Data Sufficiency

If xy is not equal to 0, is (1/x + 1/y) = 2?
(1) x = y
(2) x + y = 2xy

OA is B but i guess statement one is also sufficient. How? by putting the value of y as x we will get 1/x +1/x = 2 = 2/x = 2 and solving this we will get x = 1. thus if x and y are equal than 1 + 1 is equal to 2.

You have shown that the answer to the question could be 'yes' according to (1). But what if x=y=2 ?

If xy is not equal to 0, is (1/x + 1/y) = 2?
(1) x = y
(2) x + y = 2xy

There are three keys to data sufficiency.

Don't assume anything beyond the confines of the given info and the given statements. For example, the statement x>3 means x is greater than 3. It doesn't mean x is an integer, or x is 5, or x isn't an irrational number.

Evaluate the statements separately. Don't look at statement 2 before evaluating statement 1. Don't think about statement 1 when evaluating statement 2.

Eliminate answer choices based on evaluations. For example, if statement 1 isn't sufficient, instantly eliminate A and D.

The given info is that neither x nor y are equal to 0. If one of them is 0, or both of them are 0, then xy=0.

1. x=y tells us that they are equal. It doesn't tell us their values. That isn't sufficient to determine if (1/x + 1/y) = 2. x and y could be 1, or they could be 2, or they could be 1/2, or they could be... Eliminate A and D.

2. x + y = 2xy: Try some numbers. If x=1 and y=1 then x+y = 2 = 2(1)(1). 1/1 + 1/1 = 2. If x=1 and y=2 then x+y = 3 which does not equal 2(1)(2) = 4. You won't find any other combination of x and y (1/x + 1/y = 2)that makes x+y = 2xy . Statement 2 is sufficient. B is the answer.

That idea of rephrasing target questions is a good one. It works out perfectly for this question.

fcabanski wrote:2. x + y = 2xy: Try some numbers. If x=1 and y=1 then x+y = 2 = 2(1)(1). 1/1 + 1/1 = 2. If x=1 and y=2 then x+y = 3 which does not equal 2(1)(2) = 4. You won't find any other combination of x and y that makes x+y = 2xy. Statement 2 is sufficient.

What about (x = 2 and y = 2/3), (x = 3 and y = 3/5), (x = 4 and y = 4/7) and many more?

If you are trying to say "You won't find any other combination of x and y for which x+y = 2xy but (1/x + 1/y) ≠ 2", then also the logic is flawed because you can prove insufficiency of a statement by showing a contradiction but you cannot prove sufficiency of a statement by not showing a contradiction as you never whether you have considered all possible use cases or not. To prove sufficiency we must have unquestionable evidence like facts (like countable examples) or mathematical logic.

An weak example will be...

Is x > 0?
1. x ≥ 0
2. x > 1

For statement 1, I cannot say that "I've checked for x = 0.1, x = 1, x = 2, ..., x = 100 till x = 10,000 and I haven't found any contradiction, so statement 1 must be sufficient" as I'm not considering the possibility that x may be equal to zero.

It is true that this is not a good example as the contradiction is in front of our eyes but you get the point what I'm trying to make. I can come up with a better example also (In fact, I've one ready but that problem is not relevant to GMAT.)

sana.noor wrote:If xy is not equal to 0, is (1/x + 1/y) = 2?
(1) x = y
(2) x + y = 2xy

An alternate approach that does not require rephrasing of the question stem:

Statement 1: x=y
If x=y=1, then 1/x + 1/y = 1+1 = 2.
If x=y=2, then 1/x + 1/y = 1/2 + 1/2 = 1.
INSUFFICIENT.

Statement 2: x + y = 2xy
Plug different values for x into x + y = 2xy and solve for y.

If x=1, we get:
1 + y = 2*1*y
y = 1.
In this case, 1/x + 1/y = 1/1 + 1/1 = 2.

If x=2, we get:
2 + y = 2*2*y
2 = 3y
y = 2/3.
In this case, 1/x + 1/y = 1/2 + 3/2 = 2.

Since 1/x + 1/y = 2 in both cases, we should be pretty convinced that statement 2 is sufficient to answer the question stem.
To be safe, one more case:
If x=-2, we get:
-2 + y = 2(-2)y
-2 = -5y
y = 2/5.
In this case, 1/x + 1/y = -1/2 + 5/2 = 2.

Since 1/x + 1/y = 2 in every case, SUFFICIENT.

The correct answer is B.

Thanks Jeff for pointing out the mistake. I forgot to include "when 1/x + 1/y = 2".

As for the second issue:

"To prove sufficiency we must have unquestionable evidence like facts (like countable examples) or mathematical logic."

The Data Sufficiency section is not about proving sufficiency. It's about selecting the correct answer. If you select the correct answer but don't prove your reasoning, you still get the credit for a correct answer. Examining values for x and y, and examining trends, is good enough for proving, to the test taker, that statement 2 is sufficient.

I challenge you to find an exception for this problem (that shows statement 2 is not sufficient).

The best method is performing the algebra to see that 1/x + 1/y = 2 --> x + y = 2xy. That is unquestionable evidence. But someone had already posted that answer. I was providing an alternative to show that sometimes a test taker may not be able to perform the algebra (may not think of it), but he can still choose the correct answer.

fcabanski wrote:The Data Sufficiency section is not about proving sufficiency. It's about selecting the correct answer. If you select the correct answer but don't prove your reasoning, you still get the credit for a correct answer. Examining values for x and y, and examining trends, is good enough for proving, to the test taker, that statement 2 is sufficient.

I challenge you to find an exception for this problem (that shows statement 2 is not sufficient).

May be it was not clear from my previous post but I was trying to make a general point about DS problems not specific to this problem. And I never said that you have to show the proof of sufficiency to someone! I was trying to say that by not finding an exception you can never be 100% sure that a statement is sufficient (not only for this problem). Only by identifying a trend for some examples without any logical basis we cannot claim a statement always holds because it may very well be that there is an exception to the trend which we didn't consider.

In some cases (like this) we can get lucky and identify the correct answer but if students are taught this method of marking a statement sufficient by not finding an exception, they may not be always so lucky.

Also I've tried to provide an example of how people can make the mistake.
If that was not sufficient, here is another...

If p is a positive integer, is 2^p - 1 prime?
1. p < 5
2. p is prime

For statement 1, if p = 1, (2^p - 1) = 1 is not prime
And if p = 2, (2^p - 1) = 3 is a prime
So, statement 1 is not sufficient as we have found a contradiction.

[Now if a student is taught to mark a statement sufficient by not finding an exception, he/she will approach as follows...]
For statement 2, let's check some values for p,
for p = 2, (2^2 - 1) = 4 - 1 = 3 is a prime
for p = 3, (2^3 - 1) = 8 - 1 = 7 is a prime
for p = 5, (2^5 - 1) = 32 - 1 = 31 is a prime
for p = 7, (2^7 - 1) = 128 - 1 = 127 is a prime
...
Looks like there is a trend. So 2^p - 1 is always prime.
So, statement 2 is sufficient as we have not found a contradiction.

But in this case that logic fails!
Because for p = 11, 2^p - 1 = 2048 - 1 = 2047 = 23*89 is not prime.
So, actually statement 2 is not sufficient.


I hope now I'm being clear about the point I'm trying to make.

Your point is clear. But it's not valid. I never suggested test takers should try to prove sufficiency by excluding one or a few values.

"Only by identifying a trend for some examples without any logical basis we cannot claim a statement always holds because it may very well be that there is an exception to the trend which we didn't consider." I agree. That's why the algebraic answer is best. But that answer was already given. I gave an answer for "what if you didn't think of the algebra". It's not perfect or 100% accurate in all cases. But if the test taker is careful, it is accurate.