Proportions
An equation that expresses the equality of two rational expressions is called a
proportion. The equation
is a proportion. The terms in the position of b and c are called the means. The terms
in the position of a and d are called the extremes. If we multiply this proportion by
the LCD, bd, we get
or
ad = bc.
The equation ad = bc says that the product of the extremes is equal to the product
of the means. When solving a proportion, we can omit multiplication by the LCD
and just remember the result, ad = bc, as the extremesmeans property.
ExtremesMeans Property
If
then
ad = bc.
The extremesmeans property makes it easier to solve proportions.
Example 1
A proportion with one solution
Solve
Solution
Rather than multiplying by the LCD, we use the extremesmeans property to eliminate
the denominators:



20(x + 20) 
= 30x 
Extremesmeans property 
20x + 400
400
40 
= 30x = 10x
= x 

Check 40 in the original equation. The solution set is {40}.
Example 2
A proportion with two solutions
Solve
Solution
Use the extremesmeans property to write an equivalent equation:
x(x + 3) 
= 2 Â· 5 
Extremesmeans property 
x^{2} + 3x 
= 10 

x^{2} + 3x 
10 
= 0 

(x + 5)(x  2) 
= 0 
Factor. 
x + 5 = 0 
or 
x  2 
= 0 
Zero factor property 
x = 5 
or 
x 
= 2 

Both 5 and 2 satisfy the original equation. The solution set is {5, 2}.
Caution
Use the extremesmeans property only when solving a proportion.
It cannot be used on an equation such as
Helpful hint
The extremesmeans property
is often referred to as cross
multiplying. Whatever you call
it, remember that it is nothing
new. You can accomplish the
same thing by multiplying
each side of the equation by
the LCD.
Example 3
Ratios and proportions
The ratio of men to women at a football game was 4 to 3. If there were 12,000 more
men than women in attendance, then how many men and how many women were in
attendance?
Solution
Let x represent the number of men in attendance and x  12,000 represent the
number of women in attendance. Because the ratio of men to women was 4 to 3, we
can write the following proportion:
So there were 48,000 men and 36,000 women at the game.
