Problem Solving

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

Can someone give a little light to this problem?

Thanks.

If equal distribution was to be done, then 12000 individuals would be in each district.
To find the minimum number, we need to reduce individuals from 12000. And to be able to reduce the maximum (to reach the minimum possible for a district), we should divide the number reduced from one district equally among the remaining districts.

This leads to 11,000 as the answer.

Please comment and let me know if you think otherwise.

Sorry sethids,

Can you deduce it algebrically? I must confess that I didn't get your point. We could test the solutions, but this will take at least 5min.

Thanks

sethids wrote:If equal distribution was to be done, then 12000 individuals would be in each district.
To find the minimum number, we need to reduce individuals from 12000. And to be able to reduce the maximum (to reach the minimum possible for a district), we should divide the number reduced from one district equally among the remaining districts.

This leads to 11,000 as the answer.

Please comment and let me know if you think otherwise.

If answer is 11,000 as you say, then by your logic (that is, to equally distribute the remaining total population among the remaining districts), the other districts have a population of
(132000 - 11000)/(11-1) = 121000/10 = 12100

That means all other 10 districts have populations that are 10% greater than the district with only 11,000. This doesn't satisfy the question requirements.

However, this leads me to eliminate all answers from D and up, and just be left with E.

Thanks for the correction.
This certainly leads to 11,100 as the answer.

Here's how you do it algebraically:

First of all, we need to realize that for the minimum to occur, 10 towns will have to have the same number of people in it and 1 will have to be the smaller population.

let x be the number of people in the small town:

10(x+.1x) + x = 132000 --- the .1x is there because we know that the 10 towns will have to be 10% larger

11x + x = 132000
x = 132000/12 = 66000/6 = 11000

So the smallest town will have 11000

I disagree.. The question says: "no district is to have a population that is more than 10 percent greater than the population of any other district."

This allows for it to be 10% greater... but not more than 10%. So 11000 works.

What is the OA?

beeparoo wrote:

sethids wrote:If equal distribution was to be done, then 12000 individuals would be in each district.
To find the minimum number, we need to reduce individuals from 12000. And to be able to reduce the maximum (to reach the minimum possible for a district), we should divide the number reduced from one district equally among the remaining districts.

This leads to 11,000 as the answer.

Please comment and let me know if you think otherwise.

If answer is 11,000 as you say, then by your logic (that is, to equally distribute the remaining total population among the remaining districts), the other districts have a population of
(132000 - 11000)/(11-1) = 121000/10 = 12100

That means all other 10 districts have populations that are 10% greater than the district with only 11,000. This doesn't satisfy the question requirements.

However, this leads me to eliminate all answers from D and up, and just be left with E.

egybs wrote:I disagree.. The question says: "no district is to have a population that is more than 10 percent greater than the population of any other district."

You make a good point. What is the OA??

sethids wrote:If equal distribution was to be done, then 12000 individuals would be in each district.
To find the minimum number, we need to reduce individuals from 12000. And to be able to reduce the maximum (to reach the minimum possible for a district), we should divide the number reduced from one district equally among the remaining districts.

This leads to 11,000 as the answer.

Please comment and let me know if you think otherwise.

First of all how did u get 11000
and secondly GMAT answers have to be accurate we just can't assume.

See my post above...

Nycgrl wrote:

sethids wrote:If equal distribution was to be done, then 12000 individuals would be in each district.
To find the minimum number, we need to reduce individuals from 12000. And to be able to reduce the maximum (to reach the minimum possible for a district), we should divide the number reduced from one district equally among the remaining districts.

This leads to 11,000 as the answer.

Please comment and let me know if you think otherwise.

First of all how did u get 11000
and secondly GMAT answers have to be accurate we just can't assume.

Sorry I could not understand that explanation.......I will really appreciate if you can explain that using numbers.

suppose the min population is x
so the max population would be 1.1x

for x to be min, all other will have the same population equal to 1.1x

so x + 1.1*10*x = 132000

12x = 132000

x = 11000.

Now, all other district have a population of 12100, exactly 10% more than 11000... which is ok as per question stem .... no district has a population "more than" 10% of min .....

Roger That ! Thx

Experts, I need some help here.

I agree with durgesh's solution above. However, when I attempted this question, I setup the equation as follows:

0.9x + 10x = 132,000
x = 10,900.

What's wrong with the equation that I've setup. I'm quite sure there's a fundamental concept that I'm just not seeing here. Could someone please explain this? Thanks.

-BM-

bluementor wrote: 0.9x + 10x = 132,000

How did you come up with this equation, as in whats the logic behind it.

What Durgesh has done is that in order to minimize one variable you maximize all the other variables.

So the minimum value = x
population of the other 10 districts < 10% greater than x (1.1x)
since we need the maximum value, we'll take the population of the other 10 districts as = 1.1x

Equation should be x+ 1.1x*10 = 132,000
How did you get 0.9 and 10?