Let's solve this problem step by step.
We need to determine the rate (speed) at which each runner ran. The speed can be calculated using the formula:
[tex]\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \][/tex]
Let’s compute the speed for each runner:
1. Mallory:
[tex]\[ \text{Speed} = \frac{12 \text{ miles}}{2 \text{ hours}} = 6 \text{ miles per hour} \][/tex]
2. Justine:
[tex]\[ \text{Speed} = \frac{5 \text{ miles}}{1 \text{ hour}} = 5 \text{ miles per hour} \][/tex]
3. Henry:
[tex]\[ \text{Speed} = \frac{6 \text{ miles}}{2 \text{ hours}} = 3 \text{ miles per hour} \][/tex]
4. Adya:
Adya’s time is given as [tex]\(1 \frac{1}{2}\)[/tex] hours, which is [tex]\(1.5\)[/tex] hours.
[tex]\[ \text{Speed} = \frac{5 \text{ miles}}{1.5 \text{ hours}} = \frac{5}{1.5} = \frac{10}{3} \text{ miles per hour} \approx 3.33 \text{ miles per hour} \][/tex]
5. Lily:
Lily's distance is given as [tex]\(3 \frac{1}{2}\)[/tex] miles, which is [tex]\(3.5\)[/tex] miles.
[tex]\[ \text{Speed} = \frac{3.5 \text{ miles}}{2 \text{ hours}} = 1.75 \text{ miles per hour} \][/tex]
6. Bert:
Bert’s time is given as [tex]\(1 \frac{1}{2}\)[/tex] hours, which is [tex]\(1.5\)[/tex] hours.
[tex]\[ \text{Speed} = \frac{9 \text{ miles}}{1.5 \text{ hours}} = \frac{9}{1.5} = 6 \text{ miles per hour} \][/tex]
Now we compare the speeds to identify which two runners have the same speed.
- Mallory's speed: 6 miles per hour
- Justine's speed: 5 miles per hour
- Henry's speed: 3 miles per hour
- Adya's speed: 3.33 miles per hour
- Lily's speed: 1.75 miles per hour
- Bert's speed: 6 miles per hour
From the speeds calculated, we observe that:
- Mallory and Bert both have a speed of 6 miles per hour.
Hence, the two runners who have a proportional relationship of time and distance are Mallory and Bert.
The correct option is:
D. Mallory and Bert
