18$. To find n, divide both sides by 0.045: $n = \frac{18}{0.045} = 400$. So, 4.5% of 400 dollars is 18 dollars.