# 3.15: Percent of Change

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

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**Practice**Percent of Change

What if a printer that normally cost $125 were marked down to $100. How could you calculate the percent it was marked down by? After completing this Concept, you'll be able to determine the percent of change in problems like this one.

### Watch This

CK-12 Foundation: 0315S Percent of Change (H264)

### Guidance

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**.

#### Example A

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

**Solution**

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*}

Cross multiplying is our next step; that gives us \begin{align*}500 = 5 \times (\text{actual change})\end{align*}

**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.

#### Example B

*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?*

**Solution**

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*}

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.

**Solve Real-World Problems Using Percents**

#### Example C

*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.*

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

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

**Solution**

a) 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) 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) 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.**

Watch this video for help with the Examples above.

CK-12 Foundation: Percent of Change

### Vocabulary

- A
**percent**is simply a ratio with a base unit of 100—for example, \begin{align*}13\% = \frac{13}{100}\end{align*}. - The
**percent equation**is \begin{align*}\text{Rate} \times \text{Total} = \text{Part}\end{align*}, or R% of Total is Part. - The percent change equation is \begin{align*}\text{Percent change} = \frac{\text{final amount - original amount}}{\text{original amount}} \times 100\%.\end{align*} A
**positive**percent change means the value**increased**, while a**negative**percent change means the value**decreased**.

### Guided Practice

*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:*

a) *Population of New York*

b) *Total reported crimes*

c) *violent crimes*

**Solution**

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.

a) 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%.**

b) 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%.**

c) 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%.

### Practice

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?

### Texas Instruments Resources

*In the CK-12 Texas Instruments Algebra I FlexBook, 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.*

### Image Attributions

## Description

## Learning Objectives

Here you'll learn how to use the percent change equation to find how much a value increases or decreases.