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Question

During the Labor Day weekend, Amy and Kathleen each ran in a race. Amy ran in a 5K and completed it in 31 minutes and 15 seconds. Kathleen ran in a half-marathon, which is 21.1 kilometers, and completed it in 2 hours, 11 minutes, and 52.5 seconds.
Let d represent distance in kilometers, r represent the rate, and t represent time in minutes. The proportional relationship
can be used to determine the speed at which Amy and Kathleen ran their respective races. Using Amy's distance and time, the proportional relationship
can be used to determine her rate. Using Kathleen's distance and time, the proportional relationship
can be used to determine her rate. Based on the actual rates that they ran during their respective races, Amy ran at
Kathleen.

Answer 1

To determine the speed at which Amy and Kathleen ran their respective races, we first need to calculate their rates.

1. Amy's race:

   • Distance = 5K = 5 kilometers
   • Time = 31 minutes and 15 seconds = 31.25 minutes (since 15 seconds is 0.25 minutes)
   • Rate (r) = Distance (d) / Time (t) = 5 km / 31.25 min = 0.16 km/min

2. Kathleen's race:

   • Distance = Half-marathon = 21.1 kilometers
   • Time = 2 hours, 11 minutes, and 52.5 seconds = (2 * 60) + 11 + (52.5 / 60) = 131.875 minutes
   • Rate (r) = Distance (d) / Time (t) = 21.1 km / 131.875 min = 0.16 km/min

With both their rates calculated, we can write the statements:

Using Amy's distance and time, the proportional relationship d/t = r can be used to determine her rate. Using Kathleen's distance and time, the proportional relationship d/t = r can be used to determine her rate. Based on the actual rates that they ran during their respective races, Amy ran at the same rate as Kathleen.

The final sentence would look like: "Amy ran at the same rate as Kathleen."