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A set of nonnegative integers consists of {x, x + 7, 2x, y, y + 5}. [#permalink]
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A set of nonnegative integers consists of {x, x + 7, 2x, y, y + 5}. The numbers of this set have four distinct values. What is its average (arithmetic mean)? (1) x ≠ 5 (2) 4y + 12 = 6(y + 2)
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Originally posted by madimo on 09 Oct 2012, 08:33.
Last edited by Bunuel on 29 Mar 2017, 05:15, edited 3 times in total.
Edited the question.



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Re: A set of nonnegative integers consists of {x, x + 7, 2x, y, y + 5}. [#permalink]
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09 Oct 2012, 09:33
madimo wrote: A set of nonnegative integers consists of {x, x + 7, 2x, y, y + 5}. The numbers of this set have four distinct values. What is its average (arithmetic mean)?
(1) x ≠ 5 (2) 4y + 12 = 6(y + 2)
can anybody help? I think the answer should be E, however I saw the answer in some references as C! Hi, It goes like this: Statement 1: X not equal to 5, Which is INSUFFICIENT as we need to know the value of Y. Statement 2 says: 4y+12=6y+12 Solving STATEMENT 2, We get Y=0, Hence the series become, X, X+7, 2X, 0, 5 Now, we know that 2 numbers are identical, Which arises following possibilities: 1: x=2x Which means X=0, (But this is not possible as Three numbers in the series will be equal, which refuted the given condition) 2: X+7=2x, which means, X=7 3: X=5 4: X+7=5, (This is also not possible as it gives negative values of X) So Statement 2 says either X=7 or X=5, So Statement 2 is also Insufficient, Combining bo S 1 and 2, X=7, Hence C..



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Re: A set of nonnegative integers consists of {x, x + 7, 2x, y, y + 5}. [#permalink]
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09 Oct 2012, 09:51
madimo wrote: A set of nonnegative integers consists of {x, x + 7, 2x, y, y + 5}. The numbers of this set have four distinct values. What is its average (arithmetic mean)?
(1) x ≠ 5 (2) 4y + 12 = 6(y + 2)
can anybody help? I think the answer should be E, however I saw the answer in some references as C! A set of nonnegative integers consists of {x, x + 7, 2x, y, y + 5}. The numbers of this set have four distinct values. What is its average (arithmetic mean)?Notice couple of things: 1. We are told that all numbers in the set are integers; 2. We are told that all those integers are nonnegative; 3. We are told that the set contains four distinct values out of five (so two integers out of 5 are alike and other three are distinct). (1) x ≠ 5. Clearly insufficient. (2) 4y + 12 = 6(y + 2) > 4y+12=6y+12 > 2y=0 > y=0. So, we have that our set is {x, x + 7, 2x, 0, 5}. Since we know that two integers out of 5 are alike then: 1. x, x+7 or 2x is 0. If x=0, then 2x=0 too, so in this case we'll have three alike integers x, 2x, and y. Thus this case is out. If x+7=0, then x=7 and we know that all integers must be nonnegative so this case is out too. 2. x, x+7 or 2x is 5 If x=5, then x+7=12 and 2x=10. This scenario is OK. The set in this case would be: {5, 12, 10, 0, 5}If x+7=5, then x=2 and we know that the integers in the set must be nonnegative. Thus this case is out. If 2x=5, then x=5/2 and we know that the numbers in the set must be integers. Thus this case is out. 3. Two out of x, x+7 and 2x are alike. x=2x is not possible, since in this case x=2x=0 and in this case we'll have three alike integers x, 2x, and y. x=x+7 has no solution. 2x=x+7 > x=7. This scenario is OK. The set in this case would be: {7, 14, 14, 0, 5}.Two cases are possible. Not sufficient. (1)+(2) Since from (1) x ≠ 5, then from (2) x=7. So, the set is {7, 14, 14, 0, 5}. Sufficient. Answer: C. Hope it's clear.
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Re: A set of nonnegative integers consists of {x, x + 7, 2x, y, y + 5}. [#permalink]
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09 Oct 2012, 10:08
madimo wrote: A set of nonnegative integers consists of {x, x + 7, 2x, y, y + 5}. The numbers of this set have four distinct values. What is its average (arithmetic mean)?
(1) x ≠ 5 (2) 4y + 12 = 6(y + 2)
can anybody help? I think the answer should be E, however I saw the answer in some references as C! (1) If x = 0, the numbers are 0, 7, 0, y, y + 5. If we take y different from 0, 7 and 2, for sure we get four distinct values. Therefore, we cannot know the average of the numbers. For example y = 1: 0, 7, 0, 1, 6, for y = 10  0, 7, 0, 10, 15. Not sufficient. (2) From the given equality, y = 0. We have the numbers x, x + 7, 2x, 0, 5. x cannot be 0  0, 7, 0, 0, 5  only 3 distinct values x can be 5  5, 12, 10, 0, 5  exactly 4 distinct values x can be 7  7, 14, 14, 0, 5  exactly 4 distinct values Not sufficient. (1) and (2) together: If x is not 5, and x cannot be 0, then x, 2x, 0, and 5 are distinct numbers. It means that x + 7 must be one of them. Since x + 7 > x, x + 7 > 0, x + 7 > 5 (x is positive), the only possibility left is x + 7 = 2x, from which x = 7. The numbers are 7, 14, 14, 0, 5. Sufficient. Answer C.
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Re: A set of nonnegative integers consists of {x, x + 7, 2x, y, y + 5}. [#permalink]
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04 Jun 2013, 05:04



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Re: A set of nonnegative integers consists of {x, x + 7, 2x, y, y + 5}. [#permalink]
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05 Jun 2013, 00:29
madimo wrote: A set of nonnegative integers consists of {x, x + 7, 2x, y, y + 5}. The numbers of this set have four distinct values. What is its average (arithmetic mean)?
(1) x ≠ 5 (2) 4y + 12 = 6(y + 2) From F.S 1, all we know is that x ≠ 5. Now as there are 4 distinct values, any 2 of the elements have to be exactly same. Thus, we can have : x = 2x > x= 0 OR x+7 = 2x> x = 7. In both cases we could assume the value of y = 1 and have the set as (0,7,0,1,6)> Average = 14/5 = 2.8 OR the set would be (7,14,14,1,6) > Average = 42/5 = 8.4. Thus,even after adhering to the condition given in the F.S; as we are getting two different values for average>Insufficient. From F.S 2, we know that y=0. Thus, the set reads as (x,x+7,2x,0,5). To make the elements assume 4 distinct values, we can set x = 5, and have the set as (5,12,10,0,5) We could also assume x = 7 just as above and get the set as (7,14,14,0,5). Again, Insufficient. Taking both the fact statements together, we now know that x = 7. Sufficient. C.
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Re: A set of nonnegative integers consists of {x, x + 7, 2x, y, y + 5}. [#permalink]
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05 Jun 2013, 04:03
An interesting question. 1statement) It tells us tha x does not equal to 5, lets take x=1 and y=1 (I am taking these numbers because in the question stem i see that there are four distinct numbers but we have five numbers, this means that two out of five numbers should be identical, so it is easier to take x and y as the same numbers. But to make this question even clearer i would add the word EXACTLY or ONLY four distinct numbers) then we have 1; 8; 2; 1; 6. If we take x=2 and y=2 then we have 2; 9; 4; 2; 7. Two sets which satisfies the question conditions of having four distinct numbers. Not sufficient. 2 statement) if we solve the equation it tells us that y=0. In this case we have last two numbers: 0 and 5. Lets see what are possible other numbers. We have x; x+7 and 2x. According to the question conditions there should be four distinct numbers, in order that to happen x should be equal to 5 or to 7, because in other cases we would have more than 4 distinct numbers or less than 4 distinct numbers. Two possible options  not sufficient. Combining both statements we see that x does not equall to 5 then it means it is equall to 7. Both statements together are sufficient  choice C.
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Re: A set of nonnegative integers consists of {x, x + 7, 2x, y, y + 5}. [#permalink]
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19 Aug 2013, 08:38
I missread the second stmt, but I think this can be an alternate question where stmt 2 is alone sufficient (1) x ≠ 5 (2) 4x + 12 = 6(y + 2) Came up with answer B – please let me know if my approach is right here.
Set: {x, x + 7, 2x, y, y + 5} Stmt 2 gives: 4x=6y i.e. 2x=3y hence x ≠ y or y ≠ 2x Given that all are integers and are nonnegative, hence x ≠ 2x, so from above set only 2 possible scenarios 1. y=x+7 (when combined with 2x=3y) leads to x = 21 negative not possible 2. x=y+5 (when combined with 2x=3y) leads to y = 10 so the set becomes {15, 22, 30, 10, 15} hence answer is B



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Re: A set of nonnegative integers consists of {x, x + 7, 2x, y, y + 5}. [#permalink]
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13 Jul 2016, 06:14
madimo wrote: A set of nonnegative integers consists of {x, x + 7, 2x, y, y + 5}. The numbers of this set have four distinct values. What is its average (arithmetic mean)?
(1) x ≠ 5 (2) 4y + 12 = 6(y + 2) We are given 5 nonnegative integers and told that 4 are distinct numbers. That means there are two integers that are same. (1) x ≠ 5. We don't know about y. not sufficient. (2) 4y + 12 = 6(y + 2) Solving this equation we get the value of y=5 But we don't know the value of x. Hence not sufficient. Combining both statements: y=5 y+5=10 one value from x, x + 7, 2x is 5 or 10 We know x is not equal to 5 which means 2x is not equal to 10 x+7=10 x= 3 And now we can calculate the average. C is the answer
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Re: A set of nonnegative integers consists of {x, x + 7, 2x, y, y + 5}. [#permalink]
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29 Mar 2017, 05:11
A tough one.. Kudos to bunuel for such nice explanation.
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