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M05 #04 New Question but from Non-Adaptive GMAT Club Test

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M05 #04 New Question but from Non-Adaptive GMAT Club Test [#permalink] New post   12 Sep 2013, 13:56
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

Official Explanation:

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

102nd digit will be 8, thus 101st digit will be 0.
The correct answer is A.

It took me a while to see that 1/27 is equal to 37/999 and that 1/37 is equal to 27/999. What clue in the question indicates that you need to make that conversion? What is the shortcut for that conversion?

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Re: M05 #04 New Question but from Non-Adaptive GMAT Club Test [#permalink] New post   12 Sep 2013, 14:00
Expert Reply
Samwong 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

Official Explanation:

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

102nd digit will be 8, thus 101st digit will be 0.
The correct answer is A.

It took me a while to see that 1/27 is equal to 37/999 and that 1/37 is equal to 27/999. What clue in the question indicates that you need to make that conversion? What is the shortcut for that conversion?



\(\frac{1}{3} =\frac{1*333}{3*333}=\frac{27}{999}\).

\(\frac{1}{9} =\frac{1*111}{9*111}=\frac{27}{999}\).

\(\frac{1}{27} =\frac{1*37}{27*37}=\frac{27}{999}\).

\(\frac{1}{37} =\frac{1*27}{37*27}=\frac{27}{999}\).

Following link might help for this problem: math-number-theory-88376.html (check Converting Fractions chapter).
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Re: M05 #04 New Question but from Non-Adaptive GMAT Club Test [#permalink] New post   12 Sep 2013, 15:14
Thanks Bunuel for the quick response and link. You are awesome!
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Re: M05 #04 New Question but from Non-Adaptive GMAT Club Test [#permalink] New post   20 Dec 2013, 07:21
I do not understand what you mean by "101st digit after the decimal point" (terminology). If I follow the logic the 103rd digit would be 5. However what didn't you ask what is the "hundredths" digit? Are they equivalent terminology?
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Re: M05 #04 New Question but from Non-Adaptive GMAT Club Test [#permalink] New post   20 Dec 2013, 08:12
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Sleed wrote:
I do not understand what you mean by "101st digit after the decimal point" (terminology). If I follow the logic the 103rd digit would be 5. However what didn't you ask what is the "hundredths" digit? Are they equivalent terminology?


Consider number 0.xxxxxxxx....

101st digit after the decimal point is 101st x, while the hundredths digit is second x.

Hope it's clear.

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Re: M05 #04 New Question but from Non-Adaptive GMAT Club Test [#permalink]
20 Dec 2013, 08:12
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Bunuel
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