37% of the items in a box are green. Of those, 37% are also rectangular. Of the green rectangular items,...

1. TRANSLATE the problem information

• Given information:
  • 37% of all items are green
  • Of green items, 37% are also rectangular
  • Of green rectangular items, 42% are also metal
  • Find: percentage that are NOT rectangular green metal items

• What this tells us: We need to work through nested percentages step by step to find a specific subset, then find its complement.

2. INFER the solution approach

• Key insight: We must find the percentage that ARE rectangular green metal items first, then subtract from 100%
• Strategy: Work through the nested conditions sequentially, then find the complement

3. Calculate the specific subset through nested percentages

• Let \(\mathrm{x}\) = total items in box
• Green items: \(37\% \mathrm{\ of\ } \mathrm{x} = 0.37\mathrm{x}\)
• Green rectangular items: \(37\% \mathrm{\ of\ green} = 0.37 \times 0.37\mathrm{x} = 0.1369\mathrm{x}\)
• Green rectangular metal items: \(42\% \mathrm{\ of\ green\ rectangular} = 0.42 \times 0.1369\mathrm{x}\)

4. SIMPLIFY the final calculation

• Green rectangular metal items = \(0.42 \times 0.1369\mathrm{x} = 0.057498\mathrm{x}\) (use calculator)
• Items that are NOT green rectangular metal = \(\mathrm{x} - 0.057498\mathrm{x} = 0.942502\mathrm{x}\)
• As percentage: \((0.942502\mathrm{x}/\mathrm{x}) \times 100\% = 94.2502\%\)

5. Select closest answer choice

• 94.2502% is closest to 94.25%

Answer: C (94.25%)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret the nested conditional statements and calculate \(37\% + 37\% + 42\% = 116\%\), then try to work with this meaningless sum.

They fail to recognize that "37% of those" means 37% of the previous subset, not 37% of the original total. This leads to nonsensical calculations and confusion, causing them to guess randomly among the choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the nested percentages but make calculation errors in the sequential multiplications \(0.37 \times 0.37 \times 0.42\), leading to an incorrect final percentage.

Small errors in intermediate steps compound, and they might calculate something like 6% instead of 5.75% for the specific subset, leading them to select Choice D (98.84%) as the complement.

The Bottom Line:

This problem tests whether students can systematically work through conditional percentages without getting confused by the nested language, and whether they understand that finding "NOT in category X" means finding the complement of category X.