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Weighted averages - Fraction of combined populations

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by gmattesttaker2 » Sat Feb 08, 2014 11:41 pm
Hello,


I was wondering if the following solution is correct. Thanks for your help - Sri


What fraction of the combined populations of Towns A and B live in Town A?

(1) The average income of the population of Town A is $3,000 higher than the average
income of the populations of both towns combined.

(2) The average income of the population of Town B is $2,000 lower than the average
income of the populations of both towns combined.

OA: C


Let a = number of people living in Town A
Let b = number of people living in Town B

To find: a/(a+b) = ?

1)

a---------------a+b---------------------b
3000+x x

Hence, In-suff.


2)
a---------------a+b---------------------b
x x-2000

Hence, In-suff.


1 and 2 combined:

a---------------a+b---------------------b
3000+x x x-2000

a/b = 2000/3000
=> a/b = 2/3

Hence, a/(a+b) = 2/(2+3) = 2/5. Hence, Suff.
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Source: — Data Sufficiency |

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by [email protected] » Sun Feb 09, 2014 11:32 am
Hi gmmattesttaker2,

Yes, your approach is correct. There are actually a couple of different ways to prove the ratio that you proved. Once you have THAT deduction, you have the necessary proof to answer the given question.

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by Brent@GMATPrepNow » Sun Feb 09, 2014 1:05 pm
gmattesttaker2 wrote: What fraction of the combined populations of Towns A and B live in Town A?

(1) The average income of the population of Town A is $3,000 higher than the average
income of the populations of both towns combined.

(2) The average income of the population of Town B is $2,000 lower than the average
income of the populations of both towns combined.


Target question: What fraction of the combined populations live in Town A?

Statement 1: The average income of the population of Town A is $3,000 higher than the average income of the populations of both towns combined.
Conceptually, we should see that this does not provide enough information to answer the target question.

If we're not sure, we can look for scenarios that satisfy statement 1 yet yield conflicting answers to the target question. Here are two such scenarios:
Case a: Town A has 1 person who earns $6000 and Town B has 1 person who earns $0. Here, the average income for both towns is $3000, so the town A person's income is $3000 higher than the combined average. In this case, 1/2 of the combined populations live in Town A
Case b: Town A has 1 person who earns $6000, and Town B has 1 person who earns $0 and 1 person who earns $3000. Here, the average income for both towns is $3000, so the town A person's income is $3000 higher than the combined average. In this case, 1/3 of the combined populations live in Town A
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The average income of the population of Town B is $2,000 lower than the average income of the populations of both towns combined.
Once we've demonstrated that statement 1 is NOT SUFFICIENT, we should recognize that, since statement 2 is very similar to statement 1, it too is NOT SUFFICIENT.

Statements 1 and 2 combined
At this point, we can use the following weighted averages formula:
Weighted average = (group A proportion)(group A average) + (group B proportion)(group B average) + (group C proportion)(group C average) + ...

So, for this question, let's assign some variables:
Let a = number of people living in Town A
Let A = average income of Town A
Let b = number of people living in Town B
Let B = average income of Town B
Let T = average income of combined towns

When we plug this into the formula, we get: T = [a/(a+b)](A) + [b/(a+b)](B)

Statement 1 tells us that A = T+3000, and statement 2 tells us that B = T-3000
Plug these into the equation to get: T = [a/(a+b)](T+3000) + [b/(a+b)](T-2000)
Next, eliminate fractions by multiplying both sides by (a+b) to get: T(a+b) = a(T+3000) + b(t-2000)
Expand to get: aT + bT = aT + 3000a + bT - 2000b
Now subtract aT and bT from both sides: 0 = 3000a - 2000b
Rearrange: 3000a = 2000b
Rearrange: a/b = 2000/3000
Simplify: a/b = 2/3
In other words, the ratio of a to b is 2 to 3
So, out of every 5 people, 2 live in Town A and 3 live in Town B
This means that 2/5 of the combined populations live in Town A
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C

--------------------
Here's another Data Sufficiency question that tests weighted averages: https://www.beatthegmat.com/weighted-ave ... 74015.html

Cheers,
Brent

PS: For more information on weighted averages, you can watch this free GMAT Prep Now video: https://www.gmatprepnow.com/module/gmat- ... ics?id=805
Brent Hanneson - Creator of GMATPrepNow.com
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by 2mist » Mon Feb 10, 2014 7:19 am
Hello Brent,

In this formula :

Weighted average = (group A proportion)(group A average) + (group B proportion)(group B average) + (group C proportion)(group C average) + ...

How do we decide which quantity is proportion and which quantity is average.
I had really tough time figuring out whether average income is proportion or average and vice versa.

your insight will be highly appreciated and helpful.

Regards,
Mist
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by Brent@GMATPrepNow » Mon Feb 10, 2014 7:29 am
2mist wrote:Hello Brent,

In this formula :

Weighted average = (group A proportion)(group A average) + (group B proportion)(group B average) + (group C proportion)(group C average) + ...

How do we decide which quantity is proportion and which quantity is average.
I had really tough time figuring out whether average income is proportion or average and vice versa.

your insight will be highly appreciated and helpful.

Regards,
Mist
First we need to identify the populations that are combined. Here we have a population of residents. So, the NUMBER of residents will be the PROPORTIONS. That is, we're combining Town A residents and Town B residents in a certain proportion. In fact, we learn later on that the combined population consists of 2/5 Town A people and 3/5 Town B people.

This leaves the average income as the statistic associated with the 2 populations.

In general, the proportions will be populations or volumes.
For example, if we take 2 gallons of a 12% alcohol solution and combine it with 5 gallons of a 20% alcohol solution, then the percent alcohol in the combined solutions is as follows:

% alcohol in combined solution = (2/7)(12) + (5/7)(20)

Here, we might say we're combining populations of molecules. One population consists of 12% alcohol molecules, and the other population consists of 20% alcohol molecules.

I hope that helps.

Cheers,
Brent
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