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

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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.

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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For an evenly spaced set (arithmetic progression), the median equals to the mean. Though the reverse is not necessarily true. Consider {0, 1, 1, 2} --> median = mean = 1 but the set is not evenly spaced.

So, for the original question we cannot assume that the mean and the median are the same.
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Here is another way. We know that shortest rod is 65 so, 77-65 = 12 is the additional lenght to be distributed between other 2 rods.

There are 2 ways to do that.

1. distribute equally so data points will be 65, 83, 83 ( 6 given to each remaining data point )
2. distribute unequally 65, 65, 89 ( median will be minimized )

so, 83 is the max value of the median.
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Mean = 77
Rods = 3
Sum = 77*3
= 231
Shortest Rod's length = 65
Thus 231-65 = 166
Median max value = 166/2
=83
Value of 3 rods will be 65,83,83