A delivery truck travels exactly 315 miles on a fixed route. The driver estimates that the trip takes more than...
GMAT Algebra : (Alg) Questions
A delivery truck travels exactly \(315\) miles on a fixed route. The driver estimates that the trip takes more than \(4.5\) hours but less than \(6.0\) hours, depending on traffic conditions. Which inequality represents the truck's average speed \(\mathrm{s}\), in miles per hour, for this route?
- \(315 - 6.0 \lt \mathrm{s} \lt 315 - 4.5\)
- \(315(4.5) \lt \mathrm{s} \lt 315(6.0)\)
- \(315/6.0 \lt \mathrm{s} \lt 315/4.5\)
- \(315/4.5 \lt \mathrm{s} \lt 315/6.0\)
1. TRANSLATE the problem information
- Given information:
- Distance = 315 miles (constant)
- Time: more than 4.5 hours but less than 6.0 hours
- Need to find inequality for average speed s
- This gives us: \(\mathrm{4.5 \lt time \lt 6.0}\)
2. INFER the key relationship
- Average speed = distance ÷ time = 315 ÷ time
- Key insight: Since distance is fixed, speed and time have an inverse relationship
- When time gets larger, speed gets smaller
- When time gets smaller, speed gets larger
3. APPLY CONSTRAINTS to find the speed bounds
- At the longest possible time (approaching 6.0 hours):
- Speed approaches \(\mathrm{315 ÷ 6.0 = 52.5}\) mph (minimum speed)
- Since time \(\mathrm{\gt 4.5}\), we have \(\mathrm{s \gt \frac{315}{6.0}}\)
- At the shortest possible time (approaching 4.5 hours):
- Speed approaches \(\mathrm{315 ÷ 4.5 = 70}\) mph (maximum speed)
- Since time \(\mathrm{\lt 6.0}\), we have \(\mathrm{s \lt \frac{315}{4.5}}\)
- Combining: \(\mathrm{\frac{315}{6.0} \lt s \lt \frac{315}{4.5}}\)
Answer: C
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Not recognizing the inverse relationship between time and speed
Students may think that since \(\mathrm{4.5 \lt time \lt 6.0}\), the speed inequality should follow the same pattern: \(\mathrm{4.5 \lt s \lt 6.0}\). They fail to apply the division operation and understand that as time increases, speed decreases.
This may lead them to select Choice (D) \(\mathrm{(\frac{315}{4.5} \lt s \lt \frac{315}{6.0})}\) by incorrectly matching the inequality direction.
Second Most Common Error:
Poor TRANSLATE reasoning: Using wrong operations instead of division
Students may think speed involves addition or subtraction rather than division, leading to expressions like \(\mathrm{315 - 6.0}\) or \(\mathrm{315 × 4.5}\).
This may lead them to select Choice (A) \(\mathrm{(315 - 6.0 \lt s \lt 315 - 4.5)}\) or Choice (B) \(\mathrm{(315(4.5) \lt s \lt 315(6.0))}\).
The Bottom Line:
This problem tests whether students can work with inverse relationships in inequalities. The key breakthrough is recognizing that when the denominator (time) increases, the fraction (speed) decreases, which flips the apparent direction of the inequality.