To determine which two runners ran at the same rate, we need to find the rate at which each runner covered the distance. The rate is calculated as the distance covered divided by the time taken. Let's compute the rates:
1. Henry:
- Distance: 2 units
- Time: 6 units
- Rate: [tex]\( \frac{2 \text{ units}}{6 \text{ units}} = \frac{1}{3} \)[/tex]
2. Adya:
- Distance: [tex]\( 1 \frac{1}{2} \)[/tex] units (which is 1.5 units)
- Time: 5 units
- Rate: [tex]\( \frac{1.5 \text{ units}}{5 \text{ units}} = \frac{1.5}{5} = 0.3 \)[/tex]
3. Lily:
- Distance: 2 units
- Time: [tex]\( 3 \frac{1}{2} \)[/tex] units (which is 3.5 units)
- Rate: [tex]\( \frac{2 \text{ units}}{3.5 \text{ units}} = \frac{2}{3.5} \approx 0.571428571 \)[/tex]
4. Bert:
- Distance: [tex]\( 1 \frac{1}{2} \)[/tex] units (which is 1.5 units)
- Time: 9 units
- Rate: [tex]\( \frac{1.5 \text{ units}}{9 \text{ units}} = \frac{1.5}{9} = \frac{1}{6} \)[/tex]
Now, let's compare these rates to identify the two runners who ran at the same rate:
- Henry's Rate: [tex]\( \frac{1}{3} \approx 0.33333333 \)[/tex]
- Adya's Rate: 0.3
- Lily's Rate: [tex]\( \approx 0.571428571 \)[/tex]
- Bert's Rate: [tex]\( \frac{1}{6} \approx 0.16666667 \)[/tex]
As we see, none of the rates match exactly.
Therefore, the two runners who ran at the same rate are not present in the given set, none of the pairs have matching rates. Hence, no two runners have a proportional relationship of time and distance.
