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.

\text{Amount}=\text{Rate}\cdot\text{Base}

A=R\cdot{B}

Finding the Base After 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}.

If a number is increased by a percent, add that percent to 100\% and use that result for R.

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 12\% increase, the new price of a computer is $ 1,120. What was the original price?”

People often work this problem by finding 12\% of $ 1,120 and subtracting that away: 12\% of 1,120 is 134.40, and 1,120-134.40=985.60. It appears that the original price was $ 985.60, but if we check this result, we find that the numbers don’t add up. 12\% of 985.60 is 118.272, and 985.60+118.272=1,103.872, not 1,120.

The correct way to think about this is 1,120=1.12\cdot{B}. Dividing 1,120 by 1.12 gives us the answer 1,000, which is clearly correct because we can find that 12\% of 1,000 is 120, making the new amount 1,120. The original price was $ 1,000.

To summarize, we cannot subtract 12\% from the new amount; we must instead divide the new amount by 112\%.

Exercises

1. 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?

2. The U.S. population in 2018 was estimated to be 327.2 million, which represents a 7.6\% increase from 2008. What was the U.S. population in 2008?

Finding the Base After 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}.

If a number is decreased by a percent, subtract that percent from 100\% and use that result for R.

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 12\% decrease, the new price of a computer is $ 880. What was the original price?”

People often work this problem by finding 12\% of 880 and adding it on: 12\% of 880 is 105.60, and 880+105.60=985.60. It appears that the original price was $ 985.60, but if we check this result, we find that the numbers don’t add up. 12\% of 985.60 is 118.272, and 985.60-118.272=867.328, not 880.

The correct way to think about this is 880=0.88\cdot{B}. Dividing 880 by 0.88 gives us the answer 1,000, which is clearly correct because we can find that 12\% of 1,000 is 120, making the new amount 880. The original price was $ 1,000.

To summarize, we cannot add 12\% to the new amount; we must instead divide the new amount by 88\%.

Exercises

3. A city department’s budget was cut by 5\% this year. If this year’s budget is $ 3.04 million, what was last year’s budget?

4. CCC’s enrollment in Summer 2019 was 9,116 students, which was a decrease of 2.17\% 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 $ 5, you’ll save 69\% off the standard price. What was the standard price? (Use you best judgment when rounding your answer.)


  1. 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.

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