Question 21·Hard·Percentages
In 2023, a company’s profit was equal to of its revenue. In 2024, the company’s revenue increased by , but its profit margin (profit as a percent of revenue) fell to . If the company’s profit in 2024 was $1.8 million greater than its profit in 2023, what was the company’s revenue in 2023, in millions of dollars? (Round your answer to the nearest tenth.)
For percent word problems like this, first define a variable for the base quantity the question asks about (here, 2023 revenue). Convert all percentages to decimals, and remember that an increase of means multiplying by , not adding a flat amount. Express each year’s profit in terms of the same variable, then use the given difference (here, the $1.8 million increase in profit) to write a simple linear equation and solve. Finally, double-check that the quantity you report (2023 vs. 2024 revenue, profit vs. revenue) matches what the question is asking for.
Hints
Start by choosing a variable
Let a variable represent the 2023 revenue (in millions of dollars). How can you write the 2023 profit using 22% of that variable?
Handle the 18% revenue increase correctly
An 18% increase means the new amount is times the old amount. Express the 2024 revenue in terms of your variable using this idea.
Write the 2024 profit using the 20% margin
Once you have 2024 revenue as an expression, multiply it by 0.20 to get 2024 profit in terms of your variable.
Use the million difference
Set up an equation where (2024 profit) minus (2023 profit) equals 1.8, then solve that equation for your variable.
Desmos Guide
Enter the equation for the profit difference
In a new Desmos line, type the expression for the difference between 2024 and 2023 profits minus 1.8:
y = 0.20*1.18x - 0.22x - 1.8.
Here, represents the 2023 revenue in millions of dollars, and the zeros of this function correspond to the solution of .
Find the x-intercept
Zoom in or adjust the viewing window until you can see where the graph crosses the x-axis (where ). Tap on that intersection point; the x-coordinate shown there is the 2023 revenue in millions of dollars.
Step-by-step Explanation
Define a variable and write the 2023 profit
Let be the company's revenue in 2023, in millions of dollars.
The 2023 profit is 22% of the revenue, so
Express the 2024 revenue and profit in terms of
The revenue in 2024 increased by 18%, so the 2024 revenue is
In 2024, the profit margin is 20%, meaning profit is 20% of revenue. So the 2024 profit is
Use the million profit increase to set up an equation
We are told that the 2024 profit is $1.8 million greater than the 2023 profit.
So:
Subtract the coefficients on the left:
Solve for and interpret the result
Solve for :
Multiply top and bottom by to clear decimals:
So the company’s revenue in 2023 was $112.5 million, which corresponds to choice C.