Author 
Message 
TAGS:

Hide Tags

Intern
Joined: 24 Jan 2012
Posts: 9

The decimal d is formed by writing in succession all the pos
[#permalink]
Show Tags
Updated on: 24 Sep 2013, 11:30
Question Stats:
67% (01:56) correct 33% (01:50) wrong based on 650 sessions
HideShow timer Statistics
The decimal d is formed by writing in succession all the positive integers in increasing order after the decimal point; that is d = 0.123456789101112 What is the 100th digit of d to the right of decimal point ? a) 0 b) 1 c) 5 d) 8 e) 9 I got the answer of this question after writing the 100th term in the series  although it was correct but it was time consuming. Not able to figure out the pattern. Can someone please help ?
Official Answer and Stats are available only to registered users. Register/ Login.
Originally posted by violetsplash on 24 Sep 2013, 08:35.
Last edited by Narenn on 24 Sep 2013, 11:30, edited 1 time in total.
Pls use the question stem to name the topic




MBA Section Director
Affiliations: GMAT Club
Joined: 22 Feb 2012
Posts: 7321
City: Pune

Re: 100th digit in series
[#permalink]
Show Tags
24 Sep 2013, 08:47
Here is comparatively simpler way.... The number is 0.123456789101112......... After Decimal digit 1  9  9 Digits 10  19  20 digits 20  29  20 digits 30  39  20 digits 40  49  20 digits 50  54  10 digit 5 and 4 are the 98th and 99th digits respectively hence 100th will be 5 Alternate ApproachAssume there are 100 blank spaces to fill. First 9 places will be filled by first 9 single digit numbers i.e. 1 to 9. We are now left with 91 places which will be filled by 2 digit numbers starting from 10. TO fill 91 places we would need 91/2 = 45 numbers starting from 10. which will end on the number 54. Next number, for filling out 91th place, will be 5 from (55). Hope that helps! Narenn
_________________
2020 MBA Applicants: Introduce Yourself Here!
MBA Video Series  Video answers to specific components and questions about MBA applications.
2020 MBA Deadlines, Essay Questions and Analysis of all top MBA programs




Manager
Joined: 08 Feb 2014
Posts: 200
Location: United States
Concentration: Finance
WE: Analyst (Commercial Banking)

Re: The decimal d is formed by writing in succession all the pos
[#permalink]
Show Tags
03 Dec 2014, 17:53
They ask for the 100th term. Clearly they don't expect us to write out the next 100 terms, thus there must be a shortcut.
first write out the first ten terms (careful, not the first ten complete numbers):
1 2 3 4 5 6 7 8 9 1
next, write out the next ten numbers, and so on:
1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 3...
The 100th term will be divisible by ten, and thus will be in the farthest column to the right. Notice that this column seems to repeat each digit once, before increasing by 1 (we have a 1, followed by a 1, then a 2, which is followed by a 2, then a 3...)
The 100th term will be the tenth term in the last column, so with the pattern above in mind, write out the the first ten digits: 1122334455. Thus the answer is 5.




SVP
Joined: 06 Sep 2013
Posts: 1545
Concentration: Finance

Re: The decimal d is formed by writing in succession all the pos
[#permalink]
Show Tags
24 Feb 2014, 14:32
violetsplash wrote: The decimal d is formed by writing in succession all the positive integers in increasing order after the decimal point; that is d = 0.123456789101112
What is the 100th digit of d to the right of decimal point ?
a) 0 b) 1 c) 5 d) 8 e) 9
I got the answer of this question after writing the 100th term in the series  although it was correct but it was time consuming. Not able to figure out the pattern. Can someone please help ? Good problem + 1 Now, let's see first we have those single digit numbers that occupy 9 spaces Then the next integers from 10 to 19 will be 10 of them that will occupy 2 places each hence 20 The next ones will be from 20 to 29, 10 of them again, 2 places each hence 20 and so on ... Until we get to 5059 which will be 10 integers again, 2 places each hence 20 digits. Now the 100th digit will be in the 5059 range taking into consideration the first 9 single digit integers. We also know that the tenth digits is always in an even place while the units digit is in the odd place. Therefore, the 100th digit will be the tenth digit of the given range (5059) thus 5 Answer is C Hope its clear Cheers J



SVP
Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1727
Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)

Re: The decimal d is formed by writing in succession all the pos
[#permalink]
Show Tags
04 Mar 2014, 22:33
0.123456789101112......... We require to find the 100th digit from decimal Looking at the series carefully, 10 th digit is 1, 12th digit is 1 & so on...... means 100th digit (even) will be in the tenth place of that number......
We require to find Tenth place of the two digit number; 100th term would be the digit near to 5 ... Answer = C



Intern
Joined: 17 May 2012
Posts: 33

Re: The decimal d is formed by writing in succession all the pos
[#permalink]
Show Tags
19 Nov 2014, 23:49
Hi,
The question stem sounds way too difficult for me to comprehend. Can someone kindly explain what the question asking us to find?
Thanks



SVP
Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1727
Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)

Re: The decimal d is formed by writing in succession all the pos
[#permalink]
Show Tags
20 Nov 2014, 01:05
aj0809 wrote: Hi,
The question stem sounds way too difficult for me to comprehend. Can someone kindly explain what the question asking us to find?
Thanks d = 0.1 23 45 67 89 1011 12 ......... The above series goes on .......... infinite We require to find "100th digit" of d to the right of decimal point



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9859
Location: Pune, India

Re: The decimal d is formed by writing in succession all the pos
[#permalink]
Show Tags
03 Dec 2014, 20:17
aj0809 wrote: Hi,
The question stem sounds way too difficult for me to comprehend. Can someone kindly explain what the question asking us to find?
Thanks Look at the question one sentence at a time: "The decimal d is formed by writing in succession all the positive integers in increasing order after the decimal point; " This is how decimal d is formed  write all positive integers (i.e. 1, 2, 3, etc) in increasing order "that is d = 0.123456789101112" Here they have shown what they mean d = 0.123456789 101112... and so on goes the decimal "What is the 100th digit of d to the right of decimal point ?" If you continue putting down positive integers in increasing order, you will get a decimal with infinite digits to the right of decimal. What will be the 100th digit? d = 0.1234567891011121314... first digit after decimal is 1 second digit is 2 third digit is 3 10th digit is 1 What will be 100th digit? First 9 digit will be 1 to 9. Next 90 digits will come from the next 45 two digit numbers. The next 45 two digit numbers are 10  54. We have accounted for 99 digits. The 100th digit will be the 5 of 55.
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >



GMAT Club Legend
Joined: 12 Sep 2015
Posts: 4125
Location: Canada

Re: the decimal d is formed by writing in succession all positive integers
[#permalink]
Show Tags
15 Feb 2016, 16:43
abhasjha wrote: The decimal d is formed by writing in succession all the positive integers in increasing order after the decimal point; that is d = 0.123456789101112
What is the 100th digit of d to the right of decimal point ?
a) 0 b) 1 c) 5 d) 8 e) 9
Onedigit numbers: .123456789 9 digits down, 91 remaining Twodigit numbers: With 91 remaining, we can handle 45 TWODIGIT NUMBERS (which will accommodate 90 of the required digits) That is 10, 11, 12, ..... 54 ( 55 is next) So, the 98th digit is 5, the 99th digit is 4, which means the 100th digit will be 5Answer: C ASIDE: To determine that there are 45 digits from 10 to 54 inclusive, I used a nice rule that says: the number of integers from x to y inclusive equals y  x + 1 Cheers, Brent
_________________
Test confidently with gmatprepnow.com



Retired Moderator
Joined: 22 Aug 2013
Posts: 1409
Location: India

Re: The decimal d is formed by writing in succession all the pos
[#permalink]
Show Tags
22 May 2017, 11:11
d = 0.1234567891011121314151617...
from 1 to 9 = 9 digits.. remaining digits are 91 After that all numbers (10, 11, 12, 13, ...) are composed of 'sets of two each' , meaning they are all 2 digit numbers. So to figure this out, we can divide 91 by 2 and write it in the form: Dividend = Divisor*Quotient + Remainder (dividend is 91, divisor is 2)
Now, 91 = 2*45 + 1
This means we will have '45' sets of twodigit numbers.. and a remainder of '1' means that our required digit will be the 1st digit of the 46th number.
Once again, we need to figure out 1st digit of the 46th number After 9, i.e., once we start counting from 10 onwards.
46th number after 9 = 46+9 = 55 So our required digit is the 1st digit of 55, or 5.
Hence answer is C



Intern
Joined: 03 May 2014
Posts: 16

Re: The decimal d is formed by writing in succession all the pos
[#permalink]
Show Tags
20 Nov 2017, 08:17
VeritasPrepKarishma wrote: aj0809 wrote: Hi,
The question stem sounds way too difficult for me to comprehend. Can someone kindly explain what the question asking us to find?
Thanks Look at the question one sentence at a time: "The decimal d is formed by writing in succession all the positive integers in increasing order after the decimal point; " This is how decimal d is formed  write all positive integers (i.e. 1, 2, 3, etc) in increasing order "that is d = 0.123456789101112" Here they have shown what they mean d = 0.123456789 101112... and so on goes the decimal "What is the 100th digit of d to the right of decimal point ?" If you continue putting down positive integers in increasing order, you will get a decimal with infinite digits to the right of decimal. What will be the 100th digit? d = 0.1234567891011121314... first digit after decimal is 1 second digit is 2 third digit is 3 10th digit is 1 What will be 100th digit? First 9 digit will be 1 to 9. Next 90 digits will come from the next 45 two digit numbers. The next 45 two digit numbers are 10  54. We have accounted for 99 digits. The 100th digit will be the 5 of 55. Using your reasoning above I just wanted to check if this would be the right approach if the question asked for the 200th digit to the right of the decimal. First 9 digits is 1 to 9 The next 191 digits will come from a combination of two digit and three digit numbers because we do not have enough two digit integers to reach 200 digits. So we have to find how many digits we need to get to the 99th two digit integer. From 10  99 is 90 two digit integers which is 180 digits therefore next 180 digits will come from the next 90 two digit numbers which are 10  99. We now have 9 + 180 = 189 digits but we still need 11 to get to the 200th digit. Because the next batch of numbers are 3 digit numbers we need to take the next 3 (11/3) three digit numbers with are 100, 101, 102 with a remainder of 2 1( 0)3. Therefore 0 is the 200th digit.



Manager
Joined: 29 Sep 2017
Posts: 114
Location: United States

Re: The decimal d is formed by writing in succession all the pos
[#permalink]
Show Tags
12 May 2018, 12:25
+ 1 kudos Brent. You always have the easiest to understand explanations (which is why I also love your videos). Thanks, this makes a lot more sense to me now! GMATPrepNow wrote: abhasjha wrote: The decimal d is formed by writing in succession all the positive integers in increasing order after the decimal point; that is d = 0.123456789101112
What is the 100th digit of d to the right of decimal point ?
a) 0 b) 1 c) 5 d) 8 e) 9
Onedigit numbers: .123456789 9 digits down, 91 remaining Twodigit numbers: With 91 remaining, we can handle 45 TWODIGIT NUMBERS (which will accommodate 90 of the required digits) That is 10, 11, 12, ..... 54 ( 55 is next) So, the 98th digit is 5, the 99th digit is 4, which means the 100th digit will be 5Answer: C ASIDE: To determine that there are 45 digits from 10 to 54 inclusive, I used a nice rule that says: the number of integers from x to y inclusive equals y  x + 1 Cheers, Brent



Director
Joined: 12 Feb 2015
Posts: 961

Re: The decimal d is formed by writing in succession all the pos
[#permalink]
Show Tags
23 Jul 2018, 08:57
100th Digit will come after 9 single digits and 45 two digit numbers, i.e. tens digit of 46th two digits number. i.e. 5 (option C is the correct answer)
_________________
________________ Manish "Only I can change my life. No one can do it for me"



Intern
Joined: 03 Jul 2018
Posts: 4

The decimal d is formed by writing in succession all the pos
[#permalink]
Show Tags
28 Aug 2018, 07:24
Hi, This is the first time I'm posting a response. Only because I feel looking for patterns is an easier way to solve this problem. My approach, In the decimal d=0.1234567891011121314151617181920......, 10th digit is 1(from 10), 20th is again 1(from 15). That leads us to a pattern  the 30th and 40th digits will both begin with 2(from 20 & 25), 50th and 60th with 3, 70th and 80th with 4 and 90th and 100th with 5(from 50 & 55) =)
Hope this helps.
Anviksha



Intern
Joined: 08 Apr 2019
Posts: 8

Re: The decimal d is formed by writing in succession all the pos
[#permalink]
Show Tags
29 Aug 2019, 09:58
I got this one right but I had to write out all the numbers. Can anyone think of a way to do this algebraically or at least a timesaving approach to it?




Re: The decimal d is formed by writing in succession all the pos
[#permalink]
29 Aug 2019, 09:58






