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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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25 Oct 2005, 00:08
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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
[Reveal] Spoiler: OA

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25 Oct 2005, 00:20
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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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25 Oct 2005, 00:21
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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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21 Feb 2006, 06:43
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, 07:00
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, 07:10
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, 07:24
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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, 07:49, edited 1 time in total.

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21 Feb 2006, 07:26
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, 07:36
Thanx SimaQ for your help, what was i thinking of!!!???

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21 Feb 2006, 07:51
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, 08:33
yeah guys i calculated as 600 , it is 620 , then it will be 100... thnks
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09 Jun 2007, 21:15
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, 21:25
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

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09 Jun 2007, 22:12
Mishari wrote:
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

Perfect - Mishari - I second (B) as the answer.

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10 Jun 2007, 01:12
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

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27 Sep 2007, 09:31
5 pieces of wood have an average length of 124 cm and a median lenght 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

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27 Sep 2007, 10:58
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Fistail wrote:
ywilfred wrote:
5 pieces of wood have an average length of 124 cm and a median lenght 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

b. 100

= (124 x 5 - 140x3)/2
= 200/2
=100

yup. it's B. I forgot some pieces could be the same length

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27 Sep 2007, 12:07
Why should the 2 smallest pieces of wood be 100cm each ?
Why couldn't we have the smallest piece at 71cm and the second smallest at 139cm ??
Since the sum of the 2 smallest should be 200cm.

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27 Sep 2007, 12:26
hidalgo01
------------

We could, there are so many possibilities for the two shortest pieces. Remember always to focus on what the question is asking: THE MAXIMUM possible length of the shortest piece. Since the average is below the median, then the lower two pieces shall be of the same length to obtain maximum possible length.

On the other hand, the length of the larger two pieces should be minimzed to get larger smaller pieces and still maintaining the same average and median --> we have now { X , X , 140, 140, 140 }

2X + 3(140) / 5 = 124 --> X = 100

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27 Sep 2007, 12:53
Thanks for your axplanation Mishari !
It makes sence now.

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27 Sep 2007, 12:53

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