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# There are 5 boxes of different weights...

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Intern
Joined: 01 Nov 2015
Posts: 20
There are 5 boxes of different weights...  [#permalink]

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08 Jun 2016, 06:08
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55% (hard)

Question Stats:

64% (02:45) correct 36% (02:20) wrong based on 84 sessions

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There are 5 boxes of different weights. They are weighed in pairs taken 2 at a time and the weights obtained are 110, 112, 113, 114, 115, 116, 117, 118, 120 and 121. What is the sum of the weights of the 5 boxes ?

A) 250
B) 289
C) 290
D) 295
E) 305
Intern
Joined: 08 Jun 2016
Posts: 1
Re: There are 5 boxes of different weights...  [#permalink]

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08 Jun 2016, 06:30
1
Let the 5 boxes be a, b, c, d and e. The ten weights correspond to all the possible combinations of 2 weights out of 5.
On adding all the given weights, each boxes weight gets added 4 times:
a+b, a+c, a+d, a+e.
Similarly for b:
b+c, b+d, b+e and a+b (already included above)
Similarly for c, d and e.
Therefore, 110+112+113+114 .....+120+121 = 4* (a+b+c+d+e)
or, 4* (a+b+c+d+e) = 1156
or, (a+b+c+d+e) = 1156/4 = 289
Math Expert
Joined: 02 Aug 2009
Posts: 6785
There are 5 boxes of different weights...  [#permalink]

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08 Jun 2016, 06:52
2
aayushagrawal wrote:

There are 5 boxes of different weights. They are weighed in pairs taken 2 at a time and the weights obtained are 110, 112, 113, 114, 115, 116, 117, 118, 120 and 121. What is the sum of the weights of the 5 boxes ?

A) 250
B) 289
C) 290
D) 295
E) 305

Hi,

since the weights given are of 2 boxes taken at one time..
so the SUM of all the weights measured taking two boxes at any time will be 4 times total weight of all five combined..

We choose 2 boxes out of 5 in 5C2 ways = 5!/3!2! = 10 ways....
How many ways 1 box say A will not be included = choosing 2 out of 4 boxes other than A = 4C2 = 4!/2!2! =6..
so NUMBER of times A has been counted is 10-6 = 4.....

similarily all other boxes also are counted 4 times...
eg say A,B,C,D,E - AB,AC,AD,AE has A in it

Now since these 10 weights of pair of boxes given has 4 times each box..
we divide the SUM of these 10 weights to get SUM of single boxes

$$\frac{110+112+113+114+115+116+117+118+120+121}{4}$$...
UNITS digit is 6 of the NUMERATOR..

so when you divide by 4, 0 and 5 cannot be the UNITS digit..
ONLY B has units digit other than 0 and 5..
ans B
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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There are 5 boxes of different weights...  [#permalink]

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08 Jun 2016, 09:40
chetan2u wrote:
aayushagrawal wrote:

There are 5 boxes of different weights. They are weighed in pairs taken 2 at a time and the weights obtained are 110, 112, 113, 114, 115, 116, 117, 118, 120 and 121. What is the sum of the weights of the 5 boxes ?

A) 250
B) 289
C) 290
D) 295
E) 305

Hi,

since the weights given are of 2 boxes taken at one time..
so the SUM of all the weights measured taking two boxes at any time will be 4 times total weight of all five combined..
$$\frac{110+112+113+114+115+116+117+118+120+121}{4}$$... UNITS digit is 6 of the NUMERATOR..
so when you divide by 4, 0 and 5 cannot be the UNITS digit..
ONLY B has units digit other than 0 and 5..
ans B

Thank you for the explanation chetan2u! I was just looking to understand exactly, why is it that the number is 4 times the total weight of all five boxes combined. Would it be correct to say that every number is counted 5 times instead of one, which makes it 4 times too much. And since it applies to every single number, we divide by 4 to cancel repetition? Thank you!
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Math Expert
Joined: 02 Aug 2009
Posts: 6785
Re: There are 5 boxes of different weights...  [#permalink]

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08 Jun 2016, 09:48
1
fantaisie wrote:
chetan2u wrote:
aayushagrawal wrote:

There are 5 boxes of different weights. They are weighed in pairs taken 2 at a time and the weights obtained are 110, 112, 113, 114, 115, 116, 117, 118, 120 and 121. What is the sum of the weights of the 5 boxes ?

A) 250
B) 289
C) 290
D) 295
E) 305

Hi,

since the weights given are of 2 boxes taken at one time..
so the SUM of all the weights measured taking two boxes at any time will be 4 times total weight of all five combined..
$$\frac{110+112+113+114+115+116+117+118+120+121}{4}$$... UNITS digit is 6 of the NUMERATOR..
so when you divide by 4, 0 and 5 cannot be the UNITS digit..
ONLY B has units digit other than 0 and 5..
ans B

Thank you for the explanation chetan2u! I was just looking to understand exactly, why is it that the number is 4 times the total weight of all five boxes combined. Would it be correct to say that every number is counted 5 times instead of one, which makes it 4 times too much. And since it applies to every single number, we divide by 4 to cancel repetition? Thank you!

Hi,

It is 4 times the SUM of all five boxes because-

We choose 2 boxes out of 5 in 5C2 ways = 5!/3!2! = 10 ways....
How many ways 1 box say A will not be included = choosing 2 out of 4 boxes other than A = 4C2 = 4!/2!2! =6..
so NUMBER of times A has been counted is 10-6 = 4.....

similarily all other boxes also are counted 4 times...
eg say A,B,C,D,E - AB,AC,AD,AE has A in it

Now since these 10 weights of pair of boxes given has 4 times each box..
we divide the SUM of these 10 weights to get SUM of single boxes
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Re: There are 5 boxes of different weights...  [#permalink]

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09 Oct 2017, 17:25
1
aayushagrawal wrote:

There are 5 boxes of different weights. They are weighed in pairs taken 2 at a time and the weights obtained are 110, 112, 113, 114, 115, 116, 117, 118, 120 and 121. What is the sum of the weights of the 5 boxes ?

A) 250
B) 289
C) 290
D) 295
E) 305

When the boxes are weighted 2 at a time, the weight of each box is counted 4 times. Thus, the sum 110 + 112 + 113 + 114 + 115 + 116 + 117 + 118 + 120 + 121 = 1,156 must be 4 times the sum of the weights of the 5 boxes. Therefore, the sum of the weights of the 5 boxes is 1,156/4 = 289.

Note: If you can’t see why the weight of each box is counted 4 times, we can label the boxes as A, B, C, D, and E. Thus, when we pair them up, we have:

AB
AC
AE
BC
BD
BE
CD
CE
DE

We see that each of the letters appears 4 times in the list.

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Re: There are 5 boxes of different weights... &nbs [#permalink] 09 Oct 2017, 17:25
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