Learn on PengiThe Art of Problem Solving: Prealgebra (AMC 8)Chapter 8: Percents

Lesson 2: Word Problems

In this Grade 4 lesson from The Art of Problem Solving: Prealgebra (AMC 8), students practice applying percent calculations to real-world word problems, including computing sales tax, finding what percent one quantity is of a total, and calculating target heart rate using multi-step percent equations. A key focus is avoiding common errors, such as dividing by the wrong total when finding a percentage of a group. The lesson builds fluency in translating written scenarios into mathematical expressions involving percents.

Section 1

Translate basic percent equations

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.

Section 2

Solve percent applications

Property

To solve applications of percent, translate the application to a basic percent equation and solve it.
Solve an application.
Step 1. Identify what you are asked to find and choose a variable to represent it.
Step 2. Write a sentence that gives the information to find it.
Step 3. Translate the sentence into an equation.
Step 4. Solve the equation using good algebra techniques.
Step 5. Check the answer in the problem and make sure it makes sense.
Step 6. Write a complete sentence that answers the question.

Examples

A restaurant bill is 85 dollars. How much is a 20% tip? The tip is 20% of 85 dollars. Let $t$ be the tip. The equation is $t = 0.20 \cdot 85$ , so $t = 17$ . The tip should be 17 dollars.
A cereal provides 90 milligrams of a vitamin, which is 3% of the daily recommendation. What is the total recommended amount? 90 is 3% of the total amount $a$ . The equation is $90 = 0.03 \cdot a$ . So $a = \frac{90}{0.03} = 3000$ . The total is 3000 mg.
A shirt originally costing 40 dollars is on sale for 32 dollars. What percent of the original price is the sale price? What percent of 40 is 32? The equation is $p \cdot 40 = 32$ . So $p = \frac{32}{40} = 0.8$ . The sale price is 80% of the original.

Explanation

This step-by-step strategy turns confusing word problems into manageable tasks. By first writing a simple sentence describing the problem, you create a clear guide for setting up and solving the correct percent equation. It's a roadmap to the answer.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Translate basic percent equations

Property

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

Section 2

Solve percent applications

Property

Examples

A restaurant bill is 85 dollars. How much is a 20% tip? The tip is 20% of 85 dollars. Let $t$ be the tip. The equation is $t = 0.20 \cdot 85$ , so $t = 17$ . The tip should be 17 dollars.
A cereal provides 90 milligrams of a vitamin, which is 3% of the daily recommendation. What is the total recommended amount? 90 is 3% of the total amount $a$ . The equation is $90 = 0.03 \cdot a$ . So $a = \frac{90}{0.03} = 3000$ . The total is 3000 mg.
A shirt originally costing 40 dollars is on sale for 32 dollars. What percent of the original price is the sale price? What percent of 40 is 32? The equation is $p \cdot 40 = 32$ . So $p = \frac{32}{40} = 0.8$ . The sale price is 80% of the original.

Explanation

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Translate basic percent equations

Property

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

Section 2

Solve percent applications

Property

Examples

A restaurant bill is 85 dollars. How much is a 20% tip? The tip is 20% of 85 dollars. Let $t$ be the tip. The equation is $t = 0.20 \cdot 85$ , so $t = 17$ . The tip should be 17 dollars.
A cereal provides 90 milligrams of a vitamin, which is 3% of the daily recommendation. What is the total recommended amount? 90 is 3% of the total amount $a$ . The equation is $90 = 0.03 \cdot a$ . So $a = \frac{90}{0.03} = 3000$ . The total is 3000 mg.
A shirt originally costing 40 dollars is on sale for 32 dollars. What percent of the original price is the sale price? What percent of 40 is 32? The equation is $p \cdot 40 = 32$ . So $p = \frac{32}{40} = 0.8$ . The sale price is 80% of the original.