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Question from Gmat Prep- Help please! 3

by bobdylan » Fri Apr 13, 2012 1:21 pm
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 population that the least populated district could have?
a)10,700
b)10,800
C) 10,900
d) 11,000
e) 11,100
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by Bill@VeritasPrep » Fri Apr 13, 2012 2:31 pm
If the population is distributed equally, then each district has 132,000/11 = 12,000 residents. Let's define x as the district with the lowest population. Since no other district can exceed x by more than 10%, we can set up the following equation:

x + 1.1(10x) = 132,000
12x = 132,000
x = 11,000
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by iwillsurvive101 » Fri Apr 13, 2012 4:57 pm
Bill

Can you please clarify your reasoning behind setting up this equation?

x + 1.1(10x) = 132,000

How did you get 1.1(10x) ?

I reasoned as follows:

- Apart from x, there are 10 other districts. Each of those districts can have 10% more population than x. So 10% x X 10 (10 percent of x times 10). I know this is not right.

Thanks
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by Anurag@Gurome » Fri Apr 13, 2012 5:24 pm
bobdylan 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 population that the least populated district could have?
a)10,700
b)10,800
C) 10,900
d) 11,000
e) 11,100
Let us assume that the minimum possible population of the least populated district be P.

It is given that "no district is to have a population that is more than 10% greater than the population of any other district."

Then the population of 11 districts should be between P and (P + 10% of P) = 1.1P
Now we have minimize P. For this we should make only one district to have minimum possible population and the remaining 10 districts should have maximum possible population.
So, P + (10 * 1.1P) = 132,000
12P = 132,000
P = 11,000

The correct answer is D.
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by GMATGuruNY » Fri Apr 13, 2012 6:00 pm
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
An alternate approach is to plug in the answer choices, which represent the minimum possible population.

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: 11,000.
Each of the other 10 districts = (132,000 - 11,000)/10 = 12,100.
Difference between the populations = 12,100 - 11,000 = 1100, which is 10% of the smallest population.
Success!

The correct answer is D.
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by Deepthi Subbu » Sat Jul 27, 2013 3:01 am
Let x be the cities with maximum population. So for a city to have the least population, let me maximise x and also assume it is the same for all 10 cities.

So, 10(x) + .9(x) = 132000.

Whats wrong with my approach ?
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by Matt@VeritasPrep » Sat Jul 27, 2013 8:13 am
Deepthi Subbu wrote:Let x be the cities with maximum population. So for a city to have the least population, let me maximise x and also assume it is the same for all 10 cities.

So, 10(x) + .9(x) = 132000.

Whats wrong with my approach ?
Your logic isn't bad, but that equation won't give us an integer, so we have to go another way.

Let's take it from the top:

Given our restrictions, we have two ways of setting up the problem. We could say the other districts are all x, and the smallest district is 10% less, or .9x. Or we could say the smallest district is x, and the other districts are 10% greater, or 1.1x.

That gives us two choices of equation:

10x + .9x = 132,000

or

x + 10*(1.1x) = 132,000

So we have either 10.9x = 132,000 or 12x = 132,000. While the first equation does give us a smaller value for the population of the least populated district, it doesn't give us an integer value, so we have to throw it out and use the second equation.
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by [email protected] » Sat Jul 27, 2013 2:52 pm
Hi Deepthi Subbu,

Certain Quant questions have built in shortcuts IF the answer choices are numbers and if certain other conditions are met. In the EMPOWERgmat Course, we call this tactic TEST THE ANSWERS.

We have 5 answers that are numbers here, so I'm going to use them against the question to figure out the correct answer.

Logically, when a GMAT question asks you to figure out the LEAST or GREATEST value of something, then there are going to be restrictions on how the values will relate to one another. Here we have 11 cities; to make one as SMALL as possible, I'd think to make all of the others as LARGE as possible.

The answers:
10,700
10,800
10,900
11,000
11,100

Statistically, it's best to TEST either B or D first. D seems like an easier number to manipulate, so I'll TEST that one first.

IF the least city = 11,000
Then 10% greater would be 12,100
IF the other 10 cities are 12,100 each, then they would sum to 121,000
Add in the least city: 121,000 + 11,000 = 132,000

Since all the math that I've done matches the data that I was given....

Final Answer: D

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