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

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

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