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Does b = 2b - a ?

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Math Expert
Joined: 02 Sep 2009
Posts: 49206
Does b = 2b - a ?  [#permalink]

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02 Oct 2017, 23:49
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Difficulty:

75% (hard)

Question Stats:

51% (00:53) correct 49% (01:00) wrong based on 41 sessions

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Does $$\sqrt[3]{b} = \sqrt[3]{2b - a}$$ ?

(1) a + b = 0
(2) a = 2b

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Joined: 08 Jul 2010
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Re: Does b = 2b - a ?  [#permalink]

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02 Oct 2017, 23:59
Bunuel wrote:
Does $$\sqrt[3]{b} = \sqrt[3]{2b - a}$$ ?

(1) a + b = 0
(2) a = 2b

Question : Does $$\sqrt[3]{b} = \sqrt[3]{2b - a}$$ ?
Taking cube on both sides
Does b = 2b - a ?

Question : Does b = a ?

Statement 1: a+b = 0

i.e. a = -b

@a = 0 the answer of the question is YES
@a Not equal to 0, the answer of the question is NO

NOT SUFFICIENT

Statement 2: a = 2b

i.e. @a = 0 the answer of the question is YES
@a Not equal to 0, the answer of the question is NO

NOT SUFFICIENT

Combining the two statements

a = -b and also
a = 2b

we get b = 0 i.e. a = 0

SUFFICIENT

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Does b = 2b - a ?  [#permalink]

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04 Oct 2017, 13:17
Cubing both sides we get:

b = 2b - a --> Essentially the question can be rephrased into "is a + b = 2b?"

Statement 1: a + b=0. This doesn't mean whether a + b = 2b so NOT SUFFICIENT.

Statement 2: a = 2b --> a + b = 2b + b = 3b. So a + b = 3b answers our question.

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Re: Does b = 2b - a ?  [#permalink]

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04 Oct 2017, 21:40
hi Bunuel..
A quick question ...
In these short of question , can we do cubing both side and proceed , or it will fail as we do not know the sign of the variables...
Math Expert
Joined: 02 Sep 2009
Posts: 49206
Does b = 2b - a ?  [#permalink]

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04 Oct 2017, 21:44
sobby wrote:
hi Bunuel..
A quick question ...
In these short of question , can we do cubing both side and proceed , or it will fail as we do not know the sign of the variables...

Check below:

1. We can raise both parts of an inequality to an even power if we know that both parts of an inequality are non-negative (the same for taking an even root of both sides of an inequality).
For example:
$$2<4$$ --> we can square both sides and write: $$2^2<4^2$$;
$$0\leq{x}<{y}$$ --> we can square both sides and write: $$x^2<y^2$$;

But if either of side is negative then raising to even power doesn't always work.
For example: $$1>-2$$ if we square we'll get $$1>4$$ which is not right. So if given that $$x>y$$ then we cannot square both sides and write $$x^2>y^2$$ if we are not certain that both $$x$$ and $$y$$ are non-negative.

2. We can always raise both parts of an inequality to an odd power (the same for taking an odd root of both sides of an inequality).
For example:
$$-2<-1$$ --> we can raise both sides to third power and write: $$-2^3=-8<-1=-1^3$$ or $$-5<1$$ --> $$-5^3=-125<1=1^3$$;
$$x<y$$ --> we can raise both sides to third power and write: $$x^3<y^3$$.

Check for more here: Manipulating Inequalities (adding, subtracting, squaring etc.).

9. Inequalities

For more check Ultimate GMAT Quantitative Megathread

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Re: Does b = 2b - a ?  [#permalink]

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05 Oct 2017, 21:34
Equating two even roots means the absolute value of variables under the radical must be same.

Whereas, Equating two odd roots means variables under the radical have same absolute value and same sign +-.

So, Q stem is asking us if a=b in value and sign or a=b=0.

St.1: a=-b.
Means two options for a&b:
*same value & opposite sign
*a=b=0
So, Insufficient.

St.2: a=2b
Means two options for a&b:
*differnt value & same sign
*a=b=0
So, Insufficient.

St.1+2: Combining two statements, we can deduce that a=b=0.
Sufficient.

Ans. C

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Re: Does b = 2b - a ? &nbs [#permalink] 05 Oct 2017, 21:34
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