The everyday world is filled with so much data given in the form of **percents** (or **percentages**) that you may never stop to think much about them.

You may understand what is meant by "60 percent of Americans sing off-key." If true, it means that 60 out of every 100, or 3 in every 5, Americans cannot properly carry a tune. But what about percent differences between two data points, or between the same data point at different times?

Percent difference calculations are straightforward but can be tricky when you fail to correctly identify the starting value. This often happens when conveniently round numbers make drawing incorrect inferences especially appealing. For example, if someone told you that his income increased last week by 10 percent because it rose from $90 to $100, you should have a rebuttal prepared.

## What Is Percentage Change?

To find the percent difference between a final value and an initial value, which can also be called percent charge, you first subtract the initial value from the final value, then divide this by the initial value. After you multiply the result by 100 to convert the decimal value into a percentage, you have your final answer.

In the language of math:

Note that percent change can be negative or zero. Use the information in the words of the problem carefully so that you keep initial values and final values straight.

## Percent Difference Calculation: Clothing Sale

A particular kind of blue jeans has become popular so rapidly that their price has shot up from $39 per pair six weeks ago to $99. What is the percent price increase?

From above, you have [(99 − 39)/39] × 100 = (60/39) × 100 = 153.85 percent.

This shows that even though "per cent" means "for each 100," situations exist in which percentages can greatly exceed 100.

- The % symbol is usually reserved for formal scientific and mathematical documents and papers. In everyday use, "percent" is preferred.

As a bonus question, assume the price increased by the same *percentage* each week over the six-week period. What is the value of this percentage?

You may be tempted to notice that the price increased by $60 over six weeks, a steady percentage increase translates to a neat $10 per week. However, this is the right basic approach, but the wrong math. Instead, divide the total percentage increase, not the magnitude of the numerical change, by 6:

153.85 / 6 = 25.64 percent per week.

## Percent Difference Calculation: Mile Run

Say your physical education teacher has all of the students in her classes complete a one-mile run for time at the start of the academic year. The students complete this "diagnostic" run in an average time of 10 minutes. At the end of the spring, she has the members of the class again run an all-out mile, and this time the class average is seven minutes even. What is the percentage improvement (i.e., reduction in time)?

This time, the equation of interest is [(7 − 10)/10] × 100 = −3/10 × 100 = −30 percent.

(The negative sign here is desirable, but this is not always the case.)

Now, assume the school year ends, and some of the students keep exercising over the summer while others discontinue physical activity. On returning to school, this group of students runs a third mile test, and the average of these "slackers" is back up to 10 minutes. What is the percentage decrease in performance compared to the previous spring?

Now the equation is [(10 − 7)/7] × 100 = −3/7 × 100 = 42.9 percent.

Because the initial value for the second part of the problem is 7 rather than 10, the same absolute difference of three minutes creates a larger *percentage* difference.

## Percent Difference Calculation: A Wage Increase

Returning to your friend's boast about his wage increase, you're now prepared to tell him the news is even better than he thought, from the standpoint of percentage differences. Can you calculate the percent increase when moving from 90 to 100?