1. Ratios & Proportions
Lesson 8.1

2. Ratio: a ratio is a quotient of two numbers. a:b a to b a÷b
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.
a:b = c:d =
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 aren’t equal, then the ratios aren’t 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. Given: pq = rs Then: = = = pq = rs pq = rs pq = rs
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: = = x is the geometric mean 4 is the geometric mean
In a mean proportion, the means are the same. = =
x is the geometric mean
4 is the geometric mean

8. Find the arithmetic & geometry means between 3 and 27.
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. = x2 = 64 x =  8 Find the mean proportional(s) between 4 and 16.
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 = b(x-2y) = y(a-2b) bx-2by = ay-2by bx = ay ay = bx
Cross multiply and simplify both sets.
b(x-2y) = y(a-2b)
bx-2by = ay-2by
bx = ay
ay = bx
Yes, they are equal.