What is the optimal way to solve such problems?

If r and s are both less than 1, is r^2 + s^2 > 1?

(1)r^2 + s > 1
(2)r + s > 1/2

Please cite the source (author) of this problem. We cannot reply unless a source is cited (and, if no source is cited, we will have to delete the post!). Thanks.

_________________
Ben Ku
Instructor
ManhattanGMAT

Apologies for not citing the source. This is one of the questions that i came across while solving Belcurves exercise.

Thanks!

I'm not sure what you mean by "such problems," but let's look at this particular problem and then see what we can generalize.

Unlike many superficially similar problems, this one doesn't allow any productive rephrasing of the question. The question does, however, give us a strong idea about the sorts of values relevant: the squares of numbers less than 1. If we choose to test values, we may need to consider positive values very near 1 (such as 0.9), positive values very nearly 0 (such as 0.1), 0, and negative numbers (might as well use simple integers such as -2).

Let's suppose that we judge that (2) is easier to evaluate.

(2)r + s > 1/2

You don't have to arrange the information as I have below--I do it mostly for pedagogical purposes--but you do have to take into account the same stuff: the proposed values for the variables, whether those values are legal for the statement, and what answer they give to the question.

R S r+s>1/2? r^2 + s^2 > 1?
0.9 0.9 legal YES
0.6 0 legal NO

Statement 2 is not sufficient, so eliminate BD.
Now consider (1) r^2 + s > 1.

R S r^2 + s > 1? r^2 + s^2 > 1?
0.9 0.9 legal YES
0.6 0.7 legal NO

Let me explain why I chose the value I did for that second line. In the first line I made the value pretty nearly as large as I could, so in the second I wanted to make the value of the r^2 + s just barely greater than 1. For these values, r^2 + s=1.06. Once you choose a value for R, let S be as small a simple value as you think of, such that (1) is true. For instance, we could have chosen:
R S r^2 + s > 1? r^2 + s^2 > 1?
0.5 0.8 legal NO
0.7 0.6 legal NO
0.8 0.4 legal NO

In any event, (1) is also not sufficient, so eliminate A.

Since the values we originally considered for (2) are legal for both statements, we don't have any more work to do:

R S r^2 + s > 1? r^2 + s^2 > 1?
&
r+s>1/2?
0.9 0.9 legal YES
0.6 0.7 legal NO

So E.


Alternatively, we might have approached at least (1) algebraically/theoretically.

Hmm, let's focus on s, since it's the only difference between the equation in (1) and the equation in the question. It looks as though some rephrasing might be useful after all. Let's rewrite the question as
Is s^2 > 1-r^2?

and rewrite (1) as

(1) s > 1-r^2

The answer could be "yes," but since s^2<s for proper fractions, the answer could also be "no."


What can we generalize here? If testing numbers, consider legal values likely to give different answers to the question.

Best, Michael