How you know when quantities vary proportionally?

Answer 1

If, a unit increase in one is accompanied by the same positive increase in the other (whatever the starting value), and when one is zero the other is also zero.

In algebraic terms, if x and y are the two variables, and their relationship is given by

y = f(x)

then the first condition states that dy/dx is a positive constant, say m. Equivalently, the relationship is the linear function,

y = mx + c.

The second condition then states that the intercept term, c, is zero.

Thus the relationship reduces to y = mx.