Proportion:
“A statement of equality of two ratios is called proportion”
Four numbers a, b, c, d are said to be in proportion when the ratio first two a and b is equal to ratio of last two c and d.
i.e.  or a : b = c : d
e.g.  or 2 : 3 = 6 : 9
Some authors used the notation for proportion as a : b :: c : d, but this notation is not preferred now.
Here  or a : b = c : d
If four numbers are in proportion, then we can also derived some other proportion from it.
Let  be the given proportion, then
(1) 
(2) 
(3) 
(4) 
(5) 
are called the Derived Proportions.
Here  or a : b = c : d, then
The numbers a and d are called extremes of proportion, and the numbers b and c are called means of proportion.
Hence Product of extremes = Product of Means
To solve proportion, we use above principal, A single term in the proportion is called proportional.
“a” is the 1st proportional.
“b” is the 2nd proportional.
“c” is the 3rd proportional.
“d” is the 4th proportional.
Example:
Find the 3rd proportional in 2 : 3 =  : 15
Solution:
Let 2 : 3 =  : 15
i.e. 
 3 x  = 2 x 15 (by the principle of proportion)
 3 = 30
 
 
Example:
Find the missing value in  : 8 = 9 : 12
Solution:
Let  : 8 = 9 : 12
 
 12 = 9 x 8 (by the principle of proportion)
 12 = 72
 
 
Example:
Find the 1st proportional in 18, 8 and 6.
Solution:
Let  be the 1st proportional
  : 18 = 8 : 6
 x 6 = 18 x 8
 =  = 24
Example:
Find the 2nd proportional in 4, 20, 30.
Solution:
Let  be the 2nd proportional
 4 : =20 : 30
 x 20 = 4 x 30
 =  = 6
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