buoyant wrote:Given: a² − b² = b² − c²

Is a = |b|?

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

Given: a² − b² = b² − c²

Target question: Is a = |b|?

IMPORTANT: Since the absolute value of any value is always greater than or equal to zero, we will need to ensure that a > 0 in order for a to equal |b|. Of course, that's just one of the things we need to ensure, but it's a big one. It's a big one because, when we scan the two statements, we see nothing that ensures that a > 0. Even when we add the given information (a² − b² = b² − c²), we cannot ensure that a > 0

So, at this point, I have a very strong feeling that neither statement is SUFFICIENT. In fact, I also feel that the combined statements are not sufficient, so I'm going to head straight to examining the combined statements to see if I can find two counter-examples...

Statements 1 and 2 combined
There are several values of a, b and c that satisfy both conditions (as well as the given information). Here are two:
Case a: a = 1, b = 1 and c = 1, in which case a = |b|
Case b: a = -1, b = 1 and c = 1, in which case a ≠ |b|

Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer = E

Cheers,
Brent

We're given that a^2-b^2=b^2-c^2 , so we can rephrase it as: (a + b)(a - b) = (b + c)(b - c)

Question: Is a=|b|?

If a = |b|, then either (a + b) or (a - b) would have to equal 0. But... (a + b) or (a - b) would also have to equal 0 if b = |a|. So, if we can prove that the product (a + b)(a - b) equals 0, then we'll know that |a| = |b|, but we would further need to prove that a is POSITIVE to prove that a=|b|


Statement 1: b=|c|

If b is equal to the absolute value of c, then either (b + c) or (b - c) must equal 0. This means that the product (b + c)(b - c) definitely equals 0, and thus so does (a + b)(a - b). This tells us that either a = |b| or b = |a|, but we can't tell which. Insufficient.

Statement 2: b=|a|

This tells us that the two have the same absolute value, and that b is positive, but it does not tell us if a is positive. Insufficient.

Together:
We can prove that |a| = |b|, and that b is positive, but we still don't know if a is positive, so we can't answer the question.

The answer is E.

Hi buoyant,

DS questions are frequently built around Number Properties (sometimes obvious ones, sometimes subtle, hidden or advanced ones). If you know or can quickly figure out the Number Property involved, then great. However, staring at the screen, trying to figure it out is NOT a good use of your time. You'd be far better of TESTing Values and proving what the answer is.

A^2 - B^2 = B^2 - C^2

Is A = |B|?

This is a YES/NO question. I'm going to leave the given equation as is for now.

Fact 1: B = |C|

Let's TEST Values:
B = 0
C = 0
Using the original equation
A = 0
The answer to the question is YES

B = 1
C = -1
Using the original equation
A = 1 OR -1
The answer to the question is YES OR NO
Fact 1 is INSUFFICIENT

Fact 2: B = |A|
Let's TEST Values
A = 0
B = 0
Using the original equation
C = 0
The answer to the question is YES

A = -1
B = 1
Using the original equation
C = 1 or -1
The answer to the question is NO
Fact 2 is INSUFFICIENT

Combined, we can still TEST Values:
A = 0
B = 0
C = 0
Answer is YES

A = 1 or -1
B = 1
C = 1 or -1
Answer is YES or NO
Combined, INSUFFICIENT

Final Answer: E

GMAT assassins aren't born, they're made,
Rich