If pq 0, is p + q 1/p + 1/q? (1) p

Question

If  pg 
eq 0 , is  p + q > \frac{1}{p} + \frac{1}{q}  ?

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

M21-19

BrentGMATPrepNow · Accepted Answer

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: https://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 Cheers, Brent

BengalScientist · Answer

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...

mau5 · Answer

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.

EMPOWERgmatRichC · Answer

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:  E

GMAT assassins aren't born, they're made, 
Rich

Raksat · Answer

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

Pritishd · Answer

1. I see that two values that can be used prove and disprove both the statements are  P = \frac{1}{3}  and  Q = \frac{1}{2}  &  P = \frac{-1}{2}  and  Q = \frac{-1}{3} 
2. Plugging both sets of values into the inequality

3.a  P + Q = \frac{1}{3} + \frac{1}{2} = \frac{5}{6}

3.b  \frac{1}{P} + \frac{1}{Q} = 3 + 2 = 5

3.c In the above case,  P + Q < \frac{1}{P} + \frac{1}{Q}

4.a  P + Q = \frac{-1}{2} + (\frac{-1}{3}) = \frac{-5}{6}

4.b  \frac{1}{P} + \frac{1}{Q} = -2 + (-3) = -5

4.c In the above case,  P + Q > \frac{1}{P} + \frac{1}{Q}

Ans.  E

Bunuel · Answer

Is p + q > \frac{1}{p} + \frac{1}{q} ? (1) p < q < 1 If both p and q are positive fractions, then the left-hand side is the sum of two numbers less than 1, while the right-hand side is the sum of two numbers greater than 1. Therefore, p + q < \frac{1}{p} + \frac{1}{q} . For instance, consider p = \frac{1}{3} and q = \frac{1}{2} . However, if both are negative fractions, then the left-hand side will be the sum of two negative numbers greater than -1, whereas the right-hand side will be the sum of two negative numbers less than -1. In this scenario, p + q > \frac{1}{p} + \frac{1}{q} . For instance, consider p = -\frac{1}{2} and q = -\frac{1}{3} . Not sufficient. (2) pq < 1 Using the logic from statement (1), this statement alone also does not determine if p + q > \frac{1}{p} + \frac{1}{q} . Not sufficient. (1)+(2) We can use the same logic as for (1) and (2). Even when combined, the statements are not sufficient. Answer: E

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

If pq 0, is p + q > 1/p + 1/q? (1) p < q < 1 (2) pq < 1

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions