A certain city with a population of 132000 is to be divided in to 11 voting districts and no district is to have a population that is more than 10% percent greater than the population of any other district. What is the minimum possible population that the least populated district could have.

Answer: 11000

ok, so here's the story (which is also easily the most important takeaway you can glean from this problem):
if you want to minimize some quantity in a problem, the best way to do that is to maximize the other quantities in the problem.

in this case, if you want to minimize the population of the least populated district, you want to maximize the population of ALL the other districts. this means that the minimally populated district should have population 'x' and EACH of the other districts should have population '1.1x' (i.e., 10% greater than that of the minimally populated district, as stipulated).
this means that you have one district with population 'x', and ten districts with population '1.1x'.
therefore, x + 10(1.1x) = 132,000
12x = 132,000
x = 11,000
there you go.

again, make sure you get the generic takeaway listed at the top of this post. if you realize this rather simple fact - the way to minimize something is to maximize everything else, and the way to maximize something is to minimize everything else - you'll be well on your way to solving just about every optimization problem you can get your hands on.