A cherry pitting machine pits 12 pounds of cherries in 3 minutes. At this rate, how many minutes does it...

1. TRANSLATE the problem information

• Given information:
  • Machine pits 12 pounds of cherries in 3 minutes
  • Need to find time to pit 96 pounds at same rate

• What this tells us: We have a consistent rate that we can use to find the unknown time

2. INFER the approach

• This is a rate problem - the machine works at a constant speed
• We can solve this using either unit rate or proportional reasoning
• Since we know one rate and need to find time for a different quantity, proportion setup works well

3. TRANSLATE into a proportion

Set up the relationship: \(\frac{12 \text{ pounds}}{3 \text{ minutes}} = \frac{96 \text{ pounds}}{x \text{ minutes}}\)

4. SIMPLIFY using cross multiplication

• Cross multiply: \(12 \times x = 96 \times 3\)
• This gives us: \(12x = 288\)
• Divide both sides by 12: \(x = 288 \div 12 = 24\)

Answer: C. 24

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may set up the proportion incorrectly as \(\frac{3}{12} = \frac{x}{96}\), thinking "small time over large pounds equals unknown time over large pounds."

This flips the rate relationship and leads to: \(3 \times 96 = 12 \times x\), so \(288 = 12x\), giving \(x = 24\). Interestingly, this arithmetic error happens to give the same answer, but the logical setup is wrong.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\frac{12}{3} = \frac{96}{x}\) but make arithmetic errors during cross multiplication or division.

For example, they might calculate \(96 \times 3 = 278\) (instead of \(288\)) or divide \(288 \div 12 = 23\) (instead of \(24\)). This may lead them to select Choice B (15) if they make multiple errors, or cause confusion and guessing.

The Bottom Line:

This problem tests whether students can recognize rate relationships and maintain consistent units throughout their solution. The key insight is that the machine's efficiency stays constant, so the ratio of pounds to minutes remains the same.