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
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
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, 
.
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. CCC’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. ↵
