Final answer:
Sandy and Jessica continue running until Sandy catches up to Jessica for the third time. After calculating the difference in their lap times and the number of laps Sandy needs to catch Jessica three times, we find that Sandy ran 3.75 miles, while Jessica ran 3 miles.
Step-by-step explanation:
To solve the problem of how far Jessica and Sandy ran, we must first determine how much faster Sandy is compared to Jessica. Jessica can run the quarter-mile track once in 90 seconds, while Sandy can do it in 72 seconds. This means for every lap Sandy completes, she gains on Jessica by 90 - 72 = 18 seconds. Sandy is supposed to catch up to Jessica for the third time, which means she needs to overlap her by two laps (since they started together) plus one more lap for the third catch-up. Each lap is a quarter-mile, so catching up once requires Sandy to complete an additional lap, hence catching up three times requires three extra laps for Sandy.
To find out how long it takes for Sandy to catch up each time, we divide Jessica's lap time by Sandy's gain per lap. The number of laps for each catch-up is 90 seconds / 18 seconds = 5 laps. Because Sandy needs to catch up three times, we multiply by 3, resulting in 3 * 5 laps = 15 laps that Sandy runs more than Jessica. Jessica then runs 15 laps - 3 laps = 12 laps during this time.
Finally, to find the distance each ran, we convert laps to miles by knowing each lap is a quarter mile. Sandy runs 15 laps * 0.25 miles/lap = 3.75 miles, and Jessica runs 12 laps * 0.25 miles/lap = 3 miles.