The population of a town is currently 50{,000}, and the population is estimated to increase each year by 3% from...

1. TRANSLATE the problem information

• Given information:
  • Initial population: 50,000
  • Growth rate: 3% increase each year
  • Target population: 60,000
  • Need to find: equation for number of years (t)

• What "3% increase each year" means: Each year the population becomes \(103\%\) of the previous year

2. INFER how exponential growth works

• With exponential growth, we multiply by the same factor each time period
• A 3% increase means we multiply by \(1.03\) each year (not \(0.03\))
• After t years: Population = \(\mathrm{(Initial)} \times \mathrm{(Growth\ factor)}^\mathrm{t}\)

3. TRANSLATE this into our specific equation

• Starting population: 50,000
• Growth factor: \(1.03\)
• After t years: \(50,000(1.03)^\mathrm{t}\)
• We want this to equal 60,000

4. Set up the final equation

• Target population = Initial population × \(\mathrm{(Growth\ factor)}^\mathrm{t}\)
• \(60,000 = 50,000(1.03)^\mathrm{t}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students see "3% increase" and immediately think to use \(0.03\) as the multiplier, forgetting that a 3% increase means the new amount is \(103\%\) of the original.

With this error, they might choose Choice C (\(60,000 = 50,000(0.03)^\mathrm{t}\)), not realizing that multiplying by \(0.03\) would actually cause the population to decrease dramatically each year, not increase.

Second Most Common Error:

Poor INFER reasoning about equation setup: Students might set up the equation backwards, thinking the current population should equal some expression involving the target population.

This leads them to select Choice A (\(50,000 = 60,000(0.03)^\mathrm{t}\)) or Choice B (\(50,000 = 60,000(3)^\mathrm{t}\)), neither of which makes logical sense since the smaller current population cannot equal an expression starting with the larger target population.

The Bottom Line:

The key insight is recognizing that "3% increase" means multiplying by \(1.03\) (which is \(100\% + 3\%\)), not by \(0.03\). This fundamental translation error leads to most wrong answers on exponential growth problems.