A small business owner determines that her weekly profit \(\mathrm{P(x)}\) equals $15 times the number of whole products x sold,...
GMAT Algebra : (Alg) Questions
A small business owner determines that her weekly profit \(\mathrm{P(x)}\) equals $15 times the number of whole products \(\mathrm{x}\) sold, minus $180 in overhead costs. If this relationship is graphed with \(\mathrm{x}\) on the horizontal axis and \(\mathrm{P(x)}\) on the vertical axis, what does the x-intercept represent?
The number of products that must be sold to break even
The overhead cost when no products are sold
The profit increases by $1 for every 15 products sold
The profit increases by $15 for every additional product sold
1. TRANSLATE the problem information
- Given information:
- Weekly profit = $15 × (number of products sold) - $180 overhead
- Need to find meaning of x-intercept on the graph
- This gives us the profit function: \(\mathrm{P(x) = 15x - 180}\)
2. INFER what finding the x-intercept means
- The x-intercept occurs where the graph crosses the x-axis
- At the x-axis, the y-value (which is P(x) here) equals 0
- So we need to solve: \(\mathrm{P(x) = 0}\)
3. SIMPLIFY to solve for x
- Set up the equation: \(\mathrm{0 = 15x - 180}\)
- Add 180 to both sides: \(\mathrm{180 = 15x}\)
- Divide by 15: \(\mathrm{x = 12}\)
4. INFER the business meaning
- When \(\mathrm{x = 12}\) products are sold, profit \(\mathrm{P(x) = 0}\)
- Zero profit means the business breaks even (neither gains nor loses money)
- Therefore, the x-intercept represents the break-even point
Answer: A. The number of products that must be sold to break even
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students correctly find \(\mathrm{x = 12}\) but misinterpret what this means in the business context. They might think it represents overhead costs or rate of change instead of recognizing that when profit equals zero, this is the break-even point. This leads to confusion and they may select Choice B ($180 overhead) or Choice D (rate of change).
Second Most Common Error:
Poor TRANSLATE reasoning: Students struggle to convert "15 times the number of products minus 180 overhead" into the correct function \(\mathrm{P(x) = 15x - 180}\). They might set up the wrong equation entirely, leading them to get stuck and randomly select an answer.
The Bottom Line:
This problem tests whether students can connect the mathematical concept of x-intercept to its real-world business interpretation. The algebra is straightforward, but understanding that "profit = 0" means "break-even point" requires solid conceptual reasoning about business contexts.
The number of products that must be sold to break even
The overhead cost when no products are sold
The profit increases by $1 for every 15 products sold
The profit increases by $15 for every additional product sold