How long will it take them to run $99$ laps?

Question

Problem

Zeta always runs around the tract at a rate of $30$ laps per $75$ minutes, and Ray always runs around the track at a rate of $20$ laps per $40$ minutes. If they start at the same time, how many minutes will it take them to run a combined distance of $99$ laps?

Attempt

Zeta and Ray run $65$ laps in $75$ minutes, so therefore they run $99$ laps in $\dfrac{99}{65}*75 = 114.231$ minutes.

This is not the correct answer (the answer is $110$). What did I do wrong?

Answer 1

In $75$ minutes Ray runs $\frac {75}{40}\cdot 20=37.5$ laps, not $35$, so it takes $\frac {99}{67.5}\cdot 75=110$ minutes

Answer 2

If you're running for $t$ minutes, Zeta will cover $\frac{30}{75} t$ distance (this is rate * time). Similarly, Ray will cover $\frac{20}{40} t$ distance.

You want to solve $\frac{30}{75}t + \frac{20}{40} t = 99$ for $t$, where the left hand side is the distance zeta runs plus the distance ray runs in time $t$, and 99 is the combined distance you want them to run in time $t$.