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• ### $450 Tuition Credit & Official CAT Packs FREE December 15, 2018 December 15, 2018 10:00 PM PST 11:00 PM PST Get the complete Official GMAT Exam Pack collection worth$100 with the 3 Month Pack ($299) • ### FREE Quant Workshop by e-GMAT! December 16, 2018 December 16, 2018 07:00 AM PST 09:00 AM PST Get personalized insights on how to achieve your Target Quant Score. # M28-36  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Author Message TAGS: ### Hide Tags Math Expert Joined: 02 Sep 2009 Posts: 51218 M28-36 [#permalink] ### Show Tags 16 Sep 2014, 00:31 3 14 00:00 Difficulty: 95% (hard) Question Stats: 42% (01:37) correct 58% (02:15) wrong based on 203 sessions ### HideShow timer Statistics 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. $$8$$ _________________ Math Expert Joined: 02 Sep 2009 Posts: 51218 Re M28-36 [#permalink] ### Show Tags 16 Sep 2014, 00:31 2 3 Official Solution: 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 _________________ Manager Joined: 28 Aug 2013 Posts: 79 Location: India Concentration: Operations, Marketing Schools: Insead '14, ISB '15 GMAT Date: 08-28-2014 GPA: 3.86 WE: Supply Chain Management (Manufacturing) Re: M28-36 [#permalink] ### Show Tags 24 Oct 2014, 02:00 1 Bunuel wrote: Official Solution: 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. $$8$$ $$\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{37}=\frac{333}{999} + \frac{111}{999} + \frac{27}{999} + \frac{37}{999}=\frac{508}{999}=0.508508...$$. 102nd digit will be 8, thus 101st digit will be 0. Answer: A I was trying though a different approach, 1/3 - 0.333333....Thus 101 digit in this sequence will be 3 1/9 - 0.111111....Thus 101 digit in this sequence will be 1 1/27-0.037037....Thus 101 digit in this sequence will be 3 1/37-0.027027....Thus 101 digit in this sequence will be 2 Their sum must be 9... What's wrong with this approach ? Regards LS _________________ G-prep1 540 --> Kaplan 580-->Veritas 640-->MGMAT 590 -->MGMAT 2 640 --> MGMAT 3 640 ---> MGMAT 4 650 -->MGMAT 5 680 -- >GMAT prep 1 570 Give your best shot...rest leave upto Mahadev, he is the extractor of all negativity in the world !! Math Expert Joined: 02 Sep 2009 Posts: 51218 Re: M28-36 [#permalink] ### Show Tags 24 Oct 2014, 02:03 2 lastshot wrote: Bunuel wrote: Official Solution: 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. $$8$$ $$\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{37}=\frac{333}{999} + \frac{111}{999} + \frac{27}{999} + \frac{37}{999}=\frac{508}{999}=0.508508...$$. 102nd digit will be 8, thus 101st digit will be 0. Answer: A I was trying though a different approach, 1/3 - 0.333333....Thus 101 digit in this sequence will be 3 1/9 - 0.111111....Thus 101 digit in this sequence will be 1 1/27-0.037037....Thus 101 digit in this sequence will be 3 1/37-0.027027....Thus 101 digit in this sequence will be 2 Their sum must be 9... What's wrong with this approach ? Regards LS You'd have a carry over 1, from the sum of 102nd digits. _________________ Manager Joined: 28 Aug 2013 Posts: 79 Location: India Concentration: Operations, Marketing Schools: Insead '14, ISB '15 GMAT Date: 08-28-2014 GPA: 3.86 WE: Supply Chain Management (Manufacturing) Re: M28-36 [#permalink] ### Show Tags 24 Oct 2014, 02:05 Bunuel wrote: lastshot wrote: Bunuel wrote: Official Solution: 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. $$8$$ $$\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{37}=\frac{333}{999} + \frac{111}{999} + \frac{27}{999} + \frac{37}{999}=\frac{508}{999}=0.508508...$$. 102nd digit will be 8, thus 101st digit will be 0. Answer: A I was trying though a different approach, 1/3 - 0.333333....Thus 101 digit in this sequence will be 3 1/9 - 0.111111....Thus 101 digit in this sequence will be 1 1/27-0.037037....Thus 101 digit in this sequence will be 3 1/37-0.027027....Thus 101 digit in this sequence will be 2 Their sum must be 9... What's wrong with this approach ? Regards LS You'd have a carry over 1, from the sum of 102nd digits. Thanks ...another thing i learn today !!! Regards LS _________________ G-prep1 540 --> Kaplan 580-->Veritas 640-->MGMAT 590 -->MGMAT 2 640 --> MGMAT 3 640 ---> MGMAT 4 650 -->MGMAT 5 680 -- >GMAT prep 1 570 Give your best shot...rest leave upto Mahadev, he is the extractor of all negativity in the world !! Intern Joined: 04 Apr 2015 Posts: 40 Re: M28-36 [#permalink] ### Show Tags 03 Aug 2015, 09:51 Hi bunuel how do you come up with 333/999+111/999+27/999+37/999 can you explain please.. jimmy Veritas Prep GMAT Instructor Joined: 15 Jul 2015 Posts: 110 GPA: 3.62 WE: Corporate Finance (Consulting) Re: M28-36 [#permalink] ### Show Tags 03 Aug 2015, 10:03 1 1 jimmy02 wrote: Hi bunuel how do you come up with 333/999+111/999+27/999+37/999 can you explain please.. jimmy Jimmy02 - 999 is the common demoninator for 3, 9, 27 and 37. 3, 9 and 27 are all divisible by 3, so 27 would be a common denominator for them. 37 is prime, so the lowest common multiple between 37 and 27 is 37 x 27, or 999. As for the fractions, 333/999 = 1/3; 111/999 = 1/9; 37/999 = 1/27; and 27/999 = 1/37. _________________ Dennis Veritas Prep | GMAT Instructor Get started with Veritas Prep GMAT On Demand for$199

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Intern
Joined: 04 Apr 2015
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03 Aug 2015, 21:21
Hi

yes 999 is common denominator.. I just want to know what if i get different values, how would i deduce them to same expression.

jimmy

VeritasPrepDennis wrote:
jimmy02 wrote:
Hi bunuel

how do you come up with 333/999+111/999+27/999+37/999

jimmy

Jimmy02 -

999 is the common demoninator for 3, 9, 27 and 37. 3, 9 and 27 are all divisible by 3, so 27 would be a common denominator for them. 37 is prime, so the lowest common multiple between 37 and 27 is 37 x 27, or 999. As for the fractions, 333/999 = 1/3; 111/999 = 1/9; 37/999 = 1/27; and 27/999 = 1/37.
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06 Aug 2015, 06:35
1
jimmy02 wrote:
Hi

yes 999 is common denominator.. I just want to know what if i get different values, how would i deduce them to same expression.

jimmy

VeritasPrepDennis wrote:
jimmy02 wrote:
Hi bunuel

how do you come up with 333/999+111/999+27/999+37/999

jimmy

jimmy02 -

I am not sure what you mean. Can you give a little more information?
If 999 is the common denominator, we need to set every fraction up with 999 as the denominator.
Thus 1/3 becomes 333/999 (think of it as a proportion; if I multiply the denominator by 333, I need to multiply the numerator by 333)
1/9 becomes 111/999
1/27 becomes 37/999
and, 1/37 becomes 27/999

Is this what you were asking? Let me know.
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Intern
Joined: 14 Oct 2015
Posts: 31
GMAT 1: 640 Q45 V33

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18 Oct 2015, 06:40
3
3
Three things:
1. This question is the same as M05-04
2. In the explanation, should read 333/999 + 111/999 + 37/999 + 27/999 = 508/999 instead of 333/999 + 111/999 + 27/999 + 37/999 = 508/999 because it implies that 27/999 corresponds to 1/27 and 37/999 corresponds to 1/37 and it is the other way around. Not written incorrectly, as order does not matter, just confusing if you are trying to figure it out.
3. Would be helpful to include in explanation the following: We are dealing with a repeating decimal in this question. It's helpful to know that there's a way to write these kinds of decimals as a fraction. For example, the repeating decimal 0.444444444(4) may be written as 4/9. So, 5/9, 7/9 and 8/9 will all be repeating decimals. You might check it in your calculator. In order to make two decimal points repeat, you have to divide the two digit number by 99. For example, 23/99=0.232323232323(23). In order to make 3 decimal points repeat, you have to divide the three digit number by 999. For example, 508/999=0.508508508508(508)
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Joined: 02 Sep 2009
Posts: 51218

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19 Jan 2016, 09:30
danjbon wrote:
Three things:
1. This question is the same as M05-04
2. In the explanation, should read 333/999 + 111/999 + 37/999 + 27/999 = 508/999 instead of 333/999 + 111/999 + 27/999 + 37/999 = 508/999 because it implies that 27/999 corresponds to 1/27 and 37/999 corresponds to 1/37 and it is the other way around. Not written incorrectly, as order does not matter, just confusing if you are trying to figure it out.

3. Would be helpful to include in explanation the following: We are dealing with a repeating decimal in this question. It's helpful to know that there's a way to write these kinds of decimals as a fraction. For example, the repeating decimal 0.444444444(4) may be written as 4/9. So, 5/9, 7/9 and 8/9 will all be repeating decimals. You might check it in your calculator. In order to make two decimal points repeat, you have to divide the two digit number by 99. For example, 23/99=0.232323232323(23). In order to make 3 decimal points repeat, you have to divide the three digit number by 999. For example, 508/999=0.508508508508(508)

Updated the question.
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18 Aug 2016, 17:11
I think this is a high-quality question and I agree with explanation. good question
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28 Nov 2018, 04:20
Bunuel How did you figure out that 1/37 is the same as 27/999 ?
Intern
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03 Dec 2018, 05:59
(1/3) + (1/9) + (1/27) + (1/37) =(13716/26973)=0.508508.....
Since after the decimal point the number in the third position happens to be 8 and 99th position is a multiple of the 3rd position, it follows that - according to the rule of cyclicity - 5 will take the 100th position, 0 will take 101th position. So A is the answer.
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04 Dec 2018, 23:40
Bunuel wrote:
Official Solution:

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.

I'm a bit confused as to how you converted 508/999 to .508508... etc. Can you just assume that the 999 is close enough to 1,000?

Math Expert
Joined: 02 Sep 2009
Posts: 51218

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05 Dec 2018, 04:25
Bunuel wrote:
Official Solution:

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.

I'm a bit confused as to how you converted 508/999 to .508508... etc. Can you just assume that the 999 is close enough to 1,000?

Converting Decimals to Fractions

• To convert a terminating decimal to fraction:
1. Calculate the total numbers after decimal point
2. Remove the decimal point from the number
3. Put 1 under the denominator and annex it with "0" as many as the total in step 1
4. Reduce the fraction to its lowest terms

Example: Convert $$0.56$$ to a fraction.
1: Total number after decimal point is 2.
2 and 3: $$\frac{56}{100}$$.
4: Reducing it to lowest terms: $$\frac{56}{100}=\frac{14}{25}$$

• To convert a recurring decimal to fraction:
1. Separate the recurring number from the decimal fraction
2. Annex denominator with "9" as many times as the length of the recurring number
3. Reduce the fraction to its lowest terms

Example #1: Convert $$0.393939...$$ to a fraction.
1: The recurring number is $$39$$.
2: $$\frac{39}{99}$$, the number $$39$$ is of length $$2$$ so we have added two nines.
3: Reducing it to lowest terms: $$\frac{39}{99}=\frac{13}{33}$$.

• To convert a mixed-recurring decimal to fraction:
1. Write down the number consisting with non-repeating digits and repeating digits.
2. Subtract non-repeating number from above.
3. Divide 1-2 by the number with 9's and 0's: for every repeating digit write down a 9, and for every non-repeating digit write down a zero after 9's.

Example #2: Convert $$0.2512(12)$$ to a fraction.
1. The number consisting with non-repeating digits and repeating digits is 2512;
2. Subtract 25 (non-repeating number) from above: 2512-25=2487;
3. Divide 2487 by 9900 (two 9's as there are two digits in 12 and 2 zeros as there are two digits in 25): 2487/9900=829/3300.
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05 Dec 2018, 07:53
Bunuel wrote:
Official Solution:

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.

@

Bunuel :
really appreciate your solution to the problem , but can you please debrief on how you decided to go by taking 999 as the common denominator for all , usually in fractions we take the LCM of the terms or common multiplier term... i agree that here its 999 but how did you come up with .. any input would be great..
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Re: M28-36 &nbs [#permalink] 05 Dec 2018, 07:53
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# M28-36

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