### Key Concepts

- Understanding percent change.
- Find percent increase.
- Find percent decrease.
- Find percent error.

## What is percent change?

The percent change explains how much a quantity has changed relative to its original amount. The percent change can be an increase or decrease. The quantity here can be anything like size, weight, profit, loss, etc., to name a few.

The change in percent can be determined using the percent equation.

## Deducing the percent equation:

(part )/(whole) = p/100

(part )/(whole) = percent

(part )/(whole) x whole = percent × whole

part = percent × whole

Therefore, percent equation can be written as part = percent × whole.

**Example 1: **The length of the snake was found to be 32 inches last year. This year on measuring again, it was found to be 40 inches. What is the percent change in snake’s length?

**Solution:**

**Step 1:**

Draw a bar diagram to represent the percent change from last year to this year.

The change in length is 8 inches. The percent change is a percent increase because the snake’s length increased from last year to this year.

Step 2:** **Use the percent equation to find the percent change.

We know that, part = percent × whole

Here we understand that, part = change in length, whole = original length and p = percent change.

Let us take percent as *p*, which we are about to find.

Change in length = percent change × original length.

8 = p × 32

**Divide the equation by 32 on both sides.**

8/32 = p x 32 / 32

8/32 = p

P = 0.25

Express the decimal as a percent by multiplying by 100.

P = 25%

Therefore, the snake’s length increased by 25% this year.

**3.4.1 Find the percent increase**

**Example 1: **Emily scored 72 marks in an assessment last month. This month she scored 90 marks, what is the percent change in Emily’s marks?

**Solution:**

**Step 1:**

Draw a bar diagram to represent the percent change from last month to this month.

The change in marks is 18. The percent change is a percent increase because Emily’s marks increased from last month to this month.

**Step 2: **Use the percent equation to find the percent change.

We know that, part = percent × whole

Here we understand that, part = change in marks, whole = last month marks and p = percent change.

Let us take percent as *p*, which we are about to find.

Change in marks = percent change × last month marks.

18 = p × 90

**Divide the equation by 90 on both sides.**

18/90 = (p × 90)/90

18/90 = p

P = 0.2

Express the decimal as a percent by multiplying by 100.

P = 20%

Therefore, Emily’s marks increased by 20% this month.

**Example 2: **The cost of constructing a well was estimated to be $1200 and later increased to $1800 due to an increase in the depth of the well. What percent change is noticed in the cost?

**Solution:**

**Step 1:**

Draw a bar diagram to represent the percent change from last month to this month.

The change in cost is $600. The percent change is a percent increase because the cost has been increased.

Step 2:** **Use the percent equation to find the percent change.

We know that, part = percent × whole

Here we understand that, part = change in cost, whole = previous estimate and p = percent change.

Let us take percent as *p*, which we are about to find.

Change in cost = percent change × previous estimate.

600 = p × 1800

**Divide the equation by 1800 on both sides.**

600/1800 = (p × 1800)/1800

600/1800 = p

P = 0. 3333

Express the decimal as a percent by multiplying by 100.

P = 33.33%

Therefore, the cost of constructing the well was increased by 33.33%.

### 3.4.2 Find the percent decrease

**Example 1: **A coat was originally priced at $50. It went on sale for $35. What was the percent that the coat was discounted?

**Solution:**

**Step 1:**

Draw a bar diagram to represent the percent change from last month to this month.

The change in cost is $15. The percent change is a percent decrease because the cost of coat is decreased from $50 to $35.

Step 2:** **Use the percent equation to find the percent change.

We know that, part = percent × whole

Here we understand that, part = change in cost, whole = cost before discount and p = percent change.

Let us take percent as *p*, which we are about to find.

Change in cost = percent change × cost before discount.

15 = p × 50

**Divide the equation by 50 on both sides.**

15/50 = (p × 50)/50

15/50 = p

P = 0.3

Express the decimal as a percent by multiplying by 100.

P = 30%

Therefore, the coat was given a discount of 30%.

**Example 2: **A house was purchased in 2010 for $200,000. It is now valued at $125,000. What is the rate (percent) of depreciation for the house?

**Solution:**

**Step 1:**

Draw a bar diagram to represent the percent change from 2010 to present.

The change in cost is $75,000. The percent change is a percent decrease because the price of the house is decreased from $200,000 in the year 2010 to $150,000 presently.

**Step 2: **Use the percent equation to find the percent change.

We know that, part = percent × whole

Here we understand that, part = change in price, whole = cost of the house in 2010 and p = percent change.

Let us take percent as *p*, which we are about to find.

Change in price = percent change × cost of the house in the year 2010.

75000 = p × 200000

**Divide the equation by 200,000 on both sides.**

75000/200000 = (p × 200000)/200000

75000/200000 = p

75/200 = p

P = 0.375

Express the decimal as a percent by multiplying by 100.

P = 37.5%

Therefore, the price of the house is depreciated by 37.5%.

## 3.4.3 Find percent error

**Example 1: **The specification for the width of the aluminum sheet is 4.75 centimeters. A machinist makes a sheet that is 4.769 centimeter in width. What is the percent error of the bolt’s width?

**Solution: **Useabsolute value to find the positive difference between the estimated and actual width of the bolt. Then use the percent equation to find the percent error.

|4.75 – 4.769| = 0.019 centimeter.

We know that, part = percent × whole

Here we understand that, p = percent of error and part = difference in width and whole = estimated width.

0.019 = p × 4.75

**Divide the equation by 200,000 on both sides.**

0.019/4.75 = (p × 4.75)/4.75

0.019/4.75 = p

75/200 = p

P = 0.004.

Express the decimal as a percent by multiplying by 100.

P = 0.4%

Therefore, the error percent is 0.4%.

**Example 2: **The label on a package of bolts says each bolt has a diameter of 0.35 inch. To be in the package, the percent error must be less than 5%. One bolt has a diameter of 0.33 inch. Should it go into the package? Why or why not?

**Solution: **Useabsolute value to find the positive difference between the estimated and actual diameter of the bolt. Then use the percent equation to find the percent error.

|0.35 – 0.33| = 0.02 inch.

We know that, part = percent × whole

Here we understand that, p = percent of error and part = difference in diameter and whole = estimated diameter.

0.02 = p × 0.35

**Divide the equation by 0.35 on both sides.**

0.02/0.35 = (p × 0.35)/0.35

0.02/0.35 = p

P = 0.0571.

Express the decimal as a percent by multiplying by 100.

P = 5.71%

Therefore, the error percent is 5.71%.

## Exercise:

- In school, on Monday, there were 50 students in the class. On Tuesday, it was found to be 75 students. What is the percent change in attendance?
- What is the percent change when a ladder of 15 feet is increased to 25 feet?
- A shopkeeper earned $300 profit in the month of June, next month, his profit increased by $200. What is the percent change in his profit?
- The price of gasoline dipped from $3.71 to $3.50 in one week. By what percent did the gas price dip?
- Skis at a sports store near Snow Summit are on sale for $425. If the original price was $575, what discount rate does this represent?
- A pair of snow boots at an equipment store in Big Bear that originally cost $100 is on sale for $67. What is the rate of discount?
- Brinda estimates the weight of her cat to be 10 pounds. The actual weight of the cat is 13.75 pounds. Find the percent error.
- Marcus estimated that 230 people would attend the choir concert. The actual total that attended the concert was 300 people. Determine the percent error.
- Elena calculates the cost of the book as $50. The actual price was $56. What is Elena’s percent error?
- A manager’s salary at Green Dot increased from $48,000 to $51,360. What is the rate of increase?

### What have we learned?

- Understanding percent change.
- Finding percent increase.
- Finding percent decrease.
- Finding percent error.