# Percent Change

Hello, and welcome to this video about **percent change**! Today we’ll learn how to solve problems involving percent increases and decreases and then we’ll see how this skill can help you in the real world.

Before we get started, let’s review a few things. First, **change** is the difference between two quantities over time. A **percent** is a part-to-whole ratio that’s expressed as a fraction of 100. For example, the percentage 80% means 80 out of \(100:\frac{80}{100}\). In this case, 80 is the part and 100 is the whole.

## How to calculate percentage change?

**Percent change** is the difference between a new quantity and an old quantity, expressed as a percent. Percent change is calculated by finding the difference between the new quantity and the old quantity, and dividing the difference by the old quantity, then you multiply it by 100.

Let’s take a look at an example together.

*The price of a container of laundry detergent went from $8 to $10. What is the percent change?*

First, find the difference between the new quantity and the old quantity by subtracting. Since \(10-8=2\), the difference is 2.

Next, divide the difference, 2, by the old quantity, which is \(8: 2\div 8=0.25\).

Last, convert the decimal into a percentage by multiplying it by 100. \(0.25\times 100=25\), so the percent change is 25%. In other words, the cost of the laundry detergent changed by 25%.

Percent change can be calculated by using this formula:

\(\frac{\text{new}-\text{old}}{\text{old}}\times 100\)

Let’s use this formula to solve a percent change problem.

*A company used to sell 12 oz smoothies but recently upgraded to 16 oz smoothies. What is the percent change between the smaller and larger smoothie sizes?*

First, we identify the old value and the new value in the word problem. The old value, which is the original smoothie size given, is 12 ounces. And the new value is 16 ounces.

From here, substitute these numbers into our formula and solve.

\(\%\text{ change}=\frac{\text{new}-\text{old}}{\text{old}}\times 100\)\(\%\text{ change}=\frac{16-12}{12}\times 100\)

\(\%\text{ change}=\frac{4}{12}\times 100\)

\(\frac{4}{12}\) simplifies to \(\frac{1}{3}\) by dividing both the numerator and denominator by 4.

\(\%\text{ change}=\frac{1}{3}\times 100\)

If you plug \(\frac{1}{3}\) into a calculator, you’ll get an estimate of 0.33. It’s really 0.3, but we can round it to two digits. So we’re going to round to 0.33 and multiply by 100, which will give us 33 percent.

\(\%\text{ change}=0.33\times 100=33\%\)

So our percent change for this problem is 33%.

Now it’s your turn. Consider the following percent change problem:

*A ticket to a basketball game went from $50 to $56. What is the percent change?*

Pause the video here, substitute the values into the formula, and solve. When you’re done, resume the video and we’ll go over it together.

Now that you’ve tried this one on your own, let’s take a look at it together.

First, we want to identify the old value and the new value in the word problem. The old value, which is the original ticket price, is $50. And the new value is $56.

So from here, we’re going to substitute these numbers into our formula and solve. So remember, our percent change formula is:

\(\%\text{ change}=\frac{\text{new}-\text{old}}{\text{old}}\times 100\)

Remember, our new value is the larger price, so 56, and our old value is the original price, which is 50.

\(\%\text{ change}=\frac{56 -50}{50}\times 100\)

\(\%\text{ change}=\frac{6}{50}\times 100\)

Plug \(\frac{6}{50}\) into a calculator to get:

\(\%\text{ change}=\frac{0.12}{100}=12\%\)

So the percent change for this problem is 12%.

A percent change can indicate an increase or a decrease. A **percent increase** means that the new value is greater than the old value. All of the examples we’ve looked at so far have involved percent increases.

A **percent decrease** means that the new value is less than the original value. Let’s take a look at an example together involving percent decrease.

*Mrs. Slater started the school day with 150 pencils in her classroom. At the end of the day, there were only 120 pencils in her classroom. What is the percent change in the number of pencils from the start of the day to the end of the day?*

First, we want to identify the old value and the new value in the word problem. The old value, which is the number of pencils at the beginning of the day, is 150. And the new value, which is the amount of pencils at the end of the day, is 120.

Now, we can substitute these values into our formula and solve:

\(\%\text{ change}=\frac{\text{new}-\text{old}}{\text{old}}\times 100\)

Remember, our old value is the larger number, and the new value is the smaller number, in this one. So our new value is 120 and our old value is 150.

\(\%\text{ change}=\frac{120-150}{150}\times {100}\)

\(\%\text{ change}=\frac{-30}{150}\times {100}\)

This simplifies to:

\(\%\text{ change}=-\frac{1}{5}\times {100}\)

Which, as a decimal is:

\(\%\text{ change}=-0.20\times {100}=-20\%\)

This shows a percent change of -20%.

Let’s take a look at what’s different about this percent change problem. Notice that the new value is less than the old value, so the numerator is a negative number. This negative number results in a negative percentage, indicating that there is a percent decrease from the original value.

In the other percent change problems we’ve looked at today, the results were positive percentages, indicating that there was a percent increase from the original value.

Now it’s your turn. Consider the following percent change problem:

*A coffee shop had 550 customers on Wednesday and 300 customers on Thursday. What is the percent change in the number of customers from Wednesday to Thursday? Is the change a percent increase or decrease? Round to the nearest tenth.*

Pause the video here, substitute the values into the formula, and solve. When you’re done, resume the video and we’ll go over it together.

Now that you’ve tried this one on your own, let’s take a look at it together.

First, identify the old value and the new value in the word problem. The old value, which is the number of customers on Wednesday, is 550. The new value, which is the number of customers on Thursday, is 300.

From here, substitute these numbers into the formula and solve.

\(\%\text{ change}=\frac{\text{new}-\text{old}}{\text{old}}\times 100\)

Remember, our new value in this instance was 300, and our old value was 550.

\(\% \text{ change}=\frac{300-550}{550}\times 100\)

\(\% \text{ change}=\frac{-250}{550}\times 100\)

If we plug this number into a calculator, we’ll get -0.45, so we’re going to round this to -0.455 and then multiply by 100.

\(\% \text{ change}=-0.455\times 100=-45.5\%\)

This gives us a percent change of -45.5%. Since the percent change is negative, it indicates a percent decrease. Therefore, there was a 45.5% decrease in customers at the coffee shop from Wednesday to Thursday.

Great work!

When solving word problems involving percent change, it’s really important to consider the wording of the question being asked. Let’s take a look at why this is necessary.

Consider the following percent change problem.

*A video game usually costs $60, but Max used a coupon to get the game for $51. What is the percent decrease from the original cost to the coupon cost?*

First, we’re going to identify the old value and the new value in the word problem. The old value, which is the original cost of the video game, is $60. And the new value, which is the discounted cost of the game, is $51.

From here, substitute these numbers into the formula and solve. So let’s write out our formula again. Remember, writing out our formula helps us memorize it so it’s always helpful to do.

\(\%\text{ change}=\frac{\text{new}-\text{old}}{\text{old}}\times 100\)

Our new value is 51, because that’s the discounted price of the game, and the old value is 60.

\(\% \text{ change}=\frac{51-60}{60}\times 100\)

\(\% \text{ change}=\frac{-9}{60}\times 100\)

Which can be simplified to:

\(\% \text{ change}=-\frac{3}{20}\times 100\)

Which, when we plug this into a calculator, we get:

\(\% \text{ change}=-0.15\times 100=-15\%\)

So our percent change is -15%. But let’s take another look at the way this question is phrased.

*A video game usually costs $60, but Max used a coupon to get the game for $51. What is the percent decrease?*

This question asks us for what the percent *decrease* is. Notice that this problem is not asking for the percent change. Instead, it’s asking for the percent decrease. Since the word “decrease” already indicates that the percent change is negative, there is no need to include the negative sign in your final answer. So the percent decrease in the cost of the video game is 15%.

Percent change is something you’ll encounter often in the real world. When a quantity changes over a period of time, it can be expressed as percent change. Calculating percent change is a regular occurrence in fields such as finance, physics, and sales.

I have one more problem for you to try. This one is a little more difficult, but I know you can solve it! Consider the following percent change problem.

*On Monday, Starz Movie Theater sold 436 matinee tickets, and on Tuesday, they sold 321 tickets. Because of the matinee special on Wednesdays, Starz sold 590 tickets on Wednesday. What is the percent change from Monday to Tuesday and from Tuesday to Wednesday? Round your answers to the nearest whole number.*

Pause the video here, substitute the values into the formula, and solve to answer both questions. When you’re done, resume the video and we’ll go over it together.

Now that you’ve tried this one on your own, let’s take a look at it together.

Let’s start by finding the percent change from Monday to Tuesday. Identify the old value and the new value in the word problem. The old value, which is the number of tickets sold on Monday, is 436. And the new value, which is the number of tickets sold on Tuesday, is 321.

So, from Monday to Tuesday:

\(\%\text{ change}=\frac{\text{new}-\text{old}}{\text{old}}\times 100\)

So this is equal to the new number, which is 321, minus our old number, which is 436, because that’s the number of tickets sold on Monday.

\(\% \text{ change}=\frac{321-436}{436}\times 100\)

\(\% \text{ change}=\frac{-115}{436}\times 100\)

Which, we can then plug this into our calculator and we’ll get a decimal value that can be rounded to 0.26.

\(\% \text{ change}=-0.26\times 100=-26\%\)

So this gives us a percent change of -26%. Since the percent change is a negative value, it is a percent decrease. So this shows that from Monday to Tuesday, the movie theater had a percent decrease of 26%.

Now let’s find the percent change from Tuesday to Wednesday. Identify the old value and the new value in the word problem. The old value, which is the number of tickets sold on Tuesday, is 321. And the new value, is the number of tickets sold on Wednesday, so 590.

So if we plug this into our formula, we’ll get:

\(\% \text{ change}=\frac{ 590-321}{321}\times 100\)

\(\% \text{ change}=\frac{269}{321}\times 100\)

When we plug \(\frac{269}{321}\) into our calculator, we can round to get a decimal value of about 0.84.

\(\% \text{ change}=0.84\times 100=84\%\)

Therefore, the percent change in ticket sales from Tuesday to Wednesday is 84%. Since the percent change is a positive value, it is a percent increase.

Great work!

I hope this video about percent change was helpful. Thanks for watching, and happy studying!

## Practice Questions

**Question #1:**

What is the percent change from 180 to 189?

8% increase

5% increase

9% increase

5.5% increase

**Answer:**

The percent change formula is \(\text{% change}=\frac{\text{new value-old value}}{\text{old value}}\times 100\). In this case, the new value is 189, and the old value is 180. Substitute these values into the formula.

\(\text{% change}=\frac{\text{new value-old value}}{\text{old value}}\times 100\)

\(\text{% change}=\frac{189-180}{180}\times 100\)

\(\text{% change}=5\%\)

The percent change is positive because the new value is larger than the old value. The scenario represents a 5% increase.

**Question #2:**

What is the percent change from 12.5 to 3.6?

71.9% decrease

78.0% decrease

71.2% decrease

78.4% decrease

**Answer:**

The percent change formula is \(\text{% change}=\frac{\text{new value-old value}}{\text{old value}}\times 100\). In this scenario, 12.5 is the old value, and 3.6 is the new value. Substitute these values into the formula.

\(\text{% change}=\frac{\text{new value-old value}}{\text{old value}}\times 100\)

\(\text{% change}=\frac{3.6-12.5}{12.5}\times 100\)

\(\text{% change}=-71.2\%\)

**Question #3:**

Which set of values shows a 62.5% increase?

7 to 13

8 to 11

7 to 20

8 to 13

**Answer:**

The percent change formula is \(\text{% change}=\frac{\text{new value-old value}}{\text{old value}}\times 100\). Substitute the old value, 8, and the new value, 13, into the percent change formula.

\(\text{% change}=\frac{\text{new value-old value}}{\text{old value}}\times 100\)

\(\text{% change}=\frac{13-8}{8}\times 100\)

\(\text{% change}=62.5\%\)

The result is 62.5%. So, when 8 increases to 13 a 62.5% increase occurs.

**Question #4:**

A fundraiser earns $75 on Saturday and $525 on Sunday. What is the percent change in money earned from Saturday to Sunday?

450% increase

60% increase

500% increase

600% increase

**Answer:**

The percent change formula is \(\text{% change}=\frac{\text{new value-old value}}{\text{old value}}\times 100\). $75 represents the old value, and $525 represents the new value. When the values are substituted into the formula, it becomes:

\(\text{% change}=\frac{\text{new value-old value}}{\text{old value}}\times 100\)

\(\text{% change}=\frac{525-75}{75}\times 100\)

\(\text{% change}=600\%\)

The percent change is positive because the amount of money increases from Saturday to Sunday. The scenario represents a 600% increase.

**Question #5:**

A pair of shoes are on sale. The original price was $120. The new sale price is $90. What is the percent change for the price of the shoes?

25% increase

35% decrease

25% decrease

15% decrease

**Answer:**

The percent change formula is \(\text{% change}=\frac{\text{new value-old value}}{\text{old value}}\times 100\). The old price of the shoes was $120, and the new price of the shoes is $90. Substitute these values into the formula.

\(\text{% change}=\frac{\text{new value-old value}}{\text{old value}}\times 100\)

\(\text{% change}=\frac{90-120}{120}\times 100\)

\(\text{% change}=-25\%\)

The percent change is negative, so the scenario represents a 25% percent decrease.