A city plans to spend exactly $3,000 on park improvements, buying benches and trees.Each bench costs $150, and each tree...
GMAT Algebra : (Alg) Questions
- A city plans to spend exactly $3,000 on park improvements, buying benches and trees.
- Each bench costs $150, and each tree costs $75.
- The relationship between the number of benches x and the number of trees y that can be purchased is represented by a line in the xy-plane that satisfies \(150\mathrm{x} + 75\mathrm{y} = 3{,}000\).
- Which of the following is the best interpretation of the x-coordinate of the line's x-intercept in this context?
1. TRANSLATE the question requirement
- Question asks for: The meaning of the x-coordinate of the x-intercept
- TRANSLATE this to: Find where the line crosses the x-axis (when \(\mathrm{y} = 0\))
- Mathematical action needed: Substitute \(\mathrm{y} = 0\) into the equation
2. SIMPLIFY to find the x-intercept
- Start with: \(150\mathrm{x} + 75\mathrm{y} = 3{,}000\)
- Set \(\mathrm{y} = 0\): \(150\mathrm{x} + 75(0) = 3{,}000\)
- This gives us: \(150\mathrm{x} = 3{,}000\)
- Solve: \(\mathrm{x} = 3{,}000 \div 150 = 20\)
- The x-intercept is \((20, 0)\)
3. INFER the real-world meaning
- At the x-intercept, \(\mathrm{y} = 0\) means zero trees are purchased
- The x-coordinate (20) represents the number of benches
- Real-world interpretation: With the entire $3,000 budget spent on benches only, the city can buy 20 benches
4. APPLY CONSTRAINTS to select the best interpretation
- Check each choice against our finding:
- (A) "Number of benches if no trees bought" ✓ - Matches our interpretation
- (B) "Maximum combined number" - This isn't what intercepts represent
- (C) "Rate of decrease" - This describes slope (\(-150/75 = -2\)), not intercept
- (D) "Cost of one bench" - The cost is $150, not 20
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students often misinterpret what "x-coordinate of the x-intercept" means, particularly confusing it with the slope or another aspect of the line.
They might calculate the slope (\(-150/75 = -2\)) and think this represents how benches decrease when trees increase, leading them to select Choice C (describing the rate of decrease). While the slope calculation is correct, they've answered the wrong question.
Second Most Common Error:
Poor INFER reasoning: Students find the correct x-intercept (20) but struggle to interpret its real-world meaning within the constraint scenario.
They might think the 20 represents some kind of maximum total rather than understanding it's specifically the number of benches when \(\mathrm{y} = 0\) (no trees). This conceptual confusion can lead them to select Choice B (maximum combined number).
The Bottom Line:
This problem tests whether students can connect the abstract mathematical concept of intercepts to real-world constraints. The mathematical calculation is straightforward, but the interpretation requires understanding that intercepts represent specific scenarios where one variable equals zero.