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

Answer is C

Hope its clear
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Re: The decimal d is formed by writing in succession all the positive inte [#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
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Re: The decimal d is formed by writing in succession all the positive inte [#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
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Re: The decimal d is formed by writing in succession all the positive inte [#permalink]
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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 positive inte [#permalink]
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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.
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Re: The decimal d is formed by writing in succession all the positive inte [#permalink]
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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

Answer: 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


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Re: The decimal d is formed by writing in succession all the positive inte [#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.

Hence answer is C
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Re: The decimal d is formed by writing in succession all the positive inte [#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 positive inte [#permalink]
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0.123456789101112

=> 1 to 9 = 9
=> 10 to 19 = 20 digits
=> 20 to 29 = 20 digits
=> 30 to 39 = 20 digits
=> 40 to 49 = 20 digits

Total 89 digits.

We need 100 digit: 9 + ( 20*4) + 11

= 9+ ( 10-49) + 11 more digits from 50 onwards.

Every number has 2 digits, therefore 11 digits means ( 2*5) + 1 from 50.

So, we reach 55.

The first '5' is the 100th digit.

Answer C

= 5
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Re: The decimal d is formed by writing in succession all the positive inte [#permalink]
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1 to 10 --> 11 digit
11 to 20 --> 20 digit
21-30--> 20 digit
31--40--> 20 digit
41--50 --> 20 digit
--------------------

total = 91 digit
51-->2 digit
52--> 2 digit
53-->2 digit
54 --> 2 digit
5----> 1 ( 100th digit)

------------------

total. ==> 100 digit
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Re: The decimal d is formed by writing in succession all the positive inte [#permalink]
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 ?


Main idea: 0.1234567891011121314...
After 10, digits at even place are always the tens digit of the number you write (Bold faced above). So to find hundredth digit, you need to find the tens digit of the number that holds that place.

Now, let us see how it goes:
1 to 9: 9 digits
10th digit to 29th digit: tens place is 1
30th digit to 49th digit: tens place is 2
50th digit to 69th digit: tens place is 3
70th digit to 89th digit: tens place is 4
90th digit to 109th digit: tens place is 5

Therefore, 100th digit is 5. Therefore, (C)
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Re: The decimal d is formed by writing in succession all the positive inte [#permalink]
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Re: The decimal d is formed by writing in succession all the positive inte [#permalink]
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