Min/Max Problems, Statistics And Sets Problems

swerve wrote: ↑

Sun Jul 31, 2022 8:47 am

A certain city with a 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. \(11,100\)

The OA is D

Source: GMAT Prep

Let x = the population of the district with the LOWEST population.
To MINIMIZE the population in the smallest district, we must MAXIMIZE the population of the other 10 districts.

IMPORTANT: No other district can exceed x by more than 10%.
So 1.1x = the MAXIMUM population of each of the other 10 districts.

The TOTAL population is 132,000, so we can write:
(population of smallest district) + (population of other 10 districts) = 132,000
Rewrite as: x + [(10)(1.1x)] = 132,000
Simplify: 12x = 132,000
x = 11,000

Answer: D

Cheers,
Brent

swerve wrote: ↑

Sun Jul 31, 2022 8:47 am

A certain city with a 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. \(11,100\)

The OA is D

Source: GMAT Prep

Solution:

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