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Avg of 3 largest numbers

by anurag_7 » Mon Jul 14, 2014 9:15 am
The average of 5 different numbers is 14. What is the average (arithmetic mean) of the 3 largest numbers?

(1) The average (arithmetic mean) of the two smallest numbers is 5.

(2) The average (arithmetic mean) of the two smallest numbers is 1/4 of the average of the 3 largest numbers.

According to me ans should be A, but the correct ans is D. Can someone explain?
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by GMATinsight » Mon Jul 14, 2014 10:28 am
anurag_7 wrote:The average of 5 different numbers is 14. What is the average (arithmetic mean) of the 3 largest numbers?

(1) The average (arithmetic mean) of the two smallest numbers is 5.

(2) The average (arithmetic mean) of the two smallest numbers is 1/4 of the average of the 3 largest numbers.

According to me ans should be A, but the correct ans is D. Can someone explain?
Suppose the numbers (in ascending order of their values) are a, b, c, d and e

Then (Given), a+b+c+d+e = 5x14 = 70

c+d+e=?

Statement 1)The average (arithmetic mean) of the two smallest numbers is 5

a+b = 2x5 = 10

therefore c+d+e = 70-10 = 60
therefore, Average of largest three numbers = 60/3 = 20

SUFFICIENT

Statement 2)The average (arithmetic mean) of the two smallest numbers is 1/4 of the average of the 3 largest numbers

(a+b)/2 = (1/4)x{(c+d+e)/3}
i.e. (a+b) = (1/6)x(c+d+e)

therefore, a+b+c+d+e = 70 ===> [(1/6)x(c+d+e)]+(c+d+e) = 70

i.e. (7/6)x(c+d+e) = 70
i.e. (c+d+e) = 70x6/7 = 60

Average of largest three numbers = 60/3 = 20

SUFFICIENT

Answer: Option D
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by [email protected] » Mon Jul 14, 2014 11:37 am
Hi anurag_7,

Certain DS question can be solved with a conceptual approach, as opposed to a "math" or "tactical" one.

This prompt does NOT ask us for any of the specific numbers, it asks for the average of the 3 largest. This type of "construction" gives us some options in terms of how we can solve it.

Since your question is just about Fact 2, I'm going to focus on that piece of the question.

From the prompt, we know that the average of 5 different numbers is 14. This means that the SUM of the 5 numbers = 70.

Fact 2: The average of the two smallest numbers = (1/4)(the average of the 3 largest numbers).

Here's a way to approach Fact 2 by using 1 variable:

X = Average of the three largest numbers

Thus... (1/4)(X) = the average of the two smallest numbers

Since the sum of the 5 numbers is 70, we have....

X/4 + X/4 + X + X + X = 70

3.5X = 70

X = 20

We now know that the average of the largest three numbers is 20.
Fact 2 is SUFFICIENT

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by GMATGuruNY » Mon Jul 14, 2014 1:14 pm
anurag_7 wrote:The average of 5 different numbers is 14. What is the average (arithmetic mean) of the 3 largest numbers?

(1) The average (arithmetic mean) of the two smallest numbers is 5.

(2) The average (arithmetic mean) of the two smallest numbers is 1/4 of the average of the 3 largest numbers.
For average problems, remember the following:
sum = (number)(average).
number = sum/average.
average = sum/number.

Let a, b, c, d, and e be the 5 values, in ascending order.
a+b+c+d+e = (number)(average) = 5*14 = 70.

Statement 1: The average (arithmetic mean) of the two smallest numbers is 5.
a+b = (number)(average) = 2*5 = 10.
c+d+e = (a+b+c+d+e) - (a+b) = 70-10 = 60.
Average of c, d, and e = sum/number = 60/3 = 20.
SUFFICIENT.

Statement 2: The average (arithmetic mean) of the two smallest numbers is 1/4 of the average of the 3 largest numbers.
Statement 1 implies the following values:
Average of a and b = 5.
Average of c, d, and e = 20.
Sum of a+b+c+d+e = 70.
Notice that the two averages -- 5 and 20 -- also satisfy statement 2, since 5 is 1/4 of 20.
If these two averages both increase or both decrease, then the SUM will also increase or decrease.
Not possible, since the sum must be 70.
Implication:
Statement 2 is satisfied only by the SAME combination of values that satisfy statement 1.
Thus, the average of c, d and e = 20.
SUFFICIENT.

The correct answer is D.
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by anurag_7 » Tue Jul 15, 2014 6:06 am
Problem sorted.. Thanks everyone for helping me out.
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