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

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If ap > aq, is a/pq>0?  [#permalink]

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Updated on: 22 May 2017, 23:59
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Question Stats:

24% (02:35) correct 76% (02:33) wrong based on 96 sessions

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If $$ap > aq$$, is $$\frac{a}{pq}>0$$?

(1) $$p < q$$

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

Originally posted by niteshwaghray on 22 May 2017, 23:51.
Last edited by Bunuel on 22 May 2017, 23:59, edited 1 time in total.
Edited the question.
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If ap > aq, is a/pq>0?  [#permalink]

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23 May 2017, 00:12
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.

Hope it's clear.
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Re: If ap > aq, is a/pq>0?  [#permalink]

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28 Jun 2017, 07:02
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.

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

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Re: If ap > aq, is a/pq>0?  [#permalink]

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28 Jun 2017, 08:53
1
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.

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

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]

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Updated on: 20 Aug 2019, 04:00
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 $$\frac{a=positive}{pq=negative}<0$$

(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 $$a>0$$: $$\frac{a=positive}{pq=positive}>0$$

Originally posted by exc4libur on 20 Aug 2019, 03:21.
Last edited by exc4libur on 20 Aug 2019, 04:00, edited 1 time in total.
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Re: If ap > aq, is a/pq>0?  [#permalink]

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20 Aug 2019, 03:51
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] 20 Aug 2019, 03:51