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# M13-08

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Math Expert
Joined: 02 Sep 2009
Posts: 49915

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16 Sep 2014, 00:48
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Difficulty:

65% (hard)

Question Stats:

51% (01:11) correct 49% (01:20) wrong based on 94 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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Math Expert
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16 Sep 2014, 00:48
Official Solution:

(1) $$r \gt 1$$ and $$s \gt 1$$. Since both variables are positive, we can rewrite the question as follows: is $$\frac{r+s}{rs} \lt 2$$? Or: is $$\frac{1}{s}+\frac{1}{r} \lt 2$$? Since it is given that $$r \gt 1$$ and $$s \gt 1$$, then both $$\frac{1}{s}$$ and $$\frac{1}{r}$$ are less than 1. Therefore, their sum is definitely less than 2. Sufficient.

(2) $$r \gt 2$$ and $$s \gt 0$$. The same here: is $$\frac{1}{s}+\frac{1}{r} \lt 2$$? 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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Joined: 01 Jun 2016
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GMAT 1: 600 Q42 V31
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11 Sep 2016, 17:26
Why you wrote down (r+s/rs)<2? I couldn't understand the logic used.
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Joined: 01 Aug 2016
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Location: United States (AZ)
GMAT 1: 720 Q47 V41
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30 Sep 2016, 15:23
mounirbr wrote:
Why you wrote down (r+s/rs)<2? I couldn't understand the logic used.

I am very certain that was a typo. He meant for 2 to be (1/2)
Math Expert
Joined: 02 Aug 2009
Posts: 6956

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30 Sep 2016, 19:45
1
ddb123 wrote:
mounirbr wrote:
Why you wrote down (r+s/rs)<2? I couldn't understand the logic used.

I am very certain that was a typo. He meant for 2 to be (1/2)

Hi,
It isn't a typo....
The original equation was rs/(r+s)>1/2....
Since both r and s are > than 1, they are positive..
So, you can cross multiply..
2*rs>1*(r+s)....
2/1>(r+s)/rs ....
(r+s)/rs<2...

Hope it is clear
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Joined: 01 Aug 2016
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Location: United States (AZ)
GMAT 1: 720 Q47 V41
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30 Sep 2016, 20:17
chetan2u wrote:
ddb123 wrote:
mounirbr wrote:
Why you wrote down (r+s/rs)<2? I couldn't understand the logic used.

I am very certain that was a typo. He meant for 2 to be (1/2)

Hi,
It isn't a typo....
The original equation was rs/(r+s)>1/2....
Since both r and s are > than 1, they are positive..
So, you can cross multiply..
2*rs>1*(r+s)....
2/1>(r+s)/rs ....
(r+s)/rs<2...

Hope it is clear

Thanks for the clarification! I obviously wasn't looking at the information closely enough. Cheers.
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Posts: 9
Location: India
Schools: IIMC
GMAT 1: 560 Q46 V21
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WE: Operations (Manufacturing)

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03 Oct 2018, 17:26
let simplify the equation

2rs>r+s
rs-r>s-rs
r(s-1)>s(1-r)
r/s<(r-1)/(s-1)

stat-1

if r=3 and s=4 .... not true

but if r= 99 and s=2 ..... true

So can some one expain how the statement one itself sufficient.

Error detection will be appriciated on my expaination.......
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Joined: 02 Sep 2009
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03 Oct 2018, 21:17
rinkumaa4 wrote:
let simplify the equation

2rs>r+s
rs-r>s-rs
r(s-1)>s(1-r)
r/s<(r-1)/(s-1)

stat-1

if r=3 and s=4 .... not true

but if r= 99 and s=2 ..... true

So can some one expain how the statement one itself sufficient.

Error detection will be appriciated on my expaination.......

What is the point of algebraic manipulations you've done?

$$\frac{rs}{r + s} \gt \frac{1}{2}$$ is true for r=3 and s=4 as well as for r= 99 and s=2.
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Re: M13-08 &nbs [#permalink] 03 Oct 2018, 21:17
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# M13-08

Moderators: chetan2u, Bunuel

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