Five pieces of wood have an average (arithmetic mean) length of 124 centimeters and a median length of 140 centimeters. What is the maximum possible length, in centimeters, of the shortest piece of wood?

A. 90
B. 100
C. 110
D. 130
E. 140

Given: 5 peices of wood have an average length of 124 centimeters --> total length = 124*5=620. Also median = 140.

If a set has odd number of terms the median of a set is the middle number when arranged in ascending or descending order;
If a set has even number of terms the median of a set is the average of the two middle terms when arranged in ascending or descending order.

As we have odd # of pieces then 3rd largest piece \(x_3=median=140\).

So if we consider the pieces in ascending order of their lengths we would have \(x_1+x_2+140+x_4+x_5=620\).

Question: what is the MAX possible length of the shortest piece of wood? Or \(max(x_1)=?\)

General rule for such kind of problems:
to maximize one quantity, minimize the others;
to minimize one quantity, maximize the others.

So to maximize \(x_1\) we should minimize \(x_2\), \(x_4\) and \(x_5\). Min length of the second largest piece of wood, \(x_2\) could be equal to \(x_1\) and the min lengths of \(x_4\) and \(x_5\) could be equal to 140 --> \(x_1+x_1+140+140+140=620\) --> \(x_1=100\).

Answer: B.
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For questions with a fixed sum (here 5*avg = 620), if you need to maximize one term, then you minimize all the others. Likewise, if asked to minimize one term, you maximize all the others.

I find it useful to create placeholders for the terms on paper, filling in with numbers/variables:

(In order, smallest to largest)
x / x / 140 / 140 / 140 = sum of 620
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It is a nice question, a GMAT type question i.e. fun to work out, can be reasoned out fairly quickly but needs you to think a little. Follow my train of thought here (which finally takes just a few seconds when you start doing it on your own)

First thing that comes to mind - Median is the 3rd term out of 5 so the lengths arranged must look like:

___ _____ 140 _______ __________

The mean is given and I need to maximize the smallest number. Basically, it should be as close to the mean as possible. Which means the greatest number should be as close to the mean as possible too.
If this doesn't make sense, think of a set with mean 20:
19, 20, 21 (smallest number very close to mean, greatest very close to mean too)
10, 20, 30 (smallest number far away, greatest far away too)

Using the same logic, lets make the greatest number as small as possible. The two greatest numbers should both be at least 140 (since 140 is the median)

___ _____ 140 140 140

Since the mean is 124, the 3 greatest numbers are already 16 each more than 124 i.e. total 16*3 = 48 more than the mean. So the two smallest numbers should be a total of 48 less than mean, 124. To make the smallest number as great as possible, both the small numbers should be 24 each less than the mean i.e. they should be 100.
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Hey gmat1220,

Smallest just means the smallest element. It doesn't necessarily mean that there should be a unique 'smallest number'.

Say {1, 2, 5, 9, 1, 3, 9}
Which is the smallest number here? 1 right? It doesn't matter even if it appears twice. If I arrange them in ascending order {1, 1, 2, 3 ....} .. the first and the second both are smallest (or shortest length).
So two pieces of wood could have the shortest length. It would be maximized only if their lengths are equal and both have a length of 100.
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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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We want to maximize \(x_1\), not to minimize.

Next, \(x_4\) and \(x_5\) cannot be less than the median, which is 140.
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we get x1 + x2 = 200, now what if we put x1 = 90 and x2 = 110 (since option a = 90, cant eliminate it by elimination.
Expert Pls suggest

Your question is not clear. What are you trying to achieve? The question asks: What is the maximum possible length, in centimeters, of the shortest piece of wood? The answer is 100 centimeters. You got 90 centimeters...[/quote]:

My bad, miss read the question. Apologies