# 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.[latex] \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}[/latex]

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:[latex]25\%\,\,\text{of}\,\,14\,\,\,\text{gallons}=0.25\,\cdot \,14=3.5\,\,\,\text{gallons}[/latex]

### 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.

[latex]15\%=0.15[/latex]

Multiply the decimal form of the percent by the whole number.[latex]\begin{array}{c}0.15\cdot200\\30\end{array}[/latex]

#### Answer

15% of $200 is $30- the
**percent**, - the
**amount**, the amount is part of the whole - and the
**base,**the base is the whole 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### 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## 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

### The Percent Equation

Percent of the Base is the Amount.[latex]\text{Percent}\cdot\text{Base}=\text{Amount}[/latex]

*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.

[latex]\text{Percent}\cdot\text{Base}=\text{Amount}[/latex]

[latex]20\%\cdot{n}=30[/latex]

#### Answer

[latex-display]20\%\cdot{n}=30[/latex-display][latex]20\%\cdot{n}=30[/latex]

You can solve this by writing the percent as a decimal or fraction and then dividing.[latex]20\%\cdot{n}=30[/latex]

[latex]n=30\div20\%=30\div0.20=150[/latex]

### 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 [latex]\cdot[/latex] Base = Amount. Use *n* for the unknown (percent).

[latex]n\cdot72=9[/latex]

Divide to undo the multiplication of*n*times 72.

[latex]n=\frac{9}{72}[/latex]

Divide 9 by 72 to find the value for*n*, the unknown.

[latex] \displaystyle 72\overset{0.125}{\overline{\left){9.000}\right.}}[/latex]

Move the decimal point two places to the right to write the decimal as a percent.[latex]\begin{array}{c}n=0.125\\n=12.5\%\end{array}[/latex]

#### Answer

12.5% of 72 is 9.### 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.

[latex]\text{Percent}\cdot\text{Base}=\text{Amount}[/latex]

The amount is unknown, so use*n*.

110% · 24 = *n*

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.## 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.

[latex]\begin{array}{c}\text{Percent}\cdot\text{Base}=\text{Amount}\\15\%\cdot220=n\end{array}[/latex]

Convert 15% to 0.15, then multiply by 220. 15% of $220 is $33.[latex]0.15\cdot220=33[/latex]

#### Answer

The coupon will take $33 off the original price.[latex]\begin{array}{c}10\%\,\,\text{of}\,\,220=0.1\cdot220=22\\20\%\,\,\text{of}\,\,220=0.2\cdot220=44\end{array}[/latex]

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%.

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

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

[latex]\begin{array}{c}\text{Percent}\cdot\text{Base}=\text{Amount}\\105\%\cdot{n}=31.50\end{array}[/latex]

Convert 105% to a decimal.[latex]1.05\cdot{n}=31.50[/latex]

Divide to undo the multiplication of*n*times 1.05.

[latex]n=31.50\div1.05=30[/latex]

#### Answer

The books cost $30 before tax.### 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.

[latex]\begin{array}{c}\text{Percent}\cdot\text{Base}=\text{Amount}\\n\cdot20=35\end{array}[/latex]

Divide to undo the multiplication of*n*times 20.

[latex]n=35\div20[/latex]

Convert 1.75 to a percent.[latex]n=1.75=175\%[/latex]

#### Answer

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.”)## 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.

[latex]\begin{array}{c}\text{Percent}\cdot\text{Base}=\text{Amount}\\0.50\cdot500=n\end{array}[/latex]

Multiply.[latex]\left(0.50\right)\left(500\right) = n[/latex]

[latex]250 = n[/latex]

#### Answer

This tells us that the final is worth 250 points.## 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.## Licenses & Attributions

### CC licensed content, Original

- Revision and Adaptation.
**Provided by:**Lumen Learning**License:**CC BY: Attribution. - Screenshot: Unemployment Graph.
**Provided by:**Lumen Learning**License:**CC BY: Attribution.

### CC licensed content, Shared previously

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