A hiker walks uphill at a constant speed and downhill at a different constant speed. On one trail, the hiker...

1. TRANSLATE the problem information

• Given information:
  • Hiker has constant uphill speed and constant downhill speed
  • Trail 1: 1 hour uphill + 4 hours downhill = 21.0 miles total
  • Trail 2: 3 hours uphill + 2 hours downhill = 18.0 miles total
  • Need to find: downhill speed

• Let \(\mathrm{u}\) = uphill speed (mph) and \(\mathrm{v}\) = downhill speed (mph)

2. TRANSLATE each trail into mathematical equations

• Using Distance = Speed × Time for each trail:
  • Trail 1: (1 hour)(\(\mathrm{u}\) mph) + (4 hours)(\(\mathrm{v}\) mph) = 21.0 miles → \(\mathrm{u + 4v = 21.0}\)
  • Trail 2: (3 hours)(\(\mathrm{u}\) mph) + (2 hours)(\(\mathrm{v}\) mph) = 18.0 miles → \(\mathrm{3u + 2v = 18.0}\)

3. SIMPLIFY by solving the system of equations

• We have the system:
  • \(\mathrm{u + 4v = 21.0}\) ... (1)
  • \(\mathrm{3u + 2v = 18.0}\) ... (2)

• From equation (1): \(\mathrm{u = 21.0 - 4v}\)

• Substitute into equation (2):
  \(\mathrm{3(21.0 - 4v) + 2v = 18.0}\)
  \(\mathrm{63.0 - 12v + 2v = 18.0}\)
  \(\mathrm{63.0 - 10v = 18.0}\)
  \(\mathrm{-10v = -45.0}\)
  \(\mathrm{v = 4.5}\)

Answer: C. 4.5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to set up the correct equations from the word problem description. They might confuse which values represent time vs. distance, or incorrectly combine the uphill and downhill portions for each trail. This leads to incorrect equations like "\(\mathrm{u + v = 21.0}\)" (forgetting the time multipliers) or mixing up the trail data, making systematic solution impossible and forcing them to guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the equations but make arithmetic errors during the algebraic manipulation. Common mistakes include sign errors when distributing (\(\mathrm{63.0 - 12v}\) becomes \(\mathrm{63.0 + 12v}\)) or incorrect combination of like terms (\(\mathrm{-12v + 2v = -14v}\) instead of \(\mathrm{-10v}\)). This may lead them to select Choice B (4.2) or other incorrect values.

The Bottom Line:

This problem requires careful translation of two separate motion scenarios into a mathematical system. Students must track multiple pieces of information (two different speeds, two different time periods per trail, two total distances) and organize them systematically before any calculation can begin.