A sample of a radioactive substance initially has a mass of 80 grams. The mass of the substance decreases by...

1. TRANSLATE the problem information

• Given information:
  • Initial mass: 80 grams
  • Mass decreases by 5% each year
  • Target mass: 10 grams
  • Need to find: equation for number of years y

• What "decreases by 5% each year" means mathematically:
  • Each year, the substance retains 95% of its mass \(100\% - 5\% = 95\%\)
  • This gives us a decay factor of \(0.95\)

2. INFER the approach

• This describes exponential decay (mass decreases by a constant percentage each time period)
• We need the exponential decay formula: \(\mathrm{A = P(decay\ factor)}^t\)
• The decay factor is what remains each year: \(0.95\)

3. TRANSLATE into the equation format

• Using \(\mathrm{A = P(decay\ factor)}^t\):
  • A = 10 (final mass)
  • P = 80 (initial mass)
  • decay factor = 0.95
  • t = y (years)

• Substituting: \(10 = 80(0.95)^y\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the decay rate \(5\% = 0.05\) with the decay factor \(0.95\).

They think "decreases by 5%" means they should use 0.05 in the equation, leading to \(10 = 80(0.05)^y\). They don't realize that if something decreases by 5%, then 95% remains each year.

This leads them to select Choice A \((10 = 80(0.05)^y)\).

Second Most Common Error:

Conceptual confusion about growth vs. decay: Students see "5%" and think about growth rather than decay.

They incorrectly add the 5% to get 1.05, thinking the substance is growing rather than shrinking. This fundamental misunderstanding of the problem setup occurs when they don't carefully read "decreases by 5%."

This may lead them to select Choice C \((10 = 80(1.05)^y)\).

The Bottom Line:

The key challenge is correctly translating percentage decrease language into the mathematical decay factor. Students must understand that "decreases by 5%" means 95% remains, not that 5% remains.