1. Ratio:
The ratio of two quantities x and y in the same units, is the fractionx/yand we write it as x : y.
In the ratio x : y, we call x as the first term or antecedent and y, the second term or consequent.
Eg. The ratio 2 : 3 represents 2/3 with antecedent = 2, consequent = 3.
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 (a^1/3 : b^1/3).
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.