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.
Ex. 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.
Ex. 4: 5 = 8: 10 = 12: 15 etc. 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 * c) = (a * d).

3. (i) Fourth Proportional

If a : b = c: d, then d is called the fourth proportional
to a, b, c.

(ii) Third Proportional

If a: b = b: c, then c is called the third proportional to
a and b.

(iii) Mean Proportional

Mean proportional between a and b is square root of ab.

4. (i) COMPARISON OF RATIOS

We say that (a: b) > (c: d) <=> (a/b)>(c /d).

(ii) COMPOUNDED RATIO

The compounded ratio of the ratios (a: b), (c: d), (e : f) is (ace: bdf)

5. (i) Duplicate ratio

(a : b) is (a^{2} : b^{2}).

(ii) Sub-duplicate ratio

(a : b) is (√a : √b)

(iii)Triplicate ratio

(a : b) is (a^{3} : b^{3})

(iv) Sub-triplicate ratio

(a : b) is (a^{ ⅓} : b^{ ⅓} )

(v) If (a/b)=(c/d)

then ((a+b)/(a-b))=((c+d)/(c-d)) (Componendo and dividendo)

6. VARIATION

(i) We say that x is directly proportional to y, if x = ky for some constant k and
we write, x ∞ y.
(ii) We say that x is inversely proportional to y, if xy = k for some constant k and
we write, x∞(1/y)