Question 359·Medium·Math

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After a $20\%$ price increase, the cost of a sweater is now $54. What was the price of the sweater before the increase?

In a reverse percent change question, the price after an increase is given, but the price before it is missing. Keep the original $100\%$ , add the increase, and use a Desmos regression to find the original price. Taking the stated percent off the new price is tempting, but that percent was measured against the old price.

Hints

Hint 1
A percent increase is measured against the price before the change. You keep the full $100\%$ of that price and add $20\%$ more. What percent of the original is the new price?
Hint 2
Call the price before the increase $x$ . The new price is a percent of $x$ , not a percent of $54$ . What equation says that the new price is $54$ ?

Step-by-step

Work backward from the new price
Step 1
Find the percent of the original price
The sweater keeps its full original price and gains $20\%$ . Add the increase to the original $100\%$ :
$100\%+20\%=120\%.$
The increase is a percent of the original price, not the new price.
Step 2
Write the price relationship
Let $x$ be the price before the increase. The sweater now costs $54, so $120\%$ of $x$ equals $54:
$120\%\text{ of }x=54.$
Step 3
Solve for the original price
Type $120\%\operatorname{of}x_1\sim54$ in Desmos. The subscript in $x_1$ tells Desmos to fit a value rather than graph $x$ , and $\sim$ tells it to find a value that makes both sides match. Under PARAMETERS, Desmos shows $x_1=45$ . So the sweater cost $45 before the increase. Choice B.

Question 359·Medium·Math

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