GMATPrep Test Question

Hi G.K,

This prompt is an example of a "limit" question. Conceptually-speaking, when a question asks us for a "minimum" or "maximum" value, we have to think about the OTHER pieces of the equation and what we can do with those pieces. To make one piece really small, we normally have to make the other pieces as big as possible (and vice-versa).

You'll likely be tested on this concept at least once on the Official GMAT, but the prompt will usually be based on some type of basic arithmetic/algebra, so it shouldn't be too hard to "do the math", as long as you're clear on what you're trying to figure out.

GMAT assassins aren't born, they're made,
Rich

G.K wrote:A certain city with population of 132,000 is to be divided into 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?

A. 10,700
B. 10,800
C. 10,900
D. 11,000
E. 11,100

Anytime we are presented with a “minimum value” problem, we must “maximize” all components except for one of them, thus leaving the last component as the “minimized” component of our set.

Let’s use an easy example to test this idea. For instance, we can say that Bob and Frank have a total of 100 apples between them. What is the minimum number of apples that Frank can have? We must “maximize” the number of apples that Bob has; this number is 99. Thus, the minimum number of apples that Frank can have is 1 apple.

Similarly, in this problem we are given 11 voting districts and we must minimize the population of one of those districts. This means that we want to maximize the population of the 10 other districts. We are also given that no district is to have a population that is more than 10% greater than the population of any other district.

Thus, if we label the population of the least populous district as x, we can then say that the maximum population in any other district must be: x + 0.1x = 1.1x. This satisfies the condition that no district has a population that is more than 10% greater than that of any other district.

Because we need to maximize the population of 10 of the 11 districts, all of these 10 districts must have populations of the maximum allowed number, which is 1.1x, and thus, the total population of these 10 districts is (1.1x)(10) = 11x.

We know that the total population of all the districts is 132,000, so we can say:

10 most populous districts + 1 least populous district = 132,000

11x + x = 132,000

12x = 132,000

x = 11,000

Answer: D