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Math Expert V
Joined: 02 Sep 2009
Posts: 59674

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13 00:00

Difficulty:   85% (hard)

Question Stats: 40% (01:59) correct 60% (01:39) wrong based on 98 sessions

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$$a^2 - b^2 = b^2 - c^2$$. Is $$a = |b|$$?

(1) $$b = |c|$$

(2) $$b = |a|$$

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Math Expert V
Joined: 02 Sep 2009
Posts: 59674

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Official Solution:

Statement (1) by itself is insufficient. From S1 we know that $$a^2 - b^2 = 0$$. From here we can only conclude that $$|a| = |b|$$.

Statement (2) by itself is insufficient. From S2 we know that $$b$$ is non-negative. But whether $$a$$ is non-negative remains a question.

Statements (1) and (2) combined are insufficient. Consider $$a = b = c = 1$$ (the answer to the question is "yes") and $$a = -1$$, $$b = c = 1$$ (the answer to the question is "no").

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Intern  Joined: 03 Mar 2013
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Hi

Can the original st a = |b| translated to a=sqrt(b^2) or a^2= b^2 ?
In which case St A will suffice:
b = |c|
b^2=c^2

a^2 - b^2 = b^2 - c^2
or, a^2 - c^2 = b^2 - c^2
or, a^2=b^2
Sufficient
Math Expert V
Joined: 02 Sep 2009
Posts: 59674

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sidinsin wrote:
Hi

Can the original st a = |b| translated to a=sqrt(b^2) or a^2= b^2 ?
In which case St A will suffice:
b = |c|
b^2=c^2

a^2 - b^2 = b^2 - c^2
or, a^2 - c^2 = b^2 - c^2
or, a^2=b^2
Sufficient

No. It's possible a^2 to be equal to b^2 and a not be equal to |b|. For example, a=-1 and b=1.
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Joined: 18 Sep 2014
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Bunuel wrote:
Official Solution:

Statement (1) by itself is insufficient. From S1 we know that $$a^2 - b^2 = 0$$. From here we can only conclude that $$|a| = |b|$$.

Statement (2) by itself is insufficient. From S2 we know that $$b$$ is non-negative. But whether $$a$$ is non-negative remains a question.

Statements (1) and (2) combined are insufficient. Consider $$a = b = c = 1$$ (the answer to the question is "yes") and $$a = -1$$, $$b = c = 1$$ (the answer to the question is "no").

My doubt may sound silly but this is confusing me a lot.
what is the difference between

1. $$|a| = |b|$$.
2. $$a = |b|$$.
3. $$|a| = b$$.

Math Expert V
Joined: 02 Sep 2009
Posts: 59674

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Mechmeera wrote:
Bunuel wrote:
Official Solution:

Statement (1) by itself is insufficient. From S1 we know that $$a^2 - b^2 = 0$$. From here we can only conclude that $$|a| = |b|$$.

Statement (2) by itself is insufficient. From S2 we know that $$b$$ is non-negative. But whether $$a$$ is non-negative remains a question.

Statements (1) and (2) combined are insufficient. Consider $$a = b = c = 1$$ (the answer to the question is "yes") and $$a = -1$$, $$b = c = 1$$ (the answer to the question is "no").

My doubt may sound silly but this is confusing me a lot.
what is the difference between

1. $$|a| = |b|$$.
2. $$a = |b|$$.
3. $$|a| = b$$.

1. $$|a| = |b|$$ means that the distance from a to 0 is the same as the distance from b to 0. Or that the magnitudes of a and b are the same.

2. $$a = |b|$$ means that the distance from b to 0 is a.

3. $$|a| = b$$ means that the distance from a to 0 is b.
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Senior Manager  Joined: 31 Mar 2016
Posts: 375
Location: India
Concentration: Operations, Finance
GMAT 1: 670 Q48 V34 GPA: 3.8
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I think this is a high-quality question and I agree with explanation.
Manager  G
Joined: 18 Dec 2016
Posts: 114

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Is there a link explaining the mod theory and related problems? I have been getting most of the questions wrong in this section.
Math Expert V
Joined: 02 Sep 2009
Posts: 59674

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1
gmat2k17 wrote:
Is there a link explaining the mod theory and related problems? I have been getting most of the questions wrong in this section.

Hope it helps.
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Intern  B
Joined: 09 Sep 2015
Posts: 20

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Bunuel wrote:
Official Solution:

Statement (1) by itself is insufficient. From S1 we know that $$a^2 - b^2 = 0$$. From here we can only conclude that $$|a| = |b|$$.

Statement (2) by itself is insufficient. From S2 we know that $$b$$ is non-negative. But whether $$a$$ is non-negative remains a question.

Statements (1) and (2) combined are insufficient. Consider $$a = b = c = 1$$ (the answer to the question is "yes") and $$a = -1$$, $$b = c = 1$$ (the answer to the question is "no").

Hi Bunuel,

In Statement 1, how are we arriving at a^2-b^2=0 ? And would you please elaborate explanation for statement 2 as well.

Thanks
Amaresh
Math Expert V
Joined: 02 Sep 2009
Posts: 59674

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Bunuel wrote:
Official Solution:

Statement (1) by itself is insufficient. From S1 we know that $$a^2 - b^2 = 0$$. From here we can only conclude that $$|a| = |b|$$.

Statement (2) by itself is insufficient. From S2 we know that $$b$$ is non-negative. But whether $$a$$ is non-negative remains a question.

Statements (1) and (2) combined are insufficient. Consider $$a = b = c = 1$$ (the answer to the question is "yes") and $$a = -1$$, $$b = c = 1$$ (the answer to the question is "no").

Hi Bunuel,

In Statement 1, how are we arriving at a^2-b^2=0 ? And would you please elaborate explanation for statement 2 as well.

Thanks
Amaresh

From the stem we know that $$a^2 - b^2 = b^2 - c^2$$. From (1) $$b = |c|$$ means that $$b^2=c^2$$. Substitute: $$a^2 - b^2 = b^2 - b^2=0$$

As for (2): $$b = |a|$$. Given that b is equal to an absolute value of a number. An absolute value cannot be negative, theretofore $$b$$ is non-negative.

Hope it's clear.
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Intern  B
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$$a^2$$ -$$b^2$$ =$$b^2-c^2$$

so, $$(a-b)(a+b)=(b-c)(b+c)$$

its given that (1) $$b=|c|$$

so, $$(a-b)(a+b) = 0$$
i.e, $$a=b$$ or $$a = -b$$, or $$a= |b|$$,

So , isn't (1) sufficient ?
Manager  S
Joined: 26 Feb 2018
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I think this is a high-quality question and I agree with explanation.
Manager  S
Joined: 28 Jun 2018
Posts: 129
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GMAT 1: 490 Q39 V18 GMAT 2: 640 Q47 V30 GMAT 3: 670 Q50 V31 GMAT 4: 700 Q49 V36 GPA: 4

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samji wrote:
$$a^2$$ -$$b^2$$ =$$b^2-c^2$$

so, $$(a-b)(a+b)=(b-c)(b+c)$$

its given that (1) $$b=|c|$$

so, $$(a-b)(a+b) = 0$$
i.e, $$a=b$$ or $$a = -b$$, or $$a= |b|$$,

So , isn't (1) sufficient ?

U need one more step and u will get the correct answer.
so u found a=b OR a=-b

We HAVE to consider both these above scenarios SEPARATELY while solving the sum.
But notice that just by your first statement a=b we can prove the above as insufficient.
so if a = b and a = b = -2. Then $$a=|b|$$ is not possible.
but if a = b = 2. Then $$a=|b|$$ is possible.

You only missed considering both the OR statements SEPARATELY.

Hope it helps GMAT Club Legend  V
Joined: 12 Sep 2015
Posts: 4132

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Top Contributor
Bunuel wrote:
$$a^2 - b^2 = b^2 - c^2$$. Is $$a = |b|$$?

(1) $$b = |c|$$

(2) $$b = |a|$$

Target question: Is a = |b|?

Given: a² - b² = b² - c²

Statement 1: b = |c|
This tells us a few things, but with regard to this question, it tells us that b and c have the same magnitude
This also means that b² = c²
With this information, let's test some values that satisfy both statement 1 and the given information:
Case a: a = 0, b = 0 and c = 0. In this case, the answer to the target question is YES, a does EQUAL |b|
Case b: a = -1, b = 1 and c = -1. In this case, the answer to the target question is NO, a does NOT equal |b|
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: b = |a|
Let's test values again.
There are several values of x and y that satisfy statement 2. Here are two:
Case a: a = 0, b = 0 and c = 0. In this case, the answer to the target question is YES, a does EQUAL |b|
Case b: a = -1, b = 1 and c = -1. In this case, the answer to the target question is NO, a does NOT equal |b|
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
IMPORTANT: Notice that I was able to use the same counter-examples to show that each statement ALONE is not sufficient.
So, the same counter-examples will satisfy the two statements COMBINED.

In other words,
Case a: a = 0, b = 0 and c = 0. In this case, the answer to the target question is YES, a does EQUAL |b|
Case b: a = -1, b = 1 and c = -1. In this case, the answer to the target question is NO, a does NOT equal |b|
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Cheers,
Brent
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Intern  B
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Bunuel wrote:
Bunuel wrote:
Official Solution:

Statement (1) by itself is insufficient. From S1 we know that $$a^2 - b^2 = 0$$. From here we can only conclude that $$|a| = |b|$$.

Statement (2) by itself is insufficient. From S2 we know that $$b$$ is non-negative. But whether $$a$$ is non-negative remains a question.

Statements (1) and (2) combined are insufficient. Consider $$a = b = c = 1$$ (the answer to the question is "yes") and $$a = -1$$, $$b = c = 1$$ (the answer to the question is "no").

Hi Bunuel,

In Statement 1, how are we arriving at a^2-b^2=0 ? And would you please elaborate explanation for statement 2 as well.

Thanks
Amaresh

From the stem we know that $$a^2 - b^2 = b^2 - c^2$$. From (1) $$b = |c|$$ means that $$b^2=c^2$$. Substitute: $$a^2 - b^2 = b^2 - b^2=0$$

As for (2): $$b = |a|$$. Given that b is equal to an absolute value of a number. An absolute value cannot be negative, theretofore $$b$$ is non-negative.

Hope it's clear.

Hi Bunuel AjiteshArun chetan2u
can you please elaborate a bit.

1 - I get that b is non- negative but why is it not sufficient? Is it coz there is no new information about 'a' & 'c' (therefore we can't say whether 'a' is negative).
2 How did you combine both the statements? Is putting values the only way to do it OR is there some explanation behind it too.

Bharat
Math Expert V
Joined: 02 Sep 2009
Posts: 59674

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bratbg wrote:
Bunuel wrote:
[quote="Bunuel"

Hi Bunuel,

In Statement 1, how are we arriving at a^2-b^2=0 ? And would you please elaborate explanation for statement 2 as well.

Thanks
Amaresh

From the stem we know that $$a^2 - b^2 = b^2 - c^2$$. From (1) $$b = |c|$$ means that $$b^2=c^2$$. Substitute: $$a^2 - b^2 = b^2 - b^2=0$$

As for (2): $$b = |a|$$. Given that b is equal to an absolute value of a number. An absolute value cannot be negative, theretofore $$b$$ is non-negative.

Hope it's clear.

Hi Bunuel AjiteshArun chetan2u
can you please elaborate a bit.

1 - I get that b is non- negative but why is it not sufficient? Is it coz there is no new information about 'a' & 'c' (therefore we can't say whether 'a' is negative).
2 How did you combine both the statements? Is putting values the only way to do it OR is there some explanation behind it too.

Bharat

Have you checked the discussion above? I think your doubts should be addressed there.
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# M15-37

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