Percent
Learning Outcomes
- 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.[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, 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
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 = unknownExample
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 = 30Example
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 = 45Percent 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]
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
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.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.”)Interest
When a person takes out a loan, most lenders charge interest on the loan. Interest is a fee or change for borrowing money, typically a percent rate charged per year. We can compute simple interest by finding the interest rate percentage of the amount borrowed, then multiply by the number of years interest is earned.Simple Interest Equation
[latex]I=p\cdot{r}\cdot{t}[/latex]
Where: I is the interest paid p is the principal—the original amount of money borrowed r is the interest rate, a per-year rate, written as a decimal t is the time of the loan, expressed in years or portions of a yearExample
Treasury Notes (T-notes) are bonds issued by the federal government to cover its expenses. Suppose you obtain a $1,000 T-note with a 4% annual rate, with a maturity in 2 years. How much interest will you earn?Answer: Identify the information given in the problem. Interest, I: unknown Principal, p: $1000 Rate, r: [latex]4\%=0.04[/latex] Time, t: 2 years Put the information in the simple interest equation.
[latex]I=1000\cdot0.04\cdot2[/latex]
Multiply.[latex]I=80[/latex]
Answer
You would earn $80 in interest.Example
A friend asks to borrow $240, offering to repay you $250 in 1 month. What annual interest rate is this equivalent to?Answer: Identify the information given in the problem. Here your friend is paying back $10 more than he borrowed, so that is the interest paid. Interest, I: $10 Principal, p: $240 Rate, r: unknown Time, t: 1 month Convert the time to years.
[latex]1\,\,\text{month}=\frac{1}{12}\,\,\text{year}[/latex]
Put the information in the simple interest equation.[latex]10=240\cdot{r}\cdot\frac{1}{12}[/latex]
Regroup and simplify.[latex]\begin{array}{l}10=r\cdot240\cdot\frac{1}{12}\\\\10=r\cdot\frac{240}{12}\\\\10=r\cdot20\end{array}[/latex]
Divide to undo the multiplication.[latex]r=10\div20=0.50[/latex]
Rewrite as a percent.[latex]0.50=50\%[/latex]
Answer
This is equivalent to a 50% annual interest rate.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.
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. Put it Together
While the forgotten food in your fridge may not be what you spend your work day dreaming about devouring when you get home, those jumbled bits and pieces can be used to create something tasty enough to satisfy your hunger. Hopefully this section offered you enough tidbits to keep you full long enough to get to the next section, where you will learn how to put these forgotten skills to work as you continue to grow your mathematical prowess. Remember, you can come back to this review any time throughout the course for practice dividing fractions, simplifying with square roots, or whatever you might need to help you learn the concepts in this course more easily.
Oh, and now you know that [latex]3+5\times2[/latex] is 13!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.
- Use the Percent Equation to Find a Percent. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
CC licensed content, Shared previously
- Find the Percent of a Number. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Identify the Percent, Base, and Amount of a Percent Question. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Unit 5: Percents, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.
- Use a Percent Equation to Solve for a Base or Whole Amount. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Use a Percent Equation to Solve for an Amount or Part of a Whole. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Percent App: Find the Amount of Savings from a Discount. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Percent App: Find a Price Before Tax From Total Price. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Find a Percent of Increase Using a Percent Equation. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Determine the Amount of Interest Earned (Simple Interest). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Determine a Simple Interest Rate For a Loan with Known Interest. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Percent App: Find the Percent of An Amount After Reading a Pie Chart. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: Public Domain: No Known Copyright.
