Maya is training for a triathlon and plans to alternate between running and cycling during her workout. She intends to...

1. TRANSLATE the problem information

• Given information:
  • Running speed: 8 kilometers per hour
  • Cycling speed: 12 kilometers per hour
  • Equal time spent on each activity
  • Total distance needed: at least 30 kilometers
  • Find: minimum running time

2. INFER the mathematical approach

• Since Maya spends equal time on each activity, we can use the same variable for both time periods
• Let \(\mathrm{t}\) = time spent running = time spent cycling
• We'll use \(\mathrm{distance = speed \times time}\) to find total distance, then set up an inequality

3. TRANSLATE and calculate distances

• Distance from running = \(\mathrm{8 \times t = 8t}\) kilometers
• Distance from cycling = \(\mathrm{12 \times t = 12t}\) kilometers
• Total distance = \(\mathrm{8t + 12t = 20t}\) kilometers

4. APPLY CONSTRAINTS with the distance requirement

• 'At least 30 kilometers' means: \(\mathrm{20t \geq 30}\)
• SIMPLIFY: Divide both sides by 20
• \(\mathrm{t \geq 1.5}\)

Answer: A (1.5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret 'at least 30 kilometers' as exactly 30 kilometers, setting up \(\mathrm{20t = 30}\) instead of \(\mathrm{20t \geq 30}\).

While this still gives \(\mathrm{t = 1.5}\), they miss the conceptual understanding that this is the minimum time required. This can cause confusion about whether 1.5 is the exact answer or just a boundary, potentially leading to second-guessing and selecting Choice B (2.0) as a 'safer' option.

Second Most Common Error:

Poor INFER reasoning: Students don't recognize that equal time periods means they can use the same variable for both activities. Instead, they create separate variables (like r for running time, c for cycling time) and get stuck trying to solve a system with insufficient information.

This leads to confusion and abandoning systematic solution, often resulting in guessing among the answer choices.

The Bottom Line:

This problem tests whether students can translate a word problem with constraints into mathematical inequalities and recognize when equal conditions allow variable consolidation.