Data Sufficiency

Are x and y both positive?

1) 2x - 2y = 1

2) x/y > 1

Solution:

We are not provided any given information in the question stem, so we can immediately move to the analysis of the two statements.

Statement One Alone:

1) 2x - 2y = 1

The first thing we should do here is to simplify statement one.

2(x - y) = 1

x - y = ½

We can clearly see that x and y can both be positive to yield ½ as their difference (for example, x could be 1.5 and y could be 1)

OR

x and y could both be negative (for example, x could be -1 and y could be -1.5)

OR

x could be positive and y could be negative (for example, x could be ¼ and y could be -¼)

Thus, statement one is insufficient.

Statement Two Alone:

2) x/y > 1

Statement two does not provide enough information to determine whether x and y are either both positive or both negative. Remember that if something is > 1, that something is positive. Also, remember that a negative divided by a negative is positive, and a positive divided by a positive is also positive. We can't get anywhere with statement two alone.

Statements One and Two Together:

It's important to be cognizant of situations in which we are provided an inequality and an equation with the same two variables. In these situations, we can substitute the equation into the inequality. Doing so will allow us to simplify the inequality. In this case we need to first simplify our equation from statement one:

x - y = ½

x = ½ + y

Now we can substitute ½ + y for x into the inequality x/y > 1. Thus, we have:

(½ + y)/y > 1

½/y + y/y > 1

½/y + 1 > 1

½/y > 0

Because ½/y is greater than zero, y MUST also be greater than zero. Lastly, because we know that x = ½ + y, it follows that x MUST also be greater than zero. Thus, both x and y are positive.

Answer C.

Hi Srishti_15,

DS questions can often be solved with a variety of tactics. Beyond the "math" approaches that can be used, you can also TEST VALUES and often use Number Properties to your advantage.

Here, we're asked if X and Y are BOTH positive? This is a YES/NO question.

Fact 1: 2X - 2Y = 1

I'm going to do a little algebra to rewrite this equation....
2X = 1 + 2Y
X = 1/2 + Y

Now we know that X MUST be bigger than Y and we can find some quick values to TEST.
If.....
Y = 0
X = 1/2
Then the answer to the question is NO

Y = 1
X = 1.5
Then the answer to the question is YES
Fact 1 is INSUFFICIENT

Fact 2: X/Y > 1

Using Number Properties, we know that X and Y are either BOTH POSITIVE (YES answer) or BOTH NEGATIVE (NO answer).
Fact 2 is iNSUFFICIENT

Combined, we know....
X = 1/2 + Y
X is GREATER than Y
X/Y > 1
X and Y have the same sign

Since X is greater than Y AND they have the same sign, the ONLY way for X/Y > 1 to be possible is when they're BOTH POSITIVE. If they were both negative, then Y would be greater than X....and we know that that's not possible). The answer to the question is ALWAYS YES.
Combined, SUFFICIENT

Final Answer: C

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