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**orconsequent**.
Eg. The ratio 5 : 9 represent 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 (a

^{2}: b^{2}).
Sub-duplicate ratio of (a : b) is (√a : √b).

Triplicate ratio of (a : b) is (a

^{3}: b^{3}).
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.

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