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# M06-34

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Math Expert
Joined: 02 Sep 2009
Posts: 53796

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16 Sep 2014, 00:33
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Difficulty:

75% (hard)

Question Stats:

54% (02:12) correct 46% (02:22) wrong based on 112 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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16 Sep 2014, 00:33
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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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23 Sep 2014, 02:01
1
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?
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Joined: 02 Sep 2009
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23 Sep 2014, 02:07
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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29 Nov 2014, 22:25
Tricky question but a good one. -- I also had assumed that B = 12 , A = 11 C = 9.
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11 Apr 2015, 09:10
good question..I also did the same mistake of assuming the numbers as integers.
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27 Nov 2015, 16:17
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
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Joined: 07 Feb 2016
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18 Apr 2017, 22:24
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?
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03 Aug 2017, 23:08
I think this is a high-quality question and I agree with explanation.
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21 Oct 2017, 19:54
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
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21 Oct 2017, 22:57
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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22 Oct 2017, 09:56
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
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24 Apr 2018, 21:05
Check more solutions here: https://gmatclub.com/forum/if-a-b-and-c ... 55421.html
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14 Jun 2018, 22:11
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.
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Joined: 22 Mar 2017
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08 Jan 2019, 12:44
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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Joined: 02 Sep 2009
Posts: 53796

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08 Jan 2019, 21:23
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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25 Feb 2019, 13:30
I think this is a high-quality question and I agree with explanation.
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Re M06-34   [#permalink] 25 Feb 2019, 13:30
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# M06-34

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