Ty set a goal to walk at least 24 kilometers every day to prepare for a multiday hike. On a...

1. TRANSLATE the problem information

• Given information:
  • Goal: walk at least 24 kilometers
  • Speed: 4 kilometers per hour
  • Find: minimum number of hours needed

• What this tells us: We need to find how much time is required to cover the target distance at the given speed.

2. INFER the mathematical relationship

• This is a distance-speed-time problem
• Use the formula: \(\mathrm{Distance = Speed \times Time}\)
• Since we want "at least 24 kilometers," we need: \(\mathrm{Distance \geq 24}\)
• Let \(\mathrm{s}\) = number of hours walking

3. Set up the inequality

• Distance walked = \(\mathrm{4s}\) kilometers (\(\mathrm{4}\) km/hr \(\mathrm{\times}\) \(\mathrm{s}\) hours)
• To meet the daily goal: \(\mathrm{4s \geq 24}\)

4. SIMPLIFY to solve for s

• Divide both sides by 4: \(\mathrm{s \geq 6}\)
• This means Ty must walk for at least 6 hours

Answer: B. 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might confuse the numbers in the problem, thinking they need to multiply \(\mathrm{24 \times 4}\) instead of dividing \(\mathrm{24 \div 4}\). They might think "24 kilometers at 4 km/hr means \(\mathrm{24 \times 4 = 96}\)" or get confused about which operation to use.

This may lead them to select Choice C (20) or create confusion that leads to guessing.

Second Most Common Error:

Poor INFER reasoning: Students might not recognize this as a distance-speed-time problem or might set up the relationship backwards. They could think "4 hours at 24 km/hr" instead of "24 kilometers at 4 km/hr."

This may lead them to select Choice A (4) or Choice D (24).

The Bottom Line:

This problem requires students to correctly translate a rate word problem into mathematical language and then apply the basic distance-speed-time relationship. The key insight is recognizing that when you know distance and speed, you divide to find time.