Gary and richard are practicing tying knots. richard already has 3 knots tied when gary starts.

consider the graph below which shows the relationship between the number of knots richard and gary tie and the amount of time they take.

which of the lines shows a proportional relationship, and what is the constant of proportionality?

a. richard's line shows a proportional relationship, and the constant of proportionality is 2.

b. richard's line shows a proportional relationship, and the constant of proportionality is 0.5.

c. gary's line shows a proportional relationship, and the constant of proportionality is 1.33.

d. gary's line shows a proportional relationship, and the constant of proportionality is 0.75.

In graphs of linear relationships, you can always determine a proportional linear relationship by observing where the line crosses the y axis. If the line crosses through the origin, then the relationship is proportional. This means that the independent data(in this case time) can be multiplied by a constant factor to always get it's related dependent piece of data.

From the graph you might notice some data points for Gary: At 2mins he completes 1.5 knots or the ordered pair (2, 1.5). We see another data point at 4mins he completes 3 knots or the ordered pair (4,3). How about the ordered pair (6, 4.5)? Notice that if I multiply the "x" coordinate of all these ordered pairs by 0.75, I get the "y" coordinate. Or maybe another way to look at it, the "y" coordinate of each point on the line divided its corresponding "x" coordinate will always produce the same number, 0.75. This number, 0.75, is called the constant of proportionality.

So, the answer to this problem is choice "D"

Summary: To identify proportional linear relationship from a graph, look for the line that goes through the origin. To find the constant of proportionality, determine the coordinates of a convenient point on the line and divide the y coordinate by the x .