Area remaining after maximal number of tiles are laid on a pathway A rectangular plot measuring $30$ m $\times$ $40$ m has a $2$ m wide pathway in the middle crosswise. Tiles of dimensions $30$ cm $\times$ $50$ cm are laid on the pathway in such a way so that no portion of these tiles cross the boundary of the pathway. How much area will still remain exposed after the maximum possible numbers of tiles are laid on the pathway without breaking any tiles? This document shows answer as $1000$. However I am getting it as $4000$. Help!!!

Okay, let's see. First things first, let's put in tiles so that we fill in, as much as possible, the straight portions of the pathway. We can take four tiles the long way and they'll fill up the path width. Then we get 18.9m along the longer straight path (which is 19m long), and 13.8m along the shorter straight path (which is 14 m long). Then we have a much smaller cross: a 2m square, with 10cm extra space in each direction one way, and 20cm extra space in each direction the other way.

This thing has an area of 5.2m^2; each block has an area of 0.15m^2, which add up (if you can fit 34 in this space) to 5.1m^2. So the question is this: Can we, in fact, fit 34 pieces in here?

The answer is yes.

!34 rectangular tiles in a stubby cross

This fits nicely into the space available, leaving only $10$ $100\text{ cm}^2$ areas uncovered, for a total of $1000\text{ cm}^2$ .