Problem 5 · The skateboard wheels
Wear per kilometre: each wheel must use up exactly 100%.
Integer answer, at most 4 digitsWe've been given a four-wheeled skateboard. The four wheels are identical, but they wear differently depending on whether they're at the front or the back. Front wheels must be replaced every 300 km, rear wheels every 500 km. We want to know when to swap the front and rear wheels so that all four need replacing at the same time. How many kilometres will the skateboard have done at the moment of the swap? (If the answer is an integer, answer that number. If it has decimals, round to one decimal place and answer with all digits and no comma; e.g. if the answer is $340.17$, answer $3402$.)
Reasoned solution
Key idea: a front wheel wears $\tfrac{1}{300}$ per km and a rear one $\tfrac{1}{500}$; each wheel must accumulate exactly $1$ unit of wear by the end.
We swap at km $x$ and finish at km $D$. Wheels starting at the front: $\tfrac{x}{300} + \tfrac{D-x}{500} = 1$; wheels starting at the back: $\tfrac{x}{500} + \tfrac{D-x}{300} = 1$. Adding both equations:
And from the first equation: $5x + 3(375 - x) = 1500 \Rightarrow 2x = 375 \Rightarrow x = 187.5$ km. In the required format: