# The front wheel of a tricycle has a circumference of 63 inches, and the back wheels have a circumference of36 inches. If points P and Q are both touch

The front wheel of a tricycle has a circumference of 63 ​inches, and the back wheels have a circumference of36 inches. If points P and Q are both touching the sidewalk when Felicity starts to​ ride, how far will she have ridden when P and Q first touch the sidewalk at the same time​ again?

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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.

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