Ok, I'm not sure I fully understand the solution you provided, but this is how I think about it.
Say in 1900, there was a certain population. That amount is 100%, you agree with me? Then, to get the population in 1950, you add 50% f that amount, or, you multiply by 150% (50% added to 100%).
The reverse, is to divide by 150%, right?
Now, each year, you will have to divide by 150%. This is a geometric progression and you can use the general formula:
and convert it to:
Where P_n is the population size after going back n times (that is, n = 50 years),
p the initial population size, in this case 810.
r is the common ratio, here it's 1/150%, or 1/(150/100) = 100/150 = 2/3\
Here is where the 2/3 comes from.
Then, at n times, the population became 160:
Solve for n.
Then, multiply this by 50 years, subtract this from the year 1950.