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What is the 101st digit after the decimal point in the decimal represe

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Bunuel
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What is the 101st digit after the decimal point in the decimal represe [#permalink] New post   04 Feb 2019, 03:00
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What is the 101st digit after the decimal point in the decimal representation of \(\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{37}\)?

A. 0
B. 1
C. 5
D. 7
E. 8

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eswarchethu135
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Re: What is the 101st digit after the decimal point in the decimal represe [#permalink] New post   04 Feb 2019, 03:24
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\(\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{37}\)

\(\frac{4}{9} + \frac{1}{27} + \frac{1}{37}\)

\(\frac{13}{27} + \frac{1}{37}\)

\(\frac{13*37 + 27}{27*37}\)

\(\frac{481+27}{27*37}\)

\(\frac{508}{27*37}\)

\(\frac{508}{999}\)

If the denominator had been 1000 then \(\frac{508}{1000}\) is 0.508. As the denominator is 999 it will be a recurring decimal of 508.

\(\frac{508}{999}\) = 0.508508508508.....

Now the 101st digit is -> we have 33 pairs of 508 in this decimal, which gives 99 digits. Now the 100th digit is 5 and the 101st digit is 0.

OPTION: A
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Re: What is the 101st digit after the decimal point in the decimal represe [#permalink] New post   04 Feb 2019, 03:00
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Bunuel wrote:
What is the 101st digit after the decimal point in the decimal representation of \(\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{37}\)?

A. 0
B. 1
C. 5
D. 7
E. 8



\(\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{37}=\frac{333}{999} + \frac{111}{999} + \frac{37}{999} + \frac{27}{999}=\frac{508}{999}=0.508508...\).

102nd digit will be 8, thus 101st digit will be 0.

Answer: A.
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Re: What is the 101st digit after the decimal point in the decimal represe [#permalink] New post   Updated on: 04 Feb 2019, 03:26
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Bunuel wrote:
What is the 101st digit after the decimal point in the decimal representation of \(\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{37}\)?

A. 0
B. 1
C. 5
D. 7
E. 8



1/3= .333
1/9=.1111
1/27=.0370
1/37=.0270
sum = 0.508
101st digit =
0
IMO A

Originally posted by Archit3110 on 04 Feb 2019, 03:17.
Last edited by Archit3110 on 04 Feb 2019, 03:26, edited 1 time in total.
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Re: What is the 101st digit after the decimal point in the decimal represe [#permalink] New post   04 Feb 2019, 14:43
0.508 is the recurring value. Hence, 101st digit is 0.
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Re: What is the 101st digit after the decimal point in the decimal represe [#permalink] New post   05 Feb 2019, 12:01
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EMPOWERgmatRichC GMATPrepNow

Is there any short cut to solve the question?

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Re: What is the 101st digit after the decimal point in the decimal represe [#permalink] New post   20 Feb 2019, 07:38
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Bunuel wrote:
Bunuel wrote:
What is the 101st digit after the decimal point in the decimal representation of \(\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{37}\)?

A. 0
B. 1
C. 5
D. 7
E. 8



\(\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{37}=\frac{333}{999} + \frac{111}{999} + \frac{37}{999} + \frac{27}{999}=\frac{508}{999}=0.508508...\).

102nd digit will be 8, thus 101st digit will be 0.

Answer: A.


Seems to me like a good bail question. It takes time to reach \(\frac{508}{999}\), and then one needs to figure out that it equals 0.508508... AND THEN, finally, figure out the logics with digits.
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Re: What is the 101st digit after the decimal point in the decimal represe [#permalink] New post   19 Feb 2020, 15:05
The top answer in this will teach you how to do these problems. They are very simple to do once you know how to convert these numbers into fractions of x/99 or x/999.


https://gmatclub.com/forum/what-is-the-21st-digit-of-the-decimal-277573.html?fl=similar
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Re: What is the 101st digit after the decimal point in the decimal represe [#permalink] New post   19 Oct 2023, 04:01
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Re: What is the 101st digit after the decimal point in the decimal represe [#permalink]
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