England's stats

the answer should be 82 since it is slightly below double at year 81. and we use log for base e not 10. use lg. the solution: let the initial population be 'p'. let R(p) be a function of population that determines the ratio of population between years. hence $R(p)=p+\dfrac{p}{33}-\dfrac{p}{46}=\dfrac{1531p}{1518}$ why is the function like this. first we are obviously having a increasing function R. note the new part( change in population): every 1/33 guy is born->+1/33, but, every 1/46 dies-> -1/46. but note that this is only the change, we also need to add it to the initial population at the start of the year. so: $\begin{array}{c}a year-1:R(p)=\dfrac{1531p}{1518}\\ year-2:R(year-1)=R(\dfrac{1531p}{1518})=(\dfrac{1531}{1518})^2p\\ \cdots \cdots\cdots\\ year-n=R(year(n-1))=(\dfrac{1531}{1518})^np\end{array}$ we see that year as a function of time and population yields a geometric series. we just need $(\dfrac{1531}{1518})^np>2p$ $n=log_{\dfrac{1531}{1518}}(2)=\dfrac{log(2)}{log(1531)-log(1518)}$ plug in values to get $\lceil n\rceil=\boxed{82}$