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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? 90 100 110 130 140

Re: GMATprep practice question:wood mean [#permalink]

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07 Apr 2009, 12:01

pmal04 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? 90 100 110 130 140

Sum of lengths = 124*5=620

Number line: _ _ 140 _ _

To maximize the first from the left, you need to minimize the numbers on the right of 140. Also you need to minimize the second from the left. Ultimately, this will look like: 100 100 140 140 140

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

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04 Aug 2009, 23:38

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

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05 Aug 2009, 02:37

scarish wrote:

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.

Great Explanation....kudos to you!! and a cool signature as well...IMO 100

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

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05 Aug 2009, 05:18

[quote="robertrdzak"]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?

90 100 110 130 140

124*5 = 620 the median is the middle peice's lenght thus 2 peices are equall or bigger in length and 2 less

think of it as 2 hands of a scale and the axis is the median, one have to minimize the largest 2 values for eg: 140,140

thus the shortest 2 total lenght = 620-(140*3) = 200 , from the given the there must be a ( shortest piece) ie: the shortest 2 peices are not equal in length, from the values given , i d choose 90.

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

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05 Aug 2009, 05:47

apoorvasrivastva wrote:

Great Explanation....kudos to you!! and a cool signature as well...IMO 100

Thanks mate..

yezz wrote:

thus the shortest 2 total lenght = 620-(140*3) = 200 , from the given the there must be a ( shortest piece) ie: the shortest 2 peices are not equal in length, from the values given , i d choose 90.

Hmmm...I don't know about this. I mean we could then have 99 and 101 as the two smallest pieces. The question specifically asked for the greatest possible. Just choosing 90 because that's the lowest value available other than 100 doesn't sound right. Anyone else wanna have a crack at this.
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Re: Mean, Median and 5 pieces of wood [#permalink]

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05 Aug 2009, 13:11

my answer is still 100, since median is 140...for the shortest wood to be max is for the longest 3 to be shortest and have the remaining extra length subtracted between the last two short wood, so we have:

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

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06 Aug 2009, 16:56

scarish wrote:

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.

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

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26 Sep 2009, 18:41

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??

I solved it this way: Mean of the length of five pieces = 124 So total length = 124*5 = 620.

Median = 140, so the length of rest of the 4 pieces = 620 - 140 = 480

Assume the 5 pieces in ascending order be X1 X2 140 X3 X4 where X1 is the shortest.

For X1 to be maximum, X3 and X4 has to be minimum but we have to keep in mind the median has to be 140...HERE i had to stop cos i really dnt know how to proceed furthurserious GAPS in fundamentals.. HELP
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Re: Mean, Median and 5 pieces of wood [#permalink]

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27 Sep 2009, 02:03

I wud go with option 100.

Adding all the pieces we must get 480.So putting the larger 2 pieces as 140.We get addition of 2 smaller pieces as 200. This leaves us with the length of each piece as 100.

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

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28 Sep 2009, 15:19

tejal777 wrote:

ANYBODY?

Scarish provided a very good explanation above.

My approach was the same - maybe this one will be helpful to you:

Total sum = 124 x 5 = 620 Median is number 3 in the total order, meaning that number 4 and 5 cannot be smaller.

Now, the key here is to understand the question: it specifically asks for a maximum size of the smallest piece. They do not say that the size of piecese 4 & 5 is more than 140, nor do they say that the smaller pieces are not of equal size. If you understand this, you can make two assumptions:

1) Apply the value of the median to the pieces 4 & 5 2) Apply the same value to pieces 1 & 2

This way you maximize the size of 1 without breaking out of the terms of the question stem.

So here we go:

1) 3 + 4 + 5 = 140 x 3 = 420 => 1+2 = 620 - 420 = 200 2) Since 1+2=200 => 1 may equal maximum of 100.

Again, the key here is to really understand what is being asked. That way you can make the necessary assumptions.

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

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28 Sep 2009, 19:05

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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.

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

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

I chose (A).

Sum of 5 pieces of wood = 5(124) = 620.

a + b + 140 + d+ e = 620.

Therefore, I chose these numbers and got....

90 + 110 + 140 + 140 + 140 = 620, so hence (A) 90 was my answer. I'm not sure if this is correct. Can anyone tell me if it is or not?

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

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

I chose (A).

Sum of 5 pieces of wood = 5(124) = 620.

a + b + 140 + d+ e = 620.

Therefore, I chose these numbers and got....

90 + 110 + 140 + 140 + 140 = 620, so hence (A) 90 was my answer. I'm not sure if this is correct. Can anyone tell me if it is or not?