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

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

Hi All,

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.

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

The required answer is 101, I am not sure how to get there. Can you help identify my mistake. Thanks for your help.

2. 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

3. 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, $9 + 90 + 3 = 101$.