During a train trip, the train departed at 3:00 PM and arrived between 5:15 PM and 5:45 PM. Which inequality...

1. TRANSLATE the problem information

• Given information:
  • Departure time: 3:00 PM
  • Arrival time: between 5:15 PM and 5:45 PM
  • Need to find: inequality for travel time t in hours

• What this tells us: We need to calculate the minimum and maximum possible travel times

2. SIMPLIFY the time calculations

• Calculate minimum travel time (3:00 PM to 5:15 PM):
  • From 3:00 PM to 5:00 PM = 2 hours
  • From 5:00 PM to 5:15 PM = 15 minutes = \(\frac{15}{60} = 0.25\) hours
  • Minimum time = \(2 + 0.25 = 2.25\) hours

• Calculate maximum travel time (3:00 PM to 5:45 PM):
  • From 3:00 PM to 5:00 PM = 2 hours
  • From 5:00 PM to 5:45 PM = 45 minutes = \(\frac{45}{60} = 0.75\) hours
  • Maximum time = \(2 + 0.75 = 2.75\) hours

3. INFER the inequality structure

• Since the train arrives "between" 5:15 PM and 5:45 PM, the travel time must be between our calculated minimum and maximum values
• This creates a compound inequality: \(\mathrm{minimum} \leq \mathrm{t} \leq \mathrm{maximum}\)
• Therefore: \(2.25 \leq \mathrm{t} \leq 2.75\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly convert minutes to decimal hours, often using 15 minutes = 0.15 hours or 45 minutes = 0.45 hours instead of the correct 0.25 and 0.75 hours respectively.

This leads to incorrect time calculations like 2.15 hours and 2.45 hours, causing them to select Choice A (\(\mathrm{t} \leq 2.5\)) as it seems to encompass their incorrect range.

Second Most Common Error:

Poor INFER reasoning: Students recognize they need to find a range but incorrectly assume they only need an upper bound, focusing on "arrived by" rather than "arrived between."

This causes them to only consider the maximum time and select Choice C (\(\mathrm{t} \leq 2.75\)), missing that the problem specifies a range with both upper and lower bounds.

The Bottom Line:

This problem tests whether students can accurately convert between time units and recognize when a problem requires a compound inequality rather than a simple inequality. The key insight is that "between" times always creates a range, not just an upper limit.