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.
\(\text{Amount}=\text{Rate}\cdot\text{Base}\)
\(A=R\cdot{B}\)
Finding the Base After a Percent Increase
Suppose a 12% tax is added to a price; what percent of the original is the new amount?
Well, the original number is 100% of itself, so the new amount must be 100% + 12% = 112% of the original.
As a proportion, \(\frac{A}{B}=\frac{112}{100}\).
As an equation, \(A=1.12\cdot{B}\).
A very common error is to find 12% 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 112%.
If an unknown number is increased by a percent, add that percent to 100% and use that result for the rate.
\(\frac{\text{new amount}}{\text{unknown base}}=\frac{100+\text{percent}}{100}\)
Exercises
- A sales tax of 8% is added to the selling price of a lawn tractor, making the total price $1,402.92. What is the selling price of the lawn tractor without tax?
- Clackamas Community College’s enrollment in Fall 2023 was 17,605 students, which was an increase of 18.465% 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 $11.86, but the prices weren’t shown and they weren’t given a receipt. Suspecting that the sales tax rate was 10.1% 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 12% discount is applied to a price; what percent of the original is the new amount?
As above, the original number is 100% of itself, so the new amount must be 100% − 12% = 88% of the original.
As a proportion, \(\frac{A}{B}=\frac{88}{100}\).
As an equation, \(A=0.88\cdot{B}\).
As above, the most common error people make is finding 12% 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 88%.
If an unknown number is decreased by a percent, subtract that percent from 100% and use that result for the rate.
\(\frac{\text{new amount}}{\text{unknown base}}=\frac{100 – \text{percent}}{100}\)
Exercises
- A city department’s budget was cut by 5.00% this year. If this year’s budget is $3.04 million, what was last year’s budget?
- The estimated population of San Francisco in July 2022 was 808,400 people, which was a decrease of 7.50% 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 $5, you’ll save 69% off the standard price. What is the standard price?
