Question 12·Hard·Percentages
After a manufacturer reduced the unit price of one of its products by 20%, the number of units sold increased by 40%. As a result, the total revenue from the product rose by $24,000 compared with the revenue before the price change. What was the manufacturer's total revenue, in dollars, before the price change?
(Express the answer as an integer)
For percentage word problems, first rewrite each percent change as a decimal multiplier (for example, 20% decrease → , 40% increase → ). For revenue, profit, or similar quantities that are products (like price × quantity), multiply the corresponding multipliers to get a single overall factor on the original amount. Then use the information about how much the total increased or decreased to set up a simple linear equation (like ) and solve quickly, avoiding unnecessary algebra with separate price and quantity variables.
Hints
Turn percentage changes into multipliers
Instead of working directly with percents, convert the 20% decrease and 40% increase into decimal multipliers that you can multiply by the original price and quantity.
Express the new revenue in terms of the old revenue
Think of revenue as price × quantity. After the changes, revenue becomes (new price multiplier) × (new quantity multiplier) × (original revenue). What single number multiplies the original revenue?
Use the to set up an equation
The is the difference between the new revenue and the old revenue. Once you know how many times larger the new revenue is compared to the old, write an equation for that difference and solve for the original revenue.
Desmos Guide
Compute the original revenue directly
In Desmos, type the expression 24000/(0.8*1.4 - 1) and look at the numerical output. This value is the original total revenue before the price change.
Step-by-step Explanation
Define variables for the original situation
Let be the manufacturer's total revenue before the price change.
- Revenue is price × quantity sold.
- The exact original price and number of units are not needed separately; we only need how the product (revenue) changes.
Write the new price and new quantity as multipliers
A 20% reduction in price means the new price is of the old price:
- New price multiplier: .
A 40% increase in units sold means the new quantity is of the old quantity:
- New quantity multiplier: .
So the new revenue is the old revenue times both multipliers:
Find the overall change factor for revenue
Multiply the two multipliers:
So the new revenue is
The increase in revenue is
We are told this increase is dollars, so
Solve for the original revenue
Solve by dividing both sides by :
Compute this division:
So, the manufacturer's total revenue before the price change was $200{,}000.