A lab sample contains 4,800 microorganisms at the start of an observation. The population decreases by 25% each week. Which...

1. TRANSLATE the problem information

• Given information:
  • Initial population: 4,800 microorganisms
  • Population decreases by 25% each week
  • Time variable t is measured in days
  • Need to find equation for N (population after t days)

• Key insight: 'Decreases by 25%' means the population retains 75% each week

2. INFER the basic exponential model

• Since population decreases by a fixed percentage each week, this is exponential decay
• If we measured time in weeks (w), the equation would be: \(\mathrm{N = 4{,}800(0.75)^w}\)
• The decay factor is 0.75 because the population keeps 75% each week

3. INFER the unit conversion strategy

• Problem: Our equation uses weeks, but t is measured in days
• Solution: Convert days to weeks in the exponent
• Since 7 days = 1 week, we have: \(\mathrm{w = t/7}\) weeks

4. TRANSLATE to final equation

• Substitute \(\mathrm{w = t/7}\) into our exponential model:
• \(\mathrm{N = 4{,}800(0.75)^{(t/7)}}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the decrease percentage with the decay factor itself.

They see 'decreases by 25%' and think the decay factor should be 0.25, leading to \(\mathrm{N = 4{,}800(0.25)^{(t/7)}}\). This fundamental misunderstanding of percentage decrease versus retention percentage is a critical error in exponential decay problems.

This may lead them to select Choice C (\(\mathrm{N = 4{,}800(0.25)^{(t/7)}}\))

Second Most Common Error:

Poor INFER reasoning: Students correctly identify the 0.75 decay factor but fail to account for unit conversion.

They write \(\mathrm{N = 4{,}800(0.75)^t}\), treating t as if it represents weeks instead of days. This happens when students don't carefully track what each variable represents or rush through the setup.

This may lead them to select Choice A (\(\mathrm{N = 4{,}800(0.75)^t}\))

The Bottom Line:

This problem requires both careful language interpretation (decrease vs. retention percentages) and attention to unit consistency (days vs. weeks). Success depends on systematically translating each piece of information and ensuring all units align properly.