Math 114: Direct Variation and Indirect Variation
Direct Variation: We say that “y varies directly with x” if y  kx where k is a constant
of proportionality (“y varies directly with x” is the same as “y is directly proportional to x.”)
Example 1: If y  5 x then y varies directly with x and k  5 .
1
1
Example 2: If y    x then y varies directly with x and k  .
3
3
Example 3: Suppose y varies directly as x (or y is directly proportional to x) and
y  6 when x  3. Find k . Substitute x and y into
y  kx  6  k  3  k  2 . Use this k in the problems below.
A. Find y when x  20 . y  kx  y  2(20)  y  40
B.
Find x when y  250.62 . y  kx  250.62  2( x)  x  125.31
Indirect Variation:
We say that “y varies indirectly or inversely with x” if y 
k
where
x
k is a constant of proportionality (“y varies indirectly with x” is the same as “y is inversely
proportional to x.”)
8
, then y varies indirectly with x and k = 8.
x
1.5
Example 2: If y 
, then y varies indirectly with x and k = 1.5.
x
Example 1: If y 
Suppose y varies indirectly with x (or y is inversely proportional to x) and y = 8 when x =
y  8 when x  32 . Find k . Substitute x and y into
y
k
x
 8
k
 k  256 . Use this k in the problems below.
32
y
A.
Find y when x  96 .
B.
Find x when y  128 . y 
k
256 8
 y

x
96 3
k
256
 128 
 128 x  256  x  2
x
x
PROBLEMS
1. y varies directly with x and y = 20 when x = 4. Find:
A. y when x = 1/3
B. x when y = 200
2. y varies indirectly with x and y = 20 when x = 4. Find:
A. y when x = 1/3
B. x when y = 200
3. The recommended dosage , D, of a drug is directly proportional to a person’s weight, W.
The units for D are milligrams and the person’s weight is in pounds.
A. Write an equation using D, W and k.
B. If the recommended dosage is 3060 mg for a person who weighs 180 pounds, find the
recommended dosage for a person who weighs 140 pounds.
4. The number of painters, N, hired to paint a 2500 square foot house is inversely proportional
to the time, t, required to paint a house. t is in days
A. Write an equation with N, t, and k.
B. If it takes 5 painters 4 days to paint a 2500 square foot house, how many days will it
take 8 painters?
Answers:
1. A. 5/3 B. 40 2. A. 240 B. 0.4 A. D = kW B. 2380 mg
days
4. A. N = k/t B. 2.5