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# Are the integers x, y and z consecutive?

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Math Expert
Joined: 02 Sep 2009
Posts: 50572
Are the integers x, y and z consecutive?  [#permalink]

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11 May 2016, 02:19
1
3
00:00

Difficulty:

35% (medium)

Question Stats:

63% (01:03) correct 37% (01:08) wrong based on 92 sessions

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Are the integers x, y and z consecutive?

(1) The arithmetic mean (average) of x, y and z is y.
(2) y-x = z-y

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Are the integers x, y and z consecutive?  [#permalink]

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11 May 2016, 03:41
St1: The arithmetic mean (average) of x, y and z is y.
y = (x + y + z)/3
y = (x + z)/2
If x, y, z = 1, 2, 3 then x, y and z are consecutive
If x, y, z = 2, 4, 6 then x, y and z are not consecutive
Not Sufficient

St2: 2y = x + z
This statement is a replica of St1. Not Sufficient

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Re: Are the integers x, y and z consecutive?  [#permalink]

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11 May 2016, 19:28
In the original condition, there are 3 variables (x, y, and z). In order to match the number of equations to the number of variables, we need 3 equations. However, since the condition 1) and the condition 2) each has 1 equation, we lack 1 equation. Therefore, there is high chance that E is the correct answer. Using both the condition 1) and the condition 2), from x+z=2y, we get x=1, y=2 and z=3. The answer is yes. However, when we get x=1, y=5 and z=9. The answer becomes no. Hence, the conditions are not sufficient. Therefore, the correct answer is E.

- For cases where we need 3 more equations, such as original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 80% chance that E is the answer (especially about 90% of 2 by 2 questions where there are more than 3 variables), while C has 15% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since E is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, C or D.

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Re: Are the integers x, y and z consecutive?  [#permalink]

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15 Dec 2017, 06:34
Bunuel wrote:
Are the integers x, y and z consecutive?

(1) The arithmetic mean (average) of x, y and z is y.
(2) y-x = z-y

x,y,z WILL be consecutive Integers if y=x+1 and z=x+2

Stat 1 The arithmetic mean (average) of x, y and z is y.
i.e AM=(x+y+z)/3=y
=>x+y+z=3y
=>x+z=2y --> (1)
=>(x+z)/2=y
i.e y is avg. of x & z
so MUST that x<y<z and y-x=z-y
Therefore x,y,z are in AP
i.e x,y,z can be CONSECUTIVE Integers such as (1,2,3) or NON CONSECUTIVE integers such as (1,5,9)
So Stat 1 NOT Sufficient

Stat 2 y-x = z-y
i.e x+z=2y -> (1) - derived from statement 1
So Stat 2 ALSO NOT sufficient

BOTH 1 & 2 also does NOT give a unique solution. Therefore NOT sufficient.

So Option "E"

The detailed explanation is for understanding the concept.

BUNUEL/Experts pl. validate the above analysis.

Thanks
Re: Are the integers x, y and z consecutive? &nbs [#permalink] 15 Dec 2017, 06:34
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