Problem Solving

Brent@GMATPrepNow wrote:

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% 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

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

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% 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

Brent@GMATPrepNow wrote:

mallika hunsur wrote:
Thanks Brent, but how is 1.1x the max value..?I think I'm having trouble understanding that

Thanks,
Mallika

In my solution I let x = the population of the district with the LOWEST population.

We're told that no district is to have a population that is more than 10% greater than the population of any other district
10% of x = 0.1x

So, if x = the population of the district with the LOWEST population, then the population of the district with the GREATEST population = x + 0.1x = 1.1x

Does that help?

Cheers,
Brent

GMATGuruNY wrote:

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% 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

We can plug in the answer choices, which represent the minimum possible population of the least populated district.

To MINIMIZE the smallest population, we need to MAXIMIZE the other 10 populations.
Thus, each of the other 10 districts must have the maximum allowed population: 10% greater than the smallest population.
Since the total population of the city is 132,000 -- a multiple of 1,000 -- the correct answer choice is almost certainly a multiple of 1,000.

Answer choice D: Least populated district = 11,000.
Maximum value of each of the other 10 districts = 11,000 + .1(11,000) = 12,100.
Sum of the 11 districts = 11,000 + 10(12,100) = 132,000.
Success!

The correct answer is D.

Algebraically:
Let x = the population of the least populated district.
As noted above, to MINIMIZE the smallest population, we need to MAXIMIZE the other 10 populations.
Thus, each of the other 10 districts must have the maximum allowed population:
10% greater than the smallest population = 1.1x.
Thus, the sum of the populations in the other 10 districts = 10(1.1x) = 11x.
Since the sum of ALL the populations is equal to 132,000, we get:
x + 11x = 132,000
12x = 132,000
x = 11,000.

DavidG@VeritasPrep wrote:You have to pay close attention to the language of the prompt here. "No district is to have a population that is more than 10% greater than the population of any other district"

If some district had a population of 1.1x + .11x, that population of 1.21x would be more than 10% greater than 'x.' This would be a violation of the problem's conditions.

mallika hunsur wrote: Hi Mitch,

As I understand, x=lowest-this is fine
Next highest=1.1x
and next highest can be no more than 1.1x+.11x and so on..until term 11, as that is when we'll reach the max value for each item or district.

I just don't get how all 10 values are maxed at 1.1