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# What is the 21st digit of the decimal [

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Retired Moderator
Joined: 27 Oct 2017
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What is the 21st digit of the decimal [  [#permalink]

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29 Sep 2018, 10:19
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15
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Difficulty:

75% (hard)

Question Stats:

52% (02:23) correct 48% (02:37) wrong based on 93 sessions

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Weekly Quant Quiz Question - 5

What is the 21st digit of the decimal $$\frac{1}{111} + \frac{1}{37} + \frac{1}{27} + \frac{1}{9} + \frac{1}{3}$$ ?
a) 5
b) 6
c) 7
d) 8
e) 9

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Retired Moderator
Joined: 27 Oct 2017
Posts: 1308
Location: India
GPA: 3.64
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What is the 21st digit of the decimal [  [#permalink]

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29 Sep 2018, 22:53
1
1

Official Solution:

$$\frac{1}{111} + \frac{1}{37} + \frac{1}{27} + \frac{1}{9} + \frac{1}{3}$$
= $$\frac{9}{999} + \frac{27}{999} + \frac{37}{999} + \frac{111}{999} + \frac{333}{3999}$$
= $$\frac{517}{999}$$
= 0.517517517...
It is clear that 7 at places 3rd, 6th, 9th... multiple of 3.
Hence 21st digit is 7

Method to convert fraction (with only 9 as digits in denominator) to decimal

• If the fraction is abc/999, decimal will be 0.abcabcabc...
We need to just repeat the numerator indefinitely after the decimal.
• if the fraction is ABC/99999, decimal will be 0.00ABC00ABC
We need to add zeros after zero at the start of numerator and repeat it indefinitely.

gmatbusters wrote:

Weekly Quant Quiz Question - 5

What is the 21st digit of the decimal $$\frac{1}{111} + \frac{1}{37} + \frac{1}{27} + \frac{1}{9} + \frac{1}{3}$$ ?
a) 5
b) 6
c) 7
d) 8
e) 9

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##### General Discussion
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Joined: 18 Oct 2017
Posts: 21
Location: India
Re: What is the 21st digit of the decimal [  [#permalink]

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29 Sep 2018, 12:11
4
1
1/3 = 0.3333(non- terminating)

Thus, 21st digit is 3

1/9 = 0.11111(non-terminating)

Thus, 21st digit is 1

1/27 = 0. 037037

Thus number repeats in sequence of 3 digits - 037, thus 21st digit number is 7

1/37 = 0.027. Thus, same as above, 21st digit is 7

1/111 = 0.009, thus, for same reason as above, 21st digit is 9

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Re: What is the 21st digit of the decimal [  [#permalink]

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29 Sep 2018, 10:34
3
This can be written as -

13/27 + 4/111 = 0.0360360360... + 0.481481.. = 0.517517

Hence, 7
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Re: What is the 21st digit of the decimal [  [#permalink]

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29 Sep 2018, 15:11
2
Expressing each of the fractions in the decimal form:

1/3 = 0.333333333333.... --> 21st digit is 3
1/9 = 0.111111111111... --> 21st digit is 1
1/27 = 0.037037037037...--> 21st digit is 7
1/37 = 0.027027027027.. --> 21st digit is 7
1/111= 0.009009009...... --> 21st digit is 9

9+7+7+1+3 = 27, 7 is the 21st digit and 2 is carried over for sum in 20th digit.

C
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Re: What is the 21st digit of the decimal [  [#permalink]

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29 Sep 2018, 10:35
1
1/111+1/37=4/111=21st and 22nd digit=6,0
1/27+1/9+1/3=21st and 22nd digit 1,4
Director
Joined: 18 Dec 2017
Posts: 819
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What is the 21st digit of the decimal [  [#permalink]

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23 Nov 2019, 12:47
1
gmatbusters wrote:

Official Solution:

$$\frac{1}{111} + \frac{1}{37} + \frac{1}{27} + \frac{1}{9} + \frac{1}{3}$$
= $$\frac{9}{999} + \frac{27}{999} + \frac{37}{999} + \frac{111}{999} + \frac{333}{999}$$
= $$\frac{517}{999}$$
= 0.517517517...
It is clear that 7 at places 3rd, 6th, 9th... multiple of 3.
Hence 21st digit is 7

Method to convert fraction (with only 9 as digits in denominator) to decimal

• If the fraction is abc/999, decimal will be 0.abcabcabc...
We need to just repeat the numerator indefinitely after the decimal.
• if the fraction is ABC/99999, decimal will be 0.00ABC00ABC
We need to add zeros after zero at the start of numerator and repeat it indefinitely.

gmatbusters wrote:

Weekly Quant Quiz Question - 5

What is the 21st digit of the decimal $$\frac{1}{111} + \frac{1}{37} + \frac{1}{27} + \frac{1}{9} + \frac{1}{3}$$ ?
a) 5
b) 6
c) 7
d) 8
e) 9

gmatbusters

I guess there is a typo. The last fraction should be 333/999=1/3.

It could be confusing so if you can edit that would be great. Thank you!
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Re: What is the 21st digit of the decimal [  [#permalink]

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29 Sep 2018, 10:27
1/111+1/37+1/27+1/9+1/3= .009 + .027+.037+.111+.333 = 0.xx7 as the 3rd is 7 the 21st is also 7 ==> Ans C
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Re: What is the 21st digit of the decimal [   [#permalink] 29 Sep 2018, 10:27
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