OG 11 - DS 118

[sk818020]

If x is an integer, is y an integer?

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

(2) The average (arithmetic mean) of x and y is not an integer.

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(1) (x + y + y - 2) / 3 = x

x + 2y - 2 = 3x
2y - 2 = 2x
y - 1 = x
y = x + 1

Because x and 1 are both integers, y must be an integer. (1) is sufficient.

The OA is A. But, please look at my algebra and reasoning on (2) and tell me where I went wrong.

(2)

Any number divided by 2 that is not divisible by 2 will have a remainder of 1/2. Algebraically stated, if a number is not divisible by 2, then;

n/2 = q + 1/2

where n is any number not divisible by 2 and q is the quotient which is an integer.

Using this (2) tells us;

(x+y)/2 = q + 1/2

x+y = 2q + 1

y = 2q + 1 - x

where 2q, 1, and x are all integers. Thus, y must be an integer.

(2) should be sufficient.

Can someone please explain why my proof is incorrect?

Thanks,

Jared

[mj78ind]

@Jared

I think in 2 you have already assumed y to be an integer. Say x = 3, y =4 Now the remainder when x+y is divided by 2 is 1/2 ans (3+4)/2 can be written as q + 1/2. What if y = 7/4, then (x+y)/2 can not be written as q+1/2. Hence the logic as shown (by you) does not stand.

Here is another approach to solve this question:
Stmt 1 can be solved as you did.

Stmt 2 (x+y)/2 = non int (ni)

Thus y = ni*2 - x --------(1)
hence if we can somehow find a ni that gives and integer y and another ni which gives a non integer y, then 2 is insufficient. Say ni = 7/2, thus as per (1) y is an integer now say ni = 7/4, now per (1) y is not an integer. Hence 2 is INSUFFICIENT.

Answer A

[sk818020]

@Jared

I think in 2 you have already assumed y to be an integer. Say x = 3, y =4 Now the remainder when x+y is divided by 2 is 1/2 ans (3+4)/2 can be written as q + 1/2. What if y = 7/4, then (x+y)/2 can not be written as q+1/2. Hence the logic as shown (by you) does not stand.

Here is another approach to solve this question:
Stmt 1 can be solved as you did.

Stmt 2 (x+y)/2 = non int (ni)

Thus y = ni*2 - x --------(1)
hence if we can somehow find a ni that gives and integer y and another ni which gives a non integer y, then 2 is insufficient. Say ni = 7/2, thus as per (1) y is an integer now say ni = 7/4, now per (1) y is not an integer. Hence 2 is INSUFFICIENT.

Answer A

Actually in statement 2 I am leaving y a variable and using the other constants/variable to tell me what y is in my equation. I'm looking for why my algebra is wrong. If y = 2q + 1 - x and x, 1, and 2q are all integers how could y be a fraction? Where did I go wrong in my algebra?

[saptrishi]

Actually this expression "n/2 = q + 1/2" to represent a number which is not divisible by 2 is only applicable to integer values of n. Since y is not necessarily an integer, x+y or n will not necessarily be an integer. Hence you cannot represent n the above way.

[sk818020]

mj78ind, saptrishi. Thanks guys. Your absolutely right. I can't believe I did that. Oh well thats what good people like you are for.

Thanks again!