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# Are all of the numbers in a certain list of 15 numbers

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Are all of the numbers in a certain list of 15 numbers [#permalink]

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18 Jul 2009, 00:52
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Are all of the numbers in a certain list of 15 numbers equal?

(1) The sum of all the numbers in the list is 60
(2) The sum of any 3 numbers in the list is 12

OPEN DISCUSSION OF THIS QUESTION IS HERE: are-all-of-the-numbers-in-a-certain-list-of-15-numbers-equal-144144.html
[Reveal] Spoiler: OA

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Re: Are all of the numbers in a certain list of 15 numbers [#permalink]

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04 Mar 2013, 03:10
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pancakeFR wrote:
Hello !

Here is how I solved it. Please correct me if I'm wrong :

(1) : Since you don't have any contraints regarding the numbers : the fifteen numbers can all equal 4 or you can have fourteen 0 and one 60. Not sufficient.

(2) : There are several ways to reach 12 by adding 3 numbers together :
4 + 4 + 4 = 12
3 + 3 + 6 = 12
8 + 2 +2 = 12
etc...

Let's consider the ways where you have at least 2 different numbers. For examples : 3 + 3 + 6. Let's say your fifteen numbers are divided in 5 groups of numbers composed by 3, 3 and 6 :

Statements 2 tells us we can pick any 3 numbers and get 12 by adding them. If you pick one full group : 3+3+6, you get 12. But if you pick 3, 3 from one group and another 3 from another group, you get 3+3+3 = 9. It is therefore impossible to have different numbers, they all have to be the same. Sufficient.

Are all of the numbers in a certain list of 15 numbers equal?

(1) The sum of all the numbers in the list is 60. Clearly insufficient.

(2) The sum of any 3 numbers in the list is 12. Since the sum of ANY 3 numbers is 12 then ALL numbers must equal to 12/3=4, because if not all the numbers equal to 4, then we could pick certain set of 3 numbers so that their sum is not 12. Sufficient.

OPEN DISCUSSION OF THIS QUESTION IS HERE: are-all-of-the-numbers-in-a-certain-list-of-15-numbers-equal-144144.html
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Re: Equal number DS-find the shortcut [#permalink]

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18 Jul 2009, 14:34
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sondenso wrote:
Are all of the numbers in a certain list of 15 numbers equal?

1. The sum of all the numbers in the list is 60
2. The sum of any 3 numbers in the list is 12

Guys, can you tell me what is the logic disguided in the second stat.? Thanks!

There are a few ways to look at this. One is to reverse the problem: say they aren't all equal. Write the set in increasing order: {a, b, c, ..., m, n, o}, and while some of these might be equal, we must have a < o. Well clearly then the sum of the three smallest numbers is less than the sum of the three largest, (a+b+c < m+n+o), so the sum of any three numbers in the list isn't always the same. So the only way S2 can be true is if all the numbers are equal.
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Are all of the numbers in a certain list of 15 numbers [#permalink]

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26 Nov 2009, 12:28
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Are all of the numbers in a certain list of 15 numbers equal?

1) The sum of all the numbers in the list is 60.
2) The sum of any 3 numbers in the list is 12.

OPEN DISCUSSION OF THIS QUESTION IS HERE: are-all-of-the-numbers-in-a-certain-list-of-15-numbers-equal-144144.html

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Re: Are all of the numbers in a certain list of 15 numbers [#permalink]

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26 Nov 2009, 19:09
1
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B
Let's say the list includes a,b,c,d...
According to B, a+b+c = b+c+d = 12 => a=d.
Similarly, all numbers in the list are equal.
Hence B.
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Re: Number properties from OG 12 DS133 [#permalink]

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23 Sep 2010, 07:53
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Atrain13gm wrote:
B is the correct answer not because it allows you to demonstrate that all numbers in the set are equal, but because it allows you to prove they are NOT equal.

If all numbers are the same, then any two numbers would have to be the same. Thus, those numbers would need to be 6 an 6. But 15x6 =90, not 60.

There is a typo in statement (2). Correct question is below:

Are all of the numbers in a certain list of 15 numbers equal?

(1) The sum of all the numbers in the list is 60 --> clearly insufficient.

(2) The sum of ANY 3 numbers in the list is 12 --> as the sum of ANY 3 numbers is 12 then ALL numbers must equal to 12/3=4, because if not all the numbers equal to 4, then we could pick certain set of 3 numbers so that their sum is not 12. Sufficient.

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Re: Are all of the numbers in a certain list of 15 numbers [#permalink]

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26 Sep 2011, 05:04
1
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JimmyWorld wrote:
Are all of the numbers in a certain list of 15 numbers equal?

1) The sum of all the numbers in the list is 60.
2) The sum of any 3 numbers in the list is 12.

Its B

Data from question stem: total numbers = 15

Statement 1: If the sum of all numbers in list in 60, we can not be sure that all are equal. Lets prove it with simple examples.. If 14 numbers are 0 and 1 number is 60...the sum is still 60 but all numbers are not equal... (Answer to question NO)
or if all numbers are 4, the sum is 60 and all numbers are equal (Answer to question YES)

Hence INSUFFICIENT

Statement 2: sum of any 3 numbers is 12..all numbers have to be 4.. Hence SUFFICIENT

Time taken: 44 secs
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Re: Are all of the numbers in a certain list of 15 numbers [#permalink]

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04 Mar 2013, 01:02
1
KUDOS
Hello !

Here is how I solved it. Please correct me if I'm wrong :

(1) : Since you don't have any contraints regarding the numbers : the fifteen numbers can all equal 4 or you can have fourteen 0 and one 60. Not sufficient.

(2) : There are several ways to reach 12 by adding 3 numbers together :
4 + 4 + 4 = 12
3 + 3 + 6 = 12
8 + 2 +2 = 12
etc...

Let's consider the ways where you have at least 2 different numbers. For examples : 3 + 3 + 6. Let's say your fifteen numbers are divided in 5 groups of numbers composed by 3, 3 and 6 :

Statements 2 tells us we can pick any 3 numbers and get 12 by adding them. If you pick one full group : 3+3+6, you get 12. But if you pick 3, 3 from one group and another 3 from another group, you get 3+3+3 = 9. It is therefore impossible to have different numbers, they all have to be the same. Sufficient.

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Re: Equal number DS-find the shortcut [#permalink]

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18 Jul 2009, 03:21
sondenso wrote:
Are all of the numbers in a certain list of 15 numbers equal?

1. The sum of all the numbers in the list is 60
2. The sum of any 3 numbers in the list is 12

from one .....insuff

from 2

the only possible way to be sure of that that they are all equall

B

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Re: Equal number DS-find the shortcut [#permalink]

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18 Jul 2009, 18:29
IanStewart wrote:
sondenso wrote:
Are all of the numbers in a certain list of 15 numbers equal?

1. The sum of all the numbers in the list is 60
2. The sum of any 3 numbers in the list is 12

Guys, can you tell me what is the logic disguided in the second stat.? Thanks!

There are a few ways to look at this. One is to reverse the problem: say they aren't all equal. Write the set in increasing order: {a, b, c, ..., m, n, o}, and while some of these might be equal, we must have a < o. Well clearly then the sum of the three smallest numbers is less than the sum of the three largest, (a+b+c < m+n+o), so the sum of any three numbers in the list isn't always the same. So the only way S2 can be true is if all the numbers are equal.

Thanks IanStewart,
I got it
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Re: Are all of the numbers in a certain list of 15 numbers [#permalink]

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26 Nov 2009, 13:27
I believe it's clear, that (1) is insufficient to answer the questions. Let's analyze (2):

We have set of 15 numbers; $$x_1, x_2, ...x_{15}$$

Any 3 numbers from this set sum up to 12. For example:

$$x_1+x_2+x_3=12$$
$$x_1+x_2+x_4=12$$
. . . . . . . . .
$$x_1+x_2+x_{15}=12$$

It is clear, that $$x_3=x_4=......=x_{15}$$. This is the way to prove that in fact all the numbers in the set are equal.

Answer: (B) - Statement 2 is sufficient

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Re: Are all of the numbers in a certain list of 15 numbers [#permalink]

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27 Nov 2009, 21:28
let analyze satement 2

the sum of any 3 numbers is 12.

let assume 15 numbers are a,b,c,d,e,f,g,h,i,j,k,l,m,n,o

For example a+b+c= 4+4+4=12,d+e+f=4+4+4=12,if we assume like this then statement b is sufficient

but what can i do if a+b+c= 4+4+4=12,d+e+f=4+3+5=12,g+h+i=1+6+5=12......from this example all numbers couldnt be equal
so b is not sufficient .plz expain

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Re: Are all of the numbers in a certain list of 15 numbers [#permalink]

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27 Nov 2009, 21:33
TomB wrote:
let analyze satement 2

the sum of any 3 numbers is 12.

let assume 15 numbers are a,b,c,d,e,f,g,h,i,j,k,l,m,n,o

For example a+b+c= 4+4+4=12,d+e+f=4+4+4=12,if we assume like this then statement b is sufficient

but what can i do if a+b+c= 4+4+4=12,d+e+f=4+3+5=12,g+h+i=1+6+5=12......from this example all numbers couldnt be equal
so b is not sufficient .plz expain

if we take f,h,i from ur list the sum is not 12, from S2, any 3 numbers should total 12... so they all have to be 4
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Re: Are all of the numbers in a certain list of 15 numbers [#permalink]

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28 Nov 2009, 23:04
JimmyWorld wrote:
Are all the numbers in a certain list of 15 numbers equal?

1) The sum of all the numbers in the list of 60.
2) The sum of any 3 numbers in the list is 12.

Not sure on 2). Wonder if someone could explain? Thank you

Look for the clue: sum of "any three number" is 12.

Sum of any 3 numbers from a, b, c, d, e, f, g, h, i, j must be 12. That is possible only if all these numbers must be equall and integers i.e. 4.

TomB wrote:
let analyze satement 2

the sum of any 3 numbers is 12.

let assume 15 numbers are a,b,c,d,e,f,g,h,i,j,k,l,m,n,o

For example a+b+c= 4+4+4=12,d+e+f=4+4+4=12,if we assume like this then statement b is sufficient

but what can i do if a+b+c= 4+4+4=12,d+e+f=4+3+5=12,g+h+i=1+6+5=12......from this example all numbers couldnt be equal
so b is not sufficient .plz expain

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Are all of the numbers in a certain list of 15 numbers equal? [#permalink]

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03 Aug 2010, 13:36
Are all of the numbers in a certain list of 15 numbers equal?

1) The sum of all the numbers in the list is 60.
2) The sum of any 3 numbers in the list is 12.

OPEN DISCUSSION OF THIS QUESTION IS HERE: http://gmatclub.com/forum/are-all-of-th ... 44144.html

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Re: Are all of the numbers in a certain list of 15 numbers equal? [#permalink]

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03 Aug 2010, 13:36
Can someone explain this.

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Re: Are all of the numbers in a certain list of 15 numbers equal? [#permalink]

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03 Aug 2010, 13:47
zest4mba wrote:
Are all of the numbers in a certain list of 15 numbers
equal?
(1) The sum of all the numbers in the list is 60.
(2) The sum of any 3 numbers in the list is 12.

I think I've posted different solutions to this question elsewhere, but one way to look at Statement 2:

let a, b, c and d be four random numbers from the list. From Statement 2, since the sum of *any* three numbers is 12, we know

a + b + c = 12
d + b + c = 12

and if you subtract the second equation from the first, you find that a - d = 0, so a = d. Since a and d are just randomly chosen numbers from the list, we could do the same thing for any of the numbers in our list, and so they all need to be equal. Since Statement 1 is not sufficient, the answer is B.
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Re: Are all of the numbers in a certain list of 15 numbers equal? [#permalink]

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03 Aug 2010, 15:42
This was my approach

You have 15 numbers that add up to 60. Factoring 60 you get that the only combination of two factors that have 15 must have 4 on them. There is no other number than 4 that repeated 15 times add up to 60

1) You know that you need a 4 for the statement to be sufficient, but as it can be any other number or fraction is not sufficient.

2) Any combination of 3 numbers of the list add 12. You can see that 4 is your lucky number, therefore answer is B.

Hope Im right!

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Re: Are all of the numbers in a certain list of 15 numbers equal? [#permalink]

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04 Aug 2010, 20:09
IanStewart wrote:
zest4mba wrote:
Are all of the numbers in a certain list of 15 numbers
equal?
(1) The sum of all the numbers in the list is 60.
(2) The sum of any 3 numbers in the list is 12.

I think I've posted different solutions to this question elsewhere, but one way to look at Statement 2:

let a, b, c and d be four random numbers from the list. From Statement 2, since the sum of *any* three numbers is 12, we know

a + b + c = 12
d + b + c = 12

and if you subtract the second equation from the first, you find that a - d = 0, so a = d. Since a and d are just randomly chosen numbers from the list, we could do the same thing for any of the numbers in our list, and so they all need to be equal. Since Statement 1 is not sufficient, the answer is B.

I still didn't get you....

There are 15 different numbers, but we took only 4 for our example.

what if a + b+ c = 12
and d + e+ f = 12
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Re: Are all of the numbers in a certain list of 15 numbers equal? [#permalink]

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04 Aug 2010, 22:28
onedayill wrote:
IanStewart wrote:
zest4mba wrote:
Are all of the numbers in a certain list of 15 numbers
equal?
(1) The sum of all the numbers in the list is 60.
(2) The sum of any 3 numbers in the list is 12.

I think I've posted different solutions to this question elsewhere, but one way to look at Statement 2:

let a, b, c and d be four random numbers from the list. From Statement 2, since the sum of *any* three numbers is 12, we know

a + b + c = 12
d + b + c = 12

and if you subtract the second equation from the first, you find that a - d = 0, so a = d. Since a and d are just randomly chosen numbers from the list, we could do the same thing for any of the numbers in our list, and so they all need to be equal. Since Statement 1 is not sufficient, the answer is B.

I still didn't get you....

There are 15 different numbers, but we took only 4 for our example.

what if a + b+ c = 12
and d + e+ f = 12

Say our list is:

a, b, c, d, e, f, g, h, i, j, k, l, m, n, o

I just took the first four numbers and proved a=d. There's nothing special about the first four numbers in the list; I can use the same logic to prove that any two numbers are equal here. For example, if I want to prove that b=d, we have

b + c + a = 12
d + c + a = 12

Subtract the second equation from the first:

b - d = 0
b = d

So now we know that b = d. Since we saw that a=d as well, a, b and d are all equal. We can do this for all the letters in the list, so they all must be equal.
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Re: Are all of the numbers in a certain list of 15 numbers equal?   [#permalink] 04 Aug 2010, 22:28

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