If ap aq, is a/pq0?

Question

If ap > aq , is \frac{a}{pq}>0 ? (1) p < q (2) \frac{1}{p}<\frac{1}{q}

Bunuel · Accepted Answer

If ap > aq , is \frac{a}{pq}>0 ? Given: ap > aq --> a(p-q)>0 . So, a and p-q have the same sign. Question: is \frac{a}{pq}>0 ? --> does a and pq have the same sign? (1) p < q --> p -q < 0 , so a < 0 too. Since a < 0 , then the question becomes is pq < 0 . We know that p < q , but this is not sufficient to answer the question: one number is less than another, we cannot say from this whether their product is negative or positive. Not sufficient. (2) \frac{1}{p}<\frac{1}{q} \frac{1}{q}-\frac{1}{p}>0 \frac{p-q}{pq}>0 p-q and pq have the same sign, thus a and pq have the same sign. Sufficient. Answer: B. Hope it's clear.

jokschmer · Answer

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Hi why we can not write p1/q.... this funda used by one user in following link to solve another inequality problem https://gmatclub.com/forum/if-a-1-2-is- ... l#p1855701 Please clear this doubt...

Bunuel · Answer

We cannot cross-multiply 1/p>1/q because we don't know the sign of p and q. If p and q have the same sign then when cross multiplying we'll have q > p BUT if p and q have the opposite signs then when cross multiplying we'll have q < p. For example, if p > 0 and q < 0, we'll have 1 > p/q (keep the sign when multiplying by positive value) and then q < p (recall that we should flip the sign of an inequality if we multiply/divide it by negative value). Never multiply (or reduce) an inequality by a variable (or the expression with a variable) if you don't know its sign. Hope it helps.

exc4libur · Answer

ap > aq = a(p-q)>0 : a>0…p-q>0…p>q a<0…p-q<0…p0 then \frac{a=positive}{pq=positive}>0 if p>0…q<0 then \frac{a=positive}{pq=negative}<0 different answers, insuf. (2) \frac{1}{p}<\frac{1}{q}:…\frac{q-p}{pq}<0 means that q-p and pq are different signs; if pq<0 , then q-p>0…q>p ; , so a<0 : \frac{a=negative}{pq=negative}>0 if pq>0 , then q-p<0…q

0 : \frac{a=positive}{pq=positive}>0 same answers, sufic. Answer (B).

Kinshook · Answer

Given: ap > aq Asked: Is \frac{a}{pq}>0 ? (1) p < q ap>aq => a<0 Signs of p & q are unknown NOT SUFFICIENT (2) \frac{1}{p}<\frac{1}{q} 1/p - 1/q <0 (q-p)/pq <0 (1) ap-aq > 0 a(p-q) >0 Signs of a & (p-q) are same a/pq >0 SUFFICIENT IMO B

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If ap > aq, is a/pq>0?

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