# Module 27: Percents Part 3

**You may use a calculator throughout this module.**

There is one more situation involving percents that often trips people up: working backwards from the result of a percent change to find the original value.

*Finding the Base After Percent Increase*

*Finding the Base After Percent Increase*

Suppose a tax is added to a price; what percent of the original is the new amount?

Well, the original number is of itself, so the new amount must be of the original.

As a proportion, . As an equation, .

If a number is **increased** by a percent, add that percent to and use that result for .

The most common error in solving this type of problem is applying the percent to the new number instead of the original. For example, consider this question: “After a increase, the new price of a computer is $ . What was the original price?”

People often work this problem by finding of $ and subtracting that away: of is , and . It appears that the original price was $ , but if we check this result, we find that the numbers don’t add up. of is , and , not .

The correct way to think about this is . Dividing by gives us the answer , which is clearly correct because we can find that of is , making the new amount . The original price was $ .

To summarize, we cannot subtract from the new amount; we must instead * divide* the new amount by .

Exercises

**1.** A sales tax of is added to the selling price of a lawn tractor, making the total price $ . What is the selling price of the lawn tractor without tax?

**2.** The U.S. population in 2018 was estimated to be million, which represents a increase from 2008. What was the U.S. population in 2008?

*Finding the Base After Percent DEcrease*

*Finding the Base After Percent DEcrease*

Suppose a discount is applied to a price; what percent of the original is the new amount?

As above, the original number is of itself, so the new amount must be of the original.

As a proportion, . As an equation, .

**decreased**by a percent, subtract that percent from and use that result for .

As above, the most common error in solving this type of problem is applying the percent to the new number instead of the original. For example, consider this question: “After a decrease, the new price of a computer is $ . What was the original price?”

People often work this problem by finding of and adding it on: of is , and . It appears that the original price was $ , but if we check this result, we find that the numbers don’t add up. of is , and , not .

The correct way to think about this is . Dividing by gives us the answer , which is clearly correct because we can find that of is , making the new amount . The original price was $ .

To summarize, we cannot add to the new amount; we must instead * divide* the new amount by .

Exercises

**3.** A city department’s budget was cut by this year. If this year’s budget is $ million, what was last year’s budget?

**4.** CCCâ€™s enrollment in Summer 2019 was students, which was a decrease of from Summer 2018. What was the enrollment in Summer 2018? (Round to the nearest whole number.)^{[1]}

**5.** An educational website claims that by purchasing access for $ , you’ll save off the standard price. What was the standard price? (Use you best judgment when rounding your answer.)

- These enrollment numbers don't match those in Percents Part 2, which makes me wonder how accurate the yearly reports are. Or maybe I inadvertently grabbed data from two different ways that enrollment was being counted. ↵