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Manager  Joined: 01 Feb 2012
Posts: 85
What is the value of ab? (1) |a| + b = a (2) a + |b| = b  [#permalink]

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3
10 00:00

Difficulty:   75% (hard)

Question Stats: 55% (01:55) correct 45% (01:49) wrong based on 309 sessions

### HideShow timer Statistics What is the value of ab?

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

I cant combine the two statements and reach anywhere -
1. |a| + b = a
we wil have two cases once a positive and once a negative -
when a positive
a+b = a
implies b =0
when a -ve
-a+b=a
b=2a

so by statement -
either ab = 0 (when b is zero)
or ab= 2a² (when b=2a) (we dont know the value of a)

so we dont have one unique value of b

Statement II
a + |b| = b
either a = 0
or a= 2b
again we dont get a clear value for ab.

can someone please tell me if my approach so far is ok and how do I combine the two statements now....thanks.
Math Expert V
Joined: 02 Sep 2009
Posts: 56251
Re: What is the value of ab? (1) |a| + b = a (2) a + |b| = b  [#permalink]

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5
4
What is the value of ab?

(1) |a| + b = a. Just plug numbers to see that this statement is NOT sufficient: if a=b=0, then ab=0 but if a=-1 and b=-2, then ab=2. Not sufficient.

(2) a + |b| = b. The same here. Not sufficient.

(1)+(2) Sum (1) and (2): |a| + b + a + |b|= a + b --> |a| + |b| = 0. The sum of two non-negative values (|a| and |b|) to be 0, each must be 0, thus a=b=0 --> ab=0. Sufficient.

Hope it's clear.
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Math Expert V
Joined: 02 Sep 2009
Posts: 56251
Re: What is the value of ab? (1) |a| + b = a (2) a + |b| = b  [#permalink]

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tirbah wrote:
What is the value of ab?

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

I cant combine the two statements and reach anywhere -
1. |a| + b = a
we wil have two cases once a positive and once a negative -
when a positive
a+b = a
implies b =0
when a -ve
-a+b=a
b=2a

so by statement -
either ab = 0 (when b is zero)
or ab= 2a² (when b=2a) (we dont know the value of a)

so we dont have one unique value of b

Statement II
a + |b| = b
either a = 0
or a= 2b
again we dont get a clear value for ab.

can someone please tell me if my approach so far is ok and how do I combine the two statements now....thanks.

Similar question to practice: is-xy-132217.html

Hope it helps.
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Manager  Joined: 01 Feb 2012
Posts: 85
Re: What is the value of ab? (1) |a| + b = a (2) a + |b| = b  [#permalink]

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1
Bunuel wrote:
What is the value of ab?

(1) |a| + b = a. Just plug numbers to see that this statement is NOT sufficient: if a=b=0, then ab=0 but if a=-1 and b=-2, then ab=2. Not sufficient.

(2) a + |b| = b. The same here. Not sufficient.

(1)+(2) Sum (1) and (2): |a| + b + a + |b|= a + b --> |a| + |b| = 0. The sum of two non-negative values (|a| and |b|) to be 0, each must be 0, thus a=b=0 --> ab=0. Sufficient.

Hope it's clear.

Bunuel - Thanks for the reply. I understood the way you did that. Could you please go through my explanation once please and see if that is also fine or have I done soemthing wrong? also is there a way to combine the two statements without literally adding them...I mean the way I solved it..I reached at b=0 and 2a from Ist statement and a=0, and 2b from IInd statement....is there a way I can combine them from this point? I am usually we look for a common point from the outcomes of two statements when we combine them, can we do that here. Thanks.

Kind regards..
Math Expert V
Joined: 02 Sep 2009
Posts: 56251
Re: What is the value of ab? (1) |a| + b = a (2) a + |b| = b  [#permalink]

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1
tirbah wrote:
Bunuel wrote:
What is the value of ab?

(1) |a| + b = a. Just plug numbers to see that this statement is NOT sufficient: if a=b=0, then ab=0 but if a=-1 and b=-2, then ab=2. Not sufficient.

(2) a + |b| = b. The same here. Not sufficient.

(1)+(2) Sum (1) and (2): |a| + b + a + |b|= a + b --> |a| + |b| = 0. The sum of two non-negative values (|a| and |b|) to be 0, each must be 0, thus a=b=0 --> ab=0. Sufficient.

Hope it's clear.

Bunuel - Thanks for the reply. I understood the way you did that. Could you please go through my explanation once please and see if that is also fine or have I done soemthing wrong? also is there a way to combine the two statements without literally adding them...I mean the way I solved it..I reached at b=0 and 2a from Ist statement and a=0, and 2b from IInd statement....is there a way I can combine them from this point? I am usually we look for a common point from the outcomes of two statements when we combine them, can we do that here. Thanks.

Kind regards..

Your reasoning for (1) and (2) is correct:
(1) |a| + b = a.
a<0 --> b=2a
a>=0 --> b=0

(2) a + |b| = b
b<0 --> a=2b (b=a/2)
b>=0 --> a=0

(1)+(2) If a<0, then from b=2a, b<0 too. Thus we have that when a<0 and b<0, then b=2a and b=a/2, which cannot be true. So,we are left with a=b=0.

Hope it's clear.
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Manager  Joined: 01 Feb 2012
Posts: 85
Re: What is the value of ab? (1) |a| + b = a (2) a + |b| = b  [#permalink]

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Bunuel wrote:
tirbah wrote:
Bunuel wrote:
What is the value of ab?

(1) |a| + b = a. Just plug numbers to see that this statement is NOT sufficient: if a=b=0, then ab=0 but if a=-1 and b=-2, then ab=2. Not sufficient.

(2) a + |b| = b. The same here. Not sufficient.

(1)+(2) Sum (1) and (2): |a| + b + a + |b|= a + b --> |a| + |b| = 0. The sum of two non-negative values (|a| and |b|) to be 0, each must be 0, thus a=b=0 --> ab=0. Sufficient.

Hope it's clear.

Bunuel - Thanks for the reply. I understood the way you did that. Could you please go through my explanation once please and see if that is also fine or have I done soemthing wrong? also is there a way to combine the two statements without literally adding them...I mean the way I solved it..I reached at b=0 and 2a from Ist statement and a=0, and 2b from IInd statement....is there a way I can combine them from this point? I am usually we look for a common point from the outcomes of two statements when we combine them, can we do that here. Thanks.

Kind regards..

Your reasoning for (1) and (2) is correct:
(1) |a| + b = a.
a<0 --> b=2a
a>=0 --> b=0

(2) a + |b| = b
b<0 --> a=2b (b=a/2)
b>=0 --> a=0

(1)+(2) If a<0, then from b=2a, b<0 too. Thus we have that when a<0 and b<0, then b=2a and b=a/2, which cannot be true. So,we are left with a=b=0.

Hope it's clear.

Yes I understood it now...Thank you so much.
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Re: What is the value of ab? (1) |a| + b = a (2) a + |b| = b  [#permalink]

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2
1
Q What is the value of ab?

1. |a| + b = a
2. a + |b| = b

STAT1
Take two cases
Case 1: a is >=0
|a| = a
=> b = 0
so, ab=0

Case 2: a< 0
=> |a| = -a
-a + b = a
=> b = 2a

ab = 2a^2

Both are different so not sufficient

STAT2
Take two cases
Case 1: b>= 0
|b| = b
so, a+ b = b
=> a = 0
=> ab = 0

Case 2: b<0
=> |b| = -b
=> a - b = b
=> a = 2b
=> ab = 2b^2

Both are different so NoT sufficient

Takeing STAT1 and STAT2 together there will be 4 cases
adding both the equations we get
|a| + |b| +a +b = a+ b
=> |a| +|b| = 0
Only possible when both a=b=0
=> ab=0

So, asnwer will be C

Hope it helps!
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Posts: 56251
Re: What is the value of ab? (1) |a| + b = a (2) a + |b| = b  [#permalink]

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2
What is the value of ab?

(1) |a| + b = a --> a - |a| = b --> if a = 1 and b = 0, then ab = 0 but if a = -1 and b = -2, then ab = 2. Not sufficient.

(2) a + |b| = b --> b - |b| = a --> if b = 1 and a = 0, then ab = 0 but if b = -1 and a = -2, then ab = 2. Not sufficient.

(1)+(2) Sum (1) and (2): |a| + b + a + |b| = a + b --> |a| + |b| = 0. The sum of two nonnegative values can be 0, if and only both of them are 0 --> |a| = |b| = 0 --> ab = 0. Sufficient.

Hope it's clear.
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Re: What is the value of ab? (1) |a| + b = a (2) a + |b| = b  [#permalink]

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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

What is the value of ab?

1. |a| + b = a
2. a + |b| = b

There are 2 variables (a,b) and 2 equations are given by the 2 conditions, so there is high chance (C) will be the answer.
Looking at the conditions together,
a=b=0 --> ab=0. This is unique and sufficient, the answer becomes (C).

For cases where we need 2 more equations, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.
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Re: What is the value of ab? (1) |a| + b = a (2) a + |b| = b  [#permalink]

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Amit0507 wrote:
What is the value of ab?

1. |a| + b = a
2. a + |b| = b

1. 2 options:
a+b=a -> b=0, and ab=0
b-a=a -> b=2a - not sufficient.

2. 2 options:
a+b=b -> a=0, and ab=0
a-b=b -> a=2b - not sufficient

1+2
a=0, b=0, b=2a - true, a=2b - true
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Re: What is the value of ab? (1) |a| + b = a (2) a + |b| = b  [#permalink]

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