Hours Miles walked
1/4 1
1/2 2
3/4 3
1 4
It takes Julian 1/2 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?
Options:
(1,4)
(1/4,1)
(4,1)
(1,1/4)
The constant of proportionality can be calculated by dividing the miles walked by the hours taken. Looking at the data, we can observe that the ratio of miles walked to hours for each row is the same:
1/4 -> 1/2 = 1/2
1/2 -> 2/1 = 2
3/4 -> 3/1.5 = 2
1 -> 4/0.5 = 8
Therefore, the constant of proportionality in this data table is 2.
To determine the ratio at which the constant of proportionality appears, we can look for a pattern in the table. The table shows the relationship between the hours walked and the miles walked.
Looking at the data, we can see that each time the number of miles walked doubles, the corresponding number of hours also doubles. This means that there is a constant ratio of 1 hour for every 2 miles walked. Therefore, the constant of proportionality appears at the ratio (1,4).
Therefore, the correct option is (1,4).
To find the constant of proportionality in this situation, we need to determine the ratio between the miles walked and the corresponding amount of time taken in each row of the table.
Let's calculate the ratios for each row:
For the first row: 1 mile / (1/4) hour = 4 miles/hour
For the second row: 2 miles / (1/2) hour = 4 miles/hour
For the third row: 3 miles / (3/4) hour = 4 miles/hour
For the fourth row: 4 miles / 1 hour = 4 miles/hour
In each case, we get a ratio of 4 miles/hour. This means that the constant of proportionality is equal to 4.
Now, let's look at the options given:
(1,4) - This ratio represents 1 mile per 4 hours, which is not the constant of proportionality because it does not give a ratio of 4 miles/hour.
(1/4,1) - This ratio represents 1/4 mile per 1 hour, which is not the constant of proportionality because it does not give a ratio of 4 miles/hour.
(4,1) - This ratio represents 4 miles per 1 hour, which is not the constant of proportionality because it does not give a ratio of 4 miles/hour.
(1,1/4) - This ratio represents 1 mile per 1/4 hour, which is equal to 4 miles per hour. Therefore, this is the correct ratio for the constant of proportionality.
So, the answer is: (1,1/4).