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

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

(A) 90 (B) 100 (C) 110 (D) 130 (E) 140

total lenght of 5 pieces of wood = 124 * 5 = 620 and as median is 140, 2 numbers are below median and 2 numbers are above/equal to median as we need to find the max possible lenght of shortest piece of wood, let the 2 other long pieces of wood have lenght 140,140

so total 3 longest pieces = 420 total of 2 shortest pieces = 620-420 = 200

200/2 will be the max possible length of shortest piece of wood as any other number will be less than 100

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 _________________

Emily Sledge | Manhattan GMAT Instructor | St. Louis

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

(A) 90 (B) 100 (C) 110 (D) 130 (E) 140

total lenght of 5 pieces of wood = 124 * 5 = 620 and as median is 140, 2 numbers are below median and 2 numbers are above/equal to median as we need to find the max possible lenght of shortest piece of wood, let the 2 other long pieces of wood have lenght 140,140

so total 3 longest pieces = 420 total of 2 shortest pieces = 620-420 = 200

200/2 will be the max possible length of shortest piece of wood as any other number will be less than 100

Re: Mean, Median and 5 pieces of wood [#permalink]

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23 Sep 2010, 05:50

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Expert's post

robertrdzak wrote:

5 peices of wood have an average length of 124 inches and a median of 140 inches. What is the MAX possible length of the shortest piece of wood?

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

I see no ambiguity in this question.

Given: 5 peices of wood have an average length of 124 inches --> 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 --> \(x_1+x_1+140+140+140=620\) --> \(x_1=100\).

5 pieces of wood have an average length of 124 cm and a median of 140 cm .what is the maximum possible length in cm of the shortest piece of wood? A.90 B.100 C.110 D.130 E.140 _________________

Average length of the 5 pieces = 124. Sum = 124*5 = 620 If we arrange the woods in ascending order of their length; a,c,d,e,f to be the respective lengths. c will be 140 140 is the median; a,b,140,d,e Note: In a set with n terms, where n is odd; Median = (n+1)/2 term; Here n=5. Median will be (5+1)/2 = 3rd term.

To maximize a; we need to minimize others. Minimum values for d and e is 140 a,b,140,140,140

a+b+140+140+140=620 a+b = 200

The minimum value of b will be equal to a; a+a=200 a=100

If you get stuck on setting this type up, try the elimination method. see what would happen if you chose C for example. If the shortest was 110, and the middle is 140,then the remaining three pieces of wood would have an average length of (620-110-140)/3 = 123.333

But this is a problem because the average of all of them is 124. Logic would dictate that if you take out the smallest and the middle you would get a larger average. So this means that 110 is too big. Therefore the correct choice must be either B or A.

If you try 100, then you get the average of the remaining three as (620-100-140)/3 = 126.666. This is good because it's bigger than 124. So B is the correct choice.

I admit, it's not the most direct way of getting the answer, but it might be a useful trick if you get stuck on how to do the math.

5 pieces of wood have an average length of 124cm and median length of 140cm. what is the maximum possible length of the shortest piece of wood? 90 100 110 130 140

We know that average is 124. Therefore the sum of lengths is 620. We know median is 140. Consider the set of lengths to be {x , y , 140,140,140} We are keeping the 4th and 5th element as 140 because we are finding the max possible value of the lowest element in the set. Now the sum of last 3 elements is 420. So we are left with a sum of 200 for the first two elements. So the options for x and y would be {90,110} or {100,100} Since we are looking for maximum, the value would be 100,100. So 100 is the answer

Total value is 124*5=620 Median is 140, so there are must be three values of 140s i.e.,= 140*3=420 the sum of left values 620-420=200 least value 200/2=100 Ans. B _________________

5 pieces of wood have an average length of 124 cm and a median of 140 cm .what is the maximum possible length in cm of the shortest piece of wood? A.90 B.100 C.110 D.130 E.140

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. _________________

Karishma I don't know if the key word "shortest" means the second. It means the least. So the answer should be 99 practically because of the need to differentiate the first from the second- and be compatible with keyword. Your thoughts on this?

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gmatclubot

Re: median concept
[#permalink]
10 Apr 2011, 06:18

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