Question 25
The positive integer $k$ has $n$ digits. It is rounded to the nearest multiple of $10$, then to the nearest multiple of $100$ and so on ($n-1$ roundings in all). Numbers midway between are rounded up. For example, $1474$ is rounded to $1470$, then to $1500$, then to $2000$. Show that the final number is less than $\dfrac{18k}{13}$.