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Intern  Joined: 24 Jan 2012
Posts: 9
The decimal d is formed by writing in succession all the pos  [#permalink]

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58 00:00

Difficulty:   45% (medium)

Question Stats: 67% (01:56) correct 33% (01:50) wrong based on 664 sessions

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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 ?

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 V
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City: Pune
Re: 100th digit in series  [#permalink]

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38
23
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 Approach

Assume 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
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Re: The decimal d is formed by writing in succession all the pos  [#permalink]

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7
1
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.
##### General Discussion
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9983
Location: Pune, India
Re: The decimal d is formed by writing in succession all the pos  [#permalink]

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9
1
3
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 = 0.123456789101112... 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.
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Re: The decimal d is formed by writing in succession all the pos  [#permalink]

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3
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 50-59 which will be 10 integers again, 2 places each hence 20 digits.

Now the 100th digit will be in the 50-59 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 (50-59) thus 5

Hope its clear
Cheers
J
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Joined: 12 Sep 2015
Posts: 4214
Re: the decimal d is formed by writing in succession all positive integers  [#permalink]

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2
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

One-digit numbers:
.123456789
9 digits down, 91 remaining

Two-digit numbers:
With 91 remaining, we can handle 45 TWO-DIGIT 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 5

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
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The decimal d is formed by writing in succession all the pos  [#permalink]

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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.

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Re: The decimal d is formed by writing in succession all the pos  [#permalink]

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1
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 = 0.123456789101112 .........

The above series goes on .......... infinite

We require to find "100th digit" of d to the right of decimal point
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Re: The decimal d is formed by writing in succession all the pos  [#permalink]

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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]

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Hi,

The question stem sounds way too difficult for me to comprehend. Can someone kindly explain what the question asking us to find?

Thanks
Retired Moderator P
Joined: 22 Aug 2013
Posts: 1403
Location: India
Re: The decimal d is formed by writing in succession all the pos  [#permalink]

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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 two-digit 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.

Intern  B
Joined: 03 May 2014
Posts: 16
Re: The decimal d is formed by writing in succession all the pos  [#permalink]

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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 = 0.123456789101112... 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.
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Re: The decimal d is formed by writing in succession all the pos  [#permalink]

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+ 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

One-digit numbers:
.123456789
9 digits down, 91 remaining

Two-digit numbers:
With 91 remaining, we can handle 45 TWO-DIGIT 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 5

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
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Joined: 12 Feb 2015
Posts: 1005
Re: The decimal d is formed by writing in succession all the pos  [#permalink]

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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)
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Re: The decimal d is formed by writing in succession all the pos  [#permalink]

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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 time-saving approach to it? Re: The decimal d is formed by writing in succession all the pos   [#permalink] 29 Aug 2019, 09:58
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