A machine stamps 28 tokens every 45 seconds at a constant rate.At what rate, in tokens per hour, does the...

1. TRANSLATE the problem information

• Given information:
  • Machine stamps 28 tokens every 45 seconds
  • Rate is constant
• What we need to find: Rate in tokens per hour

2. INFER the solution approach

• Since we have a rate in tokens per 45 seconds but need tokens per hour, we need unit conversion
• Strategy: Multiply the given rate by a conversion factor that changes seconds to hours
• Key insight: \(\mathrm{1\ hour} = \mathrm{3600\ seconds}\), so we need \(\mathrm{(3600\ seconds / 1\ hour)}\) as our conversion factor

3. TRANSLATE into mathematical expression

• Set up the conversion: \(\frac{28\mathrm{\ tokens}}{45\mathrm{\ seconds}} \times \frac{3600\mathrm{\ seconds}}{1\mathrm{\ hour}}\)
• The seconds cancel out, leaving us with tokens per hour

4. SIMPLIFY the calculation

• First calculate \(\mathrm{3600 \div 45 = 80}\)
• Then: \(\mathrm{Rate} = 28 \times 80 = 2240\mathrm{\ tokens\ per\ hour}\)

Answer: 2240

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misunderstand what "tokens per hour" means or don't recognize they need to convert time units.

Some students might try to work with 45 seconds directly without converting to hours, leading to confusion about what calculation to perform. This leads to abandoning systematic solution and guessing.

Second Most Common Error:

Poor INFER reasoning: Students recognize they need conversion but use the wrong conversion factor.

For example, they might divide by 3600 instead of multiply, thinking "there are 3600 seconds in an hour, so I divide." This would give them \(\mathrm{28 \div 45 \times (1/3600) \approx 0.00017}\), which is clearly wrong and causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem tests whether students can systematically convert rates between different time units. The key is recognizing that converting from "per 45 seconds" to "per hour" requires multiplying by how many 45-second periods fit in one hour.