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Math Expert V
Joined: 02 Sep 2009
Posts: 55609

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Orange08 wrote:
Bunuel wrote:
(1) The average height of $$\frac{n}{3}$$ people is 74.5 inches and the average height of $$\frac{2n}{3}$$ people (the res of the people in the group $$n-\frac{n}{3}=\frac{2n}{3}$$) is 70 inches --> $$average \ height=\frac{sum \ of \ heights}{# \ of \ people}=\frac{74.5*\frac{n}{3}+70*\frac{2n}{3}}{n}$$ --> $$n$$ cancels out --> $$average \ height=74.5*\frac{1}{3}+70*\frac{2}{3}$$. Sufficient.

(2) Sum of heights equals to 178 feet 9 inches --> only nominator is given. Not sufficient.

Hope it helps.

Hi Bunuel,
The fact that confused me is the word tallest in statement 1.
It says "average height of $$n/3$$ [highlight]tallest[/highlight] people in the group is 6 feet 2.5 inches.

Isn't this bit ambiguous? We have no clue how many people to consider in tallest category?

It means that if we order these n people from shortest to tallest and consider $$\frac{n}{3}$$ tallest people, then their average height would be 74.5 feet.
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Re: What is the average (arithmetic mean) height of the n people in a cert  [#permalink]

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In your example set{10, 20, 30, 40, 50}

avg of first 2/5 of set = 2/5 * ((10+20)/2)
2/5 = .40
(10 + 20) / 2 = 15
.40 * 15 = 6

avg of rest 3/5 of set = 3/5 * ((30+40+50)/3)
3/5 = .60
(30 + 40 + 50) / 3 = 120/3 = 40
.60 * 40 = 24

6+24=30

you must multiply the subset average and the subset percentage

This one caught me off guard at first because of the reuse of the variable n. It should be viewed as 1n
Veritas Prep GMAT Instructor D
Joined: 16 Oct 2010
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Re: What is the average (arithmetic mean) height of the n people in a cert  [#permalink]

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trivikram wrote:
Guys a basic conceptual question.

Suppose we have a set S = {10,20,30,40,50}

If we take individual values as 10,20 and take their av = 15

and the rest 3 values 30,40,50 and avg of these is = 40

So the group average of S {10,20,30,40,50} = 30 is not equal to the sum of individual averages 15+40 = 55 or even the av of the values 55/2 =27.5

Whats the deal here? Explanations needed please

Remember, it needs to be 'weighted average', not the simple average.
Say you have two groups. Average height of one group is 5 feet and of the other group is 6 feet. What is the average of both groups combined? It will depend on how many people each group has. Hence, it is 'weighted'. If both groups have equal number of people, the average height will be 5.5 feet. If the first group has more people, the average will be closer to 5 than to 6.
In your question, the average of 2 elements is 15 and of 3 elements is 40. If you calculate their weighted average, it will be 30.
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Math Expert V
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Re: What is the average (arithmetic mean) height of the n people  [#permalink]

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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE
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Re: What is the average (arithmetic mean) height of the n people  [#permalink]

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1
Stmt 1 : Average = Sum/Total number of items

Hence, Average * Total number of items = Sum

1 foot = 12 inches, therefore the average height of n/3 people is 74.5 inches

Sum of heights of n/3 people is 74.5n/3

Rest of the people = total – n/3 = (n – n/3) = 2n/3

Similarly, the average height of the 2n/3 people is 70 inches

Hence, the sum of the heights of 2n/3 people is 140n/3

Sum of both groups = (74.5n/3)+ (140n/3) = 214.5n/3

Average = Sum/Total number of items

therefore, (214.5n/3)/n = 214.5n/3n = 71.5 inches

Hence, stmt 1 is sufficient

Stmt 2 : This statement gives us the sum of the heights of all the people in the group but the exact value of n is unknown

Since we can not determine the average height of the group, this statement is insufficient

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Re: What is the average (arithmetic mean) height of the n people  [#permalink]

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Looking at statement (2) first, we see that it is not sufficient, because the average (arithmetic mean) of a group of numbers is defined as (sum of data) / (# of data points). With statement (2), we only have the numerator of this expression (the # of people in the group is unknown), so we can't figure out the average.

Looking at statement (1) alone, we can set up the average as follows:
Average = (sum of data points) / (# of data points)
= [(n/3)(74.5) + (2n/3)(70)] / (n) <-- note that I used inches here, so I won't have to write in more fractions than necessary.
= [(1/3)(74.5) + (2/3)(70)]
There's no need to simplify further, because the 'n' is gone: you get one number. Therefore, this statement is sufficient.

Note that, if you have the averages of all the FRACTIONS or PERCENTAGES of a group, then you'll be able to calculate the overall average of the group. This is a worthwhile fact to memorize for the data sufficiency problems.
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What is the average (arithmetic mean) height of the n people  [#permalink]

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A video explanation can be found here:

Average = (SUM of terms) / (# of terms)

Need to know something about the numerator and denominator.

(1) gives us information to find a weighted average. Don't do any more math than necessary!

Average = [n/3(6ish feet) + 2n/3(5ish feet)] / n

Factor n out of the two expressions in the numerator. Then n in the numerator and n in the denominator cancel each other.

We're left with Average = 1/3(6ish) + 2/3(5ish), so statement (1) is sufficient - no need to find the exact average!

Statement (2) tells us the numerator (SUM), but not the denominator. Insufficient.

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Re: What is the average (arithmetic mean) height of the n people  [#permalink]

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I understand weighted average, but I got thrown off by the wording in the question.

I was under the impression that "a certain group" was a different variable, X, and that we were looking for a subset of this group n.
As a result for component 1 I read the "rest of the people" to be some population X - 1/3n where we don't know the remaining size of N within X. How do I avoid this mistake, would the question of stated "some group X" if that's what it was looking for?
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Re: What is the average (arithmetic mean) height of the n people  [#permalink]

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Hello Guys,

I saw this problem on Thursdays with Ron where he was explaining a number line shortcut to solve such questions on weighted averages. Even though I had understood the solution there itself, however, his shortcut was a bit puzzling to me. According to me, as per the statement A;the number of tall people should be lesser than the number of short people, as it is clearly in the ratio of 1:2 (tall : short). Then how is it possible that in his explanation it said the average of 70in and 74.5in was more towards 70 than 74.5??? Also I never got that reciprocal thing either. Re: What is the average (arithmetic mean) height of the n people   [#permalink] 01 Jun 2019, 01:11

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