M18-30

Question

Is xy > \frac{x}{y} ? (1) 0 < y < 1 (2) xy > 1

Bunuel · Accepted Answer

Official Solution: Is xy > \frac{x}{y} ? We can rephrase the question as follows: Is xy - \frac{x}{y} > 0 ? Is x(y-\frac{1}{y}) > 0 ? (1) 0 < y < 1 . Since y is a fraction between 0 and 1, y < \frac{1}{y} . Hence, y-\frac{1}{y} < 0 . Now, if x > 0 then x(y-\frac{1}{y}) < 0 and we'd have a NO answer to the question. However, if x < 0 , then x(y-\frac{1}{y}) > 0 and we'd have a YES answer to the question. Not sufficient. (2) xy > 1 . If x=y=2 then the answer is YES. However, if x=4 and y=\frac{1}{2} then the answer is NO. Not sufficient. (1)+(2) Since from (1) y > 0 and from (2) xy > 1 , then x > 0 too. Thus, we have that x > 0 and (y-\frac{1}{y}) < 0 (from 1), which means that their product x(y-\frac{1}{y}) < 0 and we have a NO answer to the question. Sufficient. Answer: C

mvictor · Answer

can we rewrite xy > x/y => xy^2 >x => y^2 > x/x => y^2 >1 or y>1?

Bunuel · Answer

No, that's wrong. We cannot multiply xy > x/y by y because we don't know its sign. If y<0, then we'd get xy^2 < x (flip the sign when multiplying by negative value) but if y > 0, then we'd have xy^2 > x (keep the sign when multiplying by positive value). For the same reason we cannot reduce by x.

Senthil7 · Answer

I think this is a high-quality question and I agree with explanation. When i solved this problem i used exactly the same thought process - absolutely no deviation- as given in the explanation! Wow kudos to me :-)

AndreiGMAT · Answer

I don't understand this rule and the logic behind it. Where can I learn more information about it.

Thank you

Bunuel · Answer

Since y is from 0 to 1, then 1/y is more than 1. Thus y - 1/y = (positive number less than 1) - (number more than 1) < 0.

Hope it's clear.

niks18 · Answer

Statement 1 : implies that  y  is positive but nothing mentioned about  x .  Insufficient

Statement 2:  we need values of  x  &  y  to determine the value of  \frac{x}{y} . this statement implies that both  x  &  y  are of same sign but their values cannot be deduced.  Insufficient

Combining 1 & 2, we know that  y  is positive so  x  is also positive and as  y<1  so it is reciprocal of any positive integer

so if  x  is integer for e.g  x=2  and  y=\frac{1}{5} , then  xy=2*\frac{1}{5}=0.4  but  \frac{x}{y}=2/\frac{1}{5}=10 . Hence  xy<\frac{x}{y}

and if  x  is not integer for e.g  x=\frac{2}{5}  and  y=\frac{1}{2} , then  xy=\frac{2}{5}*\frac{1}{2}=0.2  but  \frac{x}{y}=\frac{2}{5}/\frac{1}{2}=0.8 . Hence  xy<\frac{x}{y}

So we have a   NO   for our question stem.  Sufficient

Option  C

coreyander · Answer

I think this is a  high-quality  question and I agree with explanation.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

M18-30

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