A sample of a radioactive substance has a half-life of 20 years, meaning the amount of the substance reduces by...

1. TRANSLATE the problem information

• Given information:
  • Half-life = 20 years (mass reduces by half every 20 years)
  • Starting year: 2010 with mass = 640 grams
  • Target: Find year when mass = 40 grams

2. INFER the solution approach

• Since we need to go from 640g to 40g, we can either:
  • Track the decay step-by-step through each half-life period, OR
  • Find how many half-life periods using ratios, then calculate total time
• Let's use the ratio method first (it's more efficient)

3. SIMPLIFY to find the mass ratio

• Ratio of final to initial mass: \(\frac{40}{640} = \frac{1}{16}\)
• This tells us the mass needs to be reduced to \(\frac{1}{16}\) of its original amount

4. INFER the number of half-life periods needed

• Since mass halves each period: \((\frac{1}{2})^n = \frac{1}{16}\)
• We need to find what power of \(\frac{1}{2}\) equals \(\frac{1}{16}\)
• Since \(16 = 2^4\), we have \(\frac{1}{16} = \frac{1}{2^4} = (\frac{1}{2})^4\)
• Therefore \(n = 4\) half-life periods

5. Calculate total time and final year

• Total time = \(4 \times 20 = 80\) years
• Final year = \(2010 + 80 = 2090\)

Answer: C (2090)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students might not recognize they need multiple half-life periods and instead think one 20-year period is enough.

They might calculate: \(640 \div 2 = 320\), realize this isn't 40g, but then incorrectly think they need to divide by a different number rather than apply multiple half-life periods. This confusion often leads to guessing rather than systematic solution.

Second Most Common Error:

Poor TRANSLATE reasoning: Students might misread the starting year as something other than 2010, or confuse the target mass.

For example, if they think the starting year is 2030 instead of 2010, they would calculate \(2030 + 80 = 2110\), leading them to select Choice D (2110).

The Bottom Line:

This problem requires recognizing that radioactive decay follows a repetitive halving pattern, and students must systematically apply this pattern multiple times rather than looking for a single-step calculation.