The circumference of the front wheel is \(\displaystyle{2}\cdot{63}\cdotπ={126}π\) inches and of the back wheel is \(\displaystyle{2}\cdot{36}\cdotπ={72}π\) inches. Therefore if the front wheel does m full turns, for m a natural number, the tricycle moves \(\displaystyle{m}\cdot{126}π\) inches, and if the back wheel does n full turns the tricycle moved \(\displaystyleπ\cdot{72}π\).

The points P and Q then touch the sidewalk at the same time for m and n such that \(\displaystyle{m}\cdot{126}π={n}\cdot{72}π\), which is equivalent, by dividing both sides by \(18 \pi\), to \(\displaystyle{m}\cdot{7}=π\cdot{4}\), Since the minimal common multiple of 7 and 4 is 25 we have that the smallest m and n satisfying this equations are \(m = 4\) and \(n=7\).

Therefore Felicity will have ridden \(\displaystyle{4}\cdot{126}π={504}π\) inches when P and Q frst touch the sidewalk at the same time.