# 3.7: Percent Problems

**At Grade**Created by: CK-12

A **percent** is a ratio whose denominator is 100. Before we can use percents 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 percents, we can substitute \begin{align*}\frac{x}{100}\end{align*} for \begin{align*}x\%\end{align*}, where \begin{align*}x\end{align*} is the unknown.

**Example:** *Express* \begin{align*}\frac{3}{5}\end{align*} *as a percent.*

We start by representing the unknown as \begin{align*}x\%\end{align*} or \begin{align*}\frac{x}{100}\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 4-minute 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 percents. You have probably seen signs such as “20% more free,” or “save 35% today.” When we use percents 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, African-American 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 (non-Caucasian) employees at the USDA.

b) Calculate the percentage of African-American employees at the USDA.

c) Calculate the percentage of minority employees at the USDA who were neither African-American nor Hispanic.

a) Use the percent equation \begin{align*}\text{Rate} \times \text{Total} = \text{Part}\end{align*}. The ** total** number of employees is 112,071. We know that the number of Caucasian employees is 87,846, which means that there must be \begin{align*}(112,071 - 87,846) = 24,225\end{align*} non-Caucasian employees. This is the

**.**

*part*\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 African-American.

c) We now know there are 24,225 non-Caucasian employees. That means there must be \begin{align*}(24,225 - 11,754 - 6899) = 5572\end{align*} minority employees who are neither African-American nor Hispanic. The part is 5572.

\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 African-American 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. CK-12 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*}
- \begin{align*}\frac{5}{24}\end{align*}
- \begin{align*}\frac{6}{7}\end{align*}
- \begin{align*}\frac{11}{7}\end{align*}
- \begin{align*}\frac{13}{97}\end{align*}

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*} 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 pre-sale 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% mark-up for their prices, while store B has a 90% mark-up. 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 reality-based. How many students said their favorite show was reality-based?

**Mixed Review**

- List the property used at each step of solving the following equation:

\begin{align*}4(x-3) & = 20 \\ 4x-12 & =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*},
*where*\begin{align*}r=\end{align*}*radius and*\begin{align*}h =\end{align*}*height of the cylinder*. Determine the volume of a soup can with a 3-*inch*radius and a 5.5-*inch*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 problem-solving strategy? When may it not be such a good strategy?
- Solve for \begin{align*}w:\ \frac{10}{w} = \frac{12}{3}\end{align*}.

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