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# Five pieces of wood have an average length of 124cm and a

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Five pieces of wood have an average length of 124cm and a [#permalink]

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24 Oct 2005, 23:08
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Five pieces of wood have an average length of 124cm and a median length of 140cm. What is the maximum possible length, in cm, of the shortest piece of wood?

a) 90
b) 100
c) 110
d) 130
e) 140

OPEN DISCUSSION OF THIS QUESTION IS HERE: five-peices-of-wood-have-an-average-length-of-124-inches-and-123513.html
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Re: Mean, Median and 5 pieces of wood [#permalink]

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04 Aug 2009, 22:38
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The median of 5 pieces is 140. Therefore, there are 2 pieces >=140.
Since, we want to maximize the smallest piece, we want to limit the largest piece(s) to the lowest value possible, because the larger the largest pieces the smaller the smallest pieces will have to be. But since the median is 140, it is the floor limit on the size of the 2 largest pieces...so the two largest pieces will have to be 140.

(A+B+140+140+140)/2 = 124 [where, A and B are the smaller pieces]

Since the question is asking for the maximum size of the smallest piece while preserving the average and median, A and B must be equal, so,
(2A+140+140+140)/2 = 124

A = 100.

Another way to think about it is, how averages are distributed among numbers. For every inch more the average, their has to be an inch less than the average. So, we have 3 numbers which are 16 each more than the average...in total 48 over the average. The two smaller pieces will have to be compensate this. And to get the maximum lowest value the compensation should be distributed evenly...each member should be 24 less than the average...124-24 = 100.
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24 Oct 2005, 23:20
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B. 100

sum of all lengths of all 5 pieces of wood = 124 * 5 = 620

3rd piece (sorted in increasing length) length = 140 (median)

for sum of first 2 wood length to become max, last two should be least.
let 4th, 5th wood also have length 140 each.

total of last 3 = 140 * 3 = 420
sum of first 2 = 620 - 420 = 200

each of these 2 will have length 200/2 = 100
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Re: Five pieces of wood GMAT Prep PS [#permalink]

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18 Mar 2009, 17:52
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Accountant wrote:
Five pieces of wood have an average length of 124cm and a median lenght of 140cm. What is maximum possible length in cm of the shortest piece of wood

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

Is plugging in answers the shortest way to solve this?

Not really. That is too time consuming.

The logic is: Make the distribution of the lengths close to the median.
Since median > avg, the max. shortest valu cannot be the avg.

So it is: (124x5 - 140x3)/2 = (620 - 420)/2 = 100
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Re: Mean, Median and 5 pieces of wood [#permalink]

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23 Sep 2010, 04:50
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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 --> $$x_1+x_1+140+140+140=620$$ --> $$x_1=100$$.

Hope it's clear.
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12 Sep 2011, 19:54
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Expert's post
socalboy429 wrote:
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

Say we list the lengths of our pieces of wood in increasing order:

S, a, 140, b, L

We know that the sum of these lengths is 5*124 = 620. Now, we want to make S, the smallest length, as big as possible. To do that, we want the other unknown lengths to 'use up' as little of the sum of 620 as possible. That is, the smaller we make a, b and L, the larger we can make S. Since b and L must be at least as large as the median, the smallest possible values for b and L are 140. That gives us this set:

S, a, 140, 140, 140

The three largest values now add to 420, so the two smallest values must add to 620-420 = 200. Since making them equal will make a as small as possible (a cannot be less than S), the largest possible value of S is 200/2 = 100.
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24 Oct 2005, 23:21
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how can the shortest piece have maximum length ? when all other pieces have a min value. thats 100 100 140 140 140. B)...
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28 Jul 2008, 07:30
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singaks wrote:
. Five pieces of wood have an average length of 124 cm and a median length of 140cm. what is the maximum possible length of the shortest piece of wood?

Need help. Thanks

Five pieces of wood in order of increasing length:
A,B,C,D,E
A+B+C+D+E = 124*5 = 620
C is the median and equals to 140
Because Maximum (A+B) occurs when D and E at their minimum; and D and E cannot be lower than 140, so min. D and E = 140
So max. A+B = 620 - 3*140 = 200
A+B = 200 and B > A
So, max A = 99 when B = 101
if B=A is acceptable, max. A = 100
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Re: Mean, Median and 5 pieces of wood [#permalink]

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28 Sep 2009, 18:05
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tejal777 wrote:
Phew..NOT clear at all..
I struggled and made it through robertrdzak explanation but not clear in the last part of each pc being 100 and 100..why cant it be 90 and 110??

The smallest two pieces COULD be 90, 110. They could also be 80, 120. However, the question stem asks for what the MAX length could be for the smallest piece of wood. In both these situations the smallest piece is 90 and 80 i.e not maximised.

So {100, 100, 140, 140, 140} and {90,100,140,140,140} and many other sets satisfy the conditions for mean and median.
But in order to maximise the smallest piece 100 would be the only option. Hope that makes sense.
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10 Apr 2011, 06:51
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gmat1220 wrote:
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?

Posted from my mobile device

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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21 Feb 2006, 05:43
Five pieces of wood have an average (arithmetic mean) length of 124cm and a median length of 140 cm. What is the maximum possible length, in centimetres, of the smallest piece of wood?

a) 90

b) 100

c) 110

d) 130

e) 140
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21 Feb 2006, 06:00
A ) 90

here is my explanation

x-140 , y-140, 140 , z+140 , m+140
______________________________ = 124
5

x-140 + y-140 + 140 + z+140 + m+140 = 600

x-140 + y-140 + z+140 + m +140 = 460

now i assumed z & m == 1 ----> x-140+ y-140 = 460 - 280 = 180

so the loewst value can be 180/2 = 90
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21 Feb 2006, 06:10
think there is something wrong with this Q. If the median is 140cm and AVRG is 124 then the total lenght of the woods is 620cm.Since median is 140cm the values above the median should be > or = to the median.Their minimum value is 140x3=420cm.The max possible lenght of the shortest piece is 200-124=76cm
May be i am wrong
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21 Feb 2006, 06:24
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The average of 5 pieces is 124, this means the total length of all pieces=620

We now the median is 140, which gives us that 140 is the middle value of all the fives pieces... which means two pieces are below 140 and two pieces above 140....

To get the maximum length of the smallest piece of wood we must minimize the length of the two largest pieces....

Remember we have the total length of 620... and the middle value 140... all we need is to distribute 380 (i.e. 620-140) in such a way that two of the values would be below 140 and two at least 140... so 100+100+140+140+140=620, the maximum value of the smallest piece is 100, since if we chose 110, two other values would only yield 130 which cannot be the case...

Hope i wrote it in understandable way...

Last edited by SimaQ on 21 Feb 2006, 06:49, edited 1 time in total.
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21 Feb 2006, 06:26
BG wrote:
think there is something wrong with this Q. If the median is 140cm and AVRG is 124 then the total lenght of the woods is 620cm.Since median is 140cm the values above the median should be > or = to the median.Their minimum value is 140x3=420cm.The max possible lenght of the shortest piece is 200-124=76cm
May be i am wrong

Everything is perfect with your reasoning... you got that 3 values must be at least 140*3=420cm so we need to distribute 200 among the rest pieces (i.e 2) which would equal 100...
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21 Feb 2006, 06:36
Thanx SimaQ for your help, what was i thinking of!!!???
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21 Feb 2006, 06:51
I would select 100.

The way I solved the question as follows.
Median is 140
Average is 124
Total length is 620
Now based on the available information 2 Lengths <= 140 <= remaining two lengths.

Sum of these 4 lengths has to be 620 - 140 = 480
To maximize the shortest length, I considered that two short lengths are equal is size = x and remaining two lengths equal in size = 140 so now the addition of two short lengths is
2X = 480 - (140 + 140)
2X = 200
X = 100

I hope this makes sense.
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21 Feb 2006, 07:33
yeah guys i calculated as 600 , it is 620 , then it will be 100... thnks
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09 Jun 2007, 20:15
Five pieces of wood have an average length of 124 cm and median length 140 cm. What is the maximum length in cm of the shortest peice of wood.

1.90
2.100
3.110
4.130
5.140
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09 Jun 2007, 20:25
We already know that the middle piece, when ordering the pieces from longest to shortest, is 140 cm long [the median of five pieces is the middle one]

To result in a maximum length for the shortest piece, the two longest pieces must have minimum possible length --> both 140 cm

Now we have 3 pieces with 140 and two other pieces <140> the two short pieces must have equal length [ call it X ]

[2X + (3x140)] / 5 = 124
[2X + 420] = 5 x 124 = 620
2X = 620 - 420 = 200 --> X = 100 cm

09 Jun 2007, 20:25

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