Lesson 2: Solve General Applications of Percent

New Concept

This lesson shows you how to translate real-world percent problems, such as tips or price changes, into algebraic equations. You'll learn to solve for the unknown amount, the original base, or the percent itself.

What’s next

Next, you'll use interactive examples and practice cards to master solving for tips, discounts, and percent changes in real-world scenarios.

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 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.

Property

To find the percent increase, first we find the amount of increase, which is the difference between the new amount and the original amount. Then we find what percent the amount of increase is of the original amount.
Find Percent Increase.
Step 1. Find the amount of increase.
increase = new amount − original amount
Step 2. Find the percent increase as a percent of the original amount.

Examples

• A company's workforce grew from 200 to 250 employees. The increase is $250 - 200 = 50$. The percent increase is the increase (50) divided by the original (200), so $\frac{50}{200} = 0.25$, or a 25% increase.
• The price of a concert ticket rose from 60 dollars to 75 dollars. The increase is $75 - 60 = 15$ dollars. The percent increase is $\frac{15}{60} = 0.25$, or 25%.
• In a decade, a town's population grew from 10,000 to 11,500. The increase is $11,500 - 10,000 = 1,500$. The percent increase is $\frac{1,500}{10,000} = 0.15$, a 15% increase.

Explanation

Percent increase shows how much a value has grown relative to its starting point. First, find the simple difference (the 'increase'). Then, divide that increase by the original amount to see how big the change is proportionally.

Property

Finding the percent decrease is very similar to finding the percent increase, but now the amount of decrease is the difference between the original amount and the final amount. Then we find what percent the amount of decrease is of the original amount.
Find percent decrease.
Step 1. Find the amount of decrease.
decrease = original amount − new amount
Step 2. Find the percent decrease as a percent of the original amount.

Examples

• The price of a laptop dropped from 800 dollars to 600 dollars. The decrease is $800 - 600 = 200$ dollars. The percent decrease is $\frac{200}{800} = 0.25$, or 25%.
• A runner's time to complete a mile improved from 8 minutes to 7 minutes. The decrease is $8 - 7 = 1$ minute. The percent decrease is $\frac{1}{8} = 0.125$, or 12.5%.
• A store reduced its inventory from 500 items to 450 items. The decrease is $500 - 450 = 50$ items. The percent decrease is $\frac{50}{500} = 0.10$, or 10%.

Explanation

Percent decrease measures how much a value has shrunk compared to its original size. Calculate the amount of the drop, then divide that drop by the original value to find the percentage of value that was lost.