The length of a metal rod contracts by 20% when cooled. If the contracted length is k times the original...

1. TRANSLATE the problem information

• Given information:
  • Rod contracts by 20% when cooled
  • Contracted length = \(\mathrm{k \times original\ length}\)
  • Need to find k

• What this tells us: We need to express the contracted length in terms of the original length

2. INFER the relationship between contraction and final length

• Key insight: A 20% contraction means the rod loses 20% of its original length
• This means the final length is 80% of the original (\(\mathrm{100\% - 20\% = 80\%}\))
• Strategy: Set up the equation using this relationship

3. TRANSLATE and set up the mathematical equation

Let original length = L

• 20% contraction means: \(\mathrm{decrease = 0.20L}\)
• Contracted length = \(\mathrm{L - 0.20L = 0.80L}\)
• Given relationship: \(\mathrm{contracted\ length = k \times original\ length}\)
• Therefore: \(\mathrm{0.80L = k \times L}\)

4. SIMPLIFY to solve for k

• \(\mathrm{0.80L = k \times L}\)
• Divide both sides by L: \(\mathrm{k = 0.80}\)
• Therefore \(\mathrm{k = 0.8}\)

Answer: B (0.8)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse the percentage of contraction with the final ratio

Students see "20% contraction" and immediately think k = 0.2, missing the crucial insight that contraction by 20% means the final length is 80% of the original, not 20%. They incorrectly reason: "The rod contracts by 20%, so the new length is 20% of the original."

This leads them to select Choice A (0.2)

The Bottom Line:

The key challenge is distinguishing between the percentage decrease and the resulting percentage of the original. A 20% decrease leaves you with 80% of what you started with, not 20%.