# 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

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. 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?
3. 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

1. A city department’s budget was cut by this year. If this year’s budget is million, what was last year’s budget?
2. 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]
3. An educational website claims that by purchasing access for , you’ll save off the standard price. What is the standard price?