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# M09-17

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Math Expert
Joined: 02 Sep 2009
Posts: 52285

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15 Sep 2014, 23:39
2
4
00:00

Difficulty:

85% (hard)

Question Stats:

41% (01:03) correct 59% (01:09) wrong based on 153 sessions

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All pages of a 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

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Math Expert
Joined: 02 Sep 2009
Posts: 52285

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15 Sep 2014, 23:39
Official Solution:

All pages of a 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

Compute on how many pages 9 appears in the first 100 pages and then multiply the result by 7. In the first 100 we have 10 pages where 9 appears as a units digit and 10 pages where 9 appears as a tens digit. Page 99 is counted twice, so the total number of pages on which 9 appears in the first 100 is 19. Therefore, $$19*7 = 133$$.

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Intern
Joined: 22 Oct 2014
Posts: 4
GMAT 1: 740 Q50 V39
GPA: 4
WE: Marketing (Manufacturing)

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18 Dec 2014, 23:18
I think that the answer is wrong.
The question is asking on how many pages does number 9 appear and not how many times does number 9 appear.
Therefore page 99 has to be counted once.
Hence correct answer is 18x7= 126
Math Expert
Joined: 02 Sep 2009
Posts: 52285

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19 Dec 2014, 07:28
alemazza wrote:
I think that the answer is wrong.
The question is asking on how many pages does number 9 appear and not how many times does number 9 appear.
Therefore page 99 has to be counted once.
Hence correct answer is 18x7= 126

That's not correct.

Pages with 9 in 100: 9, 19, 29, 39, 49, 59, 69, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, and 99. Total of 19 pages. 19*7 = 133.
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Intern
Joined: 24 Jun 2015
Posts: 46

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01 Jul 2015, 07:28
Hi,

If I would like to solve with combinatorics, may be slot method _ x _ x _ ... How the method would proceed?

Thanks a lot.

Best regards.

Luis Navarro
Looking for 700
Manager
Joined: 24 Oct 2013
Posts: 134
Location: India
Concentration: General Management, Strategy
WE: Information Technology (Computer Software)

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14 Mar 2016, 06:09
Here are the list of numbers
9, 19, 29, 39, 49, 59, 69, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, and 99
Total - 20 9's in 1 hundred. Total = 20*7 = 140
Math Expert
Joined: 02 Aug 2009
Posts: 7201

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14 Mar 2016, 06:15
deepthit wrote:
Here are the list of numbers
9, 19, 29, 39, 49, 59, 69, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, and 99
Total - 20 9's in 1 hundred. Total = 20*7 = 140

the Q ask us about the PAGES on which 9 occurs, so 99 will be ONE and not two separately
so 19*7
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

GMAT online Tutor

Intern
Joined: 16 Feb 2015
Posts: 3
GMAT 1: 710 Q47 V40

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06 Jul 2016, 04:17
1
Another approach:

Pages from 1 to 705.

9 as units digit: 9- first number in set, 699-last number in set, set is inclusive. Distance between each occurrence of '9' in units digits = 10.
>> Number of occurrences = [(699-9)/10] + 1 = 70

9 as tenths digit: if hundredth digit is '0', we have 90-first number in set, 98-last number in set (omit 99 as already counted above), set is inclusive.
>> Number of occurrences: (98-90) + 1 = 9
Possible hundredth digits: 0 1 2 3 4 5 6 >> total number of occurrences = 9*7=63

Finally: total pages with '9' = 63 +70 = 133
Senior Manager
Joined: 31 Mar 2016
Posts: 384
Location: India
Concentration: Operations, Finance
GMAT 1: 670 Q48 V34
GPA: 3.8
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29 Jul 2016, 09:47
I think this is a high-quality question and the explanation isn't clear enough, please elaborate. Hi Bunuel,

Can you please take time to explain how this can be done using combination / permutations ?
Manager
Joined: 17 Aug 2015
Posts: 97
GMAT 1: 650 Q49 V29

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02 Sep 2016, 09:20
99 NEEDS to be counted as one page as question asks the number of pages on which 9 appears not the total number of 9's
Intern
Joined: 26 May 2016
Posts: 20
GMAT 1: 640 Q49 V30

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02 May 2017, 03:25
For those who want to solve this using combinatorics
>> !!!

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Re: M09-17 &nbs [#permalink] 02 May 2017, 03:25
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# M09-17

Moderators: chetan2u, Bunuel

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