Directly Proportional

Proportional Relationships

A proportional relationship is one in which two quantities vary directly with each other.

Example: The radius and circumference of a circle are proportional.

Volume and weight of a substance are also proportional (The weight of the displaced fluid is directly proportional to the volume of the displaced fluid [Archimedes‘ principle])

The equation of a proportional relationship is Y=kX, because the variable Y varies directly as X;

k is a constant, (k is called the constant of proportionality or unit rate).

Another way of writing this equation is k=Y/X    (k is the ratio of two variables, and its value is constant).

Y is called the dependent variable because it “depends” on the independent variable X.

The graph of a proportional relationship is a straight line and passes through the origin (0,0).

In a table of a proportional relationship, all the ratios (Y/X) are equivalent.

Example.  Is Y proportional to X?


0 3 6 9
Y 0 1 2







All the ratios are equivalent  (k=1/3).   Y=k X,  so  Y = ⅓ X

The two quantities always have the same relative size or “ratio”













The ordered pairs are proportional and the graph passes through (0,0), so Y = X/3  is a proportional relationship.

The conversion factor is:   1 yard = 3 feet  (there are 3 feet in each yard)



Directly Proportional


The equations of such relationships are always in the form Y = m•X, and when graphed produce a line that passes through the origin.  In this equation, “m” is the slope of the line   (or coefficient of proportionality, unit rate, the rate of change).


Proportions can be used when converting between units of measure


  1. Set up your proportion.
  2. Multiply means and extremes
  3. Solve for “x” algebraically.


ME02     Customary

ME20     Liquid

ME05     Length 1

ME06     Length 2

ME12     Time 1

ME21     Time 2

ME09     Weight 1

ME10     Weight 2

ME22     Weight 3