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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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New post Updated on: 20 May 2012, 10:29
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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

Originally posted by ajit257 on 18 Dec 2010, 07:46.
Last edited by Bunuel on 20 May 2012, 10:29, edited 1 time in total.
Edited the question and added the OA
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Re: Symbols problem  [#permalink]

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New post 18 Dec 2010, 08:02
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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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New post 05 Nov 2013, 09:39
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Hope it helps.
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Re: The operation ⊗ is defined for all nonzero numbers a and b  [#permalink]

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New post 30 Nov 2015, 18:19
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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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The operation ⊗ is defined for all nonzero numbers a and b  [#permalink]

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New post 18 May 2017, 03:15
1
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

Answer E. II and III
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Re: The operation ⊗ is defined for all nonzero numbers a and b  [#permalink]

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New post 02 Jul 2017, 06:09
Is there any short cut method for these kind of problems. I have took 3 minutes 15 sec to solve the problem. Is this the avg time required to solve the problem please...
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Re: The operation ⊗ is defined for all nonzero numbers a and b  [#permalink]

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New post 03 Jul 2017, 11:13
zonebp wrote:
Is there any short cut method for these kind of problems. I have took 3 minutes 15 sec to solve the problem. Is this the avg time required to solve the problem please...


No shortcut bro.. But you can solve it in 1:30 minutes if u solve a lots of similar problems..
I solved in 1:39 minutes...

3 minutes 15 sec is a bit high time.. So you can solve in less time if you practice more....
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Re: The operation ⊗ is defined for all nonzero numbers a and b  [#permalink]

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New post 03 Jul 2017, 17:49
shashankism wrote:
zonebp wrote:
Is there any short cut method for these kind of problems. I have took 3 minutes 15 sec to solve the problem. Is this the avg time required to solve the problem please...


No shortcut bro.. But you can solve it in 1:30 minutes if u solve a lots of similar problems..
I solved in 1:39 minutes...

3 minutes 15 sec is a bit high time.. So you can solve in less time if you practice more....


Thanks a lot mate. Im struggling to pace and just worried a lot due to the pacing issue.
Just running out of time and sometimes spending far too much time on a specific question. Slowly I started getting into countdown phobia.
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Re: The operation ⊗ is defined for all nonzero numbers a and b  [#permalink]

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New post 03 Jul 2017, 20:16
3
zonebp wrote:
Is there any short cut method for these kind of problems. I have took 3 minutes 15 sec to solve the problem. Is this the avg time required to solve the problem please...


Note that the function is symmetric (if you switch one variable with another, it doesn't change) except for the sign:

a ⊗ b = a/b – b/a

b ⊗ a = b/a – a/b = - (a/b - b/a)

Look for options which are symmetric too.

I. x ⊗ xy = x(1 ⊗ y)

This is not symmetric. You have x and xy so x is different from y. I will ignore this right now.

II. x ⊗ y = -(y ⊗ x)

This we already know is true.

III. 1/x ⊗ 1/y = y ⊗ x

Just check for this.

Answer (E).

In such questions, looking for symmetry often helps.
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Re: The operation ⊗ is defined for all nonzero numbers a and b  [#permalink]

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New post 27 May 2018, 15:03
For statement 3 I cross multiplied the LHS and was left with: y ⊗ x = y ⊗ x

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

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New post 27 May 2018, 21:36
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norovers wrote:
For statement 3 I cross multiplied the LHS and was left with: y ⊗ x = y ⊗ x

Is this operation allowed?


What do you mean by "cross multiplied the LHS"?

There are no operations you can do on 1/x ⊗ 1/y without substituting in the expression for ⊗ since you do not know how this symbol relates the two terms without the defining expression.
If say, ⊗ = *, then (1/x)*(1/y) = 1/xy
If instead ⊗ = /, then (1/x)/(1/y) = y/x
and so on.

You only know how to handle the standard operations. ⊗ is a specially defined operation and you need to convert it to standard operations to simplify it.
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Re: The operation ⊗ is defined for all nonzero numbers a and b  [#permalink]

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New post 16 Dec 2018, 05:13
Is there a quick way to answer this question? I just dumped it in the practice test as I could see it would take up too much time
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Re: The operation ⊗ is defined for all nonzero numbers a and b  [#permalink]

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Re: The operation ⊗ is defined for all nonzero numbers a and b   [#permalink] 09 Jan 2020, 17:47
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