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# Percent Review

### Learning Objectives

• Explain the basics of percents
• Find a percent of a whole
• Identify the amount, the base, and the percent in a percent problem
• Find the unknown in a percent problem
• Solve equations containing percents
• Identify the unknown in a percent problem
• Write a percent equation
• Solve equations with percent
• Solve percent change and interest problems
• Calculate discounts and markups using percent
• Calculate interest earned or owed
• Read and interpret data from pie charts as percents

## 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 $\frac{1}{4}$ of a tank, we would find $\frac{1}{4}$ of 14 gallons by multiplying:

$\frac{1}{4}\,\cdot \,14=\frac{1}{4}\,\cdot \,\frac{14}{1}=\frac{14}{4}=3\frac{2}{4}=3\frac{1}{2}\,\,\,\text{gallons}$

Likewise, if we wanted to find 25% of 14 gallons, we could find this by multiplying, but first we would need to convert the 25% to a decimal:

$25\%\,\,\text{of}\,\,14\,\,\,\text{gallons}=0.25\,\cdot \,14=3.5\,\,\,\text{gallons}$

### Finding a Percent of a Whole

To find a percent of a whole,
• Write the percent as a decimal by moving the decimal two places to the left
• Then multiply the percent by the whole amount

### Example

What is 15% of $200? Answer: Write as a decimal. Move the decimal point two places to the left. $15\%=0.15$ Multiply the decimal form of the percent by the whole number. $\begin{array}{c}0.15\cdot200\\30\end{array}$ #### Answer 15% of$200 is $30 The following video contains an example that is similar to the one above. https://youtu.be/jTM7ZMvAzsc From the previous example, we can identify three important parts to finding the percent of a whole: • the percent has the percent symbol (%) or the word “percent” • the amount, the amount is part of the whole • and the base, the base is the whole amount The following examples show how to identify the three parts: the percent, the base, and the amount. ### Example Identify the percent, amount, and base in this problem. 30 is 20% of what number? Answer: 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. #### Answer Percent = 20% Amount = 30 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? ### Example Identify the percent, amount, and base in this problem. What percent of 30 is 3? Answer: Percent: The percent is unknown, because the problem states “what percent?”. Base: The base is the whole amount, so the base is 30. Amount: The amount is a portion of the whole, which is 3 in this case. #### Answer Percent = unknown Amount = 3 Base = 30 ### Example Identify the percent, amount, and base in this problem. What is 60% of 45? Answer: Percent: The percent is known Base: The base is the whole amount, so the base is 45. Amount: The amount is a portion of the whole, which is what we want to identify. #### Answer Percent = 60% Amount = unknown Base = 45 The following video provides more examples that describe how to identify the percent, amount, and base in a percent problem. https://youtu.be/zwT58-LJCvs In the next section, you will use the parts of a percent problem to find the percent increase or decrease of a quantity by writing and solving equations. ## Percent 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. In the previous section, we identified three important parts to finding the percent of a whole: • the percent, has the percent symbol (%) or the word “percent” • the amount, the amount is part of the whole • and the base, the base is the whole amount Using these parts, we can define equations that will help us answer percent problems. ### The Percent Equation Percent of the Base is the Amount. $\text{Percent}\cdot\text{Base}=\text{Amount}$ In the examples below, the unknown is represented by the letter n. The unknown can be represented by any letter or a box □, question mark, or even a smiley face :) ### Example Write an equation that represents the following problem. 30 is 20% of what number? Answer: Rewrite the problem in the form “percent of base is amount.” 20% of what number is 30? Identify the percent, the base, and the amount. Percent is: 20% Base is: unknown Amount is: 30 Write the percent equation. using n for the base, which is the unknown value. $\text{Percent}\cdot\text{Base}=\text{Amount}$ $20\%\cdot{n}=30$ #### Answer [latex-display]20\%\cdot{n}=30[/latex-display] The following example shows how to use the percent equation to find the base in a percent equation. https://youtu.be/3etjmUw8K3A Once you have an equation, you can solve it and find the unknown value. For example, to solve [latex-display]20%\cdot{n}=30[/latex-display] you can divide 30 by 20% to find the unknown: $20\%\cdot{n}=30$ You can solve this by writing the percent as a decimal or fraction and then dividing. $20\%\cdot{n}=30$ $n=30\div20\%=30\div0.20=150$ ### Example What percent of 72 is 9? Answer: Identify the percent, base, and amount. Percent: unknown Base: 72 Amount: 9 Write the percent equation: Percent $\cdot$ Base = Amount. Use n for the unknown (percent). $n\cdot72=9$ Divide to undo the multiplication of n times 72. $n=\frac{9}{72}$ Divide 9 by 72 to find the value for n, the unknown. $\displaystyle 72\overset{0.125}{\overline{\left){9.000}\right.}}$ Move the decimal point two places to the right to write the decimal as a percent. $\begin{array}{c}n=0.125\\n=12.5\%\end{array}$ #### Answer 12.5% of 72 is 9. In the following video example, you are shown how to use the percent equation to find the base in a percent problem. https://youtu.be/p2KHHFMhJRs 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% of 72 = 0.1 · 72 = 7.2 20% of 72 = 0.2 · 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%. ### Example What is 110% of 24? Answer: Identify the percent, the base, and the amount. Percent: 110% Base: 24 Amount: unknown Write the percent equation. $\text{Percent}\cdot\text{Base}=\text{Amount}$ The amount is unknown, so use n. 110% · 24 = n Write the percent as a decimal by moving the decimal point two places to the left. Multiply 24 by 1.10 or 1.1. 1.10 · 24 = n 1.10 · 24 = 26.4 = n #### Answer 26.4 is 110% 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. The video that follows shows how top use the percent equation to find the amount in a percent equation. https://youtu.be/dO3AaW_c9s0 ## Percent Change 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 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.

Answer: Simplify the problems by eliminating extra words.

How much is 15% of $220? Identify the percent, the base, and the amount. Percent: 15% Base: 220 Amount: n Write the percent equation. $\begin{array}{c}\text{Percent}\cdot\text{Base}=\text{Amount}\\15\%\cdot220=n\end{array}$ Convert 15% to 0.15, then multiply by 220. 15% of$220 is $33. $0.15\cdot220=33$ #### Answer The coupon will take$33 off the original price.

The example video that follows shows how to use the percent equation to find the amount of a discount from the price of a phone. https://youtu.be/bC7GUc1K6LU You can estimate to see if the answer is reasonable. Since 15% is half way between 10% and 20%, find these numbers.

$\begin{array}{c}10\%\,\,\text{of}\,\,220=0.1\cdot220=22\\20\%\,\,\text{of}\,\,220=0.2\cdot220=44\end{array}$

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. ### Example 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?

Answer: In this problem, you know that the tax of 5% is added onto the cost of the books. So if the cost of the books is 100%, the cost plus tax is 105%.

$\begin{array}{c}\text{What number}\,\,+5\%\,\,\text{of that number is}\,\,\31.50?\\105\%\,\,\text{of what number}=31.50?\end{array}$

Identify the percent, the base, and the amount. Percent: 105% Base: n Amount: 31.50 Write the percent equation.

$\begin{array}{c}\text{Percent}\cdot\text{Base}=\text{Amount}\\105\%\cdot{n}=31.50\end{array}$

Convert 105% to a decimal.

$1.05\cdot{n}=31.50$

Divide to undo the multiplication of n times 1.05.

$n=31.50\div1.05=30$

The books cost \$30 before tax.

In the following video example, you will be shown how to find a price before tax using the percent equation. https://youtu.be/qgafD4z8OUE

### Example

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?

Answer: Simplify the problem by eliminating extra words.

35 is what percent of 20?

Identify the percent, the base, and the amount. Percent: n Base: 20 Amount: 35 Write the percent equation.

$\begin{array}{c}\text{Percent}\cdot\text{Base}=\text{Amount}\\n\cdot20=35\end{array}$

Divide to undo the multiplication of n times 20.

$n=35\div20$

Convert 1.75 to a percent.

$n=1.75=175\%$

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.”)

The video that follows explains how to use the percent equation to determine the percent increase of a given amount. https://youtu.be/6YYpqlSiF74

## Pie Charts

Circle graphs, or pie charts, represent data as sections of the circle (or “pieces of the pie”), corresponding to their percentage of the whole. Circle graphs are often used to show how a whole set of data is broken down into individual components. Here’s an example. At the beginning of a semester, a teacher talks about how she will determine student grades. She says, “Half your grade will be based on the final exam and 20% will be determined by quizzes. A class project will also be worth 20% and class participation will count for 10%.” In addition to telling the class this information, she could also create a circle graph.

This graph is useful because it relates each part—the final exam, the quizzes, the class project, and the class participation—to the whole.

### Example

If the total number of points possible in the class is 500, how many points is the final exam worth?

Answer: Simplify the problem by eliminating extra words.

What is 50% of 500?

Identify the percent, the base, and the amount. Percent: 50% = 0.50 Base: 500 Amount: n Write the percent equation.

$\begin{array}{c}\text{Percent}\cdot\text{Base}=\text{Amount}\\0.50\cdot500=n\end{array}$

Multiply.

$\left(0.50\right)\left(500\right) = n$

$250 = n$

This tells us that the final is worth 250 points.

In the following video, an example of using a pie chart to determine a percent of a whole is shown. https://youtu.be/TAUDMvl8Vg8

## Summary

When solving application problems with percents, it is important to be extremely careful in identifying the percent, whole, and amount in the problem. Once those are identified, use the percent equation to solve the problem. Write your final answer back in terms of the original scenario.