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# Is rs/(r + s) > 1/2 ?

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Joined: 11 May 2008
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Is rs/(r + s) > 1/2 ?  [#permalink]

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01 Sep 2008, 04:24
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Difficulty:

55% (hard)

Question Stats:

66% (02:06) correct 34% (01:35) wrong based on 116 sessions

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Is $$\frac{rs}{r + s} \gt \frac{1}{2}$$ ?

(1) $$r \gt 1$$ and $$s \gt 1$$
(2) $$r \gt 2$$ and $$s \gt 0$$
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Joined: 05 Jul 2008
Posts: 1176
Re: Is rs/(r + s) > 1/2 ?  [#permalink]

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01 Sep 2008, 13:56
Why is A incorrect? Did I miss any values?

1. r \gt 1 and s \gt 1

r=2 s=2 Y
r=2 s=3 Y
r=3 s=5 Y
r=2.5 s=3.5 value = 35/24 >1/2 Y
r=1.5 s=1.5 value = 3/4 > 1/2 Y

2. r \gt 2 and s \gt 0

r=3 s=1 Y
r=4 s=1 Y
r=3 s=3 Y
r=2.5 s=1.5 Value =15/16 Y
r=3.5 s=0.5 value = 7/16 N
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Re: Is rs/(r + s) > 1/2 ?  [#permalink]

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01 Sep 2008, 14:25
arjtryarjtry wrote:
Is $$\frac{rs}{r + s} \gt \frac{1}{2}$$ ?

1. $$r \gt 1$$ and $$s \gt 1$$
2. $$r \gt 2$$ and $$s \gt 0$$

rewrite the question stem
1/r +1/s<2

1) r>1 x>1
1/r +1/s <2 True
Suffcieint

2) r>2 and s>0

1/r +1/s can be >2 or <2

insuffcient

Will go with A.
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Re: Is rs/(r + s) > 1/2 ?  [#permalink]

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01 Sep 2008, 18:10
sorry guys.... the ans is A. i got confused with soem other question. -1 to me...
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Re: Is rs/(r + s) > 1/2 ?  [#permalink]

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29 Apr 2010, 10:17
Is rs/(r+s)>1/2 ?

1. r>1 and s>1
2. r>2 and s>0

Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient
Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
EACH statement ALONE is sufficient
Statements (1) and (2) TOGETHER are NOT sufficient
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Re: Is rs/(r + s) > 1/2 ?  [#permalink]

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29 Apr 2010, 10:58
ykaiim wrote:
Is rs/(r+s)>1/2 ?

1. r>1 and s>1
2. r>2 and s>0

Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient
Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
EACH statement ALONE is sufficient
Statements (1) and (2) TOGETHER are NOT sufficient

IMHO D

Statement 2:
Is rs/(r+s)>1/2 >>>> Is rs/(r+s)-1/2 >0 >>>> Is (2*r*s-(r+s))/(r+s) >0
Now given r and s both are positive...so r+s is positive >>> so denominator is positive..
lets have a look at numerator

so the question becomes Is (2*r*s-(r+s)) >0
Is 2rs + (r-s) > 0
Now |2rs| will be greater than |s-r|, So 2rs + (r-s) > 0. Sufficient

Statement 1:
Same way as mentioned above, this statement is sufficient.
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Re: Is rs/(r + s) > 1/2 ?  [#permalink]

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29 Apr 2010, 11:09
Try this:
S2. If r=2.1 and y=0.1, then
rs=0.21 and r+s=2.2
So, rs/(r+s)=0.21/2.2 < 1/2

What if r=4 and y=8, then
rs=32 and r+s=12
So, rs/(r+s)=32/12 > 1/2

So, Insuff.

Well, I just figured out that for r>1 and s>1 this expression holds true.

nverma wrote:
ykaiim wrote:
Is rs/(r+s)>1/2 ?

1. r>1 and s>1
2. r>2 and s>0

Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient
Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
EACH statement ALONE is sufficient
Statements (1) and (2) TOGETHER are NOT sufficient

IMHO D

Statement 2:
Is rs/(r+s)>1/2 >>>> Is rs/(r+s)-1/2 >0 >>>> Is (2*r*s-(r+s))/(r+s) >0
Now given r and s both are positive...so r+s is positive >>> so denominator is positive..
lets have a look at numerator

so the question becomes Is (2*r*s-(r+s)) >0
Is 2rs + (r-s) > 0
Now |2rs| will be greater than |s-r|, So 2rs + (r-s) > 0. Sufficient

Statement 1:
Same way as mentioned above, this statement is sufficient.

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Re: Is rs/(r + s) > 1/2 ?  [#permalink]

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29 Apr 2010, 11:11
IMO A

stmnt1 for r>1 and s>1 expression rs/(r+s) will be greater than 1/2. hence suff

stmnt2 is insuff

r=3 s = 0.1 then rs = 0.3 and r+s = 3.1
=> 0.3/3.1 <1/2

r= 3 and s = 10 then rs = 30 and r+s = 13
=> 30/13 >1/2 hence insuff
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Re: Is rs/(r + s) > 1/2 ?  [#permalink]

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04 May 2010, 13:46
kp1811 wrote:
IMO A

stmnt1 for r>1 and s>1 expression rs/(r+s) will be greater than 1/2. hence suff

stmnt2 is insuff

r=3 s = 0.1 then rs = 0.3 and r+s = 3.1
=> 0.3/3.1 <1/2

r= 3 and s = 10 then rs = 30 and r+s = 13
=> 30/13 >1/2 hence insuff

I agree with the above explanation and OA.

ykaiim - Can you confirm what is the OA for this question please?
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Re: Is rs/(r + s) > 1/2 ?  [#permalink]

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07 May 2010, 22:38
A.

St. 1 : the value of $$\frac{rs}{r+s}$$ is always > 1/2 --> suff
St. 2 : For some values of r and s, the fraction is < 1/2 and for some > 1/2 --> insuff
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Re: Is rs/(r + s) > 1/2 ?  [#permalink]

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16 May 2010, 15:31
imo A
what is the OA?
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Re: Is rs/(r + s) > 1/2 ?  [#permalink]

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12 May 2011, 14:03
Is (rs)/(r+s) > 1/2?
1. r>1 and s>1
2. r>2 and s>0

In the explanation, I don't understand how (rs)/(r+s) simplifies to 1/[(1/r) + (1/s)]
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Re: Is rs/(r + s) > 1/2 ?  [#permalink]

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12 May 2011, 17:00
Looking at it from rveverse way :

1/[(1/r) + (1/s)] = 1/[(r+s)/rs] = rs/(r+s).

Or you may divide numerator and denominator by rs:

=> rs/rs/(r+s)/rs = 1/(1/r + 1/s)

HTH.
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Re: Is rs/(r + s) > 1/2 ?  [#permalink]

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13 May 2011, 02:13
a r=s=2 gives LHS = 1 > RHS = 1/2
r=1.1=s gives LHS > RHS
a is sufficient.

b r=3 s=0.1 gives LHS = 0.3 < RHS

r=3 s=3 gives LHS > RHS.

Hence A is clean

rs/r+s = 1/[(r/rs)+s/(r+s)] = 1/[1/s+1/r]

means 1/[1/s+1/r] >1/2 means 1/s+ 1/r < 2 taking reciprocals

a for positive values of s and r >1 will give LHS < RHS always.
b for s=3 and r = 0.1 will give RHS <LHS.

a is clean
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Re: Is rs/(r + s) > 1/2 ?  [#permalink]

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13 May 2011, 05:53
Janakan wrote:
Is (rs)/(r+s) > 1/2?
1. r>1 and s>1
2. r>2 and s>0

Stmt1:r>1 and s>1. Adding, r+s>2. also, rs > 1. Since (r+s) and (rs) both are +ve, dividing rs by r+s
rs/r+s > 1/2 Sufficient.

Stmt2: r>2 and s>0. Adding, r+s>2. also, rs>0. hence rs/r+s > 0/2. or rs/r+s > 0 but can be greater than 1/2 or less than 1/2. Insufficient.

OA A.
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Re: Is rs/(r + s) > 1/2 ?  [#permalink]

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29 Aug 2011, 06:55
Is $$\frac{rs}{r + s} \gt \frac{1}{2}$$ ?

1. $$r \gt 1$$ and $$s \gt 1$$
2. $$r \gt 2$$ and $$s \gt 0$$

Hey all, can solve it by plugging in numbers. Would be happy to find an easy mathematical way if there is one.

thanks.
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Re: Is rs/(r + s) > 1/2 ?  [#permalink]

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17 Oct 2017, 12:03
x2suresh wrote:
arjtryarjtry wrote:
Is $$\frac{rs}{r + s} \gt \frac{1}{2}$$ ?

1. $$r \gt 1$$ and $$s \gt 1$$
2. $$r \gt 2$$ and $$s \gt 0$$

rewrite the question stem
1/r +1/s<2

1) r>1 x>1
1/r +1/s <2 True
Suffcieint

2) r>2 and s>0

1/r +1/s can be >2 or <2

insuffcient

Will go with A.

Why do you switch the inequality sign when you take the reciprocals in your work on statement 1?
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Re: Is rs/(r + s) > 1/2 ?  [#permalink]

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Re: Is rs/(r + s) > 1/2 ?   [#permalink] 22 Dec 2018, 19:09
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