Use the table to answer the question.
Hours Miles walked
1/4 1
1/2 2
3/4 3
1 4
It takes Julian 1/2 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?
answers
(1/4 , 1)
(1, 1/4)
(4,1)
(1,4)
The constant of proportionality is the ratio between the miles walked and the corresponding hours taken. In this case, the constant of proportionality is 2 miles per 1/2 hour, or 4 miles per hour.
Looking at the table, the ratio (1,4) represents the correct constant of proportionality, as it shows that for every 1 hour, Julian walks 4 miles.
To determine the ratio at which the constant of proportionality appears, we need to find where the hours and miles walked have a constant ratio.
Looking at the table, we can see that the ratio between the hours and miles walked is consistently 1:2. For example, when Julian walks for 1/2 hour, he walks 2 miles. When he walks for 1 hour, he walks 4 miles.
From the given answer choices, the ratio (1/4, 1) represents the constant of proportionality because it matches the ratio observed in the table - 1/4 hour and 1 mile walked. Therefore, the correct answer is (1/4, 1).
To find the ratio where the constant of proportionality appears, we need to look for a consistent relationship between the hours and miles walked in the table. In a proportional relationship, the ratio of the two values should be the same for all pairs of values.
Let's calculate the ratios for each pair of values:
- For the first pair, (1/4, 1), the ratio is 1/4 ÷ 1 = 1/4.
- For the second pair, (1/2, 2), the ratio is 2 ÷ (1/2) = 4.
- For the third pair, (3/4, 3), the ratio is 3 ÷ (3/4) = 4.
- For the fourth pair, (1, 4), the ratio is 4 ÷ 1 = 4.
As we can see, the ratio remains constant at 4 for all the pairs of values. Therefore, the constant of proportionality appears at the ratio of (4, 1).
So, the correct answer is (4, 1).