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

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Math Expert
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What is the 101st digit after the decimal point in the decimal represe  [#permalink]

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04 Feb 2019, 03:00
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Difficulty:

75% (hard)

Question Stats:

48% (02:02) correct 52% (02:34) wrong based on 67 sessions

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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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What is the 101st digit after the decimal point in the decimal represe  [#permalink]

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04 Feb 2019, 03:00
1
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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.

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What is the 101st digit after the decimal point in the decimal represe  [#permalink]

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Updated on: 04 Feb 2019, 03:26
1
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]

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04 Feb 2019, 03:24
2
1
1
$$\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{37}$$

$$\frac{4}{9} + \frac{1}{27} + \frac{1}{37}$$

$$\frac{13}{27} + \frac{1}{37}$$

$$\frac{13*37 + 27}{27*37}$$

$$\frac{481+27}{27*37}$$

$$\frac{508}{27*37}$$

$$\frac{508}{999}$$

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]

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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]

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05 Feb 2019, 12:01
1
EMPOWERgmatRichC GMATPrepNow

Is there any short cut to solve the question?

Thanks
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Re: What is the 101st digit after the decimal point in the decimal represe  [#permalink]

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20 Feb 2019, 07:38
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.

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.
Re: What is the 101st digit after the decimal point in the decimal represe   [#permalink] 20 Feb 2019, 07:38
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