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Use the table to answer the question.

Hours - Miles walked
14 - 1
12 - 2
34 - 3
1 - 4
It takes Julian 12 hour to walk 2 miles. He decides to start walking in his spare time, but because he wants to make sure he has enough time to go a desired distance he created the data table. At which ratio does the constant of proportionality appear?

(1 point)
Responses

(1/4, 1)

(4, 1)

(1, 1/4)

(1, 4)

3 answers

To find the ratio that represents the constant of proportionality for the data provided in the table, we look for a consistent relationship between the hours walked and the miles walked.

From the table:

  • For 1 mile, it takes 14 hours: \( \frac{14}{1} \)
  • For 2 miles, it takes 12 hours: \( \frac{12}{2} = 6 \)
  • For 3 miles, it takes 34 hours: \( \frac{34}{3} \approx 11.33 \)
  • For 4 miles, it takes 1 hour: \( \frac{1}{4} = 0.25 \)

To determine the constant of proportionality, we can express it as miles per hour. In the case of this provided information, we can summarize the ratios:

  • The constant ratios between hours and miles may not show consistent proportionality across all data points, but to interpret which ratio fits your choices:

  • For every mile:

    • Starting from 1 mile: \( \frac{1}{14} \)
    • Up to 2 miles: \( \frac{2}{12} = \frac{1}{6} \)
    • Up to 3 miles: \( \frac{3}{34} \)
    • Up to 4 miles: \( \frac{4}{1} = 4 \)

From your options:

  • (1/4, 1) means it takes 4 hours to walk 1 mile, which might correspond to some ratios.
  • (4, 1) suggests 1 mile over 4 hours.
  • (1, 1/4) could suggest it takes 1 mile per 4 hours.
  • (1, 4) suggests walking 1 mile in 4 hours.

Based on the most consistent and simplified ratios we can find comparing time to distance, option (4, 1) meaning it takes 4 hours to walk 1 mile seems to be one clear piece for understanding.

However, since there's no single constant of proportionality and since the data points do not maintain consistent ratios, my conclusion grants that option:

(1/4, 1) appears to be closely aligned to one of the constant proportional ratios given that it suggests 4 hours for 1 mile effectively displays a consistent figure far alongside.

Therefore, the final answer is: (1/4, 1).

double check

Let's carefully analyze the data again to determine the constant of proportionality between hours and miles based on the information given.

The values from the table are:

| Hours | Miles Walked | |-------|---------------| | 14 | 1 | | 12 | 2 | | 34 | 3 | | 1 | 4 |

To find the constant of proportionality (or the rate of miles per hour), we take the ratio of hours to miles for each entry:

  1. For 1 mile in 14 hours: \[ \text{Ratio} = \frac{14 \text{ hours}}{1 \text{ mile}} = 14 \]

  2. For 2 miles in 12 hours: \[ \text{Ratio} = \frac{12 \text{ hours}}{2 \text{ miles}} = 6 \]

  3. For 3 miles in 34 hours: \[ \text{Ratio} = \frac{34 \text{ hours}}{3 \text{ miles}} \approx 11.33 \]

  4. For 4 miles in 1 hour: \[ \text{Ratio} = \frac{1 \text{ hour}}{4 \text{ miles}} = 0.25 \]

Now let's rewrite the analysis in terms of the constant of proportionality using the correct approach:

Looking at the data, the key is to identify a consistent ratio of hours per mile within the options provided.

  • (1/4, 1) means walking 1 mile takes 4 hours, which is the same as \(4 \times 1 = 4\) hours for 1 mile.
  • (4, 1) means walking 1 mile takes 1/4 of an hour (which is 15 minutes).
  • (1, 1/4) implies that for 1 hour, you can walk 4 miles.
  • (1, 4) suggests that you take 1 hour to walk 4 miles.

Given the original table and the specified times, the best-fitting ratio comes from simplifying how long it takes to walk a set distance.

The only pairing that aligns cleanly with the earlier hours to miles consistent interpretation suggests it takes 4 hours for every mile walked as set out:

This strongly fits the full analysis through hour and distance setting, leading to the conclusion that:

(1/4, 1), because in setting forth the relationships, it manages to confirm walking 1 mile approximates a broader expectations set.

Thus, the correct answer with clarified consistency to the provided options remains indeed (1/4, 1) as the key constant of proportionality emerges directly impacted through outlined entries. Thank you for the opportunity to delve deeper into the analysis!