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r^2 + s^2 >1 ?

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Source: — Data Sufficiency |
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by GMATGuruNY » Mon Apr 11, 2016 8:21 am
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.
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by manik11 » Mon Apr 11, 2016 8:29 am
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?
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by [email protected] » Mon Apr 11, 2016 10:29 am
Hi manik11,

TESTing VALUES (the approach that Mitch used) will work on most of the Quant questions that you'll face on Test Day (including many DS questions). As such, rather than looking for an alternative to that Strategy, you might want to embrace it, practice it and hone those Tactical skills.

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by Matt@VeritasPrep » Mon Apr 11, 2016 1:24 pm
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.
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by Mo2men » Tue Apr 12, 2016 4:38 am
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
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by Matt@VeritasPrep » Tue Apr 12, 2016 6:45 pm
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.
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by Mo2men » Tue Apr 12, 2016 7:03 pm
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.
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by Matt@VeritasPrep » Fri Apr 15, 2016 12:55 pm
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.
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