8-1:Inverse
Variation
Mr. Gallo
Algebra 2
Ex. 1: It takes one student 36 working hours
to wash all of the windows at the school. If
more students helped, then each student
would work less time (the total working time
would still be 36 hours).
a). Let t = time each student works and s =
number of students washing the windows, write
an equation to show this relationship.
st  36
t
36
s
b). Use your calculator to graph the equation.
Re-size the window (y-max) and use (trace).
What happens to t as s increases?
It decreases.
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c). Create a table using your calculator
and answer the following questions
1. What happens to the time each student
works if you double the number of students
working?
Time is divided by 2
2. What happens to the time each student
works if you triple the number of students
working?
Time is divided by 3
3. What happens to the time each student
works if you halve the number of students
working?
Time is divided by ½ or multiplied by 2
Creating a Scatterplot
1.
2.
3.
4.
5.
6.
7.
Press STAT
In EDIT press 1:Editor
In L1, enter the independent variable
values
In L2, enter the dependent variable
values
Press 2nd Y= and press 1:Plot 1
Turn Plot 1 on and enter L1 in for Xlist:
and L2 in for Ylist:.
Press GRAPH
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The window washing problem is an example of
an Inverse Variation
time t varies inversely
as s the # of students.
Inverse variation (varies inversely as) is the situation that
occurs when two variables x and y are so related that when
one variable is multiplied by k, the other is divided by k.
•When the number of students
decreases
the time
increases ,
.
•When the number of students
increases
the time
decreases ,
.
Inverse Variation Function:
xy  k
x
k
y
y
k
x
where k  0
x

Independent Variable:

Dependent Variable: y

“k” is the constant of variation__
In the previous example of the window washing
36
job, t 
y = t , x = s and k =36
s
3
Write an inverse variation equation that
describes the following (let k=constant of
variation):
1.
The speed s you travel in a car varies inversely
with the time t it takes you to get there.
________________________________
2.
s
k
t
The warmer the temperature t gets on a snowy
winter day varies inversely as the amount of
snow s left on the ground.
k
t
s
________________________________
Solving Inverse Variation Problems:
Follow the following four steps:
equation that describes the
1. Write an _________
function.
2.
constant of variation - k
Solve for the _________________________.
3.
_________
Rewrite the variation function using the
constant of variation found in step 2.
4.
_________
Evaluate the function for the desired value
of the independent variable.
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Solving Inverse Variations Problems

Nancy and Sam are trying to balance the seesaw.
The distance d a person sits from the fulcrum is
inversely proportional to the person’s weight w.
Sam is sitting 2 meters from the fulcrum and
weighs 55 kilograms. How far should Nancy sit
from the fulcrum if she weighs 50 kilograms?
How do we solve this? Distance = d Weight = w
1). d 
2). Find k when d  2, w  55
k
w
k
55
110  k
3). d 
2
4). d 
110
w
110
 2.2 meters
50
Ex. 2: The time T required to do a job varies
inversely as the number of workers W. It takes 5
hours for 8 cement finishers to do a job.
1.
Write an inverse variation formula that represents
this problem.
k
T
W
2.
Find the constant of variation, k
3.
How long will it take 12 workers to do the same job?
T
k
8
40  k
5
40
1
 3 hours or 3 hours and 20 minutes.
12
3
5
Combined and Joint Variations
Combined Variation

One quantity varies
two or more
with _____________
quantities.
Joint Variation

One quantities varies
directly
___________
with
two
or
more
______________
quantities.
Equation
Form
Combined Variation
z  kxy
kxy
z
w
kx
z
wy
z varies jointly with x and y.
z varies jointly with x and y and inversely
with w.
z varies directly with x and inversely with
the product wy.
The volume of gas varies directly with its temperature and
inversely with pressure. Volume is 100 m3 when the temperature
is 150K and the pressure is 15 lb/cm2. What is the volume when
the temperature is 250K and the pressure is 20 lb/cm2?
V = volume of gas
1). V

kt
p

2). 100 

t= temperature
150k 15

15  150
10  k
3). V

p= pressure
10t
p
4). V

10  250 
20
V  125
The volume of the gas is 125 m3.
6
Homework: p. 503-505: 7-17 odd, 20, 23, 28-36 even
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