Data Sufficiency

So right off the bat, setting both P and Q equal to 1/2 isn't a good call - statement 1 says that P < Q. It's key when plugging in numbers to stick to the given constraints.

If you set P = 1/4 and Q = 1/2, you will find that P + Q > 1/P + 1/Q gives 3/4 > 6, which is not true. However, the problem never indicates that P and Q must be positive numbers. If we instead set P = -1/2 and Q = -1/4, we get -3/4 > -6, which IS true. So Statement 1 is insufficient.

On Data Sufficiency questions, it's important not to make assumptions about whether answers are positive vs. negative, integers vs. non-integers, non-zero vs. zero, etc. unless the question specifically tells us that information. It's critical to test out these options when relying on picking numbers to prove a statement false!

Is p + q > 1/p + 1/q ?

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

OA:E

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- ... cy?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

Hi VJesus12,

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