In a bio gas plant the population of yeast bacteria increases at a rate of 9%…

Concept: When a quantity grows by a fixed percentage rate r every year and the growth compounds, its value after n years is the starting value multiplied by (1 + r)n.

Concept (cont.): If two growth contributions act on the same population in the same year, their rates add to give one effective annual rate before compounding: net rate = first rate + second rate.

Application:

1. Combine the two annual contributions into one effective rate: 9% (natural growth) + 1% (other inputs) = 10% per annum.

2. Write the annual growth multiplier: 1 + 10/100 = 1.10.

3. Compound over n = 2 years: total multiplier = 1.102 = 1.21.

4. Convert the multiplier to a percentage increase: (1.21 - 1) x 100 = 21%.

Cross-check: Expand step by step on a base of 100. Year 1: 100 -> 100 x 1.10 = 110. Year 2: 110 -> 110 x 1.10 = 121. Net change = 121 - 100 = 21 per 100, i.e. a 21% increase.

Result: The yeast population increases by 21% after 2 years.