The population of a certain species of fish in a lake is decreasing exponentially. The population decreases by 50% every...

1. TRANSLATE the problem information

• Given information:
  • Population decreases by 50% every 10 years
  • Population in 2040 = 600 fish
  • Need population in 2020

• What this tells us: Every 10 years, the population becomes 50% of what it was (multiplies by 0.5)

2. INFER the time relationship and strategy

• Time difference: 2040 - 2020 = 20 years
• Number of 10-year periods: \(20 \div 10 = 2\) periods
• Key insight: Since we're working backwards in time, we need to reverse the decay process

3. INFER the mathematical relationship

• Forward in time: Population decreases by factor of 0.5 each period
• Backward in time: Population increases by factor of \(1 \div 0.5 = 2\) each period
• Alternative approach: Use the decay formula in reverse

4. SIMPLIFY the calculation

• Population in 2020 = Population in 2040 ÷ \((0.5)^2\)
• Population in 2020 = \(600 \div 0.25\)
• Population in 2020 = \(600 \times 4 = 2,400\)

Answer: C) 2,400

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students apply the decay factor in the wrong direction, multiplying by 0.5 instead of dividing by it when working backwards in time.

They think: "If it decreases by 50% every 10 years, then 20 years ago it was \(600 \times (0.5)^2 = 600 \times 0.25 = 150\)." This backwards reasoning leads to an answer much smaller than any choice, causing confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret "decreases by 50%" as meaning the population becomes 0% (completely dies out) or they confuse the direction of time.

Some students might think the population in 2020 should be smaller than in 2040, not recognizing that we're looking for the larger past population that decayed to the smaller 2040 value. This conceptual confusion may lead them to select Choice A (1,200) as it seems like a reasonable "smaller" number.

The Bottom Line:

The key challenge is understanding that exponential decay problems can work in reverse - if you know a future value, you can find the past value by "undoing" the decay process. This requires both strong translation skills to set up the relationship correctly and solid inferential reasoning to apply the math in the right direction.