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

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

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27 Nov 2018, 05:41
Quote:
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]

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14 May 2019, 10:06
If the sum of ANY 3 numbers is 12 , is not possible that the first number will be 3 the second 4 and the third 5?
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Re: Are all of the numbers in a certain list of 15 numbers equal  [#permalink]

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14 May 2019, 19:12
totoma97 wrote:
If the sum of ANY 3 numbers is 12 , is not possible that the first number will be 3 the second 4 and the third 5?

Hi totoma97,

The example that you describe does NOT fit the given information. Fact 2 tells us that the sum of ANY 3 values will equal 12. Imagine if you had one 3, one 5 and a lot of 4s. It's certainly possible that you could select the one 3, the one 5 and a 4.... but what if you DON'T (you could potentially select one 3 and two 4s... and that sum is NOT 12). Thus, this example is NOT possible given the 'restriction' in Fact 2, so you cannot use it.

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

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23 Aug 2019, 00:22
onedayill wrote:
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

but is a+b+e=12 ? or d+e+a or a+e+c or take any other combination. We need to pick a number such that no matter what, 3 numbers sum to 12. There is only one option -> 4.
e.g 5+5+2=12, 6+4+2 =12, Then is 5+6+2= 12?? or 2+4+5=12?? Nop!

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

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23 Aug 2019, 03:47
The question asks us if all the numbers in the list are equal. This is a ‘Yes-No’ type of DS question.
So, if all the numbers in the list are equal, we can answer the question with a Yes. Even if one of the numbers is unequal, the question can be answered with a No.

From statement I, we know that the sum of the numbers is 60.
If all the 15 numbers are equal to 4, the sum of the numbers will be 60. IN this case, we will be able to answer the main question with a Yes.

However, 10 of the numbers can be 5 each and 5 of them can be 2 each. In this case,

Sum of the 15 numbers = 10 * 5 + 5 * 2 = 50 + 10 = 60.

In this case, the main question can be answered with a No. Therefore, statement I alone is insufficient.
So, answer options A and D can be eliminated. Possible answer options are B, C or E.

From statement II, we know that the sum of ANY 3 numbers in the list is 12.
How do we understand this in a simple manner and not spend a lot of time on it yet??

Let’s say we pick the FIRST 3 numbers in the list, say, a,b and c. We know that a+b+c = 12. This can be satisfied by multiple values of a, b and c. One possible set is a=b=c=4.

Let us now bring in the fourth number, say d. The numbers that we now have are, a, b, c and d. You can pick any 3 of these numbers, but in all cases, the sum should be 12. This is only possible when all of the four numbers are equal to 12.

Let’s say a = 8, b = 3 and c = 1. Clearly, a + b + c = 12. Let d = 4. Out of a, b, c and d, picking b,c and d is also one way of picking any 3 numbers. But, b + c + d = 8 ≠ 12.

This is what I meant by saying that, if the constraint given in statement II has to be satisfied, it can only be done by taking all the numbers equal to 4.
This means that we can now answer the main question with “Yes, all the numbers in the list are equal”.

If you do not spend time on Statement II and try to analyse it in a hurry, you run the risk of falling for the trap answer, which is C.

The correct answer option is B.

Hope this helps!
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Re: Are all of the numbers in a certain list of 15 numbers equal   [#permalink] 23 Aug 2019, 03:47

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