"What is the ratio of r to s?"

[SarmyB]

Hello,

MY QUESTION:

Would greatly appreciate if I could please be shown a second possible pair of values for "r & s" to make STATEMENT (2) INSUFFICIENT.

On my exam, I chose ANSWER (B), as I could not find a second pair for r & s.

Explanation states the factored form "(r + s)(r – s) = 7 cannot be rewritten in the form r/s = some value". While one of our rephrased question stem states "what is the value for r/s", our second rephrased question stem states "what are the values for r and s". Looking at STATEMENT (2) exclusively (not including STATEMENT (1)), I cannot seem to find any values other than r=4 & s=3 that fit the constraints of the formula. The only slight variation I found was changing the signs: r= -4 & s= -3 (or +3), which derive to +7 on both sides of the formula.

Answering the question stem, "what is the ratio of r/s") using STATEMENT (2) alone, I can only come up with: 4/3 AND -4/3. Could these be the two values that would make STATEMENT (2) INSUFFICIENT?

Thank you in advance
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QUESTION:

What is the ratio of r to s?

(1) r + s = 7

(2) r^2 – s^2 = 7

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EXPLANATION:
To find the ratio of r to s, we need to be able to solve EITHER for r/s OR for r and s independently.

(1) INSUFFICIENT: The equation provided in statement 1 cannot be rewritten in the form r/s = some value.

(2) INSUFFICIENT: The equation provided in statement 2 can be simplified as follows:

r^2 – s^2 = 7
(r + s)(r – s) = 7.

However, this cannot be rewritten in the form r/s = some value.

(1) AND (2) SUFFICIENT: We can substitute the information from statement (1) in the equation from statement 2 as follows:

(r + s)(r – s) = 7.
(7)(r – s) = 7.
r – s = 1.

Adding this equation to the equation from the first statement allows us to solve for r.

(r – s = 1)

+

(r + s = 7)




2r = 8

Thus, r = 4. If r is 4, then s must be 3. The ratio of r to s is 4 to 3.

The correct answer is C.

[Sage Pearce-Higgins]

Your instinct to factorize the second statement is a good one. r^2 – s^2 = 7, so (r + s)(r – s) = 7 However, don't forget that it's still basically the same piece of information: it tells us that the difference between r^2 and s^2 is 7. There are an infinity of numbers that would work with this! For example r^2 could be 7 and s^2 could be 0. This will give us a different ratio from r^2 = 1007 and s^2 = 1000. As much as algebra and rephrasing are useful strategies, don't forget to understand what the statements say. I hope this answers your question.

One more point: you write "I chose ANSWER (B), as I could not find a second pair for r & s". I would say that you should revise your approach to DS on this. Just because you can't immediately see a solution doesn't mean that there isn't one. Think in slightly more general terms: "Statement 2 just gives me the difference between the squares of those numbers (and I don't know if they are integers or not); that's surely not enough information for me to find out what the numbers are."

[SarmyB]

Gotch'ya. Makes sense. Thank you very much for the detailed explanation and suggestion .

[RonPurewal]

if you have any further questions please ask. thanks.