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Three straight metal rods have an average (arithmetic mean)

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Three straight metal rods have an average (arithmetic mean) [#permalink] New post 27 Feb 2013, 21:23
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Three straight metal rods have an average (arithmetic mean) length of 77 inches and the shortest rod has a length of 65 inches. What is the maximum possible value of the median length, in inches, of the three rods?

A. 71
B. 77
C. 80
D. 83
E. 89

Source: GMAT Prep Question Pack 1
Rated: Medium

If you know of any similar problems, please post.
[Reveal] Spoiler: OA

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Re: Three straight metal rods have an average (arithmetic mean) [#permalink] New post 27 Feb 2013, 22:32
We know that there are 3 metal rods with an average length of 77 inches.
Total length=77*3=231 inches
Shortest rod length=65 inches
the sum of lengths of the two longer rods=231-65=166
Since, the longer rods have to be longer than 65, the smallest value one of these longer values could have is 66.

Now the question asks us for the longest median value. The median value is the middle value when the three rods are arranged in ascending order of lengths.

There are two ways of finding these values
1. Since we need to find the largest middle value, the two longer rods need to be of equal length. If both values are equal in length the middle value will be the largest possible value and the longest rod will be of the same length as the rod of median length. Hence, 2x =166 implies x=83.

Maximum length of the median rod=83

2. Testing the values given
Let us test the values from the bottom

e=89
166-89=77. Since 77 is less than 89, it cannot be the median value. WRONG

d=83
166-83=83. Both the rods are of the same length. Hence, this is the largest possible middle value.

DelSingh wrote:
Three straight metal rods have an average (arithmetic mean) length of 77 inches and the shortest rod has a length of 65 inches. What is the maximum possible value of the median length, in inches, of the three rods?

A.71
B.77
C.80
D.83
E.89



Source: GMAT Prep Question Pack 1
Rated: Medium





If you know of any similar problems, please post.

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Re: Three straight metal rods have an average (arithmetic mean) [#permalink] New post 28 Feb 2013, 00:17
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DelSingh wrote:
Three straight metal rods have an average (arithmetic mean) length of 77 inches and the shortest rod has a length of 65 inches. What is the maximum possible value of the median length, in inches, of the three rods?

A. 71
B. 77
C. 80
D. 83
E. 89

Source: GMAT Prep Question Pack 1
Rated: Medium

If you know of any similar problems, please post.


Say the lengths of the rods in ascending order are x_1, x_2, and x_3, where x_1\leq{x_2}\leq{x_3}.

The median of a set with odd number of terms is just the middle term, when arranged in ascending/descending order, hence the median is x_2.

Given that x_1+x_2+x_3=3*77 --> 65+x_2+x_3=3*77 --> x_2+x_3=166. We need to maximize x_2=median, so we need to minimize x_3.

The minimum value of x_3 is x_2 --> x_2+x_2=166 --> x_2=median=83.

Answer: D.

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Re: Three straight metal rods have an average (arithmetic mean) [#permalink] New post 28 Feb 2013, 03:55
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DelSingh wrote:
Three straight metal rods have an average (arithmetic mean) length of 77 inches and the shortest rod has a length of 65 inches. What is the maximum possible value of the median length, in inches, of the three rods?

A. 71
B. 77
C. 80
D. 83
E. 89

If you know of any similar problems, please post.


Because it is mentioned that the average length of the three rods is 77 inches, we can assume there are three rods each of length 77 inches. Now we have been told that the shortest rod has a length of 65 inches. Thus, we can take 12 inches from one of the rods and will have to adjust it amongst the other two remaining rods. Now, if we divide 12 in any proportion other than 6 and 6, we will not have the maximum value for the median. Imagine, we redistribute 12 by giving 5 to one and 7 to another. This gives 65, 82 and 84.
Thus, giving 6 to both the rods, we have (77+6) = 83 inches for both the rods.

D.
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Re: Three straight metal rods have an average (arithmetic mean) [#permalink] New post 21 Aug 2013, 20:41
Smallest rod = 65
Average of all 3 rods = 77

Lets consider middle rod = 77, so the largest rod (x) calculation is:
65 + x = 77*2
x = 154 - 65 = 89.
Now, considering that middle rod = 77, the max lenght of the largest rod would be = 89.
So the max length of the middle rod possible = Avg of 89 & 77
= (89 +77)/2 = 83
Answer = D = 83
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Re: Three straight metal rods have an average (arithmetic mean)   [#permalink] 21 Aug 2013, 20:41
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