Solving Percent Word Problems

\(part = \% \ rate \times base\)

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.

\(n=30\% \times 85\)

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.

\( 50 = 15\% \times w\)

Rewrite the percent as a decimal and solve.

\( \begin{align*}50 & = 0.15 \times w \\ \frac{50}{0.15} & = \frac{0.15 \times w}{0.15} \\ 333 \frac{1}{3} & = w\end{align*}\)

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 \( \frac{x}{100}\) for \(x\%\), where \(x\) is the unknown.

Example: Express \( \dfrac{3}{5}\) as a percent.

We start by representing the unknown as \(x\%\) or \(\frac{x}{100}\).

\( \begin{align*}\left (\frac{3}{5} \right) & = \frac{x}{100} && \text{Cross multiply}. \\ 5x & = 100 \cdot 3 \\ 5x & = 300 \\ x & = \frac{300}{5} = 60 && \text{Divide both sides by}\ 5\ \text{to solve for}\ x. \\ \left (\frac{3}{5} \right) & = 60\%\end{align*}\)

Now that you remember how to convert between decimals and percents, you are ready for the Percent Equation.

Source: cK-12, https://www.ck12.org/book/basic-algebra/section/3.7/
This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 License.

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:

\(\text{Percent change} = \left (\frac{\text{final amount - original amount}}{\text{original amount}} \right ) \times 100\%\)

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.

\(\text{Percent change} = \left (\frac{\text{final amount - original amount}}{\text{original amount}} \right ) \times 100\%\)

\(\begin{align*}20\% & = \left (\frac{\text{final amount} - 500}{500} \right ) \times 100\% && \text{Divide both sides by}\ 100\% .\\ & && \text{Let}\ x = \text{final amount}. \\ 0.2 & = \frac{x - 500}{500} && \text{Multiply both sides by}\ 500. \\ 100 & = x - 500 && \text{Add}\ 500\ \text{to both sides}. \\ 600 & = x\end{align*}\)

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.

\( \text{Percent change} = \left (\frac{\text{final amount - original amount}}{\text{original amount}} \right ) \times 100\%\)

\( \begin{align*}\left (\frac{x- 150} {150} \right ) \cdot 100\% & = - 30\% && \text{Divide both sides by}\ 100\%. \\ \left (\frac{x - 150}{150} \right ) &= -0.3\% && \text{Multiply both sides by}\ 150. \\ x - 150 = 150 (-0.3) &= -45 && \text{Add}\ 150\ \text{to both sides}. \\ x & = -45 + 150 \\ x & = 105\end{align*}\)

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.

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 \( \text{Rate} \times \text{Total} = \text{Part}\). 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 \( (112,071 - 87,846) = 24,225\) non-Caucasian employees. This is the part.

\(\begin{aligned}\text{Rate} \times 112,071 & = 24,225 && \text{Divide both sides by}\ 112,071. \\ \text{Rate} & \approx 0.216 && \text{Multiply by}\ 100 \ \text{to obtain percent}. \\ \text{Rate} & \approx 21.6\%\end{aligned}\)

Approximately 21.6% of USDA employees in 2004 were from minority groups.

b) \( \text{Total} = 112,071 \ \text{Part} = 11,754\)

\( \begin{align*}\text{Rate} \times 112,071 & = 11,754 && \text{Divide both sides by}\ 112,071. \\ \text{Rate} & \approx 0.105 && \text{Multiply by}\ 100 \ \text{to obtain percent}. \\ \text{Rate} & \approx 10.5\%\end{align*}\)

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 \( (24,225 - 11,754 - 6899) = 5572\) 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.