Question

Nicholas and his friends set out to sea on their annual fishing trip.
This table shows the relationship between the time (in hours) Nicholas and his friends spend sailing, x, and their distance from shore (in miles), y.
x (hours) y (miles)
1 14
2 28
3 42
5 70
According to the values in the table, do x and y have a proportional relationship?
yes
no
What is the constant of proportionality? Write your answer as a whole number or decimal?

Answer 1

To determine if \( x \) (time in hours) and \( y \) (distance in miles) have a proportional relationship, we need to see if the ratio \( \frac{y}{x} \) is constant.

Let's calculate the ratio for each pair of values:

1. For \( x = 1 \), \( y = 14 \):
\[
\frac{y}{x} = \frac{14}{1} = 14
\]

2. For \( x = 2 \), \( y = 28 \):
\[
\frac{y}{x} = \frac{28}{2} = 14
\]

3. For \( x = 3 \), \( y = 42 \):
\[
\frac{y}{x} = \frac{42}{3} = 14
\]

4. For \( x = 5 \), \( y = 70 \):
\[
\frac{y}{x} = \frac{70}{5} = 14
\]

Since all the calculated ratios are equal, \( \frac{y}{x} = 14 \), we can conclude that there is a proportional relationship between \( x \) and \( y \).

Thus, the answer is **yes**, \( x \) and \( y \) have a proportional relationship.

The constant of proportionality is 14.