Book of 2019 2019

Algebra Level 2

To number the pages of a book, the printer uses 2019 digits.

How many numbered pages are there in the book?

Note: Pages are numbered as 1, 2, 3....

710 708 709 711

Mr. India by Mr. India , 17, India

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3 solutions

Gandoff Tan
Aug 12, 2019

( 10 − 1 ) ( 1 ) + ( 99 − 9 ) ( 2 ) + ( x − 99 ) ( 3 ) = 2019 9 + 180 + 3 x − 297 = 2019 3 x = 2127 x = 709 \begin{aligned} (10-1)(1)+(99-9)(2)+(x-99)(3)&=2019\\ 9+180+3x-297&=2019\\ 3x&=2127\\ x&=\boxed{709} \end{aligned}

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Chris Lewis
May 17, 2019

Pages 1 , ⋯ , 9 1,\cdots,9 use 9 9 digits.

Pages 10 , ⋯ , 99 10,\cdots,99 use 180 180 digits.

This leaves 2019 − 180 − 9 = 1830 2019-180-9=1830 more digits, for which we need 1830 3 = 610 \frac{1830}{3}=610 three-digit page numbers, for a total of 9 + 90 + 610 = 709 9+90+610=\boxed{709} pages.

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Just counted all the digits. By the way, that is an odd number of pages, in both senses. Books usually have an even number of pages.

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