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Population distribution question

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by utkalnayak » Fri Jan 23, 2015 4:22 pm
A certain city with a population of 132,000 is to be divided into 11 voting districts, and no district is mohave 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

OA: D
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by Brent@GMATPrepNow » Fri Jan 23, 2015 4:26 pm
utkalnayak wrote:A certain city with a population of 132,000 is to be divided into 11 voting districts, and no district is mohave 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
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

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by utkalnayak » Fri Jan 23, 2015 4:43 pm
Thank you Brent.
I missed this point, somehow half way through writing these equations, I got distracted towards plugging in numbers and tried to work it out. Eventually ended up getting it wrong. Its a shame I was this close yet got it wrong.

Most of the time simplifying to equation is better than plugging in answer choices first is what my experience is.
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by [email protected] » Fri Jan 23, 2015 5:24 pm
Hi utkalnayak,

Certain Quant questions have built in shortcuts IF the answer choices are numbers and if certain other conditions are met. Here, we can TEST THE ANSWERS....

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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by Matt@VeritasPrep » Sun Jan 25, 2015 3:36 pm
utkalnayak wrote:Thank you Brent.
I missed this point, somehow half way through writing these equations, I got distracted towards plugging in numbers and tried to work it out. Eventually ended up getting it wrong. Its a shame I was this close yet got it wrong.

Most of the time simplifying to equation is better than plugging in answer choices first is what my experience is.
Plugging in answer choices is a great strategy, though it might get messy here. If I get stuck on a problem, I usually ask myself if I'm not taking the question literally enough. Sometimes the whole issue in math is taking a problem intuitively ("So what they're REALLY saying is ...") rather than literally ("So what they're ACTUALLY saying is ...").

Let's take a literal approach to this problem. We want the minimum population, so let's call that number m. (Always start with the variable you seek. Make the minimum district m, not the other districts!) We know that the OTHER districts can be AT MOST 10% greater than the minimum. Since we want the minimum for our district, we also want the MAXIMUM for the other districts, so that we have as few people as people in m. That means the other districts are 10% greater than m, or 1.1m. Success!

So our city has one district of m and 10 districts of 1.1m, for a population of m + 10*(1.1m), or 12m. Then we solve 12m = 132,000, and we're set!
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