Adam's school is a 20-minute walk or a 5-minute bus ride away from his house. The bus runs once every...

1. TRANSLATE the problem information

• Given information:
  • Walking to school: 20 minutes
  • Bus ride: 5 minutes
  • Waiting for bus: w minutes (between 0 and 30)

• What this tells us:
  • Total walking time = 20 minutes
  • Total bus time = waiting time + ride time = w + 5 minutes

2. INFER what "faster" means mathematically

• "Walking is faster than taking the bus" means:
  • Walking time < Bus time
  • \(\mathrm{20 \lt w + 5}\)

3. Recognize the inequality format needed

• The question asks for values of w where walking is faster
• We have: \(\mathrm{20 \lt w + 5}\)
• Rearranging: \(\mathrm{w + 5 \gt 20}\)

Answer: D. \(\mathrm{w + 5 \gt 20}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students incorrectly think the total bus time is \(\mathrm{w - 5}\) instead of \(\mathrm{w + 5}\), confusing whether waiting time should be added or subtracted from the bus ride time.

They might reason: "If the bus ride is 5 minutes and you wait w minutes, maybe the total is \(\mathrm{w - 5}\) because you're subtracting from your total travel time." This fundamental misunderstanding of how waiting time works leads them to select Choice A (\(\mathrm{w - 5 \lt 20}\)) or Choice B (\(\mathrm{w - 5 \gt 20}\)).

Second Most Common Error:

Poor INFER reasoning: Students correctly identify that total bus time is \(\mathrm{w + 5}\), but set up the inequality backwards, thinking "walking is faster" means \(\mathrm{w + 5 \lt 20}\).

They reason: "If walking is faster, then \(\mathrm{w + 5}\) should be less than 20." This leads them to select Choice C (\(\mathrm{w + 5 \lt 20}\)), which actually gives the values where taking the bus is faster than walking.

The Bottom Line:

This problem challenges students to carefully track what "total time" means for each option and to correctly interpret the direction of inequalities when comparing which option is "faster."