If the population of a country increases by $20$ percet in 10 years the annual growth rate of population is:
a) more than $2$ percent
b) $2%$ percent
c) less than $2%$ percent
I tried: $120=100(1+r)^{10}$ but how do i solve this further to get $r$ without a calculator?
Any help will be appreciated!
I think the idea, without using a calculator, is that if something grows 2% each year, then after 10 years you would have more total growth than merely $2\%*10years = 20\%$. This is because the growth each year is based on the new larger populations. Since this naive calculation should underestimate the growth, you can infer that the real growth rate must be less than 2% for it to result in 20% growth after 10 years.
$120=100(1+r)^{10}\qquad | :100$
$1.2=(1+r)^{10}\qquad $
Now you can use the binomial expansion
$$(1+r)^{10}=\sum_{k=0}^{10} \binom{10}k r^k$$
For a sufficiently approximation we just add up the first three summands:
$1+10r+45r^2=1.2$
$45r^2+10r-0.2=0$
This is a quadratic equation which can be solved easily.
The relevant solution is about $r^*=0.0184656$
Controlling how accuate the value of $r^*$ is:
$(1+r^*)^{10}=(1+0.0184656)^{10}=1.20078...\approx 1.2$
