To determine the constant of proportionality in the data table, we need to look at the ratio of miles walked to hours taken.
In the table provided:
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For the entry (1/4, 1): \[ \text{Ratio} = \frac{1 \text{ mile}}{1/4 \text{ hour}} = 4 \text{ miles per hour} \]
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For the entry (1/2, 2): \[ \text{Ratio} = \frac{2 \text{ miles}}{1/2 \text{ hour}} = 4 \text{ miles per hour} \]
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For the entry (3/4, 3): \[ \text{Ratio} = \frac{3 \text{ miles}}{3/4 \text{ hour}} = 4 \text{ miles per hour} \]
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For the entry (1, 4): \[ \text{Ratio} = \frac{4 \text{ miles}}{1 \text{ hour}} = 4 \text{ miles per hour} \]
In each case, the ratio of miles walked to hours taken is consistent at 4, indicating that there is a constant rate of 4 miles per hour for all entries.
However, the question asks for the ratio at which the constant of proportionality appears in the choices given:
- (1/4, 1)
- (1/2, 2)
- (3/4, 3)
- (1, 4)
All of these points satisfy the constant rate of 4 miles per hour, but to directly answer the question based on the options given, you can choose any of them as the ratio of miles walked to hours taken that reflects the constant of proportionality.
Each option indicates a point where the constant of proportionality (the ratio) appears. However, it seems the context of your question is hinting towards the fact that (1/2, 2) could be the most relevant point since it clearly shows that in half an hour, Julian walks 2 miles, which is in line with the original information given about Julian walking 2 miles in half an hour.
Therefore, all options reflect the same rate, but if a single answer must be chosen, I would recommend (1/2, 2), as it directly illustrates the initial condition described.