Proportional Relationships and Unit Rates
Name: _______________________________
proportional relationship
A ____________________________
is a relationship between two quantities in which
ratio
constant
the _________
of one quantity to the other quantity is _______________.
equation
y = kx
A proportional relationship can be described by an ________________
of the form _____________,
unit rate
constant of proportionality
where k is a number called the ___________________________________,
also known as the ______________.
Identifying Proportional Relationships
Example 1
Dylan makes $336 for 32 hours of work, and Angela makes $420 for 42 hours of work.
1) How much do Dylan and Angela each make per hour?
336
 $10.50 / hr
32
420
 $10 / hr
42
$10.50/hr
Dylan: __________________
$10/hr
Angela: _________________
2) Is Dylan’s wage for 32 hours proportional to Angela’s wage for 42 hours? Why or why not?
No. The unit rates are not equal.
Compare the unit rates
to see if they are equal
Example 2
Table 1
Number of
Hours
Find the ratio of y to x for Table 1 and Table 2, simplify the fraction to simplest form, and answer the
questions that follow.
Table 2
y
y
Total Cost
Number of
Total Cost
Ratio:
Ratio:
($)
Hours
($)
x
x
1
$75
2
$120
3
$165
4
$210
5
$255
75/1=75
they are
120/2=60
they are
165/3=55
they are
210/4=52.5
they are
255/5=51
they are
1
$45
2
$90
3
$135
4
$180
5
$225
45/1=45
they are
90/2=45
they are
135/3=45
they are
180/4=45
they are
225/5=45
they are
Table 2
3) Which table shows a proportional relationship? _____________
The ratio y/x is constant
4) What makes it a proportional relationship? ______________________________________
k = 45 Write the equation for that data: ______________
y = 45x
5) What is the constant of proportionality, k, for that data? _______
Check to see if the ratio y/x is constant
Example 3
Below are the graphs for the tables in the previous section. Use the graphs to determine proportionality.
Graph 1:
RISE/RUN = 45/1 = 45
Graph 2:
RISE/RUN = 45/1 = 45
Graph 2
6) Which graph shows a proportional relationship? _____________
( Hint: A proportional relationship is y = kx )
graph is a straight line through the origin
7) What makes it a proportional relationship? ______________________________________
k is the slope of the line
8) How can you find the constant of proportionality, k, from the graph? _________________________________
Check to see if the graph is a straight line through the origin.
(A linear equation with y-intercept = 0)
For each table below, determine whether or not it represents a proportional relationship.
If so, find k, and write the equation.
1) Yes; k = -3; y = -3x
2) Yes; k = 2; y = 2x
3) No.
4) Yes; k = 1.5; y = 1.5x
11) During her first hour on Saturday, Fiona sold 8 sweaters at the Sweater Barn. After the second hour, she had sold 15
sweaters in all. After the third hour, she had sold 22 sweaters.
a) Tell whether the relationship between the number of hours and the number of sweaters sold is proportional or
non-proportional. Explain.
# hours
1
2
3
8/1=8 , 15/2=7.5 and 22/3=7.33 are not equal, so this
# sweaters
sold
8
15
22
b) Is the relationship linear? Explain.
relationship is non-proportional.
Yes. Although the ratio of y/x is not constant,
the ratio of change in y/change in x is constant, 7/1=7.
12) Is the following relationship proportional? Explain.
y
Number of
Total Cost of
Yes. The ratio y/x is constant, so it is proportional.
Movie tickets (x)
tickets (y)
x
1
2
3
4
6
12
18
24
6
6
6
6
13) How is a proportional relationship different from a non-proportional relationship?
In a proportional relationship, the ratio y/x is constant. Its graph is a line through the origin.