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# The operation ⊗ is defined for all nonzero numbers a and b

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The operation ⊗ is defined for all nonzero numbers a and b [#permalink]

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18 Dec 2010, 08:46
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The operation ⊗ is defined for all nonzero numbers a and b by a ⊗ b = a/b – b/a. If x and y are nonzero numbers, which of the following statements must be true?

I. x ⊗ xy = x(1 ⊗ y)
II. x ⊗ y = -(y ⊗ x)
III. 1/x ⊗ 1/y = y ⊗ x

A. I only
B. II only
C. III only
D. I and II
E. II and III
[Reveal] Spoiler: OA

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Ajit

Last edited by Bunuel on 20 May 2012, 11:29, edited 1 time in total.
Edited the question and added the OA
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Posts: 39719
Re: Symbols problem [#permalink]

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18 Dec 2010, 09:02
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ajit257 wrote:
The operation ⊗ is defined for all nonzero numbers a and b by a ⊗ b = a/b – b/a. If x and y are nonzero numbers, which of the following statements must be true?
I. x ⊗ xy = x(1 ⊗ y)
II. x ⊗ y = -(y ⊗ x)
III. 1/x ⊗ 1/y = y ⊗ x

A. I only
B. II only
C. III only
D. I and II
E. II and III

Please can someone confirm the ans. I am not sure about the ans

Given: $$a@b=\frac{a}{b}-\frac{b}{a}$$, for all nonzero numbers $$a$$ and $$b$$.

I. $$x@(xy) = x(1@y)$$: $$LHS=x@(xy)=\frac{x}{xy}-\frac{xy}{x}=\frac{1}{y}-y$$ and $$RHSx(1@y)=x(\frac{1}{y}-y)$$ as you see LHS doen't equal to RHS;

II. $$x@y = -(y@x)$$: $$LHS=x@y=\frac{x}{y}-\frac{y}{x}$$ and $$RHS=-(y@x)=-(\frac{y}{x}-\frac{x}{y})=\frac{x}{y}-\frac{y}{x}$$ --> LHS=RHS;

III. $$(\frac{1}{x})@(\frac{1}{y}) = y@x$$: $$LHS=(\frac{1}{x})@(\frac{1}{y})=\frac{y}{x}-\frac{x}{y}$$ and $$RHS=y@x=\frac{y}{x}-\frac{x}{y}$$ --> LHS=RHS.

Answer: E (II and III).

Hope it's clear.
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Re: The operation ⊗ is defined for all nonzero numbers a and b [#permalink]

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05 Nov 2013, 10:38
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Re: The operation ⊗ is defined for all nonzero numbers a and b [#permalink]

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05 Nov 2013, 10:39
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Re: The operation ⊗ is defined for all nonzero numbers a and b [#permalink]

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06 Jun 2015, 14:23
Hello from the GMAT Club BumpBot!

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Re: The operation ⊗ is defined for all nonzero numbers a and b [#permalink]

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30 Nov 2015, 19:19
If you pay close attention, III is basically the operation backwards so you don't have to waste time trying it. Also, if you look at the answer choices, if III is correct then just test II to choose between C or E.
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Re: The operation ⊗ is defined for all nonzero numbers a and b [#permalink]

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18 May 2017, 02:59
Hello from the GMAT Club BumpBot!

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The operation ⊗ is defined for all nonzero numbers a and b [#permalink]

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18 May 2017, 04:15
ajit257 wrote:
The operation ⊗ is defined for all nonzero numbers a and b by a ⊗ b = a/b – b/a. If x and y are nonzero numbers, which of the following statements must be true?

I. x ⊗ xy = x(1 ⊗ y)
II. x ⊗ y = -(y ⊗ x)
III. 1/x ⊗ 1/y = y ⊗ x

A. I only
B. II only
C. III only
D. I and II
E. II and III

a ⊗ b = a/b – b/a

Lets check I option :
LHS = x ⊗ xy = x/xy -xy/x = 1/y - y
RHS = x(1 ⊗ y) = x(1/y - y)............... LHS =/= RHS

Lets check II option :
LHS = x ⊗ y = x/y - y/x
RHS = -(y ⊗ x) = -(y/x - x/y) = x/y - y/x................... LHS = RHS

Lest check III option :
LHS = 1/x ⊗ 1/y = [m]/frac{1/x}{1/y} - /frac{1/y}{1/x} = y/x - x/y
RHS = y ⊗ x = y/x - x/y ............... LHS = RHS

[Reveal] Spoiler:
Answer E. II and III

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The operation ⊗ is defined for all nonzero numbers a and b   [#permalink] 18 May 2017, 04:15
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# The operation ⊗ is defined for all nonzero numbers a and b

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