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Counting digits (m09q17)

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Re: Counting digits (m09q17) [#permalink] New post 02 Sep 2010, 02:49
Facts except 0 every digit from 1-9 comes about 19 times (need to remember only once) further apply simple mathematical formula......19*7=133
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Re: Counting digits (m09q17) [#permalink] New post 02 Sep 2010, 15:10
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19*7 = 133 that's clear.

The problem for me was how easily come to 19.

My method is maybe stupid, but it is quick and works:
1-10 - one*9
11-20 - two*9

The number of nines is exaclty the number of decimal digit + 1.
Therefore 90 there are 9+1 nines = 10. now only add 9 nines from the 91-99 and Voilà: we have 19.
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Re: Counting digits [#permalink] New post 02 Sep 2010, 15:24
Oski wrote:
Actually you don't have to make "slices" (1-9, 10-99, 100 - 999)

For instance in your first question "How many times will the digit 7 written when listing integers from 1 to 1000 ?" :

- Count the number of times 7 appears as a digits number : once every ten, that's 1000/10 = 100 times

- Count the number of times 7 appears as a tens number : ten times every hundred, that's 1000/100*10 = 100 times

- Count the number of times 7 appears as a hundreds number : a hundred times every thousand, that's 1000/1000*100 = 100 times

==> Total = 100 + 100 + 100 = 300 times


For your second question "on a book of 705 pages on how many pages does digit 9 appear in the numeration?", and as it was said above, you have to be careful not to double count.

First, notice it is the same than asking "on a book of 700 pages on how many pages does digit 9 appear in the numeration?" (since 701, 702, 703, 704 and 705 don't contain 9s)

Then same method as above :

- Count the number of times 9 appears as a digits number: once every ten, that's 700/10 = 70 times

- Count the number of times 9 appears as a tens number and not as a digits number: 9 times every hundred, that's 700/100*9 = 63 times

- Count the number of times 9 appears as a hundreds number: zero since there are less than 900 pages

==> Total = 70 + 63 + 0 = 133 pages


Very nice explanation for the first question. How would you, however, count the numbers containing 7, not how many times will 7 appear in this 1-1000 set?
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Re: Counting digits (m09q17) [#permalink] New post 17 Sep 2010, 06:46
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jallenmorris wrote:
Not many people really know the difficulty level since the authors of the question don't really know. One way we can tell the difficulty is if the question comes from the OG because the easy questions are in the front of a section and become more difficult as the question numbers get higher.

This particular question is not very difficult. Maybe around the 575 - 625 range? Anyone else want to give their opinion as to the difficulty level of this question?

I think it is of 600-650 level.

Here is a smart shortcut.
in range from 10 to 99 - you have 11 figures in "own line" and 8 "out of own line" = total 19 (7th's own line is 70 to 79)
in range from 100 to 1000 - you have 120 figures in "own line" and 20 "out of own line" = total 280.
+1 figure from 1 to 9
total 300.
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Re: Counting digits (m09q17) [#permalink] New post 17 Sep 2010, 07:14
But 300 isn't even an option in the original multiple choice answers.

Pkit wrote:
jallenmorris wrote:
Not many people really know the difficulty level since the authors of the question don't really know. One way we can tell the difficulty is if the question comes from the OG because the easy questions are in the front of a section and become more difficult as the question numbers get higher.

This particular question is not very difficult. Maybe around the 575 - 625 range? Anyone else want to give their opinion as to the difficulty level of this question?

I think it is of 600-650 level.

Here is a smart shortcut.
in range from 10 to 99 - you have 11 figures in "own line" and 8 "out of own line" = total 19 (7th's own line is 70 to 79)
in range from 100 to 1000 - you have 120 figures in "own line" and 20 "out of own line" = total 280.
+1 figure from 1 to 9
total 300.

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Re: Counting digits (m09q17) [#permalink] New post 17 Sep 2010, 08:09
:) :)
I am former big4-auditor, 300 is just a reconciliation that my counting is right.
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Re: Counting digits (m09q17) [#permalink] New post 17 Sep 2010, 08:12
Sorry, I'm not an auditory. I'm an attorney so I only think in terms of Guilty/Innocent and Liabilities.

Pkit wrote:
:) :)
I am former big4-auditor, 300 is just a reconciliation that my counting is right.

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Re: Counting digits (m09q17) [#permalink] New post 18 Sep 2010, 05:51
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jallenmorris wrote:
Sorry, I'm not an auditory. I'm an attorney so I only think in terms of Guilty/Innocent and Liabilities.

Pkit wrote:
:) :)
I am former big4-auditor, 300 is just a reconciliation that my counting is right.

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Re: Counting digits (m09q17) [#permalink] New post 11 Aug 2011, 15:23
ok, I was trying for a different method, after rectifying my silly mistake I see that, this problem can be approached in a different way

Here it is

705 - ( all cases when 9 doesnt come )

705 - [ ( one digit where 9 doesnt come ) + ( two digit where 9 doesnt come ) + (three digit where 9 doesnt come ) ]

705 - [ 8 + ( 8 X 9 ) + ( 6 X 9 X 9 ) + 6 ] ( \\ 700-705 --> 6 )

705 - [ 8 + 72 + 486 + 6 ]

= 133
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Re: Counting digits (m09q17) [#permalink] New post 11 Aug 2011, 22:37
The problem was easy to solve but real good discussion happening here..

:lol:
jallenmorris wrote:
Sorry, I'm not an auditory. I'm an attorney so I only think in terms of Guilty/Innocent and Liabilities.

Pkit wrote:
:) :)
I am former big4-auditor, 300 is just a reconciliation that my counting is right.

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Re: Counting digits [#permalink] New post 05 Sep 2011, 08:10
Oski wrote:
Actually you don't have to make "slices" (1-9, 10-99, 100 - 999)

For instance in your first question "How many times will the digit 7 written when listing integers from 1 to 1000 ?" :

- Count the number of times 7 appears as a digits number : once every ten, that's 1000/10 = 100 times

- Count the number of times 7 appears as a tens number : ten times every hundred, that's 1000/100*10 = 100 times

- Count the number of times 7 appears as a hundreds number : a hundred times every thousand, that's 1000/1000*100 = 100 times

==> Total = 100 + 100 + 100 = 300 times


For your second question "on a book of 705 pages on how many pages does digit 9 appear in the numeration?", and as it was said above, you have to be careful not to double count.

First, notice it is the same than asking "on a book of 700 pages on how many pages does digit 9 appear in the numeration?" (since 701, 702, 703, 704 and 705 don't contain 9s)

Then same method as above :

- Count the number of times 9 appears as a digits number: once every ten, that's 700/10 = 70 times

- Count the number of times 9 appears as a tens number and not as a digits number: 9 times every hundred, that's 700/100*9 = 63 times

- Count the number of times 9 appears as a hundreds number: zero since there are less than 900 pages

==> Total = 70 + 63 + 0 = 133 pages




nice way to count...will it always wrk out to be correct??...looks like lot of clever thinking is required in parallel to avoid mistake!!
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Re: Counting digits (m09q17) [#permalink] New post 05 Sep 2011, 09:16
Answer D
10 times for each set of 100's (for the 90's) [90, 91, 92....]: 10 x 7 = 70
9 times for each set of 100's for the single occurrences, excluding the 90's, [9, 19, 29...]: 9 x 7 = 63

70+63 = 133
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Re: Counting digits [#permalink] New post 05 Sep 2011, 20:44
Its D, Liked this explanation.

Oski wrote:
Actually you don't have to make "slices" (1-9, 10-99, 100 - 999)

For instance in your first question "How many times will the digit 7 written when listing integers from 1 to 1000 ?" :

- Count the number of times 7 appears as a digits number : once every ten, that's 1000/10 = 100 times

- Count the number of times 7 appears as a tens number : ten times every hundred, that's 1000/100*10 = 100 times

- Count the number of times 7 appears as a hundreds number : a hundred times every thousand, that's 1000/1000*100 = 100 times

==> Total = 100 + 100 + 100 = 300 times


For your second question "on a book of 705 pages on how many pages does digit 9 appear in the numeration?", and as it was said above, you have to be careful not to double count.

First, notice it is the same than asking "on a book of 700 pages on how many pages does digit 9 appear in the numeration?" (since 701, 702, 703, 704 and 705 don't contain 9s)

Then same method as above :

- Count the number of times 9 appears as a digits number: once every ten, that's 700/10 = 70 times

- Count the number of times 9 appears as a tens number and not as a digits number: 9 times every hundred, that's 700/100*9 = 63 times

- Count the number of times 9 appears as a hundreds number: zero since there are less than 900 pages

==> Total = 70 + 63 + 0 = 133 pages

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http://gmatclub.com/forum/massive-collection-of-verbal-questions-sc-rc-and-cr-106195.html#p832142
http://gmatclub.com/forum/1001-ds-questions-file-106193.html#p832133
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http://gmatclub.com/forum/how-to-get-6-0-awa-my-guide-64327.html
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Re: Counting digits (m09q17) [#permalink] New post 05 Sep 2011, 21:17
Hi All

Lets try a simple math formula on this-- 1 to 89 no 9's are 9 then from 90 to 99 no of 9's are 10 so 19
so these number will be repeated 7 times 1-100, 101-200, 201-300, 301-400, 401-500, 501-600, 601-705

So total pages = 19* 7 = 133
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Re: Counting digits (m09q17) [#permalink] New post 06 Sep 2011, 04:33
Did not look at all the replies but I would like to propose a simple solution:

Number of total 9s
= Number of 9s at unit place + Number of 9s at tens place
= 7*(that b/w 1-100) + 7*(that b/w 1-100)
= 7*(that b/w 90-100) +7*10*(that b/w 1-10)
= 7*10 + 7*10*1
= 140

So there are 2 mutually exclusive overlapping sets (pages with 9 at tens place AND pages with 9 at unit place - NOTE: they overlap). Imagine a venn diagram or double matrix - whichever makes it easy for you to visualize.

Number of pages where 1 or 2 9 appears
= Number of pages with 9 at tens place + Number of pages with 9 at unit place - Number of pages with 9 at both tens and unit place
= 140 - 7*Number of pages with 9 at tens and unit place b/w 1-100
= 140 - 7*1
= 133

Hope this helps.

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Re: Counting digits (m09q17) [#permalink] New post 09 Sep 2011, 12:17
----------------------7 X Y ---|-- X Y 7 --|-- X 7 Y
number of times----10 10------9 10---------9 10
-------------------------100---------90------------90 = 280

Total number of times = 280+19+1=300

Why not subtract the repeated numbers such as 777,707,770 in this situation?
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Re: Counting digits (m09q17) [#permalink] New post 09 Sep 2011, 13:24
aaron22197 wrote:
All pages of the book are numbered. If the first page is numbered 1 and the last page is numbered 705, on how many pages does digit 9 appear in the numeration?

(A) 70
(B) 77
(C) 126
(D) 133
(E) 140


1-9
9(Total=1)

10-99
Total-(Numbers without 9)=9*10-8*9=90-72=18

Total(1-100)=18+1=19

From,
(1-700)=19*7=133

Ans: "D"
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Re: Counting digits (m09q17) [#permalink] New post 20 Nov 2011, 20:46
19 times every hundred so 19*7 =133 ie D 8-)
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Re: Counting digits (m09q17) [#permalink] New post 14 Oct 2012, 07:50
Here is another way
consider numbers from 1 to 699 since after 699 no number contains 9
now, how many numbers can be formed without using 9
= 7 x 9 x 9
= 567

therefore 700- 567= 133
Re: Counting digits (m09q17)   [#permalink] 14 Oct 2012, 07:50
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