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# Five peices of wood have an average length of 124 inches and

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Director
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Five peices of wood have an average length of 124 inches and [#permalink]  20 Nov 2011, 04:49
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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

Even though I got the correct answer (B) by doing maths, but it was a total guess work. This is how I did.

Let say shortest piece be x

x+x+3*140=5*124
2x+420=620
2x=200
x=100

I have no idea how I did it. So can someone be please kind enough to let me know the mathematical approach and the concept ?
[Reveal] Spoiler: OA

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Re: 5 pieces of wood [#permalink]  20 Nov 2011, 06:21
4
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Sum of lengths is 124*5=620

Median is 140
Sum of the lengths of other four pieces= 620-140=480

The lengths of pieces-

L1, L2, 140, L4, L5

The sum of the four pieces is constant.

L4 and L5 have to be minimum for L1 to be maximum but median length must be 140.

The minimum possible values of L4 and L5 could be 140, hence the L1+L2 = 620 - 420 = 200.
The maximum possible value of L1 = 100= L2
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Director
Status: Finally Done. Admitted in Kellogg for 2015 intake
Joined: 25 Jun 2011
Posts: 552
Location: United Kingdom
Concentration: International Business, Strategy
GMAT 1: 730 Q49 V40
GPA: 2.9
WE: Information Technology (Consulting)
Followers: 29

Kudos [?]: 1143 [0], given: 217

Re: 5 pieces of wood [#permalink]  20 Nov 2011, 21:32
Apologies blink005, if I am getting this wrong. But how did you get this?

The minimum possible values of L4 and L5 could be 140, hence the L1+L2 = 620 - 420 = 200.
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MGMAT 1 --> 530
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MGMAT 3 ---> 610
GMAT ==> 730

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Re: 5 pieces of wood [#permalink]  05 Feb 2012, 14:50
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enigma123 wrote:
Apologies blink005, if I am getting this wrong. But how did you get this?

The minimum possible values of L4 and L5 could be 140, hence the L1+L2 = 620 - 420 = 200.

Below is step by step analysis of this question. Hope it helps.

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

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 --> $$x_1+x_1+140+140+140=620$$ --> $$x_1=100$$.

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Re: Five peices of wood have an average length of 124 inches and [#permalink]  14 Jun 2013, 04:27
Expert's post
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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Re: 5 pieces of wood [#permalink]  25 Aug 2013, 02:21
Bunuel wrote:
enigma123 wrote:
Apologies blink005, if I am getting this wrong. But how did you get this?

The minimum possible values of L4 and L5 could be 140, hence the L1+L2 = 620 - 420 = 200.

Below is step by step analysis of this question. Hope it helps.

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

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 --> $$x_1+x_1+140+140+140=620$$ --> $$x_1=100$$.

Why couldn't x4 and x5 be bigger than 140 and thus making x1 and x2 even smaller?
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Kudos [?]: 44880 [0], given: 6630

Re: 5 pieces of wood [#permalink]  25 Aug 2013, 05:46
Expert's post
Skag55 wrote:
Bunuel wrote:
enigma123 wrote:
Apologies blink005, if I am getting this wrong. But how did you get this?

The minimum possible values of L4 and L5 could be 140, hence the L1+L2 = 620 - 420 = 200.

Below is step by step analysis of this question. Hope it helps.

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

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 --> $$x_1+x_1+140+140+140=620$$ --> $$x_1=100$$.

Why couldn't x4 and x5 be bigger than 140 and thus making x1 and x2 even smaller?

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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Re: Five peices of wood have an average length of 124 inches and [#permalink]  12 Jul 2014, 11:22
We can find the total length to be as 620
a+b+c+d+e = 620
Since the number of pipes is 5, the median is the middle number ie c. (c=140)
a+b+d+e = 480
to find the max length of the smallest pipe. we should look to find the minimum length of other 3 pipes (we are already given length of one of the pipes ,c=140).
d=e=140 ( we assume d and e to be 140 as it is given the median is 140 )

a+b = 200
the max length will be 100.
Re: Five peices of wood have an average length of 124 inches and   [#permalink] 12 Jul 2014, 11:22
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# Five peices of wood have an average length of 124 inches and

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