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Weighted Average

by ela07mjt » Fri Jul 26, 2013 2:46 am
In a telephone survey, 72% of male respondents and 60%
of all female respondents said they intend to vote in an upcoming election. What
fraction of all people surveyed were female?

1) 450 men were surveyed in total.
2) 70% of all respondents said they intend to vote in the upcoming election.

Ans. B
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by Jim@StratusPrep » Fri Jul 26, 2013 4:31 am
B simply says the weighted average of 72 and 60 is 70. The equation would look like this:

.6x + .72(1-x) = .7

X represents the percent of the people who responded yes that are female.
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by GMATGuruNY » Fri Jul 26, 2013 6:11 am
ela07mjt wrote:In a telephone survey, 72% of male respondents and 60%
of all female respondents said they intend to vote in an upcoming election. What
fraction of all people surveyed were female?

1) 450 men were surveyed in total.
2) 70% of all respondents said they intend to vote in the upcoming election.

Ans. B
This is a WEIGHTED AVERAGE/MIXTURE problem.

Statement 1: 450 men were surveyed in total.
No way to determine the number of women.
INSUFFICIENT.

Statement 2: 70% of all respondents said they intend to vote in the upcoming election.
Of all the men, the percentage who intend to vote = 72%.
Of all the women, the percentage who intend to vote = 60%.
Of all the respondents -- the MIXTURE of men and women -- the percentage who intend to vote = 70%.

To determine the ratio of men to women, use ALLIGATION -- a very efficient way to handle mixture problems.

Step 1: Plot the 3 percentages on a number line, with the percentage for the men and women (72% and 60%) on the ends and the percentage for all the respondents (70%) in the middle.
M 72---------70----------60 W

Step 2: Calculate the distances between the percentages.
M 72----2----70----10----60 W

Step 3: Determine the ratio in the mixture.
The ratio of men to women is the RECIPROCAL of the distances in red.
M:W = 10:2 = 5:1.

Since 5+1=6, of every 6 respondents, 5 are men and 1 is a woman.
Thus, women/total = 1/6.
SUFFICIENT.

The correct answer is B.

Please note the following:
Almost NO MATH is needed here if we understand how WEIGHTED AVERAGES work.
The question stem indicates the percentages for the two INGREDIENTS (the men and the women).
Statement 2 indicates the percentage for the MIXTURE (the men and women combined).
If we know the percentages for the two ingredients and the percentage for the mixture, we can ALWAYS determine the RATIO of the two ingredients (in this case, M:W).

Thus -- without doing any math -- we can see that statement 2 is SUFFICIENT to determine women/total.

Other alligation problems:
https://www.beatthegmat.com/mixture-prob ... tml#593241
Last edited by GMATGuruNY on Fri Jul 26, 2013 7:07 am, edited 1 time in total.
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by ela07mjt » Fri Jul 26, 2013 6:55 am
Thanks Mich

Jim, I know its an easy concept. Please could you explain it in more detail?
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by Brent@GMATPrepNow » Fri Jul 26, 2013 7:37 am
Hi ela07mjt,

Jim is essentially applying the following formula:

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

For more information on weighted averages, you can watch this free GMAT Prep Now video: https://www.gmatprepnow.com/module/gmat- ... ics?id=805

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by Jim@StratusPrep » Thu Aug 01, 2013 6:24 am
Brent hit it on the head. Does that make sense?
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by Uva@90 » Sat Aug 03, 2013 8:08 pm
Brent@GMATPrepNow wrote:Hi ela07mjt,

Jim is essentially applying the following formula:

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

For more information on weighted averages, you can watch this free GMAT Prep Now video: https://www.gmatprepnow.com/module/gmat- ... ics?id=805

Cheers,
Brent
Brent,
I have been scratching my head for long time still I couldn't get it. Can you explain the problem with the formula you mentioned.
Thanks in advance.

Regards,
Uva.
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by Brent@GMATPrepNow » Sun Aug 04, 2013 5:20 am
Uva@90 wrote:
Brent@GMATPrepNow wrote:Hi ela07mjt,

Jim is essentially applying the following formula:

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

For more information on weighted averages, you can watch this free GMAT Prep Now video: https://www.gmatprepnow.com/module/gmat- ... ics?id=805

Cheers,
Brent
Brent,
I have been scratching my head for long time still I couldn't get it. Can you explain the problem with the formula you mentioned.
Thanks in advance.

Regards,
Uva.
Sure thing.

Given: 72% of males and 60% of females intend to vote.

Statement 2: 70% of all respondents intend to vote

So, among the males, 72% intend to vote, and among the females, 60% intend to vote. When you combine all males and females, we see that 70% of the entire combined population intends to vote.

Aside: this is no different from taking solution A, which is 60% alcohol and mixing it with solution B, which is 72% alcohol, and getting a new mixture that's 70% alcohol. Our goal is to determine what fraction of the new mixture consists of solution A.

Weighted Averages comes into play here, because the final (combined) group depends on the proportion of the contributing group. For example, if the final group is mainly females, then the percentage of intending voters will be closer to 60% than to 72%. Conversely, if the final group is mainly males, then the percentage of intending voters will be closer to 72% than to 60%

Let Group A be the females.
Let x = the fraction (aka proportion) of the total population who are females
60% = the average likelihood that a person in group A will vote.

IMPORTANT: If x = the fraction of the total population who are females, then 1-x must represent the fraction of the total population who are males

Let Group B be the males.
Let 1-x = the fraction (aka proportion) of the total population who are males
72% = the average likelihood that a person in group B will vote.

Once the groups are combined, we get the Weighted average.
In this combined group (Group A + Group B), 70% intend to vote. In other words, 70% = the average likelihood that a person in the combined group will vote.

Weighted average = (group A proportion)(group A average) + (group B proportion)(group B average)
70 = (x)(60) + (1 - x)(72) now solve for x
70 = 60x + 72 - 72x
70 = 72 - 12x
-2 = -12x
2/12 = x
1/6 = x
So, 1/6 of the entire survey population are females.

Cheers,
Brent
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by gmatbeater1989 » Tue Oct 20, 2015 3:08 pm
I originally got this question wrong because I thought it asked for the number of females.
Then I did the question by assuming that there are 100 people survey since that works with percents.
My question is this: when the question asks for a percent or fraction, can I always use a nice number for the total?
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by [email protected] » Tue Oct 20, 2015 7:02 pm
Hi gmatbeater1989,

When TESTing VALUES in DS questions, you can often choose whichever values you want to TEST (based on whatever 'restrictions' exist in the prompt and the two Facts). However, you also have to be open to the idea that your 'nice' number is not the only possibility (and there might be other options that are not 'so nice'). In that way, DS questions often Test the 'thoroughness' of your thinking.

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