MGMAT CAT DS

[ekattur]

I think the answer should be "C" but the explanation says E. Can anyone please help? THanks

If p < q and p < r, is (p)(q)(r) < p?
(1) pq < 0

(2) pr < 0

The question tells us that p < q and p < r and then asks whether the product pqr is less than p.
Statement (1) INSUFFICIENT: We learn from this statement that either p or q is negative, but since we know from the question that p < q, p must be negative. To determine whether pqr < p, let's test values for p, q, and r. Our test values must meet only 2 conditions: p must be negative and q must be positive.

p q r pqr Is pqr < p?
-2 5 10 -100 YES
-2 5 -10 100 NO

Statement (2) INSUFFICIENT: We learn from this statement that either p or r is negative, but since we know from the question that p < r, p must be negative. To determine whether pqr < p, let's test values for p, q, and r. Our test values must meet only 2 conditions: p must be negative and r must be positive.

p q r pqr Is pqr < p?
-2 -10 5 100 NO
-2 10 5 -100 YES

If we look at both statements together, we know that p is negative and that both q and r are positive. To determine whether pqr < p, let's test values for p, q, and r. Our test values must meet 3 conditions: p must be negative, q must be positive, and r must be positive.

p q r pqr Is pqr < p?
-2 10 5 -100 YES
-2 7 4 -56 YES

At first glance, it may appear that we will always get a "YES" answer. But don't forget to test out fractional (decimal) values as well. The problem never specifies that p, q, and r must be integers.
p q r pqr Is pqr < p?
-2 .3 .4 -.24 NO
Even with both statements, we cannot answer the question definitively. The correct answer is E.

[esledge]

"Is pqr<p?" can be rephrased VERY CAREFULLY.

Dividing both sides by p is legitimate, as a glance at the statements reveals that p cannot be 0. However, when we divide both sides of an inequality by a negative value, we must flip the sign! This is why we need to be careful.

Thus, our question can be rephrased into 2 cases:
If p is positive, the question is "Is qr<1?" (sign not flipped)
If p is negative, the question is "Is qr>1?" (sign flipped)

We don't even know which question to ask (!), making me suspect C or E as an answer already, but that's just a hunch at this point.

(1) p and q have opposite signs. But we know that p < q, so p must be negative and q must be positive. This clarifies that the question to ask is "Is qr>1?" (sign flipped), but we don't know anything about r.

(2) p and r have opposite signs. But we know that p < r, so p must be negative and r must be positive. This clarifies that the question to ask is "Is qr>1?" (sign flipped), but we don't know anything about q.

(1)&(2) We've already established the correct question is "Is qr>1?" and the statements tell us that q and r must be positive. Try to prove insufficiency by thinking of a Yes case and and No case.

Yes case: qr IS >1 when q = 5 and r = 2 (or more generally, when q and r are relatively large positive numbers)

No case: qr IS NOT >1 when q = 1/2 and r = 1/3 (or more generally, when q and r are fractions or "small enough" positive numbers)