If r and s are negative, is r/s less than 1 ?

Question

(1) r + 2s = s^2/r
(2) r/s is 2 less than s/r

Bunuel · Answer

If r and s are negative, is  \frac{r}{s}  less than 1 ?

Since r and s are negative then either  0<\frac{r}{s}<1  or  \frac{r}{s}\geq{1}

(1)  r + 2s = \frac{s^2}{r} . Divided both parts by  s :  \frac{r}{s}+2= \frac{s}{r} . We cannot have the second case because if  \frac{r}{s}\geq{1}  then its reciprocal  \frac{s}{r}\leq{1}  and it cannot equal to  \frac{r}{s}+2=positive+2 , hence  0<\frac{r}{s}<1 . Sufficient.

(2)  \frac{r}{s}  is 2 less than  \frac{s}{r}  -->  \frac{r}{s}+2= \frac{s}{r} . The same info as above. Sufficient.

Answer: D.

enigma123 · Answer

Sorry Bunuel - didn't get this buddy.

How did you get first line which says Since r & s are negative ......

Bunuel · Answer

Since r and s are negative r/s>0, and then you can consider two cases: 0=1.

srikumar · Answer

A gives us  r + 2s = s^2/r ,
dividing by r we get -> 1 + 2s/r = (s/r)^2 
call s/r = a,
 1 + 2a = a^2 
 a^2 - 2a = 1 
adding 1 to both sides we get ->  (a -1)^2 = 2 
or a -1 =  +\sqrt{2}  or a - 1 =  -\sqrt{2} 
since r and s are both -ve we know that s/r > 0 so a has to 1 + \sqrt{2}
=>  s/r = 1+\sqrt{2} .
Sufficient.

B give us same statement as A, just substitute s/r = a you will end up with (a -1)^2 = 2 and use the same logic as above. So, B is sufficient

Hence D.

KarishmaB · Answer

Even though it has already been answered by Bunuel, I would like to point out some takeaways.

1. Dont be in a hurry to bring the fractions in integer form. Look at the options and see how variables are used in the options. 
2. If r and s are both negative, it means r/s is positive i.e. > 0
3. If r/s is a positive fraction (non 1), one of r/s and s/r will be between 0 and 1 (the smaller one) and the other will be to the right of 1. (greater than 1)
4. It is possible that the two statements give the same data. It they do and if one stmnt alone is enough, both together will be enough. If one is not enough, neither are they enough together, nor is the other one enough alone.

rohit8865 · Answer

If r and s are negative, is r/s less than 1 ?

(1) r + 2s = s^2/r
(2) r/s is 2 less than s/r

(1) r + 2s - (s^2/r)=0
r^2-s^2=-2rs taking RHS part since r & s are -ve then -2rs must be negative
so r^2-s^2<0
r^2<s^2
taking root both sides IrI<IsI
as both are -ve so IrI=-r & IsI=-s
So, now we can write as -r<-s------> r>s
so r/s is always <1
eg. say r=-2 & s=-3 (as r>s and both are -ve)
now -2/-3 < 1
So suff..

(2) r/s is 2 less than s/r
r/s=s/r-2
rearranging we again get r^2-s^2=-2rs
similar reasoning as (1).
Suff..

Ans D

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If r and s are negative, is r/s less than 1 ?

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