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Math Help - Pages in a book - need to verify my solution

  1. #1
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    Pages in a book - need to verify my solution

    Hi All,

    I have solved this problem below. But the required answer and my answer don't match. Please help me find the mistake in my solution.

    The pages of a book are numbered as 1, 2, 3, .... It is found that 195 digits are used.

    (a) How many pages are numbered with a single-digit number?

    (b) How many pages are numbered with a two-digit number?

    (c) How many pages are there altogether?

    (a) I got single-digit numbers = 9

    (b) I got, two-digit numbers = 99 - 9 = 90

    (c) This is the part I need verified.

    <br />
\begin{tabular}{ l l }<br />
\textnormal{No. of 1 digit pages} &= 9 \\<br />
\textnormal{No. of 2 digit pages} &= \begin{math}\frac{90}{2}\end{math} = 45 \\<br />
\\<br />
\textnormal{No. of 3 digit numbers used} &= 195 - 99 = 96 \\<br />
\textnormal{No. of 3 digit pages} &= \begin{math}\frac{96}{3}\end{math} = 32 \\<br />
\\<br />
\textnormal{Total pages} &= 9 + 45 + 32 = 86 \\<br />
\end{tabular}<br />

    The required answer is 101, I am not sure how to get there. Can you help identify my mistake. Thanks for your help.
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  2. #2
    MHF Contributor
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    two issues here:

    first:
    The number of 2 digit pages is 90, not 45.

    If there were only 45 2 digit numbers, the pages would go
    1,2,3,4,5,6,7,8,9,10,.......58,100

    which is not correct as there would be pages missing between 58 and 100.


    second:
    The number of digits in the 3 digit pages is
    195 - (number of digits in 1 digit pages) - (number of digits in 2 digit pages)
    =195 - 9 - 90*2
    = 6
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  3. #3
    Member
    Joined
    Mar 2011
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    Thanks @SpringFan25.

    I feel really stupid now. I see how I mixed up no. of pages and no. of digits used!

    So the digits used are 9 \times 1 + 90 \times 2 + 3 \times 2 = 195

    And hence Total no of pages,  9 + 90 + 3 = 101.

    Which was the required answer.
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