Algebra

The population of a town decreases by 4% every year. If its population today is 300,000, what will its population be in two years?

Step-by-step solution with explanation

Final Answer

The population after two years will be .

Step-by-step solution

1

Identify the decay formula to use

Since the population decreases by a fixed percentage each year, we use the exponential decay formula. Here, is the starting population, is the decay rate, and is the number of years.
2

Plug in the known values

The starting population is 300,000, the decay rate is 4% which equals 0.04, and we want the population after 2 years. We substitute all three values into the formula.
3

Simplify the base of the exponent

Subtracting 0.04 from 1 gives us 0.96. This means each year the population keeps 96% of its previous value.
4

Square the decay factor

Multiply 0.96 by 0.96 to get 0.9216. This combined factor accounts for two full years of 4% decrease.
5

Multiply to find the final population

Multiplying 300,000 by 0.9216 gives the population after two years. This is our final answer.

Understanding this problem

Learning Insight

Percent decrease is multiplicative, not additive. Losing 4% two years in a row is NOT the same as losing 8% total — it is actually a loss of 7.84%, because the second year's 4% is applied to an already-smaller number. This compounding effect is the key idea behind exponential decay.

Quick Tip

For repeated percent decrease, just multiply the starting value by . Think of it as: each year you keep a fraction of what you had, and you keep multiplying that fraction together for each year.

Common Mistake

Many students subtract 4% twice in a simple additive way: . This is wrong because it ignores compounding — the second year's decrease must be applied to the reduced population, not the original.