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Question

Select the correct text in the table.
A group of friends wanted to compare their average running speeds. They recorded the distance and amount of time each person ran one Saturday morning.
Select all the runners whose speeds are in a proportional relationship with each other.

Name Time (seconds) Distance (miles)
Liam 306.6 0.5
Taylor 756.36 1.2
Sarah 504.35 0.7
Ashley 459.9 0.75
Connor 600.5 1
Nathan 942.75 1.5
Juan 827.82 1.35
Katie 429.48 0.6

Answer 1

To determine which runners have speeds that are in a proportional relationship, we need to calculate the speed (in miles per second) for each runner. The formula for speed is:

$$\text{Speed}=\frac{\text{Distance}}{\text{Time}}$$

We'll calculate the speed for each runner using the provided time (in seconds) and distance (in miles):

1. Liam: $$\text{Speed}=\frac{0.5}{306.6}\approx 0.00163\text{ miles/second}$$

2. Taylor: $$\text{Speed}=\frac{1.2}{756.36}\approx 0.00159\text{ miles/second}$$

3. Sarah: $$\text{Speed}=\frac{0.7}{504.35}\approx 0.00139\text{ miles/second}$$

4. Ashley: $$\text{Speed}=\frac{0.75}{459.9}\approx 0.00163\text{ miles/second}$$

5. Connor: $$\text{Speed}=\frac{1}{600.5}\approx 0.00167\text{ miles/second}$$

6. Nathan: $$\text{Speed}=\frac{1.5}{942.75}\approx 0.00159\text{ miles/second}$$

7. Juan: $$\text{Speed}=\frac{1.35}{827.82}\approx 0.00163\text{ miles/second}$$

8. Katie: $$\text{Speed}=\frac{0.6}{429.48}\approx 0.00140\text{ miles/second}$$

Now we can summarize the speeds:

• Liam: $\approx 0.00163$
• Taylor: $\approx 0.00159$
• Sarah: $\approx 0.00139$
• Ashley: $\approx 0.00163$
• Connor: $\approx 0.00167$
• Nathan: $\approx 0.00159$
• Juan: $\approx 0.00163$
• Katie: $\approx 0.00140$

To find runners in a proportional relationship, we look for runners where the ratio of distance to time remains constant.

From the calculations, we see:

• Liam, Ashley, and Juan all have a speed of approximately $0.00163$.
• Taylor and Nathan both have a speed of approximately $0.00159$.

Thus, the runners whose speeds are in a proportional relationship as they have the same ratio of distance to time are:

• Liam
• Ashley
• Juan

And

• Taylor
• Nathan

These pairs are in a proportional relationship based on their average speeds.