For a Direct Relationship, the ratio of the Dependent Variable to the Control Variable is constant. Another way to show this is to set the ratio equal to a constant as shown on the right.

Now, if we can write the mathematical statement we did above, we can also apply simple rules of algebra to change it into the examples on the right.

From the previous mathematical statements, we see that there is a simple algebraic statement linking the dependent to the control variable.

Often we don't worry about the actual constant, and replace the constant and the equals sign with a symbol called "is proportional to" - a.

This symbol, actually the Greek letter Alpha, is used to represent "is proportional to". It's the symbol that we use to represent all the meaning that's above and is the one you'll explore more fully in the next few cards.

And this is what we'll be working with all along !



The mathematical statement on the right states that the "Mass is directly proportional to the Volume."

This statement also has a deeper meaning in that an equation can be written which looks like the lower one.

And this is all we may need in order to work well with a mathematical relationship!

If Mass is directly proportional to the Volume, this means that should the volume be doubled (2x) then the mass will likewise be doubled. If the volume is tripled (3x) then the mass is also tripled.

See how the original data table concepts of factors would apply here with proportions. That's just what you were doing back then!!

So one way to work with proportions is to make a factor change in the control variable, and then look at the factor change in the dependent variable.

With direct proportions this is a straight-forward process.

As you'll see next, the same idea can be applied to inverse proportions, too.


In our example from the Inverse Proportion, we double the control variable, Frequency. This results in the dependent variable, the Period, being reduced to one-half.

We recognize this as the reciprocal of 2, thus the Inverse Proportion.

Here the change of control is five times the original, and the dependent variable is reduced to one-fifth the original.


When there is a Square Proportion, the factor change in the control variable is accompanied by a squared factor change in the dependent.

Note how a doubling of the length leads to a quadrupling of the area. This is characteristic of a Square Proportion.

Note how halving the length leads to a quartering of the area. This is characteristic of a Square Proportion.


In the Square Root Proportion an increase in the Control Variable will lead to an increase in the Dependent Variable, but only the square root of the size of the change.

Quadrupling the size of the Radius leads to only a doubling of the period in our example.

And that's what we mean when we say that something is "proportional to" another quantity. We use the alpha symbol to represent this relationship, but as you will find when you go to the "Solving" page, you can solve some very sophisticated problems with nothing more than proportions. Good luck!
Updated August 2001