After a manufacturer reduced the unit price of one of its products by , the number of units sold increased by . As a result, the total revenue from the product rose by $2,400 compared with the revenue before the price change. What was the manufacturer's total revenue, in dollars, before the price change?
For percentage word problems, rewrite each percent change as a decimal multiplier. For example, a decrease gives a multiplier of , and a increase gives a multiplier of . When the quantity of interest is a product, such as price multiplied by quantity, multiply the corresponding multipliers to get one overall factor. Then use the stated total increase or decrease to solve an equation such as .
Hints
Turn percentage changes into multipliers
Convert the decrease and increase into decimal multipliers for the original price and quantity.
Express the new revenue using the old revenue
Revenue is price multiplied by quantity. Multiply the new-price multiplier and the new-quantity multiplier to find the factor relating new revenue to original revenue.
Use the stated increase
The $2,400 is the difference between the new revenue and the old revenue. Use the overall revenue factor to write an equation for that difference.
Desmos Guide
Compute the original revenue directly
In Desmos, enter and inspect the numerical output. This computes the original revenue from the overall revenue increase factor.
Step-by-step Explanation
Define the original revenue
Let be the manufacturer's total revenue before the price change.
Revenue equals price multiplied by quantity sold. The original price and number of units do not need to be found separately.
Write each change as a multiplier
A reduction in price means that the new price is of the old price, so the price multiplier is
A increase in units sold means that the new quantity is of the old quantity, so the quantity multiplier is
Therefore, the new revenue is
Find the revenue increase
Multiply the two multipliers:
Thus, the new revenue is , and the increase is
The problem states that this increase is $2,400, so
Solve for the original revenue
Divide both sides by :
Therefore, the manufacturer's total revenue before the price change was $20,000.