3.7: Percent Problems
A percent is a ratio whose denominator is 100. Before we can use percent to solve problems, let's review how to convert percents to decimals and fractions and vice versa.
To convert a decimal to a percent, multiply the decimal by 100.
Example: Convert 0.3786 to a percent.
\begin{align*}0.3786 \times 100=37.86\%\end{align*}
To convert a percentage to a decimal, divide the percentage by 100.
Example: Convert 98.6% into a decimal.
\begin{align*}98.6 \div 100 = 0.986\end{align*}
When converting fractions to percent, we can substitute \begin{align*}\frac{x}{100}\end{align*}
Example: Express \begin{align*}\frac{3}{5}\end{align*}
We start by representing the unknown as \begin{align*}x\%\end{align*}
\begin{align*}\left (\frac{3}{5} \right) & = \frac{x}{100} && \text{Cross multiply}. \\
5x & = 100 \cdot 3 \\
5x & = 300 \\
x & = \frac{300}{5} = 60 && \text{Divide both sides by}\ 5\ \text{to solve for}\ x. \\
\left (\frac{3}{5} \right) & = 60\%\end{align*}
Now that you remember how to convert between decimals and percents, you are ready for the Percent Equation.
The Percent Equation
\begin{align*}part = \% \ rate \times base\end{align*}
The key words in a percent equation will help you translate it into a correct algebraic equation. Remember the equal sign symbolizes the word “is” and the multiplication symbol symbolizes the word “of.”
Example 1: Find 30% of 85.
Solution: You are asked to find the part of 85 that is 30%. First, translate into an equation.
\begin{align*}n=30\% \times 85\end{align*}
Convert the percent to a decimal and simplify.
\begin{align*}n & =0.30 \times 85 \\
n & =25.5\end{align*}
Example 2: 50 is 15% of what number?
Solution: Translate into an equation.
\begin{align*}50 = 15\% \times w\end{align*}
Rewrite the percent as a decimal and solve.
\begin{align*}50 & = 0.15 \times w \\
\frac{50}{0.15} & = \frac{0.15 \times w}{0.15} \\
333 \frac{1}{3} & = w\end{align*}
For more help with the percent equation, watch this 4minute video recorded by Ken’s MathWorld. How to Solve Percent Equations (4:10)
Finding the Percent of Change
A useful way to express changes in quantities is through percent. You have probably seen signs such as “20% more free,” or “save 35% today.” When we use percent to represent a change, we generally use the formula:
\begin{align*}\text{Percent change} = \left (\frac{\text{final amount  original amount}}{\text{original amount}} \right ) \times 100\%\end{align*}
A positive percent change would thus be an increase, while a negative change would be a decrease.
Example 3: A school of 500 students is expecting a 20% increase in students next year. How many students will the school have?
Solution: Using the percent of change equation, translate the situation into an equation. Because the 20% is an increase, it is written as a positive value.
\begin{align*}\text{Percent change} = \left (\frac{\text{final amount  original amount}}{\text{original amount}} \right ) \times 100\%\end{align*}
\begin{align*}20\% & = \left (\frac{\text{final amount}  500}{500} \right ) \times 100\% && \text{Divide both sides by}\ 100\% .\\
& && \text{Let}\ x = \text{final amount}. \\
0.2 & = \frac{x  500}{500} && \text{Multiply both sides by}\ 500. \\
100 & = x  500 && \text{Add}\ 500\ \text{to both sides}. \\
600 & = x\end{align*}
The school will have 600 students next year.
Example 4: A $150 mp3 player is on sale for 30% off. What is the price of the player?
Solution: Using the percent of change equation, translate the situation into an equation. Because the 30% is a discount, it is written as a negative.
\begin{align*}\text{Percent change} = \left (\frac{\text{final amount  original amount}}{\text{original amount}} \right ) \times 100\%\end{align*}
\begin{align*}\left (\frac{x 150} {150} \right ) \cdot 100\% & =  30\% && \text{Divide both sides by}\ 100\%. \\
\left (\frac{x  150}{150} \right ) &= 0.3\% && \text{Multiply both sides by}\ 150. \\
x  150 = 150 (0.3) &= 45 && \text{Add}\ 150\ \text{to both sides}. \\
x & = 45 + 150 \\
x & = 105\end{align*}
The mp3 player will cost $105.
Many real situations involve percents. Consider the following.
In 2004, the US Department of Agriculture had 112,071 employees, of which 87,846 were Caucasian. Of the remaining minorities, AfricanAmerican and Hispanic employees had the two largest demographic groups, with 11,754 and 6899 employees respectively.\begin{align*}^*\end{align*}
a) Calculate the total percentage of minority (nonCaucasian) employees at the USDA.
b) Calculate the percentage of AfricanAmerican employees at the USDA.
c) Calculate the percentage of minority employees at the USDA who were neither AfricanAmerican nor Hispanic.
a) Use the percent equation \begin{align*}\text{Rate} \times \text{Total} = \text{Part}\end{align*}
\begin{align*}\text{Rate} \times 112,071 & = 24,225 && \text{Divide both sides by}\ 112,071. \\
\text{Rate} & \approx 0.216 && \text{Multiply by}\ 100 \ \text{to obtain percent}. \\
\text{Rate} & \approx 21.6\%\end{align*}
Approximately 21.6% of USDA employees in 2004 were from minority groups.
b) \begin{align*}\text{Total} = 112,071 \ \text{Part} = 11,754\end{align*}
\begin{align*}\text{Rate} \times 112,071 & = 11,754 && \text{Divide both sides by}\ 112,071. \\
\text{Rate} & \approx 0.105 && \text{Multiply by}\ 100 \ \text{to obtain percent}. \\
\text{Rate} & \approx 10.5\%\end{align*}
Approximately 10.5% of USDA employees in 2004 were AfricanAmerican.
c) We now know there are 24,225 nonCaucasian employees. That means there must be \begin{align*}(24,225  11,754  6899) = 5572\end{align*}
\begin{align*}\text{Rate} \times 112,071 & = 5572 && \text{Divide both sides by}\ 112,071. \\
\text{Rate} & \approx 0.05 && \text{Multiply by}\ 100 \ \text{to obtain percent}. \\
\text{Rate} & \approx 5\%\end{align*}
Approximately 5% of USDA minority employees in 2004 were neither AfricanAmerican nor Hispanic.
Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK12 Basic Algebra: Percent Problems (14:15)
Express the following decimals as percents.
 0.011
 0.001
 0.91
 1.75
 20
Express the following fractions as a percent (round to two decimal places when necessary).

\begin{align*}\frac{1}{6}\end{align*}
16 
\begin{align*}\frac{5}{24}\end{align*}
524 
\begin{align*}\frac{6}{7}\end{align*}
67 
\begin{align*}\frac{11}{7}\end{align*}
117 
\begin{align*}\frac{13}{97}\end{align*}
1397
Express the following percentages as reduced fractions.
 11%
 65%
 16%
 12.5%
 87.5%
Find the following.
 32% of 600 is what number?

\begin{align*}\frac{3}{4}\%\end{align*}
34% of 16 is what number?  9.2% of 500 is what number
 8 is 20% of what number?
 99 is 180% of what number?
 What percent of 7.2 is 45?
 What percent of 150 is 5?
 What percent of 50 is 2500?
 A Realtor earns 7.5% commission on the sale of a home. How much commission does the Realtor make if the home sells for $215,000?
 The fire department hopes to raise $30,000 to repair a fire house. So far the department has raised $1,750.00. What percent is this of their goal?
 A $49.99 shirt goes on sale for $29.99. By what percent was the shirt discounted?
 A TV is advertised on sale. It is 35% off and has a new price of $195. What was the presale price?
 An employee at a store is currently paid $9.50 per hour. If she works a full year, she gets a 12% pay raise. What will her new hourly rate be after the raise?
 Store A and Store B both sell bikes, and both buy bikes from the same supplier at the same prices. Store A has a 40% markup for their prices, while store B has a 90% markup. Store B has a permanent sale and will always sell at 60% off those prices. Which store offers the better deal?
 788 students were surveyed about their favorite type of television show. 18% stated that their favorite show was realitybased. How many students said their favorite show was realitybased?
Mixed Review
 List the property used at each step of solving the following equation:
\begin{align*}4(x3) & = 20 \\
4x12 & =20 \\
4x & = 32 \\
x & = 8\end{align*}
 The volume of a cylinder is given by the formula \begin{align*}Volume = \pi r^2 h\end{align*}
Volume=πr2h , where \begin{align*}r=\end{align*}r= radius and \begin{align*}h =\end{align*}h= height of the cylinder. Determine the volume of a soup can with a 3inch radius and a 5.5inch height.  Circle the math noun in this sentence: Jerry makes holiday baskets for his youth group. He can make one every 50 minutes. How many baskets can Jerry make in 25 hours?
 When is making a table a good problemsolving strategy? When may it not be such a good strategy?
 Solve for \begin{align*}w:\ \frac{10}{w} = \frac{12}{3}\end{align*}
w: 10w=123 .
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