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!

  • $\begingroup$ "The best I got was near 30 when I took the geometric mean but obviously it's incorrect. " Could you show us *how * you took the geometric mean of? You do need to apply a geometric mean but I wonder if you applied it to the wrong numbers. $\endgroup$
    – Silverfish
    Oct 5, 2016 at 18:15
  • $\begingroup$ I presume this a question from a course or textbook? If the latter, you should probably give credit with a reference/citation. Either way, please add the [self-study] tag & read its wiki to see how we handle self-study questions (slightly differently to how we handle most other questions). $\endgroup$
    – Silverfish
    Oct 5, 2016 at 18:18
  • $\begingroup$ well i applied GM on the percentages ((20)(30)(45))^1/3 $\endgroup$ Oct 6, 2016 at 2:50
  • $\begingroup$ I don't know whether the question is from a textbook or not because this is one of the question from within the other questions that our professor has uploaded for our self assessment. $\endgroup$ Oct 6, 2016 at 2:53

1 Answer 1


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

$$1+R = \sqrt[3]{(1+r_{2,3})(1+r_{1,2})(1+r_{0,1})}$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.