what is the relationship in which the ratio of the manipulated variable and the responding variable is constant
since the ratio is constant, it must be a direct variation.
Which means it is linear, but also goes through (0,0).
I would call it a linear relationship
The relationship you are referring to is called a proportional relationship. In a proportional relationship, the ratio of the manipulated variable (also known as the independent variable) to the responding variable (also known as the dependent variable) remains constant. In mathematical terms, this means that if you increase or decrease the manipulated variable by a certain amount, the responding variable will increase or decrease by the same ratio.
To determine if a relationship is proportional, you need to examine a set of data or an equation representing the relationship. If the ratio between the manipulated variable and the responding variable remains constant throughout the data or equation, then it is a proportional relationship.
For example, let's consider a scenario where the manipulated variable is the number of hours worked, and the responding variable is the amount of money earned. If the hourly wage is constant, then the relationship between the number of hours worked and the amount of money earned is proportional. If you work twice as many hours, you will earn twice as much money.
To explore if a relationship is proportional using a set of data, you can select different values for the manipulated variable and calculate the corresponding values for the responding variable. If the ratio between the values remains the same, then the relationship is proportional.