Direct, Inverse, and Joint Variation
Type of Variation
Direct Variation
Equation
y  kx
or
Meaning
k
y
x
y varies directly as x
k is the constant of variation
Ex. Write the equation being described by each of the following statements.
1. G varies directly as t squared.
2. The volume, v, of a balloon is directly proportional to the cube of the balloon’s radius, r.
To solve, set up your problem as follows:
y1 y2

x1 x2
Ex. Write an equation for and solve each of the following word problems.
3. Find x when y = 8 if x varies directly as y and y = 2 when x = 5.
4. The cost, c, in cents of lighting a 100-watt bulb varies directly as the time, t, in hours,
that the light is on. The cost of using the bulb for 1,000 hours is $0.15. Determine the
cost of using the bulb for 2,400 hours.
Direct, Inverse, and Joint Variation
Type of Variation
Inverse Variation
Equation
y
k
x
or
Meaning
k  yx
y varies inversely as x
k is the constant of variation
Ex. Write the equation being described by each of the following statements.
1. y varies inversely as the square root of x
2. The number, n, of grapefruit that can fit into a box is inversely proportional to the cube of
the diameter, d, of each grapefruit.
To solve, set up your problem as follows:
y1 x1  y2 x2
Ex. Write an equation for and solve each of the following word problems.
3. If y varies inversely as x and y = 5 when x = 2 find y when x = 12.
4. Bob’s dentist determined that the number of cavities developed in his patient’s mouth
each year is inversely proportional to the total number of minutes spent brushing during
each session. If Bob developed 4 cavities during the year and spent only 30 seconds
brushing his teeth each time, how many annual cavities will Bob develop if he increases
his brushing time to 2 minutes per session?
Direct, Inverse, and Joint Variation
Type of Variation
Joint Variation
Equation
y  kxz
or
k
Meaning
y
xz
y varies jointly as x and z
k is the constant of variation
Ex. Write the equation being described by each of the following statements.
1. Y varies jointly as W and X
2. The area, A, of a triangle is jointly proportional to its base, b, and height, h.
To solve, set up your problem as follows:
y1
y
 2
x1 z1 x2 z2
Ex. Write an equation for and solve each of the following word problems.
3. Find y when x = 12 and z = 2. if x varies jointly as y and z and y = 24 when z =3 and x =1.
4. The power, P, in watts of an electrical circuit varies jointly as the resistance, R, and the
square of the current, C. For a 240-watt refrigerator that draws a current of 2 amperes, the
resistance is 60 ohms. What is the resistance of a 600-watt microwave oven that draws a
current of 5 amperes?
Direct, Inverse, and Joint Variation
Combined Variation – Direct, Inverse, and Joint
Ex. Write the equation being described by each of the following statements.
1. G varies inversely as d and jointly as b and c
2. The weight, w, that a column of a bridge can support varies directly as the fourth power of
its diameter, d, and inversely as the square of its length, l.
Ex. Write an equation for and solve each of the following word problems.
3. y varies directly as x and inversely as z, y = 3 when x = 6 and z = -4. Find z when y = -5
and x = 10.
4. The force needed to keep a car from skidding on a curve varies directly as the weight of the
car and the square of the speed and inversely as the radius of the curve. Suppose a 3,960 lb.
force is required to keep a 2,200 lb. car traveling at 30 mph from skidding on a curve of a
radius 500 ft. How much force is required to keep a 3,000 lb. car traveling at 45 mph from
skidding on a curve of a radius 400 ft.?