Module 1: Solving Linear Equations with One Variable
1.8: Applications Involving Percent
SECTION 1.8 Learning Objectives
1.8: Applications Involving Percent
Use an algebraic equation to solve a basic percent question
Use an algebraic equation to solve application problems involving percent
Percent of a Whole
Percents are the ratio of a number and 100. Percents are used in many different applications. Percents are used widely to describe how something changed. For example, you may have heard that the amount of rainfall this month had decreased by 12% from last year, or that the number of jobless claims has increase by 5% this quarter over last quarter.
Unemployment rate as percent by year between 2004 and 2014.
We regularly use this kind of language to quickly describe how much something increased or decreased over time or between significant events.
Before we dissect the methods for finding percent change of a quantity, let’s learn the basics of finding percent of a whole.
For example, if we knew a gas tank held 14 gallons, and wanted to know how many gallons were in [latex]\frac{1}{4}[/latex] of a tank, we would find [latex]\frac{1}{4}[/latex] of 14 gallons by multiplying:
Notice in the example above that 10% of 200 is 20. Finding 10% of a number is one of the easiest percentages to calculate. To calculate 10 percent of a number, we always simply divide it by 10 or move the decimal point one place to the left. For example, 10% of 325 is 32.5. Being able to recall what 10% of a value is might often be helpful when reflecting on the reasonableness of your answers to problems involving different percentages as you will see in some examples in this section.
The following video contains an example that is similar to the one above.
Use an algebraic equation to solve a basic percent question
From the previous examples, we can identify some important parts to finding the percent of a whole. We are going to focus on the equation method to solve problems involving percent. To translate the following problems into equations, we need to know a few key words and what they mean.
“of” means multiply
“is” means equals
“what” can be represented by the variable
The following examples show how to use an algebraic equation to solve a basic percent question:
Example 2
What is 12% of 270?
Show Answer
In this question, we will replace:
“what” with the variable [latex]x[/latex],
“is” with the equals sign,
“of” with the multiplication symbol.
We also need to convert the percentage into a decimal number before writing it in our equation. To convert a percentage to a decimal number, we move the decimal two places to the left. Why is it two places? Percent means “per hundred.” Therefore, 12% means 12/100. Dividing 12 by 100 gives the result of 0.12. Since we will always be dividing by 100 to convert percentages to decimal numbers, we will always move the decimal two places to the left.
*Reflect on your answer and make sure it is reasonable! We can recall that 10% of 270 would be 27.0, so since 12% is just a slightly larger percentage than 10%, it makes sense that 32.5 (which is slightly bigger than 27) is 12% of the number 270.
*Reflect on your answer to see if it is reasonable! The problem stated that 30 is 20% of some number. If 30 is a percentage of a number, that means it is a portion of the whole. The number we are finding must be larger than 30. We can recall that 30 is 10% of 300. It makes sense then that is 30 equates to 20% of 150. Our answer of 150 is reasonable. 30 is 20% of 150.
The previous problem states that 30 is a portion of another number. Note that this problem could be rewritten: 20% of what number is 30?
Example 4
What percent of 72 is 9?
Show Solution
In this question, we will replace:
“what percent” with the variable [latex]x[/latex],
Divide 9 by 72 to find the value for x, the unknown.
[latex]x=0.125[/latex]
Since we are looking for a percent, we need to convert the decimal answer we got for x into a percentage. We do that by moving the decimal two places to the right. In order to convert a decimal to a percent, we multiply by 100. Multiplying by 100 always moves the decimal two places to the right.
*Reflect on your answer to see if it is reasonable! From what is given, we know that 9 is a portion of 72. Use 10% to see if it gets you close to the answer. We might recall that 7.2 is 10% of 72 so it makes sense that 9 (which is slightly larger that 7.2) would be a slightly larger percentage of 72, at 12.5%. Our answer makes sense!
Below are some additional video examples to view:
Try some of the examples on your own before revealing the answer.
Write the percent as a decimal by moving the decimal point two places to the left.
[latex]x = 1.10 \cdot 24[/latex]
[latex]x = 1.10 \cdot 24[/latex]
[latex]x = 26.4[/latex]
Answer
26.4 is 110% of 24.
* Reflect on your answer to see if it is reasonable! 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.
Example 6
What percent of 70.52 is 13.27? Round answer to the nearest tenth of a percent.
Show Answer
In this question, we will replace:
“what percent” with the variable [latex]x[/latex],
Since we are trying to find “what percent”, convert our answer to a percentage BEFORE rounding. We do that by moving the decimal two places to the right. In order to convert a decimal to a percent, we multiply by 100. Multiplying by 100 always moves the decimal two places to the right.
[latex]x=18.81735..\%[/latex]
Round to the nearest tenth of a percent
[latex]x=18.8\%[/latex]
Note: Often times, problems like the ones in this section will ask you to round your final answer. If you need a review on rounding, refer back to Module 0 of this textbook: Module 0: Rounding Decimals Review
Use an algebraic equation to solve application problems involving percent
Percents have a wide variety of applications to everyday life, showing up regularly in taxes, discounts, markups, and interest rates. We will look at several examples of how to use percent to calculate markups, discounts, and interest earned or owed.
Example 7
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.
Show Solution
First, we need to identify what it is we are being asked to find. We can summarize the problem with the question,
What is 15% of $220?
Once we identify that question, we can translate it into an equation.
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.
The answer, 33, is between 22 and 44. So $33 seems reasonable.
Example 8
Tracey paid $185 for an item that was originally priced at $390. What percent of the original price did Tracey pay? Round answer to the nearest tenth of a percent.
Show Answer
We can summarize the problem with the question,
What percent of $390 is $185?
Once we identify that question, we can translate it into an equation.
In this question, we will replace:
“what percent” with the variable [latex]x[/latex],
Since we are looking for a percent, we still need to convert this decimal answer into a percent. Remember we do that by moving the decimal two places to the right.
[latex]x=47.4358974..\%[/latex]
Round to the nearest tenth of a percent
[latex]x=47.4\%[/latex]
Tracey paid 47.4% of the original price.
*Reflect if your answer is reasonable! We can find 50% fairly easily by dividing $390 by 2. We see that $195 is 50% of $390, so it makes sense that $185 (which is a little less than $195), is about 47.4% of $390 (which is a little less than 50%).
