A company calculates the total daily revenue, R, in dollars, from selling a product using the expression \(\mathrm{n(24 - n)}\)....
GMAT Advanced Math : (Adv_Math) Questions
A company calculates the total daily revenue, \(\mathrm{R}\), in dollars, from selling a product using the expression \(\mathrm{n(24 - n)}\). In this expression, \(\mathrm{n}\) is the number of units sold that day, and the price per unit decreases as more units are sold. Which of the following best describes the quantity represented by the expression \(\mathrm{(24 - n)}\)?
1. TRANSLATE the problem information
- Given information:
- Total daily revenue: \(\mathrm{R = n(24 - n)}\) dollars
- \(\mathrm{n}\) = number of units sold that day
- Price per unit decreases as more units are sold
- What this tells us: We have a revenue expression with two factors, and we need to figure out what \(\mathrm{(24 - n)}\) represents.
2. INFER the mathematical relationship
- The key insight: Total revenue is always calculated the same way
- \(\mathrm{Revenue = (Number\,of\,Units\,Sold) \times (Price\,per\,Unit)}\)
- Since we're told \(\mathrm{R = n(24 - n)}\), we can match up the parts:
- \(\mathrm{n}\) matches "Number of Units Sold"
- Therefore \(\mathrm{(24 - n)}\) must match "Price per Unit"
3. Verify this makes sense
- The problem states "the price per unit decreases as more units are sold"
- In the expression \(\mathrm{(24 - n)}\): as \(\mathrm{n}\) gets larger, \(\mathrm{(24 - n)}\) gets smaller
- This confirms that \(\mathrm{(24 - n)}\) represents the price per unit
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE reasoning: Students might focus too much on the specific numbers or the decreasing price description and lose sight of the basic revenue structure. They may not clearly translate that \(\mathrm{R}\) represents total revenue calculated as units times price per unit.
This confusion about the fundamental revenue relationship can lead them to misinterpret what \(\mathrm{(24 - n)}\) represents, potentially selecting Choice C (total daily profit) or Choice D (total daily revenue) instead of recognizing it as the price per unit.
The Bottom Line:
This problem tests whether students can connect a given algebraic expression to the fundamental business formula for revenue. The key is recognizing that any revenue expression must follow the pattern (units) × (price per unit), regardless of how complex the individual parts might be.