1. Ratio:

The ratio of two quantities a and b in the same units, is the fraction
 a b
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 9
with antecedent = 5, consequent = 9.
Rule: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio.
Eg. 4 : 5 = 8 : 10 = 12 : 15. Also, 4 : 6 = 2 : 3
2. 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).
3. 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
4. Comparison of Ratios:

We say that (a : b) > (c : d)
 a b
>
 c d

Compounded Ratio:
The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).
5. Duplicate Ratios:
Duplicate ratio of (a : b) is (a2 : b2).

 Sub-duplicate ratio of (a : b) is ( a : b )
Triplicate ratio of (a : b) is (a3 : b3).
Sub-triplicate ratio of (a : b) is (a1/3 : b1/3).

If
 a b
=
 c d
, then
 a + b a - b
=
 c + d c - d
. [componendo and dividendo]

6. 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