1.27: Percents Part 3

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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, $\dfrac{A}{B}=\dfrac{112}{100}$.

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.

$\dfrac{\text{new amount}}{\text{unknown base}}=\dfrac{100+\text{percent}}{100}$

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, $\dfrac{A}{B}=\dfrac{88}{100}$.

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.

$\dfrac{\text{new amount}}{\text{unknown base}}=\dfrac{100-\text{percent}}{100}$

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?

Source: www.census.gov/quickfacts/fact/table/sanfranciscocitycalifornia/PST045222↵

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