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Is the product of x and y greater than the sum of x and y?

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by Vincen » Tue Jan 09, 2018 5:56 am
Is the product of x and y greater than the sum of x and y?

(1) xy < 0

(2) x > -y

The OA is C.

I don't know how to use both statements together to get an answer. What can I conclude from the statement (2)? Experts, I'd appreciate your help. Thanks.
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Source: — Data Sufficiency |

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by Jay@ManhattanReview » Wed Jan 10, 2018 12:42 am
Vincen wrote:Is the product of x and y greater than the sum of x and y?

(1) xy < 0

(2) x > -y

The OA is C.

I don't know how to use both statements together to get an answer. What can I conclude from the statement (2)? Experts, I'd appreciate your help. Thanks.
We have to determine if xy > x + y.

(1) xy < 0

=> One between x and y is negative and the other is positive.

Case 1: Say x = 1 and y = -1. Thus xy = -1*1 = - 1; x + y = -1 + 1 = 0. We see that xy < x + y. The answer is No.
Case 2: Say x = 1/10 and y = -10. Thus xy = -10*(1/10) = - 1; x + y = -10 + 1/10 = -9.90. We see that xy > x + y. The answer is Yes.

No unique answer. Insufficient.

(2) x > -y

Case 1: Say x = 2 and y = -1; (x > -y). Thus xy = -1*2 = - 2; x + y = -1 + 2 = 1. We see that xy < x + y. The answer is No.
Case 2: Say x = 10 and y = -1/10; (x > -y). Thus xy = -10*(1/10) = - 1; x + y = -1/10 + 10 = -9.90. We see that xy > x + y. The answer is Yes.

No unique answer. Insufficient.

(1) and (2) together

From (1), we have xy < 0 => xy is negative; from (2), we have x > -y => x + y > 0 => x + y is positive.

Thus, xy < x + y. The answer is No. Sufficient.

The correct answer: C

Hope this helps!

-Jay
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by [email protected] » Wed Jan 10, 2018 5:36 pm
Hi Vincen,

We're asked if the PRODUCT of X and Y is greater than the SUM of X and Y. This is a YES/NO question. It can be solved in a number of different ways, including by TESTing VALUES and using Number Properties.

1) XY < 0

Fact 1 tells us that ONE of the variables is POSITIVE and ONE is NEGATIVE.

IF....
X = +1, Y = -1, then the product (-1) is NOT greater than the sum (0) and the answer to the question is NO.
X = -2, Y = +1/2, then the product (-1) IS greater than the sum (-1.5) and the answer to the question is YES.
Fact 1 is INSUFFICIENT.

2) X > -Y

Fact 2 'limits' the value of X based on the value of Y.
IF....
Y = 0, then X is positive, the product (0) is NOT greater than the sum (positive) and the answer to the question is NO.
X = +2, Y = -1/2, then the product (-1) IS greater than the sum (-1.5) and the answer to the question is YES.
Fact 2 is INSUFFICIENT.

Combined, we know...
-ONE of the variables is POSITIVE and ONE is NEGATIVE.
X > -Y

The inequality can be rewritten as X+Y > 0. Since the PRODUCT of the two terms will ALWAYS be NEGATIVE and the SUM of the two terms will ALWAYS be POSITIVE, the answer to the question is ALWAYS YES.
Combined, SUFFICIENT

Final Answer: C

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Rich
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