A person burns calories at a rate of 8 calories per minute while cycling and 12 calories per minute while...

1. TRANSLATE the problem information

• Given information:
  • Cycling rate: 8 calories per minute
  • Rowing rate: 12 calories per minute
  • Time cycling: c hours
  • Time rowing: r hours
  • Total calories burned: 960

2. INFER the approach

• Key insight: The rates are given per minute, but our time variables are in hours
• We need to convert the time units to minutes to match the rate units
• Strategy: Convert hours to minutes, then set up calories equation

3. Convert time units to minutes

• c hours cycling = \(60\mathrm{c}\) minutes
• r hours rowing = \(60\mathrm{r}\) minutes

4. Calculate calories burned from each activity

• Calories from cycling = Rate × Time = \(8 \times 60\mathrm{c} = 480\mathrm{c}\) calories
• Calories from rowing = Rate × Time = \(12 \times 60\mathrm{r} = 720\mathrm{r}\) calories

5. Set up the total calories equation

• Total calories = Calories from cycling + Calories from rowing
• \(480\mathrm{c} + 720\mathrm{r} = 960\)

6. SIMPLIFY to match answer choices

• The equation \(480\mathrm{c} + 720\mathrm{r} = 960\) needs to be simplified
• Find the greatest common divisor: \(\mathrm{GCD}(480, 720, 960) = 240\) (use calculator)
• Divide each term by 240:
  • \(480\mathrm{c} \div 240 = 2\mathrm{c}\)
  • \(720\mathrm{r} \div 240 = 3\mathrm{r}\)
  • \(960 \div 240 = 4\)
• Simplified equation: \(2\mathrm{c} + 3\mathrm{r} = 4\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students miss the unit mismatch between rates (per minute) and time variables (hours). They directly multiply the rates by the hour variables without converting units.

This leads to the equation: \(8\mathrm{c} + 12\mathrm{r} = 960\), which matches Choice C (\(8\mathrm{c} + 12\mathrm{r} = 960\)). While this equation looks reasonable at first glance, it's dimensionally incorrect since it mixes calories per minute with hours.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up \(480\mathrm{c} + 720\mathrm{r} = 960\) but don't simplify it to match the answer format. They look for this exact equation among the choices and don't find it, leading to confusion and guessing between the available options.

The Bottom Line:

This problem tests whether students can maintain dimensional consistency in rate problems. The key insight is recognizing when units don't match and knowing how to convert them properly before setting up equations.