Is x = y? (1) x + y = 100 (2) (x 50)^2 = (y 50)^2

is x=y?

1. x+y=100
x=y=50 , =>x=y
or x and y can take any other value
x= 100 , y = 0 =>x ≠ y
Not sufficient

2. \((x-50)^2\)=\((y-50)^2\)
x=y=0 => x=y
x and y can take other values as well , equation 2 will hold as long as x=y
x=25 , y=75 =>x ≠ y

Not sufficient
Combining ,
We still get x=y=50 =>x=y
or x=25 , y=75 =>x ≠ y

Not sufficient

Answer E
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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

Is x = y ?

(1) x + y = 100

(2) (x−50) 2 =(y−50) 2

There are 2 variables (x,y) and 2 equations are given from the conditions, so there is high chance (C) will be our answer.
Combining the 2 conditions,
1) x+y=100, 2) x-50=y-50, or x-50=-y+50, that is, x+y=100. The only satisfying situation is x+y=100.
the answer is 'yes' for x=y=50, but 'no' for x=40, y=60. Therefore the answer is E

For cases where we need 2 more equation, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% 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 C 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, D or E.

Why is B not the answer for this problem?

Given information in 2), I solved as follows:

Step 1. Square root both sides -> x - 50 = y - 50
Step 2. Add 50 to both sides -> x=y
Therefore 2 is sufficient and B is the answer

Hi,

You are only considering the result to be positive when doing the square root on both sides e.g. sqrt(4) can be either 2 or -2.

Although I think the method you have chosen is not the quickest one, if you still want to go with that you will have to consider all the cases.

Case 1: sqrt of (x-50)^2 = +(x-50) and sqrt of (y-50)^2 = +(y-50) => x = y
Case 2: sqrt of (x-50)^2 = -(x-50) and sqrt of (y-50)^2 = +(y-50) => x + y = 100 ( x may or may not be equal to y)
Case 3: sqrt of (x-50)^2 = +(x-50) and sqrt of (y-50)^2 = -(y-50) => x + y = 100 ( x may or may not be equal to y)
Case 4: sqrt of (x-50)^2 = -(x-50) and sqrt of (y-50)^2 = -(y-50) => x = y

Therefore, statement 2 is insufficient.

Hope this helps.

I would suggest to simplify the equation.

(x-50)^2 = (y-50)^2
x^2 -100x + 2500 = y^2 -100y + 2500
x^2 - 100x = y^2 -100y
x^2 - y^2 -100x -100y = 0
x+y-100 = 0 or x-y = 0
x+y = 100 or x = y

Therefore, insufficient.

Thank you! I see what I wasn't thinking though now.

Hi All,

This DS prompt is based around the 'thoroughness' of your thinking. You can likely see an 'obvious' answer when dealing with each of the two Facts, but can you see any OTHER possibiilties... We can TEST VALUES to get to the solution.

We're asked if X equals Y. This is a YES/NO question.

Fact 1: X + Y = 100

IF...
X = 50
Y = 50
Then the answer to the question is YES.

IF...
X = 0
Y = 100
Then the answer to the question is NO.
Fact 1 is INSUFFICIENT.

Fact 1:\((X - 50)^{2}\) = \((Y - 50)^{2}\)

IF...
X = 50
Y = 50
Then the answer to the question is YES.

IF...
X = 0
Y = 100
Then the answer to the question is NO.
Fact 2 is INSUFFICIENT.

Combined, we have two TESTS that already fit BOTH Facts and provide 2 different answers (one YES and one NO), so there's no more work required.
Combined, INSUFFICIENT.

Final Answer:

GMAT assassins aren't born, they're made,
Rich

St1. x+y=100, INSUFF

St2. just say |x-50|=|y-50|, so distance from x to 50 is equal to distance from y to 50, so INSUFF.

St1+St2. Can be multiple options where x=y and x not equal y. INSUFF

E

Because A can be 75 and B 25, still the squares will match.

Kindly help me understand where am I going wrong here?

For (2)

(x-50)^2=(y-50)^2

or, (x-50)^2-(y-50)^2=0
or, (x-50+y-50) (x-50-y+50)=0
or, (x+y-100)(x-y)=0

so, either x=y or x=100-y.

From (1) & (2) X=100-y. Suff (C).

Where am I going wrong?

Please Help!!

Hi SOUMYAJIT,

In DS questions, you have to make sure that you're not missing any of the 'possibilities' and that you're answering the question that is ASKED.

You did enough work to determine that Fact 1 and Fact 2 were both insufficient on their own, but when combined, there is more than one answer to the question "Is X = Y?"

Here are two solutions to the equations given:
X=50
Y=50
and the answer to the question is YES.

X=0
Y=100
and the answer to the question is NO.

The are two different answers to the given question, so combined/insufficient (E) is the answer.

GMAT assassins aren't born, they're made,
Rich

Got it thanks!!

I seem to have this problem, where in :

(1) I do not forget the first options condition when I goto solve the second DS point, thereby actually checking for C and not B.


or


(2) In the process of thinking, I forget the question that was asked to me in the first place.

Sometimes, even in geometry when I make transformations in a figure to solve a question, I forget the real figure that was there in the question in the first place.

Do you have a way that I can handle this?

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