# 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 a 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, .

A very common error is to find of the new amount and subtract that from the new amount. However, this doesn’t give the correct result. Instead, we must * divide* the new amount by .

If an unknown number is increased by a percent, add that percent to and use that result for the rate.

Exercises

- 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?
- Clackamas Community College’s enrollment in Fall 2023 was students, which was an increase of from Fall 2022. What was the enrollment in Fall 2022?
- While your author and his children were stuck overnight at the Seattle airport, they bought three 20-ounce bottles of Sprite for , but the prices weren’t shown and they weren’t given a receipt. Suspecting that the sales tax rate was from previous experience, your author was able to figure out the cost of one Sprite. Can you?

# Finding the Base After a 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, .

As above, the most common error people make is finding of the new amount and adding that to the new amount, but this doesn’t give the correct result. Instead, we must * divide* the new amount by .

If an unknown number is **decreased** by a percent, subtract that percent from and use that result for the rate.

Exercises

- A city department’s budget was cut by this year. If this year’s budget is million, what was last year’s budget?
- The estimated population of San Francisco in July 2022 was people, which was a decrease of from April 2020. What was the estimated population in April 2020? (Round to the nearest hundred people.)
^{[1]} - An educational website claims that by purchasing access for , you’ll save off the standard price. What is the standard price?