
Ratio:
The ratio of two quantities a and b in the same units, is the fraction and we write it as a : b.
In the ratio a : b, we call a as the first term or antecedent and b, the second term or consequent.
Eg. The ratio 5 : 9 represents 
5 
with antecedent = 5, consequent = 9. 
9 
Rule: The multiplication or division of each term of a ratio by the same nonzero number does not affect the ratio.
Eg. 4 : 5 = 8 : 10 = 12 : 15. Also, 4 : 6 = 2 : 3.

Proportion:
The equality of two ratios is called proportion.
If a : b = c : d, we write a : b :: c : d and we say that a, b, c, d are in proportion.
Here a and d are called extremes, while b and c are called mean terms.
Product of means = Product of extremes.
Thus, a : b :: c : d (b x c) = (a x d).

Fourth Proportional:
If a : b = c : d, then d is called the fourth proportional to a, b, c.
Third Proportional:
a : b = c : d, then c is called the third proportion to a and b.
Mean Proportional:
Mean proportional between a and b is ab.

Comparison of Ratios:
We say that (a : b) > (c : d) 
a 
> 
c 
. 
b 
d 
Compounded Ratio:
The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).

Duplicate Ratios:
Duplicate ratio of (a : b) is (a^{2} : b^{2}).
Subduplicate ratio of (a : b) is (a : b).
Triplicate ratio of (a : b) is (a^{3} : b^{3}).
Subtriplicate ratio of (a : b) is (a^{1/3} : b^{1/3}).
If 
a 
= 
c 
, then 
a + b 
= 
c + d 
. [componendo and dividendo] 
b 
d 
a  b 
c  d 

Variations:
We say that x is directly proportional to y, if x = ky for some constant k and we write, x y.
We say that x is inversely proportional to y, if xy = k for some constant k and
we write, x 
1 
. 
y 