Question:A laboratory sample initially contains 2,500 bacteria. Each hour, the population decreases by 60% relative to the population at the...

1. TRANSLATE the problem information

• Given information:
  • Initial bacteria population: 2,500
  • Population decreases by 60% each hour
  • Find population after 3 hours

• What this tells us: If population decreases by 60%, then 40% remains each hour (100% - 60% = 40%)

2. INFER the mathematical approach

• This is an exponential decay problem because the population changes by a constant percentage each time period
• We need the exponential decay formula: \(\mathrm{P(t) = P(0) \times r^t}\)
• The decay factor \(\mathrm{r = 0.4}\) (since 40% remains each hour)

3. SIMPLIFY using the formula

• Set up: \(\mathrm{P(3) = 2,500 \times (0.4)^3}\)
• Calculate the exponent: \(\mathrm{(0.4)^3 = 0.4 \times 0.4 \times 0.4 = 0.064}\) (use calculator)
• Final calculation: \(\mathrm{P(3) = 2,500 \times 0.064 = 160}\)

Answer: 160

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "decreases by 60%" and use 0.6 as their decay factor instead of 0.4.

Their reasoning: "The population decreases by 60%, so I multiply by 0.6 each hour."
This leads them to calculate: \(\mathrm{2,500 \times (0.6)^3 = 2,500 \times 0.216 = 540}\)

This causes confusion since 540 doesn't match typical answer patterns, leading to guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that 40% remains (decay factor = 0.4) but make calculation errors when computing \(\mathrm{(0.4)^3}\).

They might calculate \(\mathrm{(0.4)^3}\) incorrectly, such as getting 0.012 instead of 0.064, leading to a final answer of 30 instead of 160.

The Bottom Line:

Success on this problem hinges on correctly translating percentage decrease language into the mathematical decay factor. Students must understand that "decreases by 60%" means "retains 40%," not "multiply by 60%."