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A book has 1000 pages numbered 1, 2, 3, …, and so on. How many times d

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A book has 1000 pages numbered 1, 2, 3, …, and so on. How many times d  [#permalink]

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New post 07 May 2019, 01:03
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Question Stats:

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[GMAT math practice question]

A book has 1000 pages numbered 1, 2, 3, …, and so on. How many times does the digit 2 appear on the page numbers?

A. 200
B. 250
C. 300
D. 400
E. 500

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Re: A book has 1000 pages numbered 1, 2, 3, …, and so on. How many times d  [#permalink]

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New post 07 May 2019, 01:26
From 1 to 100 total 2s=20
Hence frm 1 to 1000 2s=20×10=200
imo
option A

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Re: A book has 1000 pages numbered 1, 2, 3, …, and so on. How many times d  [#permalink]

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New post 07 May 2019, 02:57
Hope in real GMAT I wont face question like that because I have no idea how to solve it in even 4 minutes :)

Happy learning to you all
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Re: A book has 1000 pages numbered 1, 2, 3, …, and so on. How many times d  [#permalink]

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New post 07 May 2019, 03:35
MathRevolution wrote:
[GMAT math practice question]

A book has 1000 pages numbered 1, 2, 3, …, and so on. How many times does the digit 2 appear on the page numbers?

A. 200
B. 250
C. 300
D. 400
E. 500


Ignore page 1000, since it does not include the digit 2.
To make the calculation easier, consider the remaining pages numbered as 3-digit integers, beginning with 000 and ending with 999:
000, 001, 002...997, 998, 999

There are 1000 options between 000 to 999, inclusive.
Since each option is composed of 3 digits, the total number of digits = 3*1000 = 3000.
The probability that 2 will appear in any given position is the same as the probability that 3 will appear in any given position.
Implication:
Each of the 10 digits 0 through 9 will appear the SAME NUMBER OF TIMES.
Thus, the number of times that 2 will appear \(= \frac{3000}{10} = 300\).


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Re: A book has 1000 pages numbered 1, 2, 3, …, and so on. How many times d  [#permalink]

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New post 07 May 2019, 09:03
3
MathRevolution wrote:
[GMAT math practice question]

A book has 1000 pages numbered 1, 2, 3, …, and so on. How many times does the digit 2 appear on the page numbers?

A. 200
B. 250
C. 300
D. 400
E. 500


from 1 to 100 ; 2 will come ; 20 times
so except 200 to 299 series ; we get 20*9 ; 180 times
for 200 to 299 series
two will come in hundreds place; 100 times
tens place ; 220-229 ; 10 times
units place ; 202-292 ; 10 times
so total 180+100+10+10; 300
IMO C :cool:
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Re: A book has 1000 pages numbered 1, 2, 3, …, and so on. How many times d  [#permalink]

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New post 07 May 2019, 15:38
1
MathRevolution wrote:
[GMAT math practice question]

A book has 1000 pages numbered 1, 2, 3, …, and so on. How many times does the digit 2 appear on the page numbers?

A. 200
B. 250
C. 300
D. 400
E. 500


In my humble opinion, the best approach is to analyze the probability for each digit class.
First of all, let's eliminate the thousands class since there are no 2's in this class. We are left with:

Ones: 1 out of 10 (numbers in the xx2 form), so \(\frac{1}{10} * 1000 = 100\)

Tens: 10 out of 100 (numbers in the x2x form), so \(\frac{10}{100} * 1000 = 100\)

Hundreds: 100 out of 1000 (numbers from 200 to 299), so \(\frac{100}{1000} * 1000 = 100\)

Which will give us: \(100 + 100 + 100 = 300\)

Answer: C
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Re: A book has 1000 pages numbered 1, 2, 3, …, and so on. How many times d  [#permalink]

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New post 09 May 2019, 02:44
=>

The page numbers \(2, 12, 20, 21, 22, … , 29, 32, 42, … , 92\) contain \(19+1 = 20\) copies of the digit, \(2\).

The page numbers \(102, 112, 120, 121, … , 129, 132, 142, … , 192\) similarly contain \(20\) copies of the digit \(2\).

The page numbers \(200, 201, … , 299\) contain \(120\) copies of the digit, \(2\).

Page numbers \(302, 312, 320, 321, … , 329, 332, 342, … , 392\) contain \(20\) copies of the digit, \(2\).

Page numbers \(902, 912, 920, 921, 922, … , 929, 932, 942, … , 992\) contain \(20\) copies of the digit, \(2\).

The total number of copies of the digit, \(2\) printed on the pages is
\(9*20 + 120 = 300.\)

Therefore, the answer is C.
Answer: C
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Re: A book has 1000 pages numbered 1, 2, 3, …, and so on. How many times d   [#permalink] 09 May 2019, 02:44
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