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Math Expert V
Joined: 02 Sep 2009
Posts: 58464

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Difficulty:   75% (hard)

Question Stats: 54% (02:09) correct 46% (02:20) wrong based on 117 sessions

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If $$a$$, $$b$$, and $$c$$ are positive and $$a^2 + c^2 = 202$$, what is the value of $$b - a - c$$?

(1) $$b^2 + c^2 = 225$$

(2) $$a^2 + b^2 = 265$$

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Math Expert V
Joined: 02 Sep 2009
Posts: 58464

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2
Official Solution:

(1) $$b^2+c^2=225$$. Not sufficient on its own.

(2) $$a^2+b^2=265$$. Not sufficient on its own.

(1)+(2) Subtract $$a^2+c^2=202$$ from $$b^2+c^2=225$$: $$b^2-a^2=23$$.

Now, sum this with $$a^2+b^2=265$$: $$2b^2=288$$.

So, $$b^2=144$$, giving $$b=12$$ (since it is given that $$b$$ is a positive number). Since $$b=12$$, then from $$b^2-a^2=23$$ we get that $$a=11$$ and from $$a^2+c^2=202$$ we get that $$c=9$$. Sufficient.

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Intern  Joined: 06 Aug 2014
Posts: 15
Concentration: Entrepreneurship, Marketing
Schools: ISB '17 (A)
GMAT 1: 720 Q50 V37 GPA: 3.3

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Hi Bunuel,

In this question, if I take only the first statement and subtract the two equations given we get:-

b^2-a^2=23.
Thus possible values positive values will be- b=12, a=11.
Upon substitution in any of the two equations we get c=9.
Thus can we not answer the question using only statement 1.

Please kindly point out the flaw in this reasoning. Is it because the question does not give us that only integer values are possible?
Math Expert V
Joined: 02 Sep 2009
Posts: 58464

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Hi Bunuel,

In this question, if I take only the first statement and subtract the two equations given we get:-

b^2-a^2=23.
Thus possible values positive values will be- b=12, a=11.
Upon substitution in any of the two equations we get c=9.
Thus can we not answer the question using only statement 1.

Please kindly point out the flaw in this reasoning. Is it because the question does not give us that only integer values are possible?

The problem with your solution is that you assume, with no ground for it, that variables represent integers only. From b^2 - a^2 = 23 you cannot say that b = 12 and a = 11. For example b could be $$\sqrt{24}$$ and a could be 1.
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Intern  Joined: 30 Jun 2012
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Tricky question but a good one. -- I also had assumed that B = 12 , A = 11 C = 9.
Intern  Joined: 05 Feb 2015
Posts: 49
Concentration: Finance, Entrepreneurship
Schools: ISB '16, IIMA , IIMB, IIMC
WE: Information Technology (Health Care)

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good question..I also did the same mistake of assuming the numbers as integers.
Intern  B
Joined: 09 Sep 2015
Posts: 2

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To solve 3 different variables, you need 3 different equations. hence it was simple to identify 'C' as the option. Is the approach correct
? I did this question in less than 30 seconds
Intern  B
Joined: 07 Feb 2016
Posts: 20
GMAT 1: 650 Q47 V34 GMAT 2: 710 Q48 V39 Show Tags

HK17 wrote:
To solve 3 different variables, you need 3 different equations. hence it was simple to identify 'C' as the option. Is the approach correct
? I did this question in less than 30 seconds

I did it the same way without calculating the individual values. Is there a possiblity that this assumption will not hold?
Intern  B
Joined: 29 Mar 2016
Posts: 1

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I think this is a high-quality question and I agree with explanation.
Intern  B
Joined: 06 Jun 2017
Posts: 12
Location: India
GMAT 1: 610 Q49 V25 GMAT 2: 720 Q49 V38 GPA: 2.9

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In my onion A is the answer

1) a^2 + c^2 = 202 ( given )
2) b^2 + c^2 = 225 ( statement 1 )

subtracting 1 from 2, we get b^2 - a^2 = 23 ( prime number )

its given that a,b and c are positive. therefore the only solution is b+c = 23 and b-c = 1 -> b= 12, c = 11

we can substitute the value of b in equation 2 -> 12^2 + c^2 = 225 -> c = 9 ( since -9 can be negated since c has to be +ve )

also similarly for statement 2, b^2 - c^2 = 63; but 63 is not prime -> therefore we cant get a solution ( it can be 21 * 3 or 9*7 )

I think the answer is A. High quality question -> but wrong answer marked.

Ashwin
Math Expert V
Joined: 02 Sep 2009
Posts: 58464

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TheGMATcracker wrote:
In my onion A is the answer

1) a^2 + c^2 = 202 ( given )
2) b^2 + c^2 = 225 ( statement 1 )

subtracting 1 from 2, we get b^2 - a^2 = 23 ( prime number )

its given that a,b and c are positive. therefore the only solution is b+c = 23 and b-c = 1 -> b= 12, c = 11

we can substitute the value of b in equation 2 -> 12^2 + c^2 = 225 -> c = 9 ( since -9 can be negated since c has to be +ve )

also similarly for statement 2, b^2 - c^2 = 63; but 63 is not prime -> therefore we cant get a solution ( it can be 21 * 3 or 9*7 )

I think the answer is A. High quality question -> but wrong answer marked.

Ashwin

The problem with your solution is that you assume that the variables are integers. We are not given that. b^2 - a^2 = 23 has infinitely many solutions for b ans a.
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Intern  B
Joined: 26 Nov 2014
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Bunuel wrote:
Official Solution:

(1) $$b^2+c^2=225$$. Not sufficient on its own.

(2) $$a^2+b^2=265$$. Not sufficient on its own.

(1)+(2) Subtract $$a^2+c^2=202$$ from $$b^2+c^2=225$$: $$b^2-a^2=23$$.

Now, sum this with $$a^2+b^2=265$$: $$2b^2=288$$.

So, $$b^2=144$$, giving $$b=12$$ (since it is given that $$b$$ is a positive number). Since $$b=12$$, then from $$b^2-a^2=23$$ we get that $$a=11$$ and from $$a^2+c^2=202$$ we get that $$c=9$$. Sufficient.

If it would've been given that a,b,c are integers then can we mark D?

Explanation:
if a,b,c are integers-

a^2+C^2=202 => a=9/11, C=11/9

or
Solving with b^2+C^2=225 => b^2-a^2=23 => (b-a)(b+a)=23, 23 is prime number so b and a must be consecutive numbers.=> b=12,a=11 for c=9(integer value)

Similarly, we can do for a^2+b^2=265
Math Expert V
Joined: 02 Sep 2009
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Check more solutions here: https://gmatclub.com/forum/if-a-b-and-c ... 55421.html
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Intern  B
Joined: 17 Sep 2016
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For this question no need of solving. AS we are given that positive values only, we know that only c will work. no need what the solution is as long as we can confirm that a solution does exist. saves time.
Intern  B
Joined: 22 Mar 2017
Posts: 4

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Hi Bunuel,

I don't agree with the solution to this question. In the question stem, a^2+c^2 = 202 can have only two sets of values

a= 11
c= 9

OR
a= 9
c= 11

Each of the statements would then be sufficient to solve the question by giving separate values for a, b and c.

Please let me know if there is something incorrect with this approach.

Thanks :D
Math Expert V
Joined: 02 Sep 2009
Posts: 58464

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samsam00 wrote:
Hi Bunuel,

I don't agree with the solution to this question. In the question stem, a^2+c^2 = 202 can have only two sets of values

a= 11
c= 9

OR
a= 9
c= 11

Each of the statements would then be sufficient to solve the question by giving separate values for a, b and c.

Please let me know if there is something incorrect with this approach.

Thanks :D

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Manager  G
Joined: 22 Jun 2017
Posts: 168
Location: Argentina
Schools: HBS, Stanford, Wharton
GMAT 1: 630 Q43 V34 Show Tags

I think this is a high-quality question and I agree with explanation.
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VP  P
Joined: 14 Feb 2017
Posts: 1220
Location: Australia
Concentration: Technology, Strategy
Schools: LBS '22
GMAT 1: 560 Q41 V26 GMAT 2: 550 Q43 V23 GMAT 3: 650 Q47 V33 GMAT 4: 650 Q44 V36 WE: Management Consulting (Consulting)

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Fairly meaty question. The amount of quadratics present can be overwhelming. I got this one incorrect.

Once you get one value of any variable you can find the rest. That saves a bit of time!
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+1 Kudos if I have helped you Re: M06-34   [#permalink] 02 Jul 2019, 01:40
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