Question

A printer numbers the pages of a book starting with 1 and uses 3189 digits in all. How many pages does the book have?
A. 1000
B. 1074
C. 1075
D. 1080

Answer 1

Hint: First of all we have to calculate the number of digits used by pages numbered from $1-9$, $10\text{ to }99$ and from $100\text{ to 999}$. Then by adding all digits we find the difference and by assuming the number of digits used by a four digit number to be $x$, we calculate the total number of pages a book has.

Complete step-by-step answer:
We have been given that a printer numbers the pages of a book starting with 1 and uses 3189 digits in all.
We have to find the number of pages the book has.
Now, first of all we calculate the number of digits used to number the pages as the number starts from $1$ .
Now, we have a total $9$ pages from numbers $1-9$ as these are single digits.
So, the $1$ digit numbered pages used digits $=9$ digits
Now, two digit numbered pages are from $10\text{ to }99=90$ pages.
So, the total number of digits used $=90\times 2=180$ digits
Now, three digit numbered pages are from $100\text{ to 999}$.
So, the total number of digits used $=900\times 3=2700$ digits.
So, the total digits used from page number $1\text{ to 999}$ will be
$\begin{align}
  & \Rightarrow 9+180+2700 \\
 & =2889 \\
\end{align}$
But we have $3189$ digits in all.
So, the remaining digits will be $3189-2889=300$.
Now, let us assume that $x$ digits are used of $4$ digit number.
So, we have
$\begin{align}
  & 4x=300 \\
 & x=\dfrac{300}{4} \\
 & x=75 \\
\end{align}$
Now, $75$ will be added to the $999$, we have $999+75=1074$.
So, the book has $1074$ pages.

So, the correct answer is “Option B”.

Note: The confusion is created while calculating the total number of digits and number of pages. As we have been given in the question pages starting from $1$, so we have total $9$ pages from numbers $1-9$ instead of $10$. Also, be careful while calculating the value of $x$. Students may equate the $4x$ to $3189$ and get incorrect answers.