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Is P + Q > 1/P + 1/ Q?

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Is P + Q > 1/P + 1/ Q? [#permalink]

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Is P + Q > 1/P + 1/ Q?

(1) P < Q < 1
(2) PQ < 1

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[Reveal] Spoiler: OA

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Re: Is P + Q > 1/P + 1/ Q? [#permalink]

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New post 12 Jun 2013, 18:36
BengalScientist wrote:
Is P + Q > 1/P + 1/ Q?

1) P < Q < 1
2) PQ < 1


I simplified the equation as P + Q > (Q + P)/PQ , PQ>1

Statement 2 is sufficient.

What am i missing here? can some one unfreeze my brain please...

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Re: Is P + Q > 1/P + 1/ Q? [#permalink]

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New post 12 Jun 2013, 20:19
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BengalScientist wrote:
BengalScientist wrote:
Is P + Q > 1/P + 1/ Q?

1) P < Q < 1
2) PQ < 1


I simplified the equation as P + Q > (Q + P)/PQ , PQ>1

Statement 2 is sufficient.

What am i missing here? can some one unfreeze my brain please...


Firstly,you cannot cancel out the factor (P+Q) as because it can be zero also. Secondly, even if (P+Q) is not zero,you cannot cross multiply the term PQ as because you don't know about its sign. When PQ<1, PQ can be positive or negative. Had the 2nd fact statement read as PQ>1, then we could have easily cross multiplied. Thus, the correct inequality to solve would be -->Is (P+Q)(1-1/PQ)>0.

From F.S 1, for P = -1/2 and Q = 1/2, P+Q = 1/P+1/Q = 0. Thus, we get a NO for the question stem. But for P=-1/2 and Q = -1/4, P+Q(-3/4) is greater than 1/P+1/Q(-6). Thus , we get a YES for the question stem. Thus, this fact statement is Insufficient.

From F. S 2, we can take the exact 2 sets of values for P and Q as above and clearly this fact statement is also Insufficient.

Taking both together, we again have no NEW information.Insufficient.

E.
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Re: Is P + Q > 1/P + 1/ Q? [#permalink]

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BengalScientist wrote:
Is p + q > 1/p + 1/q?

(1) p < q < 1
(2) pq < 1


Target question: Is p + q > 1/p + 1/q ?

This is a good candidate for rephrasing the target question.

Aside: Here’s a video with tips on rephrasing the target question: http://www.gmatprepnow.com/module/gmat-data-sufficiency?id=1100

Let's rewrite 1/p + 1/q
Find a common denominator of pq to get: q/pq + p/pq
Add to get: (p + q)/pq

REPHRASED target question: Is p + q > (p + q)/pq?
Notice that (p + q) appears on both sides of the inequality.
Also notice that if pq = 1, the two quantities, (p+q) and (p + q)/pq, will be equal.
Also notice that if p+q is positive AND pq is between 0 and 1, then (p+q) < (p + q)/pq
Also notice that if p+q is negative AND pq is between 0 and 1, then (p+q) > (p + q)/pq

These observations will help up TEST VALUES

Statement 1: p < q < 1
There are several values of p and q that satisfy statement 1. Here are two:
Case a: p = 1/4 and q = 1/2. In which case, p + q = 1/4 + 1/2 = 3/4, AND (p + q)/pq = (3/4)/(1/8) = 6. In other words, (p+q) < (p + q)/pq
Case b: p = -1/2 and q = -1/4. In which case, p + q = (-1/2) + (-1/4) = -3/4, AND (p + q)/pq = (-3/4)/(1/8) = -6. In other words, (p+q) > (p + q)/pq
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: pq < 1
There are several values of p and q that satisfy statement 2. Here are two:
Case a: p = 1/4 and q = 1/2. In which case, p + q = 1/4 + 1/2 = 3/4, AND (p + q)/pq = (3/4)/(1/8) = 6. In other words, (p+q) < (p + q)/pq
Case b: p = -1/2 and q = -1/4. In which case, p + q = (-1/2) + (-1/4) = -3/4, AND (p + q)/pq = (-3/4)/(1/8) = -6. In other words, (p+q) > (p + q)/pq
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Notice that I used the SAME values for p and q in my earlier work. So, the same values satisfy BOTH statements. That is....
There are several values of p and q that satisfy BOTH statements. Here are two:
Case a: p = 1/4 and q = 1/2. In which case, p + q = 1/4 + 1/2 = 3/4, AND (p + q)/pq = (3/4)/(1/8) = 6. In other words, (p+q) < (p + q)/pq
Case b: p = -1/2 and q = -1/4. In which case, p + q = (-1/2) + (-1/4) = -3/4, AND (p + q)/pq = (-3/4)/(1/8) = -6. In other words, (p+q) > (p + q)/pq
Since we cannot answer the REPHRASED target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

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Brent
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Re: Is P + Q > 1/P + 1/ Q? [#permalink]

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New post 21 Nov 2017, 14:42
Hi All,

We're asked if P + Q > (1/P) + (1/Q). This is a YES/NO question. We can answer it by TESTing VALUES, although we'll have to think in terms of some rarer examples (fractions) to get the correct answer.

1) P < Q < 1

IF....
P=1/4, Q=1/2.... then 1/4 + 1/2 is NOT greater than (4) + (2)... and the answer to the question is NO.
P= -1/2, Q= -1/4.... then -1/2 + -1/4 IS greater than (-2) + (-4)... and the answer to the question is YES.
Fact 1 is INSUFFICIENT

2) (P)(Q) < 1
Both of the examples that we used in Fact 1 also 'fit' Fact 2:
IF....
P=1/4, Q=1/2.... then 1/4 + 1/2 is NOT greater than (4) + (2)... and the answer to the question is NO.
P= -1/2, Q= -1/4.... then -1/2 + -1/4 IS greater than (-2) + (-4)... and the answer to the question is YES.
Fact 2 is INSUFFICIENT

Combined, we have already have 2 examples that 'fit' both Facts and produce different answers (one 'NO' and one 'YES'), so no additional work is required.
Combined, INSUFFICIENT

Final Answer:
[Reveal] Spoiler:
E


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Re: Is P + Q > 1/P + 1/ Q? [#permalink]

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New post 21 Nov 2017, 20:59
instead of going through all these we can always assume p=0 or q=0 or both , then it will not define the cases anyhow..so E

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Re: Is P + Q > 1/P + 1/ Q?   [#permalink] 21 Nov 2017, 20:59
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