A quantity VARIES JOINTLY as two or more quantities, if it equals a constant times their product. For example, ff x, y, and z are variables and k is a constant, x varies jointly as y and z, if x - kyz. Note that this is similar to direct variation, except that there are two variable factors and the constant with which to contend in the one number; whereas in direct variation, we had only one variable and the constant. The equality, x = kyz, is equivalent to
 

If a quantity varies jointly as two or more other quantities, the ratio of the first quantity to the product of the other quantities is a constant.

The formula for the area of a rectangle is an example of joint variation. If A is allowed to vary, rather than being constant as in the example used earlier in this chapter, then A varies jointly as L and W. When the formula is written for general use, it is not commonly expressed as A = kLW, although this is a mathematically correct form. Since the constant of proportionality in this case is 1, there is no practical need for expressing it.

Using the formula A = LW, we make the following observations: If L = 5 and W = 3, then A = 3(5) = 15. If L = 5 and W = 4, then A = 4(5)= 20, and so on. Changes in the area of a rectangle depend on changes in either the length or the width or both. The area varies jointly as the length and the width.

As a general example of joint variation, consider the expression a bc. Written as an equation, this becomes a = kbc. If the value of a is known for particular values of b and c, we can find the new value of a corresponding to changes in the values of b and c. For example, suppose that a is 12 when b is 3 and c is 2. What is the value of a when b is 4 and c is 5?

Since quantities equal to the same quantity are equal to each other, we can set up the following proportion:

Practice problems. Using k as the constant of proportionality, write equations that express the following statements:

3. The length, W, of a radio wave varies jointly as the square root of the inductance, L, and the capacitance, C.

is an example of combined variation and is read, "E varies jointly as L and the square of W, and inversely as the square of p." Likewise,

is read, "V varies jointly as r and s and inversely as t."