Lesson 8.1
Ratio: a ratio is a quotient of two
numbers.
a
b
a:b
a to b
Always given in lowest terms.
 Slope of a line is a ratio between
two points. (rise over run)
a÷b
Proportions: two or more ratios
set equal to each other.
a
c
=
b
d
a:b = c:d
a is the first term

b is the second term
c is the third term
d is the fourth term
Product and Ratio Theorems
In a product containing four terms:
First and fourth terms are the extremes.
Second and third terms are the means.
Theorem 59: In a proportion, the
product of the means is equal to
the product of the extremes.
(means-extremes product
theorem.)

a
c
=
b
d
 ad = bc
If they aren’t equal, then the ratios
aren’t in proportion.
Theorem 60: If the product of
 of non-zero numbers is
a pair
equal to the product of another
pair of non-zero numbers, then
either pair of numbers may be
made the extremes, and the
other pair the means, of a
proportion. (means-extremes
ratio theorem.)
This theorem is harder to state than to use!
Given: pq = rs
Then:

s
p
=
q
r
r
p
=
q
s
r
q
=
p
s
pq = rs
pq = rs
pq = rs

These proportions are all
 their cross
equivalent
 since
products are equivalent
equations.
In a mean proportion,
the means are the same.
1 4
=
4 16

4 is the
geometric
 mean 
a
x
=
x
r
x is the
geometric
 mean
Definition: If the means in a proportion are equal,
either mean is called a geometric mean or mean
proportional between the extremes.
Find the arithmetic & geometry means
between 3 and 27.
Arithmetic mean:
3  27
2
= 15
Geometric mean:
3
x
=
x
27
x2 = 81
x=9
Solve:
Find the fourth term
(sometimes called the
fourth proportional) of a
proportion if the first
three terms are 2, 3,
and 4.
3
7
=
x
14
You might want
to reduce the
fraction first.

7x = 42
2
4
=
3
x
x=6
2x = 12
x=6


Find the mean proportional(s)
between 4 and 16.
4
x
=
x
16
x2 = 64


x=8
If we are looking for the
length of a segment,
then only the positive
number works.
If 3x = 4y, find the ratio of x to y.
Make x and 3 the extremes and y
and 4 the means.
3x = 4y
x
4
=
y
3



x a
Is =
y b
x  2y
a  2b ?
equal to
=
y
b
Cross multiply and simplify both sets.
ay = bx 
b(x-2y) = y(a-2b)

bx-2by = ay-2by
bx = ay
Yes, they are equal.