The 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

b^2 + c^2 = 370
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

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

Considering a and b not essentially integers, there are infinite possibilities of b and c for which b^2+c^2 = 370

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

Thank you Brent!!!

Necessary corrections made.

I read question in jiffy I guess.

I believe that the problem should read as follows:

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

Since the statements are in terms of a and b, rephrase the question stem in terms of a and b.
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.

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.