Final answer:
Joan passes Jim every 5.625 minutes. This is determined by calculating the point at which Joan, running at a speed of 1 circuit per 3 minutes, is exactly one circuit ahead of Jim, who completes the circuit in 7.5 minutes.
Step-by-step explanation:
To solve the mathematical problem of finding when Joan passes Jim, we need to consider the speed at which each of them completes a circuit. Joan takes 3 minutes to complete a circuit, which means her speed is $rac{1}{3}$ circuits per minute. Jim takes 7.5 minutes to complete a circuit, meaning his speed is $rac{1}{7.5}$ circuits per minute.
To find when Joan catches up with Jim again (the value of x), we measure when they have covered the same number of additional circuits, starting from the initial pass.
To calculate x, we set up the equation $rac{1}{3}x - rac{1}{7.5}x = 1$. This represents the point where Joan is one whole circuit ahead of Jim, which corresponds to the second time she passes him. Solving the equation for x gives us 5.625 minutes. Thus, Joan passes Jim every 5.625 minutes
To further illustrate this, if Joan has run for 5.625 minutes, she has completed $rac{5.625}{3} = 1.875$ circuits. During the same time, Jim has completed $rac{5.625}{7.5} = 0.75$ circuits. The difference of 1.125 circuits between them accounts for Joan completing one more circuit than Jim and therefore passing him.