Data Sufficiency

I saw the following question posted on the website but, for some reason, I could not find the method used to solve the problem. I have the following concerns with this question:

- Please let me know if what I have stated below is correct.
- If so, is there a faster way to understand this problem? This definitely took me more than 2 minutes, and I am not sure if what I have is the right method.

The numbers a, b, and c are all positive. If b^2 + c^2 = 17, what is the value of a^2 + c^2?
(1) a-b=3
(2) (a+b)/(a-b)=7

- First, the question is asking for a value. Therefore, either statement must result in ONE value in order to be sufficient.
- Second, I need the numerical value of c^2 AND a^2 to obtain sufficiency.

(1) a-b=3 -> b=a-3 -> b^2=a^2-6a+9.
b^2+c^2=17 -> a^2-6a+9+c^2=17 -> a^2+c^2=17+6a-9 -> a^2+c^2=8+6a
Insufficient. Values of c^2 & a^2 are not given. Also, 2 variables & 1 equation = no good. A&D are eliminated, BCE remain.

(2) (a+b)/(a-b) = 7 -> a+b=7(a-b) -> a+b=7a-7b -> -6a=-8b -> b=(3a)/4
b^2+c^2=17 -> (((3a)^2)/(4^2))+c^2=17 -> ((9a^2)/16)+c^2=17.
Insufficient. Values of c^2 & a^2 are not given. Also, 2 variables & 1 equation = no good. B is eliminated. C&E remain.

(1)&(2)
From (1): a^2+c^2=8+6a. From (2): ((9a^2)/16)+c^2=17.
Together: 8+6a-a^2=17-((9a^2)/16). 1 equation, 1 variable = good. We can solve for a and we can solve for a^2. Plug in a^2 to get c^2. c^2 AND a^2 are determined. Sufficient. Answer is C.

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.

GMATGuruNY,

Thank you very much for your quick reply.

I like the following conclusion you made: "Since the statements are in terms of a and b, rephrase the question stem in terms of a and b." That's a nice little trick that I will keep in mind for future DS problems.

Regards,
Ricardo