manik11 wrote: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

Both statements are satisfied by the following cases:
r=2/3 and s=2/3
r=3/4 and s=3/4

In the first case, r² + s² < 1.
In the second case, r² + s² > 1.
Thus, the two statements combined are INSUFFICIENT.

The correct answer is E.

GMATGuruNY wrote:

manik11 wrote: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

Both statements are satisfied by the following cases:
r=2/3 and s=2/3
r=3/4 and s=3/4

In the first case, r² + s² < 1.
In the second case, r² + s² > 1.
Thus, the two statements combined are INSUFFICIENT.

The correct answer is E.

Thanks Mitch!
How can I solve such questions under 2 mins if I can't quickly come up with such values of r and s that satisfy both statements?

I'd go for algebra here.

Is r² > 1 - s² ?

S1:

r² > 1 - s

So we want to know if

1 - s² > 1 - s

or

s > s²

This is true if and only if 1 > s > 0. But we only know (from the prompt) that 1 > s, so we this is INSUFFICIENT.

S2:

r + s > 1/2

(r + s)² > 1/4

r² + s² + 2rs > 1/4

r² + s² > 1/4 - 2rs

So we want to know if

1/4 - 2rs > 1

or

-3/8 > rs

But we can't answer this either, since we only know 1 > r and 1 > s.

With the two together, we only need to know whether 1 > s > 0. But S2 allows for s to be negative OR positive, so we can't answer.

Matt@VeritasPrep wrote:I'd go for algebra here.

Is r² > 1 - s² ?

S1:

r² > 1 - s

So we want to know if

1 - s² > 1 - s

or

s > s²

This is true if and only if 1 > s > 0. But we only know (from the prompt) that 1 > s, so we this is INSUFFICIENT.

S2:

r + s > 1/2

(r + s)² > 1/4

r² + s² + 2rs > 1/4

r² + s² > 1/4 - 2rs

So we want to know if

1/4 - 2rs > 1

or

-3/8 > rs

But we can't answer this either, since we only know 1 > r and 1 > s.

With the two together, we only need to know whether 1 > s > 0. But S2 allows for s to be negative OR positive, so we can't answer.

Hi Matt,

I could not follow you in Statement 2.

How do you conclude that r² + s² > 1/4 - 2rs ?? why not r² + s² < 1/4 - 2rs

Thanks

Because we know that (r + s) > 0, so when we square both sides, we're squaring two positive values.

It's similar to, say, 5 > 3. If we square both sides, we have 25 > 9, which is also true.

Matt@VeritasPrep wrote:Because we know that (r + s) > 0, so when we square both sides, we're squaring two positive values.

It's similar to, say, 5 > 3. If we square both sides, we have 25 > 9, which is also true.

Thanks Matt for your response. I copied my question wrong.

My question is how you concluded that 1/4 - 2rs > 1? why not 1/4 - 2rs < 1 ??

My apology for mistake.

Mo2men wrote:

Matt@VeritasPrep wrote:Because we know that (r + s) > 0, so when we square both sides, we're squaring two positive values.

It's similar to, say, 5 > 3. If we square both sides, we have 25 > 9, which is also true.

Thanks Matt for your response. I copied my question wrong.

My question is how you concluded that 1/4 - 2rs > 1? why not 1/4 - 2rs < 1 ??

My apology for mistake.

Oh, I see! It isn't a conclusion, it's a way of restating the question. Since we know r² + s² > 1/4 - 2rs and we want to know if r² + s² > 1, we can use what we do know to test what we want to know.

So if we know

r² + s² > 1/4 - 2rs

and we somehow discover that

1/4 - 2rs > 1

we can then conclude that

r² + s² > 1/4 - 2rs > 1

So S2 lets us rephrase the question as "Is 1/4 - 2rs > 1?" If it is, then we can also answer the original question, and we're done.