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

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Manager
Joined: 12 Sep 2010
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M05 #04 New Question but from Non-Adaptive GMAT Club Test [#permalink]  12 Sep 2013, 12:56
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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

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.

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?
Math Expert
Joined: 02 Sep 2009
Posts: 29679
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Kudos [?]: 53183 [0], given: 8043

Re: M05 #04 New Question but from Non-Adaptive GMAT Club Test [#permalink]  12 Sep 2013, 13:00
Expert's post
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.

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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Manager
Joined: 12 Sep 2010
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Re: M05 #04 New Question but from Non-Adaptive GMAT Club Test [#permalink]  12 Sep 2013, 14:14
Thanks Bunuel for the quick response and link. You are awesome!
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Status: Preparing for the GMAT
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Re: M05 #04 New Question but from Non-Adaptive GMAT Club Test [#permalink]  20 Dec 2013, 06:21
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?
Math Expert
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Posts: 29679
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Kudos [?]: 53183 [1] , given: 8043

Re: M05 #04 New Question but from Non-Adaptive GMAT Club Test [#permalink]  20 Dec 2013, 07:12
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Expert's post
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, 07:12
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M05 #04 New Question but from Non-Adaptive GMAT Club Test

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