# Percent of Change: Definition, Formula & Examples

Instructor: Joseph Vigil
In this lesson, you'll learn what percent of change is and discover the formula to determine percent of change. You'll also see the formula in action in a few examples. Then, you can test your new knowledge with a brief quiz.

## Shoes on Sale

Dan's had his eye on a new pair of shoes for a while. He's been waiting because, at \$100, they're a little pricey for him. They're on sale now, though, for \$75. If he bought the shoes now, what percentage of the original price is he saving?

Before Dan figures out the percentage of the original price he's saving, he'll have to figure out the difference between the sale price and the original price.

The original price was \$75, and the sale price is \$100.

75 - 100 = -25

Although the difference is negative, this makes sense because there was a drop in price, so it's a negative difference.

But Dan wants to know what percentage of the original price he's saving. In other words, \$25 is what percentage of \$100?

To find that percentage, Dan will need to divide -25 by 100, which gives him a quotient of -0.25. That's a decimal number, though. To convert a decimal number to a percentage, Dan will need to multiply it by 100 because percent of change is a comparison between two values expressed in hundredths.

-0.25 * 100 = -25

So, in comparison to the shoes' original price, the sale price is 25% lower. Not bad! We know the sale price is lower because Dan's final answer was negative, indicating a decrease.

If his answer had been positive, that would have indicated an increase in price.

## Building the Formula

Let's slow down Dan's process and see what he did, step by step.

Step 1: Subtract the original value from the second value.

In this case, the original value was the shoes' original price (\$100), and the second value was the sales price (\$75).

When he subtracted the original value from the second value, he got a difference of -25. That's because there was a decrease between the first and second values. If there had been an increase, Dan would have gotten a positive difference between the two.

We can write this step as a formula:

P = s - f

Where P is percent of change, s is the second value, and f is the first value.

Step 2: Divide the difference between the values by the original value.

Because percent of change is a comparison between the first and second values expressed in hundredths, Dan divided the difference by the original value (-25 / 100).

In essence, he took s - f from the previous step and divided it by f. So our formula now looks like this:

P = (s - f) / f

Step 3: Multiply the quotient by 100.

The quotient from step 2 was a decimal number (-0.25). But percentage is expressed in hundredths, so Dan needed to multiply that quotient by 100.

He took our formula so far and multiplied it by 100. So it looks like this now:

P = 100 (s - f) / f

That finally gave Dan a solution of -25%, or a decrease of 25%. Again, if the final solution had been positive, the change would have been an increase rather than a decrease.

## Speeding Things Up

Sandra is driving home from college for summer break. The speed limit was 45 miles per hour until she hit the highway, where it became 60 miles per hour. What is the percent of change between the two speeds?

Our original value here is 45, and our second value is 60. We have all we need to calculate the percent of change.

Remember, our formula for percent of change is

P = 100 (s - f) / f

Where s is the second value and f is the first value. Let's plug those values in!

P = 100 (60 - 45) / 45

P = (100 * 15) / 45

P = 1500 / 45

P = 33.3

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