JOINT VARIATION
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 µ
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?
Rewriting the proportion,
Thus
Also,
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:
1. 2 varies jointly as x and y.
2. S varies jointly as b times the square of r.
3. The length, W, of a radio wave varies
jointly as the square root of the inductance,
L, and the capacitance, C.
Answers:
COMBINED VARIATION
The different types of variation can be combined. This is frequently the case
in applied problems. The equation
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."
