Topic: Direct,
Inverse, and Joint
Variation
Questions/Main
Ideas:

How can
variables be
related to each
other?
Name: ________________________________ Date: _______________________
Objective: We will write equations of direct, inverse, and joint variation. We will use
these equations to find other values including real-life situations.
Variation is when two or more quantities are related to each other are said to vary
 _________________________________
 _________________________________
 _________________________________
All variation problems involve a ________________________________________ but whether the two
quantities grow or decrease together determines what type of variation you will use.

What are the
equations of
direct, inverse,
or joint
variation?
Variation
Equation
How are the quantities related to
each other?
Direct
Inverse
Joint
Based on the chart, determine which type of variation best describes each scenario.
Points for a touchdown
in football

What are some
examples of
variation?
Price of a movie ticket
versus attendance
Refreshment income for
# of stands, # of people,
and $2.00 hotdogs
Examples:
Ex 1: If y varies directly as x, and y = 12 when x = 15, find x when y = 21.
Ex 2: If g is inversely proportional to the square of m, and g = 3 when m = 4,
find g when m = 6.
Ex 3: The area, A, of a triangle varies jointly as the base, b, and the height, h.
find the equation of joint variation if A = 100, b = 25, and h = 8.
Ex 4: Determine from the tables below if the data shows a direct, inverse, or no
variation. If so, write an equation relating y and x.
A)
X
Y
3
-1
6
-2
9
-3
12
-4
15
-5
B)
X
Y
1
7
2
9
3
11
4
13
5
15
C)
X
Y
1.5
40
2.5
24
4
15
7.5
8
10
6
Ex6: Write the equation for each variation described.
A) w varies directly with the cube of m
Be sure to pay attention to your variables!
B) q varies jointly with a and b
C) z varies directly with x and inversely with the product of w and y
Ex7: For each of the questions below, write the equation and then solve:

How can you
solve world
problems
based on
variation?
A) In a certain manufacturing process, the cost of producing a single item varies
inversely as the square of the number of items produced. If 100 items are
produced, each costs $2. Find the cost per item if 400 items are produced.
B) The number of vibrations per second (the pitch) of steel guitar string varies directly
as the square root of the tension and inversely as the length of the string. If the
number of vibrations per second is 5 when the tension is 225 kilograms and the
length is 0.60 meter, find the number of vibrations per second when the tension is
196 kilograms and the length is 0.65 meter.
Summary: