Question
A population of a town is 20,000. It grows at the rate of 5% per year. Find the population after 3 years.
Solution — Step by Step
Population growth at a fixed annual rate is compound growth — each year’s growth is calculated on the new (increased) population, not just the original. The formula is:
where = initial population, = rate of growth per year, = number of years.
Since population must be a whole number, population after 3 years ≈ 23,153 (or 23,152 if truncating — check what your textbook expects).
Why This Works
Compound growth means the “base” increases every year. After year 1, the population is 21,000 (not still 20,000). In year 2, 5% is calculated on 21,000, giving 22,050. In year 3, 5% is on 22,050, giving 23,152.5.
This is different from simple interest where the rate is always applied to the same base (20,000 each year). With simple growth of 5% per year, the total increase would be , giving only 23,000 — less than compound growth.
Alternative Method — Year by Year
| Year | Population |
|---|---|
| Start | 20,000 |
| After year 1 | |
| After year 2 | |
| After year 3 |
This confirms our answer. The formula is simply a shortcut for doing this multiplication three times.
Common Mistake
The most common error is treating growth as simple (not compound). Students calculate: of per year, then . This is wrong for population growth because each year the base changes. The correct answer, 23,152 or 23,153, is higher because the 2nd and 3rd year growth is on a larger base.
For CBSE Class 8, if the question says “compounded annually” or simply “grows at r% per year,” use the compound formula. If it says “simple growth” or the context is simple interest, use .