Lesson 4: The Percent Equation

Property

We will solve percent equations by using the methods we used to solve equations with fractions or decimals. As a prealgebra student, you can translate word sentences into algebraic equations, and then solve the equations. To solve a basic percent problem, we translate it into a percent equation: we find the amount by multiplying the percent by the base. We must be sure to change the given percent to a decimal when we translate the words into an equation.

Examples

• What number is 40% of 150? First, translate this into an equation: $n = 0.40 \cdot 150$. Then, multiply to find the answer: $n = 60$.
• 50 is 25% of what number? First, translate this into an equation: $50 = 0.25 \cdot b$. Then, divide both sides by 0.25 to solve for $b$: $b = 200$.
• What percent of 80 is 16? First, translate this into an equation, letting $p$ be the percent: $p \cdot 80 = 16$. Then, divide by 80 to solve for $p$: $p = 0.20$, which is 20%.

Explanation

Think of every percent problem as a simple sentence: Part is Percent of Whole. By identifying these three pieces and using a variable for the unknown, you can write a simple multiplication or division equation to find your answer.

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.