Strange Operator

[Brent@GMATPrepNow]

Please note that this is not an official GMAT question; it’s my attempt to create difficult (650+ level) GMAT-style questions for this forum.

[dmateer25]

x/(x+y) = 6
x = 6x+6y
-5x = 6y
x=6y/-5


y/(y+x) = ?
y/((y/1) + (6y/-5)) = ?
(y/1)/(y/-5) = ?
-5y/y = ?
-5 = ?

I choose A

[piyush_nitt]

Please note that this is not an official GMAT question; it’s my attempt to create difficult (650+ level) GMAT-style questions for this forum.

IMO A

Given eqn = x/x+y = 6 ...(1)

reqd = y/x+y - (2)

eqn (1) can be re-written as

1/6 = x+y/x

1/6 = 1+ y/x

1/6 -1 = y/x

-5/6 = y/x

x/y = -6/5

Adding 1 to both sides

x/y + 1 = -6/5 + 1

x+y/y = -6+5/5

x+y/y = -1/5

-5 = y/y+x (same as eqn(2))

Hence A

[Brent@GMATPrepNow]

Nice work; the answer is A.

I have two different solutions:

Algebraic:
Given x ¤ y= x/(x+y) = 6
This means that the reciprocal (x+y)/x = 1/6

Aside: Why find the reciprocal? Doing so allows us to use the property that (a+b)/c = a/c + b/c. This property often comes in handy with GMAT questions involving fractions.

If (x+y)/x = 1/6, then x/x + y/x = 1/6, which gives us 1 + y/x = 1/6, which tells us that y/x = -5/6
(or that x/y=-6/5)

y ¤ x= y/(y+x)
Let’s determine (y+x)/y and then find the reciprocal

(y+x)/y = 1 + x/y
= 1+(-6/5)
= -1/5
The reciprocal of -1/5 is -5 (answer choice A)

Plug in numbers:
Given x ¤ y= x/(x+y) = 6
Find values for x and y that make this true. How about x=6 and y=-5
Then y ¤ x= y/(y+x) = -5/(-5+6) = -5

[aroon7]

x/(x+y) = 6
adding y/(x+y) on both sides,

x/(x+y) + y/(x+y) = 6 + y/(x+y)
1 = 6 + y/(x+y)
implies, y/(x+y) = -5

[rolrol19]

x/x+y=6 then x+y=x/6 and y=-5x/6

y*x= y/x+y and we replace with the above findings

-5x/6 divided by x/6 = -5

[krisraam]

x/(x+y) + y/(y+x) = 1

--> y/(y+x) = 1 - 6 = -5