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Minimum possible population question

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by muhammedz786 » Sun Nov 15, 2015 5:15 pm
I came across this question on the GMAT PREP Test 1. I have no idea how to solve it. Can anyone help?
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by Brent@GMATPrepNow » Sun Nov 15, 2015 6:35 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 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 MartyMurray » Sun Nov 15, 2015 9:29 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 possible population that the least populated district could have?
a) 10,700
b) 10,800
c) 10,900
d) 11,000
e) 11,100
Here's an alternate approach.

You have eleven districts and you want one to have the smallest possible population. The way to do that is to make the rest use up as much population as possible. Doing that is constrained by the fact that the highest population can't be more than 10% greater than the lowest.

So the best you can do is to have ten populations that are 10% greater than the smallest one.

Now just try each answer choice starting with the middle one, adding up the following.

(10 x 1.1 x the choice) + the choice.

10,900 x 1.1 = 11,990

10 x 11,990 = 119,900

Without even adding you can tell that 119,900 + 10,900 < 132,000.

So try another answer choice that will generate a larger number, and a rounder one!

11,000 x 1.1 = 12,100

12,100 x 10 = 121,000

121,000 + 11,000 = 132,000

We have our answer.

Choose D.
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by [email protected] » Mon Nov 16, 2015 12:19 pm
Hi muhammedz786,

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

This matches perfectly with what we were told, so this MUST be the answer.

Final Answer: D

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