Problem Solving

This is a tricky example of a maximization/minimization problem, a type of problem that seems to be increasingly common on the GMAT. If you haven't solved many problems of this type before, it's worthwhile understanding how to solve a simpler version first. Suppose you are asked:

T is a data set containing five positive integers. If the median of T is 70, and the arithmetic mean of T is 90, what is the smallest possible value of the largest element in T?

We can write our set in increasing order, using the fact that 70 is our median (I'll use S and L to represent the smallest and largest elements)

S, a, 70, b, L

Using the mean, we also know that the sum of our elements is 5*90 = 450. So S + a + 70 + b + L = 450, and S + a + b + L = 380. Now this is the key conceptual point: notice that the larger we make S, a and b, the more of this sum they will 'use up', and the smaller that will force L to be. So to make L as small as possible, we want to make all of the other elements as large as possible. Since S and a cannot be larger than the median, they can be at most 70, and our set looks like:

70, 70, 70, b, L

Now, since the sum of our five elements is 450, we know that b+L = 450 - 3*70 = 240. Since b < L, the largest b can be is 120, and the smallest L can be is 120.

So the key point is this: if you have a fixed sum, then to make one value as small as possible, you want to make all of the other values as large as possible. And in the equally common situation where you want to make one value as large as possible, you'd want to make all of the other values as small as possible.

So in the question above, suppose the least populated district contains s people. Then no district can have more than 1.1s people (no district's population can be more than 10% greater than the smallest district's population). We know that the sum of all district populations is 132,000, and we want to make the smallest district as small as possible. We thus want to make the size of every other district as large as possible, just as in the example I gave above. So we should imagine that all ten of the other districts have exactly 1.1s people. Then when we add the populations of all the districts, we must get 132,000, so we have:

s + 10(1.1s) = 132,000
s + 11s = 132,000
12s = 132,000
s = 11,000