The numbers a, b, and c are all positive. If b2 + c2 = 370, then what is the value of a2 + c2?
Statement #1: a - b = 3
Statement #2: (a + b)/(a - b) = 7
Best way to solve this
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b^2 + c^2 = 370The numbers a, b, and c are all positive Integers. If b2 + c2 = 370, then what is the value of a2 + c2?
Statement #1: a - b = 3
Statement #2: (a + b)/(a - b) = 7
Which means that b and c have to be 17 and 9 (NOT necessarily in the same order)
Statement 1)
a-b = 3
But @b=17, a=20
and @b=9, a=12
Therefore values of a and c are not fixed, due to variable values of b therefore INSUFFICIENT
Statement 2)
a+b/a-b = 7 ==> a+b = 7a-7b ==> 6a = 8b ==? a/b = 4/3 {this means that b will be a multiple of 3}
But as the question mentions that b can be either 17 or 9 therefore b will be 9 (being a multiple of 3
Therefore c = 7
and a = 12
therefore a^2+b^2 = 12^2 + 9^2 = 144+81 = 225 SUFFICIENT
Answer Option B
Last edited by GMATinsight on Fri Jul 04, 2014 7:35 am, edited 3 times in total.
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Be careful, Bhoopendra. The question doesn't say that a, b and c are integers.GMATinsight wrote: b^2 + c^2 = 370
Which means that b and c have to be 17 and 9 (NOT necessarily in the same order)
There are infinitely many solutions to the equation b² + c² = 370. For example, b = √368 and c = √2
Cheers,
Brent
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Considering a and b not essentially integers, there are infinite possibilities of b and c for which b^2+c^2 = 370The numbers a, b, and c are all positive. If b2 + c2 = 370, then what is the value of a2 + c2?
Statement #1: a - b = 3
Statement #2: (a + b)/(a - b) = 7
Statement 1) a-b = 3
Infinite values of a and b are possible with this relation
e.g a=4 and b=1 and c=sqrt(370-1)
e.g a=5 and b=2 and c=sqrt(370-4)
e.g a=6 and b=3 and c=sqrt(370-9) and so on....
Various values of a, b and c. Therefore Inconsistent answers INSUFFICIENT
Statement 2) a+b/a-b = 7 ==> a+b = 7a-7b ==> 6a = 8b ==? a/b = 4/3
but if a=4 then b=3 and c=sqrt(370-9)
and if a=8 then b=6 and c=sqrt(370-36)
and if a=12 then b=9 and c=sqrt(370-81) and so on...
Various values of a, b and c. Therefore Inconsistent answers INSUFFICIENT
Combining Statement 1 and 2
a/b = 4/3 and a-b=3
if a=4x then b=3x and 4x-3x = 3 which means x=3
therefore a=12, b=9 and c=sqrt(370-81) = 17
therefore a^2+c^2 = 8^2+17^2 = 225 SUFFICIENT
Correct answer Option C
Last edited by GMATinsight on Fri Jul 04, 2014 7:40 am, edited 1 time in total.
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Thank you Brent!!!
Necessary corrections made.
I read question in jiffy I guess.
Necessary corrections made.
I read question in jiffy I guess.
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I believe that the problem should read as follows:
Substituting c²=17-b² into a²+c², we get:
a² + (17-b²)
a² - b² + 17.
(a+b)(a-b) + 17.
To determine the value of (a+b)(a-b) + 17, we need to know the value of (a+b)(a-b).
Question stem, rephrased:
What is the value of (a+b)(a-b)?
Statement 1: a-b = 3
No information about a+b.
INSUFFICIENT.
Statement 2: (a + b)/(a - b) = 7
Thus, a+b = 7(a-b).
Case 1: a-b = 1 and a+b = 7
In this case, (a+b)(a-b) = 7*1 = 7.
Case 2: a-b = 2 and a+b = 14
In this case, (a+b)(a-b) = 14*2 = 28.
INSUFFICIENT.
Statements combined:
Statement 1: a-b=3
Statement 2: a+b = 7(a-b) = 7*3 = 21.
Thus, (a+b)(a-b) = 21*3 = 63.
SUFFICIENT.
The correct answer is C.
Since the statements are in terms of a and b, rephrase the question stem in terms of a and b.The numbers a, b, and c are all positive. If b²+c²=17, then what is the value of a²+c²?
Statement #1: a - b = 3
Statement #2: (a + b)/(a - b) = 7
Substituting c²=17-b² into a²+c², we get:
a² + (17-b²)
a² - b² + 17.
(a+b)(a-b) + 17.
To determine the value of (a+b)(a-b) + 17, we need to know the value of (a+b)(a-b).
Question stem, rephrased:
What is the value of (a+b)(a-b)?
Statement 1: a-b = 3
No information about a+b.
INSUFFICIENT.
Statement 2: (a + b)/(a - b) = 7
Thus, a+b = 7(a-b).
Case 1: a-b = 1 and a+b = 7
In this case, (a+b)(a-b) = 7*1 = 7.
Case 2: a-b = 2 and a+b = 14
In this case, (a+b)(a-b) = 14*2 = 28.
INSUFFICIENT.
Statements combined:
Statement 1: a-b=3
Statement 2: a+b = 7(a-b) = 7*3 = 21.
Thus, (a+b)(a-b) = 21*3 = 63.
SUFFICIENT.
The correct answer is C.
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Another way of thinking about this is as an equation in two variables.
If c² = 370 - b², then a² + c² = a² + (370 - b²) = a² - b² + 370 = (a+b)(a-b) + 370
So we only need (a+b)(a-b), and we're done.
S1 tells us a = 3 + b, INSUFFICIENT.
S2 tells us a + b = 7a - 7b, or a = (4/3)b, INSUFFICIENT.
Together we have a = 3 + b and a = (4/3)b, so we know that (3+b) = (4/3)b, which allows us to solve for b. Once we have b, we can substitute to find a, so the two statements together are SUFFICIENT.
If c² = 370 - b², then a² + c² = a² + (370 - b²) = a² - b² + 370 = (a+b)(a-b) + 370
So we only need (a+b)(a-b), and we're done.
S1 tells us a = 3 + b, INSUFFICIENT.
S2 tells us a + b = 7a - 7b, or a = (4/3)b, INSUFFICIENT.
Together we have a = 3 + b and a = (4/3)b, so we know that (3+b) = (4/3)b, which allows us to solve for b. Once we have b, we can substitute to find a, so the two statements together are SUFFICIENT.