lesson 4 problem solving practice the percent equation

Chapter 5, Lesson 4: The Percent Equation

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5.2.1: Solving Percent Problems

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• The NROC Project

Learning Objectives

• Identify the amount, the base, and the percent in a percent problem.
• Find the unknown in a percent problem.

Introduction

Percents are a ratio of a number and 100, so they are easier to compare than fractions, as they always have the same denominator, 100. A store may have a 10% off sale. The amount saved is always the same portion or fraction of the price, but a higher price means more money is taken off. Interest rates on a saving account work in the same way. The more money you put in your account, the more money you get in interest. It’s helpful to understand how these percents are calculated.

Parts of a Percent Problem

Jeff has a coupon at the Guitar Store for 15% off any purchase of $100 or more. He wants to buy a used guitar that has a price tag of $220 on it. Jeff wonders how much money the coupon will take off the original $220 price.

Problems involving percents have any three quantities to work with: the percent , the amount , and the base .

• The percent has the percent symbol (%) or the word “percent.” In the problem above, 15% is the percent off the purchase price.
• The base is the whole amount. In the problem above, the whole price of the guitar is $220, which is the base.
• The amount is the number that relates to the percent. It is always part of the whole. In the problem above, the amount is unknown. Since the percent is the percent off , the amount will be the amount off of the price.

You will return to this problem a bit later. The following examples show how to identify the three parts: the percent, the base, and the amount.

Identify the percent, amount, and base in this problem.

30 is 20% of what number?

Percent: The percent is the number with the % symbol: 20%.

Base : The base is the whole amount, which in this case is unknown.

Amount: The amount based on the percent is 30.

Percent=20%

Base=unknown

The previous problem states that 30 is a portion of another number. That means 30 is the amount. Note that this problem could be rewritten: 20% of what number is 30?

Identify the percent, base, and amount in this problem:

What percent of 30 is 3?

The percent is unknown, because the problem states " What percent?" The base is the whole in the situation, so the base is 30. The amount is the portion of the whole, which is 3 in this case.

Solving with Equations

Percent problems can be solved by writing equations. An equation uses an equal sign (=) to show that two mathematical expressions have the same value.

Percents are fractions, and just like fractions, when finding a percent (or fraction, or portion) of another amount, you multiply.

The percent of the base is the amount.

Percent of the Base is the Amount.

\[\ \text { Percent } {\color{red}\cdot}\text { Base }{\color{blue}=}\text { Amount } \nonumber \]

In the examples below, the unknown is represented by the letter \(\ n\). The unknown can be represented by any letter or a box \(\ \square\) or even a question mark.

Write an equation that represents the following problem.

\(\ 20 \% \cdot n=30\)

Once you have an equation, you can solve it and find the unknown value. To do this, think about the relationship between multiplication and division. Look at the pairs of multiplication and division facts below, and look for a pattern in each row.

Multiplication and division are inverse operations. What one does to a number, the other “undoes.”

When you have an equation such as \(\ 20 \% \cdot n=30\), you can divide 30 by 20% to find the unknown: \(\ n=30 \div 20 \%\).

You can solve this by writing the percent as a decimal or fraction and then dividing.

\(\ n=30 \div 20 \%=30 \div 0.20=150\)

What percent of 72 is 9?

\(\ 12.5 \% \text { of } 72 \text { is } 9\).

You can estimate to see if the answer is reasonable. Use 10% and 20%, numbers close to 12.5%, to see if they get you close to the answer.

\(\ 10 \% \text { of } 72=0.1 \cdot 72=7.2\)

\(\ 20 \% \text { of } 72=0.2 \cdot 72=14.4\)

Notice that 9 is between 7.2 and 14.4, so 12.5% is reasonable since it is between 10% and 20%.

What is 110% of 24?

\(\ 26.4 \text { is } 110 \% \text { of } 24\).

This problem is a little easier to estimate. 100% of 24 is 24. And 110% is a little bit more than 24. So, 26.4 is a reasonable answer.

18 is what percent of 48?

• \(\ 0.375 \%\)
• \(\ 8.64 \%\)
• \(\ 37.5 \%\)
• \(\ 864 \%\)

Incorrect. You may have calculated properly, but you forgot to move the decimal point when you rewrote your answer as a percent. The equation for this problem is \(\ n \cdot 48=18\). The corresponding division is \(\ 18 \div 48\), so \(\ n=0.375\). Rewriting this decimal as a percent gives the correct answer, \(\ 37.5 \%\).

Incorrect. You may have used \(\ 18\) or \(\ 48\) as the percent, rather than the amount or base. The equation for this problem is \(\ n \cdot 48=18\). The corresponding division is \(\ 18 \div 48\), so \(\ n=0.375\). Rewriting this decimal as a percent gives the correct answer, \(\ 37.5 \%\).

Correct. The equation for this problem is \(\ n \cdot 48=18\). The corresponding division is \(\ 18 \div 48\), so \(\ n=0.375\). Rewriting this decimal as a percent gives \(\ 37.5 \%\).

Incorrect. You probably used 18 or 48 as the percent, rather than the amount or base, and also forgot to rewrite the percent as a decimal before multiplying. The equation for this problem is \(\ n \cdot 48=18\). The corresponding division is \(\ 18 \div 48\), so \(\ n=0.375\). Rewriting this decimal as a percent gives the correct answer, \(\ 37.5 \%\).

Using Proportions to Solve Percent Problems

Percent problems can also be solved by writing a proportion. A proportion is an equation that sets two ratios or fractions equal to each other. With percent problems, one of the ratios is the percent, written as \(\ \frac{n}{100}\). The other ratio is the amount to the base.

\(\ \text { Percent }=\frac{\text { amount }}{\text { base }}\)

Write a proportion to find the answer to the following question.

30 is 20% of 150.

18 is 125% of what number?

• \(\ 0.144\)
• \(\ 694 \frac{4}{9}\) (or about \(\ 694.4\))

Incorrect. You probably didn’t write a proportion and just divided 18 by 125. Or, you incorrectly set up one fraction as \(\ \frac{18}{125}\) and set this equal to the base, \(\ n\). The percent in this case is 125%, so one fraction in the proportion should be \(\ \frac{125}{100}\). The base is unknown and the amount is 18, so the other fraction is \(\ \frac{18}{n}\). Solving the proportion \(\ \frac{125}{100}=\frac{18}{n}\) gives \(\ n=14.4\).

Correct. The percent in this case is 125%, so one fraction in the proportion should be \(\ \frac{125}{100}\). The base is unknown and the amount is 18, so the other fraction is \(\ \frac{18}{n}\). Solving the proportion \(\ \frac{125}{100}=\frac{18}{n}\) gives \(\ n=14.4\).

Incorrect. You probably put the amount (18) over 100 in the proportion, rather than the percent (125). Perhaps you thought 18 was the percent and 125 was the base. The correct percent fraction for the proportion is \(\ \frac{125}{100}\). The base is unknown and the amount is 18, so the other fraction is \(\ \frac{18}{n}\). Solving the proportion \(\ \frac{125}{100}=\frac{18}{n}\) gives \(\ n=14.4\).

Incorrect. You probably confused the amount (18) with the percent (125) when you set up the proportion. The correct percent fraction for the proportion is \(\ \frac{125}{100}\). The base is unknown and the amount is 18, so the other fraction is \(\ \frac{18}{n}\). Solving the proportion \(\ \frac{125}{100}=\frac{18}{n}\) gives \(\ n=14.4\).

Let’s go back to the problem that was posed at the beginning. You can now solve this problem as shown in the following example.

Jeff has a coupon at the Guitar Store for 15% off any purchase of $100 or more. He wants to buy a used guitar that has a price tag of $220 on it. Jeff wonders how much money the coupon will take off of the $220 original price .

The coupon will take $33 off the original price.

You can estimate to see if the answer is reasonable. Since 15% is half way between 10% and 20%, find these numbers.

\(\ \begin{array}{l} 10 \% \text { of } 220=0.1 \cdot 220=22 \\ 20 \% \text { of } 220=0.2 \cdot 220=44 \end{array}\)

The answer, 33, is between 22 and 44. So $33 seems reasonable.

There are many other situations that involve percents. Below are just a few.

Evelyn bought some books at the local bookstore. Her total bill was $31.50, which included 5% tax. How much did the books cost before tax?

The books cost $30 before tax.

Susana worked 20 hours at her job last week. This week, she worked 35 hours. In terms of a percent, how much more did she work this week than last week?

Since 35 is 175% of 20, Susana worked 75% more this week than she did last week. (You can think of this as, “Susana worked 100% of the hours she worked last week, as well as 75% more.”)

Percent problems have three parts: the percent, the base (or whole), and the amount. Any of those parts may be the unknown value to be found. To solve percent problems, you can use the equation, \(\ \text { Percent } \cdot \text { Base }=\text { Amount }\), and solve for the unknown numbers. Or, you can set up the proportion, \(\ \text { Percent }=\frac{\text { amount }}{\text { base }}\), where the percent is a ratio of a number to 100. You can then use cross multiplication to solve the proportion.

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CHAPTER 4 Ratio, Proportion, and Percent

4.4 Solve General Applications of Percent

Learning Objectives

By the end of this section, you will be able to:

• Translate and solve basic percent equations
• Solve applications of percent
• Find percent increase and percent decrease

Translate and Solve Basic Percent Equations

In the last section, we solved percent problems by setting them up as proportions. That is the best method available when you did not have the tools of algebra. Now, in this section we will translate word sentences into algebraic equations, and then solve the percent equations.

We’ll look at a common application of percent—tips to a server at a restaurant—to see how to set up a basic percent application.

In the next examples, we will find the amount. We must be sure to change the given percent to a decimal when we translate the words into an equation.

In the next examples, we are asked to find the base.

In the next examples, we will solve for the percent.

Solve Applications of Percent

Many applications of percent occur in our daily lives, such as tips, sales tax, discount, and interest. To solve these applications we’ll translate to a basic percent equation, just like those we solved in the previous examples in this section. Once you translate the sentence into a percent equation, you know how to solve it.

We will update the strategy we used in our earlier applications to include equations now. Notice that we will translate a sentence into an equation.

HOW TO: Solve an Application

• Identify what you are asked to find and choose a variable to represent it.
• Write a sentence that gives the information to find it.
• Translate the sentence into an equation.
• Solve the equation using good algebra techniques.
• Check the answer in the problem and make sure it makes sense.
• Write a complete sentence that answers the question.

Now that we have the strategy to refer to, and have practiced solving basic percent equations, we are ready to solve percent applications. Be sure to ask yourself if your final answer makes sense—since many of the applications we’ll solve involve everyday situations, you can rely on your own experience.

Find Percent Increase and Percent Decrease

People in the media often talk about how much an amount has increased or decreased over a certain period of time. They usually express this increase or decrease as a percent .

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.

HOW TO: Find Percent Increase

Step 1. Find the amount of increase.

Step 2. Find the percent increase as a percent of the original amount.

TRY IT 10.1

TRY IT 10.2

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.

HOW TO: Find Percent Decrease

• Find the percent decrease as a percent of the original amount.

TRY IT 11.1

TRY IT 11.2

Access Additional Online Resources

• Percent Increase and Percent Decrease Visualization

Key Concepts

• Find the percent increase as a percent of the original amount.

Practice Makes Perfect

In the following exercises, translate and solve.

Solve Applications of Percents

In the following exercises, solve the applications of percents.

In the following exercises, find the percent increase or percent decrease.

Everyday Math

Writing exercises, attributions.

This chapter has been adapted from “Solve General Applications of Percent” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence . Adapted by Izabela Mazur. See the Copyright page for more information.

Introductory Algebra Copyright © 2021 by Izabela Mazur is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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Chapter 7, Lesson 4: Algebra: The Percent Equation

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