The table below shows the production rates of three different manufacturing machines, measured in both parts per minute and parts...

1. TRANSLATE the problem information

• Given information:
  • Table showing production rates for 3 machines in both parts/minute \(\mathrm{(m)}\) and parts/hour \(\mathrm{(h)}\)
  • Need to find the mathematical relationship between \(\mathrm{m}\) and \(\mathrm{h}\)

• What this tells us: We need to find how these two rate measurements relate to each other

2. INFER the approach

• Since we have corresponding values for \(\mathrm{m}\) and \(\mathrm{h}\), we can find their relationship by examining the ratio
• Strategy: Calculate \(\mathrm{h \div m}\) for each machine to see if there's a consistent pattern

3. SIMPLIFY by calculating the conversion factor

• Machine A: \(\mathrm{150 \div 2.5 = 60}\)
• Machine B: \(\mathrm{240 \div 4.0 = 60}\)
• Machine C: \(\mathrm{216 \div 3.6 = 60}\)

• All ratios equal 60, confirming: \(\mathrm{h = 60m}\)

4. APPLY CONSTRAINTS to verify against answer choices

• (A) \(\mathrm{h = 0.017m}\) would make \(\mathrm{h}\) smaller than \(\mathrm{m}\) (doesn't match our data)
• (B) \(\mathrm{h = 60m}\) matches our calculated relationship ✓
• (C) \(\mathrm{m = 60h}\) would make \(\mathrm{m}\) larger than \(\mathrm{h}\) (opposite of what we see)
• (D) \(\mathrm{mh = 60}\) gives \(\mathrm{2.5 \times 150 = 375 \neq 60}\) (doesn't work)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students may not recognize they need to find the conversion factor by calculating ratios. Instead, they might try to work backwards from the answer choices or make assumptions about time conversions.

Without a systematic approach, they may guess or select Choice A (\(\mathrm{h = 0.017m}\)) thinking it looks mathematically sophisticated, or Choice D (\(\mathrm{mh = 60}\)) because 60 appears in the time conversion.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify they need ratios but make arithmetic errors in their divisions, especially with \(\mathrm{216 \div 3.6}\), leading to incorrect conversion factors.

This calculation error causes them to lose confidence in the systematic approach and resort to guessing among the remaining choices.

The Bottom Line:

This problem tests whether students can systematically analyze tabular data to identify mathematical relationships rather than memorizing conversion formulas. The key insight is recognizing that consistent ratios reveal the underlying relationship.