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

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niteshwaghray
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If ap > aq, is a/pq>0? [#permalink] New post   Updated on: 23 May 2017, 00:59
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If \(ap > aq\), is \(\frac{a}{pq}>0\)?

(1) \(p < q\)

(2) \(\frac{1}{p}<\frac{1}{q}\)
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B

Originally posted by niteshwaghray on 23 May 2017, 00:51.
Last edited by Bunuel on 23 May 2017, 00:59, edited 1 time in total.
Edited the question.
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If ap > aq, is a/pq>0? [#permalink] New post   23 May 2017, 01:12
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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.
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Re: If ap > aq, is a/pq>0? [#permalink] New post   28 Jun 2017, 08:02
Bunuel wrote:
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.


^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Hi why we can not write p<q as 1/p>1/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...
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Re: If ap > aq, is a/pq>0? [#permalink] New post   28 Jun 2017, 09:53
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jokschmer wrote:
Bunuel wrote:
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.


^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Hi why we can not write p<q as 1/p>1/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...


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.
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If ap > aq, is a/pq>0? [#permalink] New post   Updated on: 20 Aug 2019, 05:00
niteshwaghray wrote:
If \(ap > aq\), is \(\frac{a}{pq}>0\)?

(1) \(p < q\)
(2) \(\frac{1}{p}<\frac{1}{q}\)


\(ap > aq = a(p-q)>0\):
[case 1] \(a>0…p-q>0…p>q\)
[case 2] \(a<0…p-q<0…p<q\)

(1) \(p < q\): [case 1] \(a<0\)
if \({p,q}>0\) 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 \(q-p>0…q>p\); [case 2], so \(a<0\): \(\frac{a=negative}{pq=negative}>0\)
if \(pq>0\), then \(q-p<0…q<p\); [case 1], so \(a>0\): \(\frac{a=positive}{pq=positive}>0\)
same answers, sufic.

Answer (B).

Originally posted by exc4libur on 20 Aug 2019, 04:21.
Last edited by exc4libur on 20 Aug 2019, 05:00, edited 1 time in total.
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Re: If ap > aq, is a/pq>0? [#permalink] New post   20 Aug 2019, 04:51
niteshwaghray wrote:
If \(ap > aq\), is \(\frac{a}{pq}>0\)?

(1) \(p < q\)

(2) \(\frac{1}{p}<\frac{1}{q}\)


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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Re: If ap > aq, is a/pq>0? [#permalink] New post   11 May 2023, 23:42
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Re: If ap > aq, is a/pq>0? [#permalink]
11 May 2023, 23:42
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Bunuel
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