Let's say you play the game of collecting the maximum amount of coins along a path in the $(x,y)$ plane. The total amount of coins collected $S$ is given by $S = 5x + 7y$ where $5$ and $7$ represents the number of coins collected by walking respectively one unit in the $x$ and $y$ direction. However I can only go as far as one unit of distance ($x^2+y^2=1$).
It turns out that the solution to the problem is so such that $y/x = 7/5$ or said in other words, the ratio of how much you have to walk in the $y$ direction compared to in the $x$ direction is the same as the ratio of the rates at which you collect the coins in the $y$ direction compared to in the $x$ direction.
My question : suppose I have to explain this result to a middle school student who does not know any calculus of trigonometry. He is also quite shaky on describing equations of lines in the plane so I would not rely on that. How can I convince him with intuition that the two ratios are the same? (I'm not looking for a series of algebraic equations but rather for some kind of a visual proof).