2, 10, 3, 7, 6 The mean of the list of numbers above is what fraction of the sum of...

1. TRANSLATE the problem information

• Given information:
  • Five numbers: 2, 10, 3, 7, 6
  • Need to find: "The mean is what fraction of the sum?"
• What this tells us: We need \(\mathrm{Mean/Sum}\) as our final answer

2. INFER the key relationship

• Don't get trapped calculating actual values!
• Key insight: If there are 5 numbers, then \(\mathrm{Mean = Sum \div 5}\)
• This means: \(\mathrm{Mean/Sum = (Sum \div 5)/Sum = 1/5}\)

3. SIMPLIFY the fraction relationship

• \(\mathrm{Mean/Sum = (Sum/5)/Sum}\)
• Multiply by reciprocal: \(\mathrm{(Sum/5) \times (1/Sum) = Sum/(5 \times Sum) = 1/5}\)
• The Sum terms cancel out completely!

Answer: \(\mathrm{1/5}\) (or 0.2 or .2)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students calculate the actual sum (28) and mean (5.6), then struggle to express \(\mathrm{5.6/28}\) as a simple fraction.

They compute: \(\mathrm{5.6/28 = 56/280 = 14/70 = 1/5}\), making the problem much harder than necessary and increasing chances for arithmetic errors. While they might eventually reach the correct answer, they've taken the long, error-prone path instead of recognizing the elegant algebraic relationship.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret the question direction and find \(\mathrm{Sum/Mean}\) instead of \(\mathrm{Mean/Sum}\).

They correctly recognize that \(\mathrm{Mean = Sum/5}\), but then calculate \(\mathrm{Sum/Mean = Sum/(Sum/5) = 5}\), completely missing that the question asks for the opposite ratio. This may lead them to guess or abandon the problem if "5" isn't among the answer choices.

The Bottom Line:

This problem rewards students who can step back from calculations and see the underlying mathematical relationship. The numbers themselves are irrelevant - any five numbers would give the same answer of \(\mathrm{1/5}\).