A runner maintains a steady pace, taking an average of 8 minutes to run each kilometer on a treadmill. Which...

1. TRANSLATE the problem information

• Given information:
  • The runner takes 8 minutes to run each kilometer
  • We need a function t(d) for the time to run d kilometers

• This tells us we have a rate: 8 minutes per kilometer

2. INFER the mathematical relationship

• To find total time from a rate, we multiply:

\(\mathrm{Total\ time = Rate \times Distance}\)

• Since rate = 8 minutes per kilometer and distance = d kilometers:

\(\mathrm{t(d) = 8 \times d = 8d}\) minutes

3. Check our function against the answer choices

• (A) \(\mathrm{t(d) = \frac{d}{8}}\) → This would decrease time as distance increases (wrong direction)
• (B) \(\mathrm{t(d) = d + 8}\) → This adds 8 minutes regardless of distance (ignores the rate)
• (C) \(\mathrm{t(d) = d - 8}\) → This could give negative time for small distances (impossible)
• (D) \(\mathrm{t(d) = 8d}\) → This matches our reasoning

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "8 minutes to run each kilometer" and think they need to divide distance by 8 instead of multiply.

They might reason: "If it takes 8 minutes for 1 kilometer, then for d kilometers it takes d/8 minutes." This backwards thinking confuses the relationship between rate and total.

This may lead them to select Choice A (d/8).

Second Most Common Error:

Poor INFER reasoning about rates: Students recognize they need to use 8 and d together but don't understand how rates work with totals.

They might think: "I need 8 and d in my answer" and choose addition or subtraction instead of multiplication, not recognizing that rates require multiplication to find totals.

This may lead them to select Choice B (d + 8) or causes confusion leading to guessing.

The Bottom Line:

Rate problems require understanding that "per unit" rates must be multiplied by the number of units to find totals - this is a fundamental relationship that students often reverse or replace with simpler operations.