M 1310
3.3 Variation
1
Direct Variation: If a situation is described by an equation in the
form y = kx where k is a nonzero constant, we say that y varies
directly as x or y is directly proportional to x. The number k is
called the constant of variation or the constant of proportionality.
Example: Sales Tax varies directly as the sale price of
an item.
The larger the sale, the more tax to be paid.
The smaller the sale, the less tax to be paid.
Other ways to say direct variation:
y varies directly as x.
y varies with x.
y is directly proportional to x.
y is proportional to x.
Inverse Variation: If a situation is described by an equation in
the form y = k/x where k is a nonzero constant, we say that y
varies inversely as x or y is inversely proportional to x. The
number k is called the constant of variation or the constant of
proportionality.
Example: In the formula, T = D/R, time varies inversely
as the rate, given that D is a constant.
The faster you go, less time it takes to get there.
The slower you go, the more time it takes to get there.
Joint Variation: If a situation is described by an equation in the
form y = kxy where k is a nonzero constant, we say that y varies
jointly as x and y. The number k is called the constant of variation
or the constant of proportionality.
M 1310
3.3 Variation
2
Example: In the formula, I = PRT (Simple Interest
Formula), the simple interest, I, varies jointly as the
simple interest rate, R, and time, T, given that the
principal, P, is a constant.
Example 1:
P varies directly as t. If t = 6, then P = 120. Find the
constant of proportionality and write the formula to express
this statement.
Example 2:
y varies inversely as the square of x. If x = 5, then y = –3.
Find the constant of proportionality and write the formula to
express this statement.
Example 3:
C varies directly with x and inversely with the square root
of w. If x = 4 and w = 36, then C = 10. Find the constant of
proportionality and write the formula to express this
statement.
M 1310
3.3 Variation
3
Example 4:
T varies jointly with x and then inversely with the cube
of z. If x = –2 and z = 3, then
T = 6. Find the constant of proportionality and write
the formula to express this
statement.
Example 5:
The cost of printing a magazine is jointly proportional to
the number of pages in the magazine and the number of
magazines printed. Find the expression to express this
statement.
Find the constant of proportionality if the printing cost is
$60,000 for the 4,000 copies of a 120 page magazine.
How much would the printing cost be for 5,000 copies of a
92 page magazine?
M 1310
3.3 Variation
4
Example 6:
The distance that an object falls varies directly as the
square of the time it has been falling. An object falls 16
feet in 1 second. How far will it fall in 3 seconds? How
long will it take an object to fall 64 feet?
Example 7:
The time required to assemble computers varies directly as
the number of computers assembled and inversely as the
number of workers. If 30 computers can be assembled by 6
workers in 10 hours, how long would it take 5 workers to
assemble 40 computers.