In a biogas plant the population of yeast bacteria at a rate of 4% per annum,…

Concept

When a quantity grows by a fixed percentage each period, the growth is compound: each year's increase is computed on the new (larger) total, not on the original. If the single combined annual growth rate is r%, then after n years the quantity is multiplied by (1 + r/100)n, and the overall percentage increase is [(1 + r/100)n − 1] × 100%.

Application

Two separate yearly increases act on the same population, so first combine them into a single annual rate, then compound it over 2 years:

1. Combine the rates: the base growth is 4% per year and the additional input adds 1% per year, giving a single effective rate r = 4% + 1% = 5% per annum.

2. Write the 2-year multiplier: with r = 5%, the factor is (1 + 0.05)2 = (1.05)2.

3. Expand: (1.05)2 = 1 + 2(0.05) + (0.05)2 = 1 + 0.10 + 0.0025 = 1.1025.

4. Convert the multiplier to a percentage increase: (1.1025 − 1) × 100% = 10.25%.

Cross-check

Apply the two successive 5% steps directly. Start with 100 units: after year 1 it becomes 100 × 1.05 = 105; after year 2 it becomes 105 × 1.05 = 110.25. The rise from 100 to 110.25 is 10.25 units, i.e. 10.25%. The extra 0.25% over a naive 5% + 5% = 10% is exactly the compounding term 5% of 5%, which confirms 10.25%.