If p and q are negative, is p / q > 1

Given: \(p<0\) and \(q<0\). Question: is \(\frac{p}{q}>1\) --> multiply both sides by \(q\) and as it's negative flip the sign: is \(p<q\)? or is \(p-q<0\)?

(1) The positive diff erence between p and q is 2 --> \(|p-q|=2\): either \(p-q>0\) (answer NO) and \(p-q=2\) or \(p-q<0\) (answer YES) and \(p-q=-2\). Not sufficient.

(2) q - p < 1 (\(p-q>-1\)) --> if \(q=-1\) and \(p=-1\) then the answer will be NO but if \(q=-1\) and \(p=-1.5\) then the answer will be YES. Not sufficient.

(1)+(2) As from (2) \(p-q>-1\) then from (1) \(p-q=2\) so \(p-q>0\) and we have the answer NO. Sufficient.

Answer: C.
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I'm very confused. First why do you have absolute value? How did you derive p-q>0? p-q<0? p-q=-2? Any way to demonstrate? or explain the concepts? Thank you very much.



How can q and p both equal -1?

[/quote]

C

bookmarking

given p<0 & q<0

find
p/q>1 or p < q

(A) The positive diff erence between p and q is 2.

there |p-q| = 2
so we have

if (p-q) = 2 then p > q

(p-q) = -2 then q > p

datanot sufficient

(B) q - p < 1 or p-q > -1 (multiply both sides by -ve and flip the sign)

if(p>q)
p = -0.5
q = -1

then p-q > -1

if(p<q)
p = -1
q = -0.5

then p-q > -1

datanot sufficient

(C)

only possibility
if(p>q)
p = -0.5 / -3
q = -1 / -5
then p-q > -1
also if (p-q) = 2 then p > q


opposite not true
if(p<q)
p = -5
q = -3

(p-q) = -2 then q > p but p-q not > -1

then p-q > -1 [not possible]

st. (1) the +ve difference = |p-q| = 2
implies p-q > 0 or p-q < 0
if p-q > 0 then
p>q then p/q> 1

But if p-q < 0 then p<q or p/q cannot be greater than 1
anyway st. (1) gives two options which leads "insufficient"

st. (2) q-p<1 (this could be p-q < -1 which mean p-q> -1 )
implies q-p = 0 or q-p is -ve
if q-p=0then p/q > 1 is not possible

But if q-p is -ve then it gives different values of p and q which says both -
p/q>1 or p/q<1
however st.(2) insufficient

Combining together st. (1) and st. (2)
p-q > -1 and p-q =2 implies
p>q or we can say p/q>1
Sufficient

hope i got it correct frm bunnel and karishma

Luckily I went for the diagram and was able to do it in under 2 minutes.

Draw a line with 0 in the middle. p and q are both to the left of 0. We only know this much. We don't know their position with respect to 0, i. e we don't know whether q or p is closer to 0 or even whether p and q are the same number, both negative. Question asks whether p/q>1 meaning is p more negative than q? This suggests that the question is about the position of p and q with respect to each other and 0.

1. the positive difference between them is 2 suggests that the distance between p and q is 2 units. this only tells us the distance and not which one is more negative than the other. Not Sufficient.
2. q-p<1 suggests that the difference between the two is less than 1. but their degree of negativity is not clear. by itself, this statement points to the possibility that q and p may be the same number; -2-(-2)=0<1, or one of them could be slightly more negative and still have satisfy q-p<1. so Not Sufficient.

when you take 1 and 2 together, the possibility that the two are the same number is eliminated because 1 says that the two numbers are 2 units apart. so now, the number line will have p and q standing at 2 units apart and based on statement 2, q has to be more negative than p. Hence C.

As @grumpytesttaker said the best way to solve such problems is to use the number line.

The question stem says that p and q are negative so we can have 2 scenarios (p to the left of q or p to the right of q)



Now to definitively say whether p/q>1 we need to find if |p| > |q| as both are negative so there is no question of signs. Since both are negative nos |p| > |q| only if p is to the left of q on the Number Line. So we just need to find if p is to the left or right of q.

(1) |p - q| = 2, this means p and q have a separation of 2. But this is possible if p is to the left of q or p is to the right of q. So this statement doesn't help us. Not Sufficient.



(2) q - p < 1, Since both nos are negative we can rewrite this statement as |p| - |q| < 1. Now if p is to the left of q (|p| > |q|) then the separation between p and q have to be less than 1.
But if p is to the right of q (|p| < |q|) then the separation can be anything. Since this statement doesn't say if p is to the left or right of q, it is Not Sufficient.



(1) & (2) Now if we combine the 2 statements we can see that p cannot be to the left of q because (1) -- |p| - |q| = 2 and (2) -- |p| - |q| < 1 together is not possible.
So the only possibility is p is to the right of q, which answers the question, since |p| < |q| hence p/q < 1



So (1) & (2) put together answers the question. Sufficient. Answer C
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