Proportions
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A proportion is a comparison of ratios.

A proportion is an equation that states that two ratios are equal, such as

Proportions always have an EQUAL sign!

 

A proportion can be written in two ways:

 Both are read "4 is to 8 as 1 is to 2".


In each proportion the first and last terms (4 and 2) are called the extremes. 
The second and third terms (8 and 1) are called the means.

 

You can tell if a simple proportion is true by just examining the fractions.  If the fractions both reduce to the same value, the proportion is true.


This is a true proportion, since both fractions reduce to 1/3.

You can often use this same approach when solving for a missing part of a simple proportion.  Remember that both fractions must represent the same value.  Notice how we solve this problem by getting a common denominator for the two fractions.  

                
To change the denominator of 3 to 15 requires multiplying by 5.  The SAME must be done to the top to keep the fractions equal. 
Answer:  x = 5
 

This simple approach may not be sufficient
when working with more complex proportions.
You need a rule:

Some people call this rule
 Cross Multiply!!

A more precise statement of the rule is:
RULE:  In a true proportion, the product of the means equals the product of the extremes.

Proportions can also be solved by multiplying each side of the proportion by
the common denominator for both fractions.

 Example 1:

Solve for x algebraically in this proportion:

Solution: 

Method 1:
 Appling the rule that the "in a true proportion, the product of the means equals the product of the extremes".

 5x = (25)(2)
5x = 50
               x = 10  Answer

 Method 2:
Multiplying by the common
denominator, 2x.

2•25 = 5•x
50 = 5x
10 = x Answer
 



Example 2:
 The length of a stadium is 100 yards and its width is 75 yards.  If 1 inch represents 25 yards, what would be the dimensions of the stadium drawn on a sheet of paper?

Solution: This problem can be solved by an intuitive approach, such as:
                                100 yards by 75 yards
                                100 yards = 4 inches  (HINT:  100 / 25)
                                75 yards = 3 inches (HINT:  75 / 25)
                  Therefore, the dimensions would be 4 inches by 3 inches.

 Solution by proportion:  (Notice that the inches are all on the top and the yards are all on the bottom for this solution.  Other combinations are possible.) 

Length:

25x = 100
x = 4 inches

 Width:

25y = 75
y = 3 inches