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Which of the following is closest to 1/9+1/99+1/999

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by Anurag@Gurome » Mon Jul 09, 2012 10:46 am
gmatter2012 wrote:Which of the following is closest to 1/9+1/99+1/999
Approximation Approach:
1/9 = 0.111...
1/99 ≈ 1/100 = 0.01
1/999 ≈ 1/1000 = 0.001

So, (1/9 + 1/99 + 1/999) ≈ 0.111... + 0.01 + 0.001 = 0.122

Now, 1/8 = 0.125 and 1/6 ≈ 0.166...
Hence, option C is the closest.

The correct answer is C.
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by eagleeye » Mon Jul 09, 2012 3:46 pm
gmatter2012 wrote:Which of the following is closest to 1/9+1/99+1/999

A. 1/10
B. 1/9
C. 1/8
D. 1/6
E. 1/5
Couple of ways of doing this:
1.
1/9+1/99+1/999 =~ 1/999*(111+10+1)= 122/999 ~= 125/1000 = 1/8.

2.
Multiply the expression by 99, we get: 99*(1/9+1/99+1/999) = 11+1+99/999 ~= 12.1

Now check the options, since the expression is definitely greater than 1/9, start with 1/9
99*1/9 = 11
99*1/8 = 12+3/8 = 12.375
99*1/6 = 15+3/6 = 15.5

1/8 gives the closest value to 12.1. Go with 1/8.


Hence C. :)
Last edited by eagleeye on Tue Jul 10, 2012 5:46 pm, edited 1 time in total.
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by gmatter2012 » Tue Jul 10, 2012 1:36 am
Anurag@Gurome wrote:
gmatter2012 wrote:Which of the following is closest to 1/9+1/99+1/999
Approximation Approach:
1/9 = 0.111...
1/99 ≈ 1/100 = 0.01
1/999 ≈ 1/1000 = 0.001

So, (1/9 + 1/99 + 1/999) ≈ 0.111... + 0.01 + 0.001 = 0.122

Now, 1/8 = 0.125 and 1/6 ≈ 0.166...
Hence, option C is the closest.

The correct answer is C.
Ok, I was just wondering why we didn't take 1/9 = 1/10 ( approx) like we took 1/99 = 1/100 approx
and 1/999 = 1/1000 approx

I know its easy to divide 1/9 and to get as close as possible to the actual answer we should approximate as less as possible.

But in case I took 1/9 = 1/10( approx ) then I would get
(1/9 + 1/99 + 1/999) = (1/10 +1/100 +1/1000)approx

= .1+.01+.001= .111

and then the answer turns out to be B = 1/9 = .111

so we cannot approximate 1/9 as 1/10 as that will give a different answer so does that mean we should approximate only when division is impossible

another way to look at this sum approximation wise would be since 1/999 is very small compared to 1/99 we can ignore 1/999 so we have 1/9 +1/99 = .111 + .01( approx) = .121 = 1/8 = option B

please share your thoughts.
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by GMATGuruNY » Tue Jul 10, 2012 2:14 am
gmatter2012 wrote:Which of the following is closest to 1/9+1/99+1/999

A. 1/10
B. 1/9
C. 1/8
D. 1/6
E. 1/5
An alternate approach would be to use the answer choices.

The sum clearly cannot be less than 1/9.
Eliminate A.
The difference between B and C = 1/8 - 1/9 = 1/72.
Since 1/99 is more than 1/2 of this difference -- (1/2)(1/72) = 1/144 -- the sum is closer to 1/8 than to 1/9.
Eliminate B.
Since 1/999 has an relatively insignificant effect on the sum, the sum cannot be as great at 1/6 or 1/5.
Eliminate D and E.

The correct answer is C.
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by gopal.prashad » Tue Jul 10, 2012 5:38 pm
gmatter2012 wrote:
Anurag@Gurome wrote:
gmatter2012 wrote:Which of the following is closest to 1/9+1/99+1/999
Approximation Approach:
1/9 = 0.111...
1/99 ≈ 1/100 = 0.01
1/999 ≈ 1/1000 = 0.001

So, (1/9 + 1/99 + 1/999) ≈ 0.111... + 0.01 + 0.001 = 0.122

Now, 1/8 = 0.125 and 1/6 ≈ 0.166...
Hence, option C is the closest.

The correct answer is C.
Ok, I was just wondering why we didn't take 1/9 = 1/10 ( approx) like we took 1/99 = 1/100 approx
and 1/999 = 1/1000 approx

I know its easy to divide 1/9 and to get as close as possible to the actual answer we should approximate as less as possible.

But in case I took 1/9 = 1/10( approx ) then I would get
(1/9 + 1/99 + 1/999) = (1/10 +1/100 +1/1000)approx

= .1+.01+.001= .111

and then the answer turns out to be B = 1/9 = .111

so we cannot approximate 1/9 as 1/10 as that will give a different answer so does that mean we should approximate only when division is impossible

another way to look at this sum approximation wise would be since 1/999 is very small compared to 1/99 we can ignore 1/999 so we have 1/9 +1/99 = .111 + .01( approx) = .121 = 1/8 = option B

please share your thoughts.
Hi Anurag,
I thought of the same reason for not approximating 1/9 to 1/10 as it will be bigger approximation and will not land us to closest answer.
:wink:
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