6.5: Percent of Change
Introduction
The Trouble with Sales
Taylor’s Mom is confused about their sales. In one month, the sales have decreased instead of increased in the store. This decrease has happened despite the fact that they have added many new products to their inventory.
“I just don’t understand it,” her Mom said at breakfast. “Last month, we had a terrific month with $15,000 in sales. This month it was only $12,500.”
“We can’t keep losing business,” her Dad said, drinking his coffee.
“I wonder how much of a loss it actually is,” thought Taylor. “Maybe if we can figure out why the sales decreased, then we can increase them again.
Taylor began figuring out the percent of the decrease. She started by subtracting, but then got stuck.
Taylor needs some help and this lesson will do the trick. You will learn all about calculating the percent of increase and the percent of decrease in this lesson. Pay attention and you can help Taylor with her arithmetic at the end of the lesson.
What You Will Learn
In this lesson, you will learn how to execute the following skills:
- Find a percent of increase given the original amount and the amount of increase.
- Find a percent of decrease given the original amount and the amount of decrease.
- Use percent of change to find a new amount.
- Solve real-world problems involving percent of change.
Teaching Time
I. Find a Percent of Increase Given the Original Amount and the Amount of Increase
Sometimes, we have a price that has been increased a specific amount, or we can observe that over time a price has increased. We think of past pricing in this way.
The cost of a postage stamp has increased over time. In fact, some people think that the cost has increased too much. When we compare a past price and an increased current price, we can figure out the percent that a price has increased. We call this the percent of increase.
How do we calculate the percent of increase?
The percent of increase from one amount to another is the ratio of the amount of increase to the original amount.
To find the percent of increase, follow these steps
Step 1: Find the amount of increase by subtracting the original price from the new price.
Step 2: Write a fraction in which the numerator is the amount of increase and the denominator is the original amount.
\begin{align*}\text{Percent of increase} = \frac{\text{Amount of increase}} {\text{Original amount}}\end{align*}
Step 3: Write the fraction as a percent.
Take a few minutes to write down these steps for finding the Percent of Increase.
Let’s look at an example.
Example
Find the percent of increase from 20 to 35.
Step 1: Subtract 20 from 35. \begin{align*}35 - 20 = 15\end{align*}
Step 2: \begin{align*}\text{Percent of increase} = \frac{\text{Amount of increase}}{\text{Original amount}} = \frac{15}{20}\end{align*}
Step 3:
One Way | Another Way |
---|---|
\begin{align*}\frac{15}{20} = \frac{x}{100}\end{align*} | \begin{align*}\frac{15}{20} = \frac{15 \div 5}{20 \div 5} = \frac{3}{4}\end{align*} |
\begin{align*}20 x = 1,500\end{align*} | \begin{align*}\overset{ \ \ 0.75}{4 \overline{ ) {3.00 \;}}} \leftarrow \ \text{Divide to 2 decimal places.}\end{align*} |
\begin{align*}\frac{\cancel{20} x} {\cancel{20}} = \frac{1,500}{20}\end{align*} | \begin{align*}0.75 = 75 \%\end{align*} |
\begin{align*}x = 75\end{align*} | |
\begin{align*}\frac{75}{100} = 75 \%\end{align*} |
The percent of increase from 20 to 35 is 75%.
Notice that we could solve for the percent in two different ways. One was to use a proportion and the other was to simply divide. Either way, you will get the same answer.
Example
Find the percent of increase from 24 to 72.
Step 1: Subtract 24 from 72. \begin{align*}72 - 24 = 48\end{align*}
Step 2: \begin{align*}\text{Percent of increase} = \frac{\text{Amount of increase}}{\text{Original amount}} = \frac{48}{24}\end{align*}
Step 3:
One Way | Another Way |
---|---|
\begin{align*}\frac{48}{24} = \frac{x}{100}\end{align*} | \begin{align*}\frac{48}{24} = 2\end{align*} |
\begin{align*}24 x = 4,800\end{align*} | \begin{align*}2 = 200 \%\end{align*} |
\begin{align*}\frac{\cancel{24} x}{\cancel{24}} = \frac{4,800}{24}\end{align*} | |
\begin{align*}x = 200\end{align*} | |
\begin{align*}\frac{200}{100} = 200 \%\end{align*} |
The percent of increase from 24 to 72 is 200%.
Yes. Sometimes the percent of increase can be greater than 100%!!
6M. Lesson Exercises
Find the percent of increase. You may round to the nearest whole percent when needed.
- From 45 to 50
- From $1.00 to $1.75
- From 34 to 60
Check your work. Did you remember to write each answer as a percent?
II. Find a Percent of Decrease Given the Original Amount and the Amount of Decrease
Prices can increase. Costs can increase. Numbers can increase. We can find the percent of increase when dealing with an increase. All of these things can also decrease. When there has been a decrease from an original amount to a new amount, we can find the percent of decrease.
How do we find the percent of decrease?
The percent of decrease from one amount to another is the ratio of the amount of decrease to the original amount.
To find the percent of decrease, follow these steps
Step 1: Find the amount of decrease by subtracting the two numbers.
Step 2: Write a fraction in which the numerator is the amount of decrease and the denominator is the original amount.
\begin{align*}\text{Percent of decrease} = \frac{ \text{Amount of decrease}}{\text {Original amount}}\end{align*}
Step 3: Write the fraction as a percent.
Take a few minutes to write the steps to finding the percent of decrease in your notebook.
Now let’s look at how to apply these steps in an example.
Example
Find the percent of decrease from 50 to 40.
Step 1: Subtract 40 from 50. \begin{align*}50 - 40 = 10\end{align*}
Step 2: \begin{align*}\text{Percent of decrease} = \frac{\text{Amount of decrease}}{\text{Original amount}} = \frac{10}{50}\end{align*}
Step 3:
One Way | Another Way |
---|---|
\begin{align*}\frac{10}{50} = \frac{x}{100}\end{align*} | \begin{align*}\frac{10}{50} = \frac{10 \div 10}{50 \div 10} = \frac{1}{5}\end{align*} |
\begin{align*}50 x = 1,000\end{align*} | \begin{align*}\overset{ \ \ 0.20}{5 \overline{ ) {1.00 \;}}} \leftarrow \ \text{Divide to 2 decimal places.}\end{align*} |
\begin{align*}\frac{\cancel{50} x} {\cancel{50}} = \frac{1,000}{50}\end{align*} | \begin{align*}0.20 = 20 \%\end{align*} |
\begin{align*}x = 20\end{align*} | |
\begin{align*}\frac{20}{100} = 20 \%\end{align*} |
The percent of decrease from 50 to 40 is 20%.
Example
Find the percent of decrease from 200 to 170.
Step 1: Subtract 170 from 200. \begin{align*}200 - 170 = 30\end{align*}
Step 2: \begin{align*}\text{Percent of decrease} = \frac{\text{Amount of decrease}}{\text{Original amount}} = \frac{30}{200}\end{align*}
Step 3:
One Way | Another Way |
---|---|
\begin{align*}\frac{30}{200} = \frac{x}{100}\end{align*} | \begin{align*}\frac{30}{200} = \frac{30 \div 10}{200 \div 10} = \frac{3}{20}\end{align*} |
\begin{align*}200 x = 1,800\end{align*} | \begin{align*}\overset{ \ \ 0.15}{20 \overline{ ) {3.00 \;}}}\end{align*} |
\begin{align*}\frac{\cancel{200} x} {\cancel{200}} = \frac{3,000}{200}\end{align*} | \begin{align*}0.15 = 15 \%\end{align*} |
\begin{align*}x = 15\end{align*} | |
\begin{align*}\frac{15}{100} = 15 \%\end{align*} |
The percent of decrease from 200 to 170 is 15%.
6N. Lesson Exercises
Find the percent of each decrease.
- From 10 to 5
- From 25 to 15
- From 125 to 70
Take a few minutes to check your work with a friend.
III. Use Percent of Change to Find a New Amount
If we are given the percent of increase or the percent of decrease and the original amount, we can find the new amount by using the following formula.
Amount of change = percent of change \begin{align*}\times\end{align*} original amount
Let’s look at an example.
Example
Find the new number when 75 is decreased by 40%.
First find the amount of change.
\begin{align*}\text{Amount of change} & = \text{percent of change} \times \text{original amount}\\ & = 40 \% \times 75\\ & = 0.40 \times 75\\ & = 30\end{align*}
Since the original number is being decreased, we subtract the amount of change from the original number to find the new number. \begin{align*}75 - 30 = 45\end{align*}
When 75 is decreased by 40%, the new number is 45.
Example
Find the new number when 28 is increased by 125%.
First find the amount of change.
\begin{align*}\text{Amount of change} & = \text{percent of change} \times \text{original amount}\\ & = 125 \% \times 28\\ & = 1.25 \times 28\\ & = 35\end{align*}
Since the original number is being increased, we add the amount of change to the original number to find the new number. \begin{align*}28 + 35 = 63\end{align*}
When 28 is increased by 125%, the new number is 63.
6O. Lesson Exercises
Find each new amount.
- Find the new number when 45 is increased by 10%.
- Find the new number when 80 is decreased by 15%.
- Find the new number when 50 is increased by 25%.
Take a few minutes to check your work with a partner.
IV. Solve Real-World Problems Involving Percents of Change
Remember the postage stamp? The increase in price is an example of a percent of change that can be calculated. We can apply all of the things that we have learned when solving real-world problems involving increases and decreases.
Example
The population of Westville grew from 25,000 to 27,000 in two years. What was the percent of increase for this period of time?
First, find the amount of the change by subtracting.
The amount of change is \begin{align*}27,000 - 25,000 = 2,000\end{align*}
Next, find the percent of the increase.
\begin{align*}\text{Percent of increase or decrease} & = \frac{\text{Amount of change}}{\text{Original amount}}\\ & = \frac{2,000}{25,000} = \frac{2}{25} = 0.08 = 8 \%\end{align*}
The population increased 8% over this period of time.
Example
At noon the temperature was \begin{align*}40^\circ F\end{align*}. At 6:00 P.M. the temperature dropped to \begin{align*}32^\circ F\end{align*}. What was the percent of change in the temperature?
First, figure out the amount of the change.
The amount of change is \begin{align*}40 - 32 = 8.\end{align*}
\begin{align*}\text{Percent of increase or decrease} & = \frac{\text{Amount of change}}{\text{Original amount}}\\ & = \frac{8}{40} = \frac{1}{5} = 0.20 = 20 \%\end{align*}
The temperature decreased by 20%.
Remember the problem with sales at the candy store? Well let’s go back and see if we can help figure out the percent of the decrease.
Real Life Example Completed
The Trouble with Sales
Here is the original problem once again. Reread it and underline any important information.
Taylor’s Mom is confused about their sales. In one month, the sales have decreased instead of increased in the store. This decrease has happened despite the fact that they have added many new products to their inventory.
“I just don’t understand it,” her Mom said at breakfast. “Last month, we had a terrific month with $15,000 in sales. This month it was only $12,500.”
“We can’t keep losing business,” her Dad said, drinking his coffee.
“I wonder how much of a loss it actually is,” thought Taylor. “Maybe if we can figure out why the sales decreased, then we can increase them again.
Taylor began figuring out the percent of the decrease. She started by subtracting, but then got stuck.
First, we have to figure out the amount of the decrease.
\begin{align*}15,000 - 12,500 = 2,500\end{align*}
Next, we need to figure out the percent of the decrease by dividing the amount of the change by the original amount.
\begin{align*}\frac{2,500}{15,000} = .166\end{align*}
The sales in the candy store decreased by 16.6 or 17%
As Taylor began thinking about this she remembered something. She ran in to tell her Mom and Dad.
“I just remembered something that could have affected sales. Remember, we started closing an hour earlier,” Taylor said.
“Maybe, that could be it,” her Dad said, smiling. “Good thinking.”
The family readjusted the closing time and within a month the sales were back to normal.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Percent of Increase
- the percent that a price or cost or number has increased.
- Percent of Decrease
- the percent that a price or cost or number has decreased.
Technology Integration
Khan Academy Growing By a Percentage
James Sousa, Percent of Change
James Sousa, Example of Determining a Percent of Change
James Sousa, Another Example of Determining a Percent of Change
Other Videos:
- http://www.mathplayground.com/mv_percent_change.html – This is a Brightstorm video on calculating a percent change.
- http://www.wonderhowto.com/how-to-find-percent-change-using-proportions-319019 – This video shows how to calculate the percent change using a proportion.
Time to Practice
Directions: Find the percent of increase given the original amount. You may round to the nearest whole percent when necessary.
1. From 25 to 40
2. From 15 to 30
3. From 18 to 50
4. From 22 to 80
5. From 16 to 18
6. From 3 to 10
7. From 85 to 100
8. From 75 to 90
9. From 26 to 36
10. From 100 to 125
Directions: Find the percent of decrease given the original amount. You may round to the nearest whole percent when necessary or leave your answer as a decimal.
11. From 25 to 10
12. From 30 to 11
13. From 18 to 8
14. From 30 to 28
15. From 12 to 8
16. From 90 to 85
17. From 200 to 150
18. From 97 to 90
19. From 56 to 45
20. From 15 to 2
Directions: Solve each problem.
21. Zak earns $960 per week in a brokerage firm and he just received a 5% increase in salary. What will Zak’s new salary be?
22. The price of a share of stock went from $40 to $37. What was the percent of decrease?
23. Alicia had 150 shares of XYZ stock. She just bought more shares of the same stock and now has a total of 225 shares. What is the percent of increase in the number of shares?