Ratios

1
Ratios Proportions

• Lesson 8.1

2
Ratio a ratio is a quotient of two
numbers. ab a to b ab
Always given in lowest terms.
Slope of a line is a ratio between two points.
(rise over run)
3
Proportions two or more ratios set equal to
each other.
ab cd

a is the first term b is the second term c is the
third term d is the fourth term
4
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.)
5

? ad bc
If they arent equal, then the ratios arent in
proportion.
Theorem 60 If the product of a pair of non-zero
numbers is 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.)
6
This theorem is harder to state than to use!
Given pq rs
Then



pq rs pq rs pq rs
These proportions are all equivalent since their
cross products are equivalent equations.
7
Geometric Mean
In a mean proportion, the means are the same.


x is the geometric mean
4 is the geometric mean
8
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
Geometric mean

x2 81 x ? 9
15
9
Solve
Find the fourth term (sometimes called the fourth
proportional) of a proportion if the first three
terms are 2, 3, and 4.

You might want to reduce the fraction first.

7x 42 x 6
2x 12 x 6
10
Find the mean proportional(s) between 4 and 16.

x2 64
x ? 8
If we are looking for the length of a segment,
then only the positive number works.
11
If 3x 4y, find the ratio of x to y.
Make x and 3 the extremes and y and 4 the means.
3x 4y

12
Is

?
equal to

Cross multiply and simplify both sets.
b(x-2y) y(a-2b) bx-2by ay-2by bx ay
ay bx
Yes, they are equal.