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# M13 Q8

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Senior Manager
Joined: 28 Dec 2010
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09 Jun 2012, 22:00
Is $$\frac{rs}{r + s} \gt \frac{1}{2}$$ ?

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

Can some one pls explain the ans to this question?
Senior Manager
Joined: 28 Dec 2010
Posts: 334
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Kudos [?]: 152 [0], given: 33

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09 Jun 2012, 22:02
When i try the nos r=2, s= 5, the fraction becomes 10/7 which is less than 1/2! so how is the OA justified?
Intern
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10 Jun 2012, 02:12
rs/(r+s) >1/2

(r+s)/rs < 2

(1/r) + (1/s) < 2

Statement-1

r>1 implies (1/r)<1
s>1 implies (1/s)<1

so, (1/r)+(1/s)<2
statement-1 is sufficient.

Whereas in statement-2,
s>0 , so if 's' lies between 0 and (1/2) then (1/r)+(1/s) would be greater than 2, otherwise less than 2.
so statement-2 is not sufficient.
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10 Jun 2012, 02:24
vibhav wrote:
Is $$\frac{rs}{r + s} \gt \frac{1}{2}$$ ?

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

Can some one pls explain the ans to this question?

Given the statements they both have one thing in common, that both r and s are positive that is > 0

Let us rephrase the question Is $$\frac{rs}{r + s} \gt \frac{1}{2}$$

to 2 ( rs) > (r + s)
=> 2 > (r+s) / rs
=> 2 > 1/s + 1/r

Statement 1:
If r=s=1
Then 1/r = 1/s = 1
or 1/r + 1/s = 2

But as r and s >1
we get 1/r <1
adding both we get 1/r + 1/s < 2...........Sufficient.

Statement 2
r > 2
1/r < 1/2

s > 0
1/s can be anything > 0
for eg s = 1/100
1/s = 100 .................1/r + 1/s is more than 2. NO

s = 2
1/s = 1/2

1/r + 1/s < 2 ...Yes

Statement 2 is Insuff.

Ans A.
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10 Jun 2012, 02:54
1
KUDOS
Expert's post
Is $$\frac{rs}{r+s}>\frac{1}{2}$$ ?

(1) $$r>1$$ and $$s>1$$ --> since both variables are positive we can rewrite the question as follows: is $$\frac{r+s}{rs}<2$$? --> is $$\frac{1}{s}+\frac{1}{r}<2$$? Since given that $$r>1$$ and $$s>1$$ then both $$\frac{1}{s}$$ and $$\frac{1}{r}$$ are less than 1 therefore their sum is definitely less than 2. Sufficient.

(2) $$r>2$$ and $$s>0$$ --> the same here: is $$\frac{1}{s}+\frac{1}{r}<2$$? If $$r=3$$ and $$s=1$$ then the answer is YES but if $$r=3$$ and $$s=0.1$$ then the answer is NO. Not sufficient.

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11 Jun 2012, 07:04
A: rs/(r+s) > (1*1)/(1+1) = 1/2

B) rs/ (r+s) > (2*0)/(2+0) = 0, could be <1/2
Re: M13 Q8   [#permalink] 11 Jun 2012, 07:04
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# M13 Q8

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