Often the value of a variable quantity depends on the values of several other quantities for instance the amount of simple interest on an investment depends on the interest rate, the amount invested, and the period of time involved. For compound interest the amount of interest depends on an additional variable how often the compounding takes place. A situation in which one variable depends on several others is called combined variation. An important type of combined variation is defined as follows:

Joint Variation:
            If a variable quantity  is proportional to the product of two or more variable quantities we say that  is jointly proportional to these quantities, or the   varies jointly as (or with) these quantities.
            For instance, if  is jointly proportional to and , then  is related to and  by the formula
                                                            ,
Where  is a constant. Sometimes the word “Jointly” is omitted and we simply say that  is proportional to and .

Example:
            In chemistry, the absolute temperature of a perfect gas varies jointly as its pressure  and its volume . Given that  Kelvin when  pounds per square inch andcubic inches, find a formula for  in terms of  and  find when pounds per square inch and  cubic inches.

Solution:
            Since varies jointly as  and , there is a constant  such that . Putting ,  and , we find that
                                       or   
Thus, the desired formula is
                                                           
When  and , we have
                                                 Kelvin.

We some times have joint variation together with inverse variation. Perhaps the most important historical discovery of this type of combined variation is Newton’s law of universal gravitation, which states that the gravitation force of attraction between two particles is jointly proportional to their masses and , and inversely proportional to the square of the distance  between them. In other words, is related to , and  by the formula
                                               
Where is constant of proportionality.

Example:
            Careful measurements shows that two kilogram masses  meter a part exact a mutual gravitational attraction of Newton. (One pound of force is approximately  Newton.) the earth has a mass of kilograms. Find the earth’s gravitational force on a space capsule that has a mass of kilograms and that is meters from the center of the earth.

Solution:
            Putting  Newton, kilogram, kilogram, and  meter in the formula , we find that
                                                ,
So that, for arbitrary values of ,  and ,
                                               
Now we substitute ,, and  to obtain
                        Netwons.
(or less than  pound of gravitational force).