### Percent of Change

A useful way to express changes in quantities is through percents. You’ve probably seen signs such as “20% extra free,” or “save 35% today.” When we use percents to represent a change, we generally use the formula

\begin{align*}\text{Percent change} = \frac{\text{final amount - original amount}}{\text{original amount}} \times 100\%\end{align*}

or

\begin{align*}\frac{\text{percent change}}{100} = \frac{\text{actual change}}{\text{original amount}}\end{align*}

This means that a **positive** percent change is an **increase**, while a **negative** change is a **decrease**.

#### Real-World Application: School Enrollment

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

First let’s solve this using the first formula. Since the 20% change is an increase, we represent it in the formula as 20 (if it were a decrease, it would be -20.) Plugging in all the numbers, we get

\begin{align*}20\% = \frac{\text{final amount} - 500}{500} \times 100\%\end{align*}

Dividing both sides by 100%, we get \begin{align*}0.2 = \frac{\text{final amount} - 500}{500}\end{align*}.

Multiplying both sides by 500 gives us \begin{align*}100 = \text{final amount} - 500\end{align*}.

Then adding 500 to both sides gives us 600 as the final number of students.

How about if we use the second formula? Then we get \begin{align*}\frac{20}{100} = \frac{\text{actual change}}{500}\end{align*}. (Reducing the first fraction to \begin{align*}\frac{1}{5}\end{align*} will make the problem easier, so let’s rewrite the equation as \begin{align*}\frac{1}{5} = \frac{\text{actual change}}{500}.\end{align*}

Cross multiplying is our next step; that gives us \begin{align*}500 = 5 \times (\text{actual change})\end{align*}. Dividing by 5 tells us the change is equal to 100. We were told this was an increase, so if we start out with 500 students, after an increase of 100 we know there will be a total of 600.

**Markup**

A **markup** is an increase from the price a store pays for an item from its supplier to the retail price it charges to the public. For example, a 100% mark-up (commonly known in business as *keystone*) means that the price is doubled. Half of the retail price covers the cost of the item from the supplier, half is profit.

#### Real-World Application: Furniture Store

A furniture store places a 30% markup on everything it sells. It offers its employees a 20% discount from the sales price. The employees are demanding a 25% discount, saying that the store would still make a profit. The manager says that at a 25% discount from the sales price would cause the store to lose money. Who is right?

We’ll consider this problem two ways. First, let’s consider an item that the store buys from its supplier for a certain price, say $1000. The markup would be 30% of 1000, or $300, so the item would sell for $1300 and the store would make a $300 profit.

And what if an employee buys the product? With a discount of 20%, the employee would pay 80% of the $1300 retail price, or \begin{align*}0.8 \times \$1300 = \$1040\end{align*}.

But with a 25% discount, the employee would pay 75% of the retail price, or \begin{align*}0.75 \times \$1300 = \$975\end{align*}.

So with a 20% employee discount, the store still makes a $40 profit on the item they bought for $1000—but with a 25% employee discount, the store loses $25 on the item.

Now let’s use algebra to see how this works for an item of any price. If \begin{align*}x\end{align*} is the price of an item, then the store’s markup is 30% of \begin{align*}x\end{align*}, or \begin{align*}0.3x\end{align*}, and the retail price of the item is \begin{align*}x + 0.3x\end{align*}, or \begin{align*}1.3x\end{align*}. An employee buying the item at a 20% discount would pay \begin{align*}0.8 \times 1.3x = 1.04x\end{align*}, while an employee buying it at a 25% discount would pay \begin{align*}0.75 \times 1.3x = 0.975x\end{align*}.

So the manager is right: a 20% employee discount still allows the store to make a profit, while a 25% employee discount would cause the store to lose money.

It may not seem to make sense that the store would lose money after applying a 30% markup and only a 25% discount. The reason it does work out that way is that the discount is bigger in absolute dollars after the markup is factored in. That is, an employee getting 25% off an item is getting 25% off the original price *plus* 25% off the 30% markup, and those two numbers together add up to more than 30% of the original price.

#### Real-World Application: Employee Demographics

In 2004 the US Department of Agriculture had 112071 employees, of which 87846 were Caucasian. Of the remaining minorities, African-American and Hispanic employees had the two largest demographic groups, with 11754 and 6899 employees respectively.\begin{align*}^*\end{align*}

a) 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 112071. We know that the number of Caucasian employees is 87846, which means that there must be \begin{align*}112071 - 87646 = 24225\end{align*} non-Caucasian employees. This is the *part*. Plugging in the total and the part, we get \begin{align*}\text{Rate} \times 112071 = 24225\end{align*}.

Divide both sides by 112071 to get \begin{align*}\text{Rate} = \frac{24225}{112071} \approx 0.216\end{align*}. Multiply by 100 to get this as a percent: 21.6%.

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

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

Here, the total is still 112071 and the part is 11754, so we have \begin{align*}\text{Rate} \times 112071 = 11754\end{align*}. Dividing, we get \begin{align*}\text{Rate} = \frac{11754}{112071} \approx 0.105\end{align*}, or 10.5%.

**10.5% of USDA employees in 2004 were African-American.**

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

Here, our total is just the number of non-Caucasian employees, which we found out is 24225. Subtracting the African-American and Hispanic employees leaves \begin{align*}24225 - 11754 - 6899 = 5572\end{align*} employees in the group we’re looking at.

So with 24225 for the whole and 5572 for the part, our equation is \begin{align*}\text{Rate} \times 24225 = 5572\end{align*}, or \begin{align*}\text{Rate} = \frac{5572}{24225} \approx 0.230\end{align*}, or 23%.

**23% of USDA minority employees in 2004 were neither African-American nor Hispanic.**

### Examples

In 1995 New York had 18136000 residents. There were 827025 reported crimes, of which 152683 were violent. By 2005 the population was 19254630 and there were 85839 violent crimes out of a total of 491829 reported crimes. (Source: New York Law Enforcement Agency Uniform Crime Reports.) Calculate the percentage change from 1995 to 2005 in:

This is a percentage change problem. Remember the formula for percentage change:

\begin{align*}\text{Percent change} = \frac{\text{final amount - original amount}}{\text{original amount}} \times 100\% \end{align*}

In these problems, the final amount is the 2005 statistic, and the initial amount is the 1995 statistic.

#### Example 1

Calculate the percentage change from 1995 to 2005 in the population of New York

Population:

\begin{align*}\text{Percent change} &= \frac{19254630 - 18136000}{18136000} \times 100\%\\ &= \frac{1118630}{18136000} \times 100\%\\ &\approx 0.0617 \times 100\%\\ &= 6.17\%\end{align*}

**The population grew by 6.17%.**

#### Example 2

Calculate the percentage change from 1995 to 2005 in the total reported crimes.

Total reported crimes:

\begin{align*}\text{Percent change} &= \frac{491829 - 827025}{827025} \times 100\%\\ &= \frac{-335196}{827025} \times 100\%\\ &\approx -0.4053 \times 100\%\\ &= -40.53\%\end{align*}

**The total number of reported crimes fell by 40.53%.**

#### Example 3

Calculate the percentage change from 1995 to 2005 in violent crimes.

Violent crimes:

\begin{align*}\text{Percent change} &= \frac{85839 - 152683}{152683} \times 100\%\\ &= \frac{-66844}{152683} \times 100\%\\ &\approx -0.4377 \times 100\%\\ &= -43.77\%\end{align*}

The total number of violent crimes fell by 43.77%.

### Review

For questions 1-3, a hair stylist charges $70 for a haircut. Depending on how much you tip, what will be the total cost of the haircut?

- You tip 15%.
- You tip 20%.
- You tip 25%.
- 250 is what percentage of 195?
- 0.0032 is what percentage of 0.045?
- 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?
- A TV is advertised on sale. It is 35% off and now costs $195. What was the pre-sale price?
- A TV was advertised on sale. If you saved $40, and bought it for $160, what percentage off was it?
- Another TV is advertised on sale. If this TV is also $40 cheaper than the pre-sale price, was it also the same percentage off as the TV in the question above? Explain!
- Store \begin{align*}A\end{align*} and Store \begin{align*}B\end{align*} both sell bikes, and both buy bikes from the same supplier at the same prices. Store \begin{align*}A\end{align*} has a 40% mark-up for their prices, while store \begin{align*}B\end{align*} has a 250% mark-up. Store \begin{align*}B\end{align*} has a permanent sale and will always sell at 60% off the marked-up prices. Which store offers the better deal?

### Review (Answers)

To view the Explore More answers, open this PDF file and look for section 3.15.

### Texas Instruments Resources

*In the CK-12 Texas Instruments Algebra I FlexBook® resource, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9613.*