What is the value of ab? (1) |a| + b = a (2) a + |b| = b

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.

Check out these posts to understand Absolute Value in Detail

How to Solve: Absolute Value (Basics)
How to Solve: Absolute Value Problems

Q What is the value of ab?

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

STAT1
Take two cases
Case 1: a >=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

Taking 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, Answer will be C
Hope it helps!

Watch the following video to learn the Basics of Absolute Values

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.

Answer: C.

Hope it's clear.

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.

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
answer is C.

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