Final answer:
Two quantities are proportional if they maintain a constant ratio. Directly proportional variables are represented by a linear relationship with a graph passing through the origin, while inversely proportional variables result in a curved graph. To evaluate a set of values, check if the ratio remains constant.
Step-by-step explanation:
Two quantities are considered proportional if they have a constant ratio, which means that as one quantity changes, the other changes in a way that maintains the same proportion. For example, if x is directly proportional to y, this relationship can be represented by the equation y = kx, where k is the constant of proportionality. When plotted on a graph, directly proportional relationships yield a straight line that passes through the origin (0, 0), indicating that as x increases, y increases at a constant rate.
On the other hand, an inversely proportional relationship means that as one variable increases, the other decreases, and vice versa, which is represented by the equation y = k/x. When graphed, inversely proportional quantities produce a curve that approaches the axes but never intersects them.
To determine if the given pairs of values represent proportional quantities, one would typically calculate the ratio y/x for each pair and see if it remains constant. In cases where there's an added constant, as in the linear equation y = mx + b where b is nonzero, the relationship is not purely proportional, as the line does not pass through the origin.