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# Three boxes have an average weight of 7kg and a median weigh

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Three boxes have an average weight of 7kg and a median weigh [#permalink]

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22 Aug 2010, 20:19
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Three boxes have an average weight of 7kg and a median weight of 9kg. What is the maximum weight of the smallest box?

A. 1
B. 2
C. 3
D. 4
E. 5
[Reveal] Spoiler: OA
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Re: Box Weight word problem [#permalink]

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22 Aug 2010, 20:29
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kwhitejr wrote:
Three boxes have an average weight of 7kg and a median weight of 9kg. What is the maximum weight of the smallest box?

A. 1
B. 2
C. 3
D. 4
E. 5

Given that the median is 9 that leaves us with the other two boxes with a combined weight of 12 kg. [Average * number = total --- 7 * 3 = 21. And total - median = sum of heaviest and smallest box = 12kg]

Now 9 should be the median and the smallest box has to have the max weight. To leave the median as 9, the heaviest box should be at least 9 kg ... which gives us 3 kg for the smallest box.

Hence the weight of the box should be 3, 9, 9. Hence answer is C.
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Re: Box Weight word problem [#permalink]

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22 Aug 2010, 20:30
Hi,

Start with assuming that the smallest is a little less than 9 kg. Say, it is 8 kg.

Now 8 + 9 + Largest = (3*3) = 21. This is a contradiction because 'Largest' becomes 4.

Hence, reduce the 'smallest' to 7 kg. Now, 7+9+Largest = 21. This will also result in a contradiction.

Continue doing this until the 'Largest' is indeed the 'Larger' than 9 kgs (which is the median).

Hope this helps. Thanks.
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Re: Box Weight word problem [#permalink]

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22 Aug 2010, 21:48
@4gmatmumbai

Thanks for your suggestion. That is actually what I did originally to solve the question, but I came up with the max weight for the smallest box to be 2. Thus the three boxes would be 2, 9 and 10. However, if the boxes are 3, 9 and 9 then the median is still 9? This is what throws me off.
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Re: Box Weight word problem [#permalink]

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23 Aug 2010, 04:56
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kwhitejr wrote:
Three boxes have an average weight of 7kg and a median weight of 9kg. What is the maximum weight of the smallest box?

A. 1
B. 2
C. 3
D. 4
E. 5

Three boxes have an average weight of 7kg --> $$a+b+c=3*7=21$$;
Three boxes have a median weight of 9kg --> median of a set with odd terms is middle term, hence $$b=9$$;

So we have a, 9, c. Question: $$a_{max}=?$$

General rule for such kind of problems:
to maximize one quantity, minimize the others;
to minimize one quantity, maximize the others.

So to maximize $$a$$ we should minimize $$c$$ --> minimum value of $$c$$ is 9 ($$c$$, the third term, can not be less than median, the second term) --> so $$a_{max}+9+9=21$$ --> $$a_{max}=3$$.

Hope it's clear.
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Re: Box Weight word problem [#permalink]

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23 Aug 2010, 09:51
lets assume 3 boxes are X,Y,Z
x+y+z= 21
Median = 9 ( middle number )
By back solving
smallest maximum weight
X Y(median) Z
1 9 11
2 9 10
3 9 9
4 9 8 ( it cant be the answer because Z value is below the Y (median value))
5 9 7 (it cant be the answer because Z value is below the Y (median value))

so the smallest maximum weight is 9 (option C)
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Re: Box Weight word problem [#permalink]

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23 Aug 2010, 11:28
kwhitejr wrote:
@4gmatmumbai

Thanks for your suggestion. That is actually what I did originally to solve the question, but I came up with the max weight for the smallest box to be 2. Thus the three boxes would be 2, 9 and 10. However, if the boxes are 3, 9 and 9 then the median is still 9? This is what throws me off.

Hi kwhitejr,

Definitely - the median of 3,9 and 9 is indeed the 2nd term -> 9.

Just pick the middle term after sorting them in ascending order - no matter what.

Hope this helps. Thanks.
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07 May 2013, 06:50
14. Three boxes of supplies have an average (A.M.) weight of 7 kg and median of 9 kg. What is the maximum
possible weight in kg of the lightest box?
a)1 b)2 c)3 d)4 e)5
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07 May 2013, 07:58
Option C. 3Kg.

Since we need to maximize the minimum weight , let the weights of the 2 heaviest boxes be equal to the median = (kg.

So total weight of all boxes is 3*7 =21 and the combined weight of the 2 heaviest boxes = 9+9 = 18Kg.

so the maximum weight of the remaining box can be 3 KG
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07 May 2013, 09:32
Option C. 3Kg.

Since we need to maximize the minimum weight , let the weights of the 2 heaviest boxes be equal to the median = (kg.

So total weight of all boxes is 3*7 =21 and the combined weight of the 2 heaviest boxes = 9+9 = 18Kg.

so the maximum weight of the remaining box can be 3 KG

This solution is correct, and the whole question hinges only on whether you have the correct definitions for median and average. If the average is 7 then the total weight must be 7x3=21 kg.

The median of a 3-term set is the middle term, so the middle term must be 9. The biggest term can be no smaller than 9, so at a minimum it's 9, leaving only 3 kg for the smallest entry.

My question here is then about the difficulty level. I'm not sure I'd put this as a 600-700 level question as this is easily solvable in ~1 minute for most people. Thoughts?

Thanks!
-Ron
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07 May 2013, 09:55
rochak22 wrote:
14. Three boxes of supplies have an average (A.M.) weight of 7 kg and median of 9 kg. What is the maximum
possible weight in kg of the lightest box?
a)1 b)2 c)3 d)4 e)5

I also agree with md .... I got the answer as C..........
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08 May 2013, 05:27
rochak22 wrote:
14. Three boxes of supplies have an average (A.M.) weight of 7 kg and median of 9 kg. What is the maximum
possible weight in kg of the lightest box?
a)1 b)2 c)3 d)4 e)5

Merging similar topics.
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Re: Three boxes have an average weight of 7kg and a median weigh [#permalink]

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09 May 2013, 11:14
3 Boxes: S, M, L
M=9 Therefore L≥M, L≥9

Average= (S+M+L)/3 = 7
Sum= S+M+L = 21
S+L = 12

S must be 3 since L must be at least 9 and we are looking for the maximum S.

Must be C
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Re: Three boxes have an average weight of 7kg and a median weigh [#permalink]

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11 Sep 2014, 12:07
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11 Jul 2016, 00:29
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Three boxes have an average weight of 7kg and a median weigh [#permalink]

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24 Sep 2016, 14:52
C.

x1 + x2 + x3 = 21 --> plug in x2 = 9. We want to minimize x3 in order to maximize x1, so we'll keep x3 @ 9

x2 + x3 = 18 --> x1 = 3
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Re: Three boxes have an average weight of 7kg and a median weigh [#permalink]

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18 Dec 2016, 10:24
Nice One.
Here is my approach to this one ->

Let the boxes be
w1
w2
w3

In increasing order

w1+w2+w3=21
w2=9

To maximises w1 -> w2=9 And w2=9

Hence w1=> 21-18=3

Hence C

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Re: Three boxes have an average weight of 7kg and a median weigh   [#permalink] 18 Dec 2016, 10:24
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