# 5.4: Percent Increase and Decrease

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

## Introduction

*An Attendance Prediction*

Cameron and Carla decided to bring their proposal of increased attendance to the student council. They thought that if the student council suggested an increase in attendance to the student body that students might really want to help out.

“We know that 100% attendance is nearly impossible, but we could get to 55 or 60%, I think,” Cameron suggested.

“Then we could really show the high school that we support their efforts,” Carla chimed in.

“I think it is a great idea,” Avery, the President of the student council said. “We could conduct a survey and see how many students anticipate attending. Then we could try to predict the number of students who will attend next fall.”

That is exactly what they did. The students conducted a survey and asked students if they would be willing to attend next year’s football games to support the high school team. Out of their survey, they learned that 198 students said they planned on attending. That is an increase from 152 students. What is the percent of the increase?

**This is the question for this lesson. Percents of increase and decrease are often used in real – life situations such as the example presented in this lesson. Work through the content of this lesson and by the end you will be able to figure out the percent of the increase if the survey proves true.**

*What You Will Learn*

In this lesson you will learn how to do the following skills.

- Find a percent of increase given an original amount and amount of increase.
- Find percent of decrease given an original amount and amount of decrease.
- Use percent of change to find a new amount.
- Solve real-world problems involving percents of change.

*Teaching Time*

I. **Find a Percent of Increase Given an Original Amount and Amount of Increase**

Many times in the real world, things change—prices go up and down, your bank account balances go up and down, your weight goes up and down, businesses get more or less business, we have more hurricanes and typhoons than before, etc. Interpreting the amount of change as a percent is oftentimes useful to understand a situation and compare it to others.

**Let’s look at percent of increase first.**

Example

Two years ago, Mingh was 116 cm tall. She is now 132 cm tall. Her height increased by 16 cm in two years. Two years ago, her little brother Charlie was only 80 cm tall. He has grown to 95 cm... almost a full meter! He grew 15 cm. Which one grew more?

**Let’s start to work on this problem. In the beginning, it might seem obvious that Mingh grew more—she grew 16 cm and Charlie only grew 15 cm. But if we consider the percent of increase, we might have a different argument.**

**What is the percent of increase?**

**The** *percent of increase***is the percent that something value increased.**

**How does that apply to this problem?**

First, we have to figure out what the percent of increase was in growth for both children. Mingh grew 16cm. Her original height was 116cm. What percent did she increase?

If we consider 16cm the part of her original height that she increased, we can find percent of increase by using the ratio \begin{align*}\frac{16}{116}=.138\end{align*} or 13.8%. We divide the amount of increase by the original amount and change to a percent by multiplying by 100 (or moving the decimal point two places to the right). Mingh’s height increased by 13.8%.

What was Charlie’s percent of increase? His height increased by 15cm but his original height was only 95cm. So his percent of increase was \begin{align*}\frac{15}{95}=.158\end{align*} or 15.8%. So although Mingh grew 1cm more than Charlie, Charlie increased by 15.8% while Mingh increased by only 13.8%.

**So we might argue that Charlie grew more than Mingh because his percent of increase was greater.**

**We can find any** *percent of increase***by dividing the amount of increase by the original amount and then multiplying by 100.**

*Write this down in your notebook.*

Now let’s look at another example.

Example

In the last 3 years, the price of gas has risen from an average of $1.89 per gallon to an average of $2.95 per gallon. This is an increase of $1.06 per gallon. What is the percent of increase?

**To solve for this percent of increase, we divide the amount of the increase by the original amount and multiply by 100. In this case, the increase was $1.06.**

\begin{align*}\frac{1.06}{1.89} = .561 = 56.1 \% \ increase\end{align*}

**56.1% is our answer.**

*Notice that we multiply by 100 to convert the decimal into a percent since we are looking for the “percent of increase.”*

II. **Find a Percent of Decrease Given the Original Amount and the Amount of Increase**

Many numbers in the real world increase. You know that many numbers decrease, too. Calculating the percent of decrease will be almost the same procedure as calculating the percent of increase. **The** *percent of decrease***is the percent that a value decreases.**

**To find the** *percent of decrease,***divide the amount of decrease by the original amount and multiply by 100.**

*Write down this information about the percent of decrease in your notebooks.*

Now let’s look at an example.

Example

A helicopter’s altitude went from 350 feet to 38 feet. This was a different of 312 feet. By what percent did the altitude decrease?

\begin{align*}\frac{312}{350} = .891 = 89.1 \%\end{align*}

**The percent of the decrease is 89.1%.**

*Notice that if you haven’t been given the actual amount of the increase or decrease that you may need to subtract to figure this out before you divide.*

III. **Use Percent of Change to Find a New Amount**

We can find the percent of change if we know an original amount and how much it either increased or decreased. At times, however, we are given the percent of increase or decrease and need to calculate a new amount. Let’s look at how we can calculate this new amount.

Example

A restaurant manager has noticed an increase in the cost of utilities of 4%. In order to pay for the increased costs, he decides to increase prices by 4% as well. Not all items are priced the same. The chicken platter currently costs $5.99 and the steak platter cost $7.99. If the prices are increased by 4%, what will the new prices be?

**First, we can notice that we are going to create two new amounts. We are going to create a new cost for the chicken platter, and we are going to create a new cost for the steak platter. We have original amounts and the percent of the increase, so now we need to calculate a new amount.**

**First we must calculate how much the change will be. The prices are increasing by 4% so we must know how much 4% is of each price.**

\begin{align*}& \text{Chicken Platter} && \text{Steak Platter}\\ & \text{Price}: \$ 5.99 && \text{Price}: \$ 7.99\\ & 4 \% \ \text{of price} = 5.99 \times 4 \% && 4 \% \ \text{of price} = 7.99 \times 4 \%\\ & or \ 5.99 \times .04 && or \ 7.99 \times .04\\ & \text{Price change} = .24 && \text{Price change} = .32\end{align*}

Now we know how much each platter’s price is going to change? Since this is an increase, we will add the price change to the price. If it were a decrease, we would subtract the decrease from the price.

\begin{align*}& \text{Previous price}: \quad \$ 5.99 && \text{Previous price}: \quad \$ 7.99\\ & \qquad \qquad \qquad \quad \quad \ \underline{\;\;\; .24 \;} && \qquad \qquad \qquad \quad \quad \ \underline{\;\;\; .32 \;}\\ & \text{New Price} \qquad \quad \ \ \$ 6.23 && \text{New Price} \qquad \quad \ \ \$ 8.31\end{align*}

**Let’s summarize. In order to find the new amount, we calculate the change amount by multiplying the original amount by the percent of change. We then add the change amount to the original amount for an increase or we subtract the change amount from the original amount for a decrease.**

Example

Find the new amount if 60 is decreased by 27%.

amount of change: \begin{align*}60 \times .27 = 16.2\end{align*}

subtract the amount of change from the original amount: \begin{align*}60 - 16.2= 43.8\end{align*}

**The answer is 43.8.**

IV. **Solve Real – World Problems Involving Percents of Change**

Throughout this lesson, you have been working with many real – world problems. Let’s continue working with percents of change in some practical dilemmas. Make sure to understand the situation completely and respond to the question that is being posed.

Example

A pair of sneakers costs $165. They’re on sale for 30% off. What is the sale price of the sneakers?

**First, let’s look at this problem carefully to understand what they are asking us to find. Because the shoes are 30% off, this is a percent decrease.**

**Now we find the amount of change.**

Amount of change: \begin{align*}165 \times .30 = 49.5\end{align*}

**For decrease, we subtract, so we subtract the amount of change from the original amount.**

\begin{align*}165 - 49.5 = 115.5\end{align*}

**The sale price is $115.50.**

Example

A gym’s membership went from 2100 members one year to 2410 members the next. This is a difference of 310 members. What was the percent of change?

**This percent is increasing, so we want to find the percent of the increase. We know the difference so now we can divide and multiply.**

\begin{align*}\frac{310}{2100}=.148=14.8 \%\end{align*}

**The gyms membership increased by 14.8%.**

**Now let’s go back to the problem from the introduction and apply these skills to solving that problem.**

## Real-Life Example Completed

*An Attendance Prediction*

**Here is the original problem once again. Reread it and then figure out the percent of the increase.**

Cameron and Carla decided to bring their proposal of increased attendance to the student council. They thought that if the student council suggested an increase in attendance to the student body that students might really want to help out.

“We know that 100% attendance is nearly impossible, but we could get to 55 or 60%, I think,” Cameron suggested.

“Then we could really show the high school that we support their efforts,” Carla chimed in.

“I think it is a great idea,” Avery, the President of the student council said. “We could conduct a survey and see how many students anticipate attending. Then we could try to predict the number of students who will attend next fall.”

That is exactly what they did. The students conducted a survey and asked students if they would be willing to attend next year’s football games to support the high school team. Out of their survey, they learned that 198 students said they planned on attending. That is an increase from 152 students. What is the percent of the increase?

*Now solve for the percent of increase.*

*Solution to Real – Life Example*

**To solve for the percent of increase, we first need to figure out the difference between the old attendance and the predicted new attendance.**

**The old attendance = 152 students**

**The new attendance = 198 students**

\begin{align*}198 - 152 = 46 \end{align*}

**Next, we put that difference over the original attendance.**

\begin{align*}\frac{46}{152}\end{align*}

**Now we divide.**

.302 = 30.2%

**If the prediction of the students is true, than it would be a 30% increase in attendance.**

## Vocabulary

Here are the vocabulary words found in this lesson.

- Percent
- a part of a whole out of 100.

- Percent of Increase
- the percent of change that a value increased.

- Percent of Decrease
- the percent of change that a value decreased.

## Time to Practice

Directions: Calculate the percent of increase. You may round to the nearest tenth.

- From 7 to 12, an increase of 5
- From 31 to 50, an increase of 19
- From 7805 to 10510, an increase of 2705
- From 16 to 30, an increase of 14
- From 200 to 230, an increase of 30
- From 180 to 200
- From 330 to 400
- From 695 to 1000
- From 1200 to 1500
- From 90 to 120

Directions: Calculate the percent of decrease. You may round to the nearest tenth.

- From 74 to 35, a decrease of 39
- From 4 to 1, a decrease of 3
- From 576 to 476, a decrease of 100
- From 200 to 175, a decrease of 25
- From 150 to 100, a decrease of 50
- From 325 to 290, a decrease of 35
- From 45 to 18, a decrease of 27
- From 19 to 1, a decrease of 18
- From 22 to 10
- From 34 to 20

Directions: Use percent to find the new amount.

- 82 increased by 90%
- 64 decreased by 10%
- 9 increased by 55%
- 25,470 decreased by 77%

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