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8.1 Ratio and Proportion
Objective:
After studying this section, you will be able to
recognize and work with ratios and proportions. You
will be able to apply the product and ratio theorems
and calculate geometric means.
Definition
A ratio is a quotient of two
numbers
Ratios can be written in the following ways
the ratio of 5 meters to 3 meters
5
3
5:3
5 to 3
53
Unless otherwise specified, a ratio is given
in lowest terms.
The slope of a line is the ratio of the rise
between any two points on the line to the
run between to two points.
(4,11)
(1,6)
Definition
1 5

2 10
A proportion is an equation
stating that two or more
ratios are equal.
5:15 = 15:45
4 10 x 2

 
6 15 y 3
Most proportions will contain only two
ratios and will be written in one of the
following equivalent forms.
a c

a:b  c: d
b d
In both of these forms,
a is called the first term
b is called the second term
c is called the third term
d is called the fourth term
a c

b d
The Product and Ratio Theorem
In a proportion containing four terms.
The first and fourth terms are called the extremes
The second and third terms are called the means
Theorem
In a proportion, the product
of the means is equal to the
product of the extremes.
(Means-Extremes Product Theorem)
Theorem
If the product of a pair of
nonzero numbers is equal
to the product of another
pair of nonzero numbers,
then either pair of numbers
may be made the extremes,
and the other pair the
means, of a proportion.
(Means-Extremes Ratio Theorem)
What did that theorem say????
pq = rs can be written as any of
the following:
p s p r
r q
 ,  , and 
r q s q
p s
The Geometric Mean
In a mean proportion, the means are the same.
1
4 a r

,

4 16 r q
Definition
4 is the geometric mean
between 1 and 16 in the first
example and r is the geometric
mean between a and q in the
second example.
If the means in a proportion are
equal, either mean is called a
geometric means, or mean
proportional, between the
extremes.
The Arithmetic Mean
The arithmetic mean is the average between two
numbers.
Find the geometric and arithmetic means between
3 and 27
3  27
2
3
x

x 27
1. Solve for x
3
7

x 14
2. Find the fourth term (sometimes called the fourth
proportional) of a proportion if the first three terms
are 2, 3, and 4.
3. Find the mean proportional(s) between 4 and 16
4. If 3x = 4y, find the ratio of x to y.
x a
x  2 y a  2b

5. Is  equivalent to
y b
y
b
a c
ab cd

are equivalent proportion s.
6. Show  and
b d
b
d
Summary:
a
c

Show that if b d ,
then a  2b  c  2d
b
d
Homework: worksheet