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abhasjha: Master | Next Rank: 500 Posts; Posts: 379; Joined: Tue Sep 30, 2008 7:17 amFollowed by: 11 members; Location: NY; Thanked: 28 times

by abhasjha » Tue Aug 05, 2014 10:56 am

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

any one having a quicker method than how i approached this problem ?

this is how i did it :

d= .12345678910( this zero = digit number 11)
continuing on
11121314151617181920( this zero = digit 31)

so .30 here the 0 will be at digit number 51

similarly .40 - zero at digit 71
.50 - zero at digit 91
now the series continues .50515253545 the hundreth digit ....
answer C

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Source: Beat The GMAT — Problem Solving | Tue Aug 05, 2014 10:56 am

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Tue Aug 05, 2014 11:14 am

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

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Wed Aug 06, 2014 2:26 am

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

Since we need to determine the 100th digit, proceed as follows:
Write out the first 100 integers, digit by digit, so that there are 10 DIGITS PER ROW.
LOOK FOR A PATTERN.

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

The digits in red imply the following pattern:
10th digit -->1
20th digit -->1
30th digit -->2
40th digit -->2
50th digit -->3
60th digit -->3
70th digit -->4
80th digit -->4
90th digit -->5
100th digit -->5

The correct answer is C.

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GMATinsight: Legendary Member; Posts: 1100; Joined: Sat May 10, 2014 11:34 pmFollowed by: 24 members; Location: New Delhi, India; Thanked: 205 times

by GMATinsight » Wed Aug 06, 2014 9:46 pm

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

any one having a quicker method than how i approached this problem ?

There are 9 Single-digit numbers:
.123456789
9 digits Counted, 91 remaining

Two-digit numbers:
With 91 remaining, we can handle 45 TWO-DIGIT NUMBERS (which will accommodate 90 of the required digits)

First 10 two digit numbers will have 1 on their tens place and 0 to 9 consecutively on their units place
Next 10 two digit numbers will have 2 on their tens place and 0 to 9 consecutively on their units place
Next 10 two digit numbers will have 3 on their tens place and 0 to 9 consecutively on their units place
Next 10 two digit numbers will have 4 on their tens place and 0 to 9 consecutively on their units place
now 6th Two digit number will have 5 at the tens place which will be the 91st Digit

Answer: Option C

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