From any two numbers x and y, we define x* y = x + 0.5y - xy. Suppose

Question

From any two numbers x and y, we define x* y = x + 0.5y - xy. Suppose that both x and y are greater
than 0.5. If x*x<y* y then which of the following must be true : 
A. 1 > x > y 
B. x > 1 > y 
C. 1 > y > x 
D. y > 1 > x
E. none of the above

nick1816 · Answer

x*x = x+0.5x-x^2 = 1.5x-x^2 y*y= y+0.5y-y^2 = 1.5y-y^2 x*x0 1.5(y-x)-(y^2-x^2) > 0 1.5(y-x) - (y+x)(y-x) > 0 (1.5-y-x) (y-x) > 0 2 cases possible. Case 1- y>x and y+x<1.5 Case 2- x>y and y+x>1.5 Option A- x=0.8 and y=0.6; x*x>y*y ( Eliminate ) Option B - Since x>1 and y>0.5; x+y>1.5 Also x>y ( Always true ) Case 2 ] We don't need to check further tho. Option C- y=0.9 and x=0.8; x*x>y*y ( Eliminate ) Option D- x+y>1.5 and y>x Opposite of Case 2 ( Eliminate )

Kinshook · Answer

Given: From any two numbers x and y, we define x* y = x + 0.5y - xy. Suppose that both x and y are greater than 0.5. Asked: If x*x 0 (x-y){(x+y) - 1.5} > 0 Case 1: x-y>0 & x+y-1.5>0; Case 2: x-y<0 & x+y-1.5<0; A. 1 > x > y B. x > 1 > y ; x>y & x+y>1.5; MUST BE TRUE C. 1 > y > x D. y > 1 > x E. none of the above IMO B

chetan2u · Answer

A faulty question... Let us remember that it is 'MUST be true"question... Now, x#x0.........(x-y)(x+y)+1.5(y-x)>0........(x-y)(x+y-1.5)>0 Two cases, as also shown in posts above Case I :- x>y, then x+y-1.5>0 or x+y>1.5 >1.5 could mean x+y=1000 too. So, in this case, MUST be true is x>y>0.5 Case II :- x1, so 11>y If x=2, y=1.5......x>y and x+y>1.5. So, x>1>y need not be true. Now, when could be B the answer If question was.. Here any values of x and y in the range x>1>y will always give x#x

Kinshook · Answer

From any two numbers x and y, we define x* y = x + 0.5y - xy. Suppose that both x and y are greater
than 0.5. If x*x<y* y then which of the following must be true :
A. 1 > x > y
B. x > 1 > y
C. 1 > y > x
D. y > 1 > x
E. none of the above

Alasoy · Answer

almost made it, but forgot to apply the condition where x,y must be > 0.5. because if you put (1/3, 2/3) in the equation, it will lands you to the C answer where 1>x>y (have tried it and made the wrong answer haha)

I think it's a very challenging question to strengthen your inequality concept. However, does this type of question is realistic for GMAT question? seems a bit too hard

Regor60 · Answer

As has been pointed out this is a faulty question as is easily proven.

The given answer B says in part that X must be greater than Y given that X*X<Y*Y.

There are no restrictions on what can be substituted for Y or X as long as each is greater than .5.

Substituting X=.6 and Y=.7 yields X*X=.54 and Y*Y=.56.

Clearly, X does not have to be greater than Y to satisfy the problem statement.

From any two numbers x and y, we define x* y = x + 0.5y - xy. Suppose

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From any two numbers x and y, we define x* y = x + 0.5y - xy. Suppose

Prep Toolkit

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My Rewards