Is x < y ? (1) y/x > x/y (2) y > 100

1.
y/x - x/y >0

y^2-x^2/xy >0

i. y^2-x^2>0 and xy>0
a. if xy>0 then x>0 and y>0 and the values of x,y should satisfy (x+y)(y-x)>0 => since x,y>0 x+y>0 => y-x >0 => y>x
b. if xy>0 then x<0 and y<0 and the values of x,y should satisfy (x+y)(x-y)>0 => since x,y<0 x+y<0 => x-y <0 => x<y

ii. y^2-x^2<0 and xy<0
a. if xy<0 then x>0 and y<0 => y<x always
b. if xy<0 then x<0 and y>0 => x<y always

first option insufficient

2.y>0 => insufficient

1 and 2. y>100 => y>0
from ia,ib,iia,iib

i. y^2-x^2>0 and xy>0
a. if xy>0 then x>0 and y>0 and the values of x,y should satisfy (x+y)(y-x)>0 => since x,y>0 x+y>0 => y-x >0 => y>x
b. if xy>0 then x<0 and y<0 and the values of x,y should satisfy (x+y)(x-y)>0 => since x,y<0 x+y<0 => x-y <0 => x<y

ii. y^2-x^2<0 and xy<0
a. if xy<0 then x>0 and y<0 => y<x always
b. if xy<0 then x<0 and y>0 => x<y always


ia x<y
iib x<y
hence sufficeint

answer is C

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Please leave kudos

This is a moderately difficult question on Inequalities. Beware of the trap element though – you cannot cross multiply the terms in an inequality until unless you are absolutely sure of the signs of the variables.

Let’s look at statement I alone. To simplify the inequality, let us take all the variable terms on to the LHS and keep the RHS as 0 (this is a sure shot strategy that you can adopt in inequalities’ questions to make your life simpler). When we do that, we have,

\(\frac{y}{x}\) – \(\frac{x}{y}\) > 0 which gives

\(\frac{y^2 – x^2}{xy}\) > 0

\(\frac{(y-x)(y+x)}{xy}\) > 0

If the LHS is positive, it only means:

Either, (y-x), (y+x) and xy should all be positive OR negative.

If we take y = 5 and x =4 , then \(\frac{(5-4) (5+4)}{20}\) > 0. Here, x < y.

If we take y = -5 and x = -4, then \(\frac{(-5+4) (-9)}{20}\) > 0. Here x > y.

This data is insufficient. Answer options A and D can be eliminated. Possible answer options are B, C and E.

From statement II alone, we have data about y only. We do not have any information about x. Therefore, this statement is insufficient. Answer option B can be eliminated.

Let’s combine both the statements now, by taking some simple values that satisfy both statements together.

If y = 200 and x = -400, \(\frac{y}{x}\) = \(\frac{-1}{2}\) and \(\frac{x}{y}\) = -2. \(\frac{y}{x}\) > \(\frac{x}{y}\). Here, x < y ( remember, x is negative and y is positive).

If y = 200 and x = 10, \(\frac{y}{x}\) = 20 and \(\frac{x}{y}\) = \(\frac{1}{20}\).\(\frac{y}{x}\) > \(\frac{x}{y}\). Here also, x<y.

Therefore, we get a definite YES as an answer when we combine the statements. So, the correct answer option is C.

The average time for this kind of a question can be about 2 minutes.

Hope this helps!