The population of a country increased by 20% in the first decade, 30% in the second decade and 45% in the third decade. What is the average rate of increase per decade in the population?
The answer is 31.28. The best I got was near 30 when I took the geometric mean $(20\times30\times45)^{1/3}$ but obviously it's incorrect. Please explain briefly your answers thank you!
Suppose an initial (i.e. at time zero) population $P_0$ rises by $25\%$ so one year later is $P_1$. What is the relationship between $P_1$ and $P_0$? Since $P_1$ is $25\%$ higher than $P_0$, i.e. has added on $25\%$ of the value of $P_0$ onto $100\%$ of the value of $P_0$, then $P_1$ must be $125\%$ of the value of $P_0$. Written as a decimal, we can say that $P_1 = 1.25 P_0$. This factor of $1.25$ is called — among other names — the scale factor, decimal multiplier or decimal growth factor.
Suppose from year one to year two the population rises by $33.1\%$, then we could write (as a decimal) that the growth rate is $r_{1,2} = 0.331$ (here I am writing subscripts to indicate this is the growth from period one to period two) while the decimal multiplier is $1.331$; the decimal multiplier is simply $1 + r_{1,2}$. If in the third year the population falls by $20\%$, then we would write the growth rate as $r_{2,3}=-0.2$ and the decimal multiplier as $1 + r_{2,3} = 0.8$. Note that a decimal multiplier below one indicates a reduction rather than growth; from time two to time three the population has fallen from $100\%$ of its value at time two to $80\%$ of its value at time two.
What is the overall effect of these changes? Well $P_3 = 0.8P_2$, $P_2 = 1.331P_1$ and $P_1 = 1.25P_0$ so we can write that:
$$P_3 = 0.8 \times 1.331 \times 1.25 \times P_0 = 1.331 P_0$$
So we see the overall decimal multiplier is $1.331$ which indicates a $33.1\%$ increase from year zero to year three. (Effectively the $20\%$ decrease and the $25\%$ increase have cancelled each other out. This might surprise you since the $25\%$ increase sounds "bigger", but if you try a few numbers you will see that e.g. a $10\%$ increase followed by a $10\%$ decrease, or vice versa, actually results in a $1\%$ reduction overall. This is because the $10\%$ fall and $10\%$ rise were not $10\%$ of the same quantity, so will not cancel out. Note that $0.9 \times 1.1 = 1.1 \times 0.9 = 0.99$.)
Suppose there were an "average growth rate" $R$ such that the effect of that growth rate, if it acted over three years, were the same as the effect of these individual growth rates. Then we would need:
$$P_3 = (1+R)(1+R)(1+R)P_0 = (1+R)^3 P_0 = 1.331 P_0$$
From which we deduce $1+R = 1.331^{1/3} = 1.10$ and hence $R = 10\%$. We are applying the geometric mean to the decimal multipliers, not to the percentage changes themselves.
In general, if we seek an $R$ such that
$$P_3 = (1+r_{2,3})(1+r_{1,2})(1+r_{0,1})P_0 = (1+R)^3 P_0$$
then we must take
$$1+R = \sqrt[3]{(1+r_{2,3})(1+r_{1,2})(1+r_{0,1})}$$
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