Have you ever heard of the stock market? You can buy a share of a company, and its value can go up or down. Suppose a share of your favorite company has risen in price by 40% this year and is now worth $200. Do you know how to find the price of a share at the beginning of the year? What if you knew the price at the beginning of the year and the percent increase? Could you find the current price? How about if you knew the price at the beginning of the year and the current price? Could you find the percent increase?

### 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.**

#### Let's solve the following percent of change problems:

- A school of 500 students is expecting a 20% increase in students next year. How many students will the school have?

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.

- A $150 mp3 player is on sale for 30% off. What is the price of the player?

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.

- A shirt is marked down to $20 from its original price of $30. Additionally, all sale items are another 20% off. What is the cost of the shirt? What percent is this cost from the original price?

In this case, $20 is our original price, and the percent change will be a decrease by 20%.

\begin{align*}\left (\frac{x- 20} {20} \right ) \cdot 100\% & = - 20\% && \text{Divide both sides by}\ 100\%. \\ \left (\frac{x - 20}{20} \right ) &= -0.2 && \text{Multiply both sides by}\ 20. \\ x - 20 &= 20(-0.2) \\ x-20 &= -4 && \text{Add}\ 20\ \text{to both sides}. \\ x & = -4 + 20 \\ x & = 16\end{align*}

The shirt will cost $16 after the additional 20% off.

Now we need to calculate what percentage $16 is of the original price, $30.

\begin{align*} \left (\frac{16- 30} {30} \right ) \cdot 100\% & = \text{Percent Change} && \text{Simplify the left side.} \\ \left (\frac{-14} {30} \right ) \cdot 100\% & = \text{Percent Change} && \text{Simplify the left side.} \\ \left (\frac{-7} {15} \right ) \cdot 100\% & = \text{Percent Change} && \text{Simplify the left side.} \\ -0.47 \cdot 100\% & = \text{Percent Change} && \text{Replace with an approximate decimal.} \\ -47\% & = \text{Percent Change} && \text{Multiply to find the final answer.} \end{align*}

The shirt was purchased at about 47% off the original price of $30.

### Examples

#### Example 1

Earlier, you were told that your share of your favorite company has risen in price by 40% this year and is now worth $200. What was the price at the beginning of the year? If you knew the price at the beginning of the year, could you find the current price? If you knew the price at the beginning of the year and the current price, could you find the percent increase?

All the questions in this example can be solved using the percent change formula:

The percent change is 40% and the final amount is $200. To find the price at the beginning of the year, substitute in the values and solve.

\begin{align*}40\% &= (\frac{200-x}{x}) \times 100\% && \text {Divide both sides by 100%.}\\
.4 &= (\frac{200-x}{x}) && \text{Multiply both sides by x.}\\
.4x&=200-x && \text{Add x to both sides.}\\
1.4x &=200 && \text{Divide both sides by 1.4.}\\
x &\approx 142.86
\end{align*}

\begin{align*}40\%&=\left (\frac{200- x} {x} \right ) \cdot 100\% && \text{Divide both sides by}\ 100\%. \\
.4&=\left (\frac{200 - x}{x} \right ) &&\text{Multiply both sides by}\ x.\\
.4x&=200-x && \text{Add x to both sides.}\\
1.4x&=200 &&\text{Divide both sides by 1.4}\\
x &\approx 142.86
\end{align*}

The price at the beginning of the year is approximately $142.86.

If you were given the price at the beginning of the year and the percent change, you could find the current price using the same formula but substituting the values in for the percent change and original amount.

If you were given the price at the beginning of the year and the current price, you could find the percent increase by using the same formula but substituting the values in for the final amount and original amount.

#### For Examples 2-4, use the following information:

#### 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 were the two largest demographic groups, with 11,754 and 6899 employees, respectively.

#### Example 2

Calculate the total percentage of minority (non-Caucasian) employees at the USDA.

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.

#### Example 3

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

\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.

#### Example 4

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

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.

### Review

Find the following.

- 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*}the radius and\begin{align*}h =\end{align*}the 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*}.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 3.11.