6.4 The Constant of Proportionality in
Complex Proportions
Common Core Standards
7.RP.1. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas
and other quantities measured in like or different units. For example, if a person walks 1/2
mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour,
equivalently 2 miles per hour.
7.RP.2. Recognize and represent proportional relationships between quantities. Decide whether
two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table
or graphing on a coordinate plane and observing whether the graph is a straight line through
the origin.
b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and
verbal descriptions of proportional relationships.
c. Represent proportional relationships by equations. For example, if total cost t is proportional to
the number n of items purchased at a constant price p, the relationship between the total cost
and the number of items can be expressed as t = pn.
d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the
situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
7.G.1. Solve problems involving scale drawings of geometric figures, including computing actual
lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
WARM-UP
1) Which pair of fractions is equivalent?
3 5
=
6 10
3 5
=
6 11
2) Find the Constant of Proportionality (k)
In one bag there were 3 cashews for every 6 peanuts.
In the other there were 5 cashews for every 10 peanuts.
3) Find the constant of proportionality and write an
equation for the ratio of cashews (C) to peanuts (p).
Peanuts
Cashews
6
3
8
4
10 12
5 6
The Constant of Proportionality in
Complex Proportions
Are the fractions proportional? If so, what is the
constant of proportionality?
1
3
1
2
=
2
7
x
y
2
1
7
3½
12
6
17
8½
NOTES
In a complex proportion one or more of the numerators
or denominators is a decimal or fraction. So, we need
to divide both sides to test for equivalency.
Examples
Are the fractions proportional?
8
12
=
1
1
3
2
1
3
= 2
5 5
6
EXAMPLES
Stan walked ¾ of a mile in 6 minutes. Karen walked
½ a mile in 4 minutes. Are the rates proportional? If
so, find the constant of proportionality.
EXAMPLES
A photo lab enlarged a 5 ½ inch by 7 inch photo so that
it is now 33 by 45. Was the enlargement proportional?
33
5½
7
45
EXAMPLES
Are the ratios of the sides of the triangles a
proportional relationship? Explain your reasoning.
2
4½
3
1½
4
6
6
8
EXAMPLES
"y to x" is in a proportional
relationship. Find the
constant of proportionality
and write an equation.
Is the ratio of feet traveled
(F) to seconds (t) in the table
a proportional relationship?
If so, find the constant of
proportionality and write an
equation.
x
y
1
1
6
Sec Feet
(t)
(F)
2
1
3
1
2
3
1
2
1
11
2
1
4
1
2
3
4
EXAMPLES
Is the ratio of y to x a proportional relationship? If so, find
the constant of proportionality and write an equation.
x
2
y
6
21
2
71
2
3
9
EXAMPLES
A cake recipe calls for 1¼ cups of milk and 2 3 of a cup
of butter.
Sara wanted to make a larger batch. She put 2½ cups of
milk and 1 13 cup of butter. Find the ratio of milk to
butter to determine if the batches are proportional?
EXAMPLES
Choose all that apply. Which combinations of length to
width would be proportional to the given rectangle?
W = 1in
L = 2½ in
a)  Width = 2 in, Length = 5 in
b) Width = 3 in, Length = 7 ½ in
c) Width = 4 in , Length = 10 in
d) Width = 5 in , Length = 12 in
PRACTICE
Which pair is proportional?
5
3
=
1
1
3
4
5
3
=
1
1
3
5
PRACTICE
Susan typed 130 words in 3¼ minutes. Paul typed
150 words in 3¾ minutes. Are the rates proportional?
If so, find the constant of proportionality.
PRACTICE
Find the constant of proportionality and write an
equation in terms of y to x.
x
y
2
2
3
3
1
4
4
3
FINAL QUESTION
Which shape is not proportional to the other three?
Explain your reasoning.
Width
Length
2
3
3
4½
5
10
4½
5
4
3
2
3
4
6
6
10