Lena lives 1.0 mile from the library. She can either jog there at a constant speed of 6 miles per...

1. TRANSLATE the problem information

• Given information:
  • Lena lives 1.0 mile from library
  • Jogging speed: 6 mph (constant)
  • Shuttle arrives every 15 minutes
  • Boarding time: 2 minutes
  • Shuttle travel speed: 30 mph
  • Wait time: w minutes \(\mathrm{(0 ≤ w ≤ 15)}\)
• What this tells us: We need to compare total travel times for both options

2. INFER the approach

• "Jogging is faster" means jogging takes less time than the shuttle
• We need to calculate both times, then set up an inequality
• The shuttle time has three components: waiting + boarding + traveling

3. TRANSLATE and calculate jogging time

• Using \(\mathrm{Time = Distance ÷ Rate}\):
• Jogging time = \(\mathrm{1.0\text{ mile} ÷ 6\text{ mph} = \frac{1}{6}\text{ hour}}\)
• Convert to minutes: \(\mathrm{\frac{1}{6} × 60 = 10\text{ minutes}}\)

4. TRANSLATE and calculate shuttle time

• Wait time: w minutes (given)
• Board time: 2 minutes (given)
• Travel time = \(\mathrm{1.0\text{ mile} ÷ 30\text{ mph} = \frac{1}{30}\text{ hour}}\) = \(\mathrm{\frac{1}{30} × 60 = 2\text{ minutes}}\)
• Total shuttle time = \(\mathrm{w + 2 + 2 = w + 4\text{ minutes}}\)

5. INFER the inequality setup

• Jogging is faster when: jogging time \(\mathrm{\lt}\) shuttle time
• This gives us: \(\mathrm{10 \lt w + 4}\)

6. SIMPLIFY the inequality

• \(\mathrm{10 \lt w + 4}\)
• Subtract 4 from both sides: \(\mathrm{10 - 4 \lt w}\)
• \(\mathrm{6 \lt w}\)
• Rewrite: \(\mathrm{w \gt 6}\)

Answer: (A) \(\mathrm{w \gt 6}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students incorrectly set up the inequality direction, thinking "jogging is faster" means jogging time \(\mathrm{\gt}\) shuttle time instead of jogging time \(\mathrm{\lt}\) shuttle time.

They reason: "If jogging is faster, then the jogging time should be the bigger number." This backward logic leads them to write \(\mathrm{10 \gt w + 4}\), which simplifies to \(\mathrm{w \lt 6}\).

This may lead them to select Choice (B) (\(\mathrm{w \lt 6}\)).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{10 \lt w + 4}\) but make an algebra error when isolating w.

They might subtract 4 from the left side only, getting \(\mathrm{6 \lt w + 4}\), then subtract 4 again to get \(\mathrm{6 \lt w}\), which is actually correct. Or they might add 4 to both sides instead of subtracting, getting \(\mathrm{14 \lt w + 8}\), leading to confusion about the final answer.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem requires careful attention to what "faster" means mathematically (less time, not more time) and systematic organization of the multiple time components for the shuttle option.