Difference Between Proportional & Linear Relationships

Difference Between Proportional & Linear Relationships
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Mathematicians, physicists and engineers have many terms to describe mathematical relationships. There is usually some logic to the names chosen, although this is not always apparent if you are not aware of the math behind it. Once you understand the concept involved the connection to the chosen words becomes obvious.

TL;DR (Too Long; Didn't Read)

The relationship between variables can be linear, non-linear, proportional or non-proportional. A proportional relationship is a special kind of linear relationship, but while all proportional relationships are linear relationships, not all linear relationships are proportional.

Proportional Relationships

If the relationship between “​x​” and “​y​” is proportional, it means that as “​x​” changes, “​y​” changes by the same percentage. Therefore, if “​x​” grows by 10 percent of “​x​,” “​y​” grows by 10 percent of “​y​.” To put it algebraically:

y = mx

where “​m​” is a constant.

Consider a non-proportional relationship. Children look different than adults, even in photographs where there is no way to tell exactly how tall they are, because their proportions are different. Children have shorter limbs and bigger heads compared to their bodies than adults do. Children's features, therefore, grow at disproportionate rates as they become adults.

Linear Relationship

Mathematicians love to graph functions. A linear function is very easy to graph, because it is a straight line. Expressed algebraically, linear functions take the form

y = mx + b

where “​m​” is the slope of the line and “​b​” is the point where the line crosses the “​y​” axis. It is important to note that “​m​” or “​b​” or both constants can be zero or negative. If “​m​” is zero, the function is simply a horizontal line at a distance of “​b​” from the “​x​” axis.

The Difference

Proportional and linear functions are almost identical in form. The only difference is the addition of the “​b​” constant to the linear function. Indeed, a proportional relationship is just a linear relationship where ​b​ = 0, or to put it another way, where the line passes through the origin (0, 0). So a proportional relationship is just a special kind of linear relationship, i.e., all proportional relationships are linear relationships (although not all linear relationships are proportional).

Examples of Proportional and Linear Relationships

A simple illustration of a proportional relationship is the amount of money you earn at a fixed hourly wage of $10 an hour. At zero hours, you have earned zero dollars, at two hours, you have earned $20 and at five hours you have earned $50. The relationship is linear because you get a straight line if you graph it, and proportional because zero hours equals zero dollars.

Compare this with a linear but non-proportional relationship. For example, the amount of money you earn at $10 an hour in addition to a $100 signing bonus. Before you start working (that is, at zero hours) you have $100. After one hour, you have $110, at two hours $120, and at five hours $150. The relationship still graphs as a straight line (making it linear) but is not proportional because doubling the time you work does not double your money.

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