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.