Question 32·Medium·Percentages
After a price increase, the cost of a sweater is now $54. What was the price of the sweater before the increase?
For questions where you know the price after a percent change and must find the original price, first express the relationship as “new price = (1 ± percent as decimal) × original price.” Here, a 20% increase means new price original, so set up and solve by dividing: . As a quick check, make sure your answer is smaller than 54 (since it increased to 54), and see if increasing your answer by 20% actually gives 54.
Hints
Define a variable for the original price
Let represent the price of the sweater before the increase. How can you express the new price in terms of and the 20% increase?
Convert the percent increase to a decimal
A 20% increase means the new price is 120% of the original. What decimal number represents 120%?
Set up and solve the equation
Write an equation where (decimal for 120%) times equals 54, then solve for by dividing both sides by that decimal.
Desmos Guide
Use Desmos to compute the original price
In Desmos, type 54/1.2 into a new expression line. The value that Desmos outputs is the original price of the sweater before the 20% increase.
Step-by-step Explanation
Translate the percent increase into a multiplier
A 20% increase means the new price is 120% of the original price.
In decimal form, 120% is , so:
- new price original price.
Write an equation using the given new price
Let be the original price of the sweater.
The problem says that after a 20% increase, the price is now , so:
Solve the equation for the original price
To find , divide both sides of the equation by :
Compute this division:
So, the price of the sweater before the increase was $45, which corresponds to choice B.