Question

mia, sam, and ben are cross-country runners on a team. their coach recorded the time it took each runner to complete their first serious segment of the loop, mix, which covers 5 miles of varying terrain.

• mia ran 3/4 of a mile in 1/6 of an hour.
• sam ran 5/6 of a mile in 2/9 of an hour.
• ben ran 1 1/2 miles in 3/5 of an hour.

1. if mia, sam, and ben all run the course at a constant pace, which runner covered the greatest distance in the first full hour of running?
2. the coach wants to compare the runners’ distances at a specific time. how far (in miles) would the slowest runner have traveled after they had run for 1 1/4 hours?
3. if all three runners maintain their pace, how many minutes will it take sam and ben to complete the entire 5-mile trail mix course? what is the difference in their finish times (in minutes)?

Explanation:

Step1: Find each runner's speed

Mia's speed: She runs $\frac{3}{4}$ mile in $\frac{1}{6}$ hour. Speed = distance/time, so $\frac{\frac{3}{4}}{\frac{1}{6}}=\frac{3}{4}\times6 = \frac{18}{4}=4.5$ mph.
Sam's speed: He runs $\frac{5}{6}$ mile in $\frac{2}{9}$ hour. Speed = $\frac{\frac{5}{6}}{\frac{2}{9}}=\frac{5}{6}\times\frac{9}{2}=\frac{45}{12}=3.75$ mph.
Ben's speed: He runs $1\frac{1}{2}=\frac{3}{2}$ miles in $\frac{3}{5}$ hour. Speed = $\frac{\frac{3}{2}}{\frac{3}{5}}=\frac{3}{2}\times\frac{5}{3}=\frac{5}{2}=2.5$ mph. Wait, no, wait—wait, the problem says "Ben ran 1½ miles in 3/5 of an hour". Wait, maybe I miscalculated. Wait, $1\frac{1}{2}$ is $\frac{3}{2}$, divided by $\frac{3}{5}$: $\frac{3}{2}\div\frac{3}{5}=\frac{3}{2}\times\frac{5}{3}=\frac{5}{2}=2.5$? But wait, the first question is which runner covered the greatest distance in 1 hour. Wait, speed is distance per hour, so their speed is the distance they cover in 1 hour. Wait, Mia: $\frac{3}{4}$ mile in $\frac{1}{6}$ hour, so in 1 hour, she runs $\frac{3}{4}\times6 = 4.5$ miles. Sam: $\frac{5}{6}$ mile in $\frac{2}{9}$ hour, so in 1 hour, $\frac{5}{6}\div\frac{2}{9}=\frac{5}{6}\times\frac{9}{2}=\frac{45}{12}=3.75$ miles. Ben: $1\frac{1}{2}$ miles in $\frac{3}{5}$ hour, so in 1 hour, $\frac{3}{2}\div\frac{3}{5}=\frac{5}{2}=2.5$? Wait, that can't be. Wait, maybe I misread Ben's distance. Wait, the text says "Ben ran 1½ miles in 3/5 of an hour". So speed is (3/2) / (3/5) = (3/2)(5/3) = 5/2 = 2.5 mph? But Mia's speed is 4.5, Sam's 3.75, Ben's 2.5. So Mia has the greatest speed, so in 1 hour, Mia covers the most distance. Wait, but let's check again. Mia: 3/4 mile in 1/6 hour. So per hour, (3/4) 6 = 18/4 = 4.5. Correct. Sam: 5/6 mile in 2/9 hour. So per hour, (5/6) / (2/9) = (5/6)*(9/2) = 45/12 = 3.75. Correct. Ben: 3/2 miles in 3/5 hour. (3/2)/(3/5) = 5/2 = 2.5. So yes, Mia's speed is highest. So the answer is Mia.

Step2: Confirm the speeds

• Mia: $\frac{3}{4} \div \frac{1}{6} = \frac{3}{4} \times 6 = \frac{18}{4} = 4.5$ mph.
• Sam: $\frac{5}{6} \div \frac{2}{9} = \frac{5}{6} \times \frac{9}{2} = \frac{45}{12} = 3.75$ mph.
• Ben: $1\frac{1}{2} \div \frac{3}{5} = \frac{3}{2} \times \frac{5}{3} = \frac{5}{2} = 2.5$ mph.

Since 4.5 > 3.75 > 2.5, Mia covers the greatest distance in 1 hour.

Answer:

Mia