** A Ratio is an expression of comparing two similar quantities.**

To know or find, how many times one thing of one kind is greater (or smaller) than another of the same kind, a ratio is used.

Suppose today, John is 50 years old and his son Tom 25 years.

Then, to compare the ages of the father and son, a ratio may be applied as:

50: 25

And,

50: 25 = 2: 1

From the ratio 2: 1, we say John is twice as old as his son today.

*Note: *

*1. Two dissimilar quantities are not compared using a ratio. *

*For example,*

*if John is 150 cms tall and Tom 120 lbs heavy, then height and weight are not compared using a ratio as they are not similar.*

*2. Ratios are only expressed on similar quantities having same units. *

*For example,*

*The length of one line segment AB is 5 cms and the length of another line segment CD is 2 inches.*

*Then, the ratio of the lengths of the two line segments is not 5: 2 as the units of length are not the same. *

*First of all, express both the quantities in same units. *

*Now, 1 inch is 2.5 cms. *

*So, 2 inches is 2 × 2.5 cms, i.e 5 cms. *

*Therefore, the ratio of the lengths of the two line segments is 5 cms: 5 cms = 1: 1*

*3. Ratios are always expressed in the lowest terms. *

*(numbers in a ratio are in lowest terms, if their HCF is 1) *

*The HCF of 150 and 120 is 30. *

*After removing this HCF 30 (by dividing both the terms 150 and 120 with 30), the ratio in lowest terms is 5: 4*

The two numbers in ratio are called terms of the ratio.

In a ratio, the first term is called the antecedent and the second, the consequent.

In the ratio 2: 3, 2 is the antecedent and 3 is the consequent.

**1.Duplicate Ratio:**

If a: b is a ratio, then its duplicate ratio is a2: b2

**Example: **

If 2: 3 is a ratio, then its duplicate ratio is 22: 32 i.e. 4:9

**2.Sub-duplicate Ratio:**

If a: b is a ratio, then its sub-duplicate ratio is √ a: √b

Example:

If 16: 25 is a ratio, then its sub-duplicate ratio is √16: √25 = 4: 5

3.Triplicate Ratio:

If a: b is a ratio, then its triplicate ratio is a3: b3

Example:

If 2: 3 is a ratio, then its triplicate ratio is 23: 33 = 8: 27

4.Sub-triplicate Ratio:

If a: b is a ratio, then its sub-triplicate ratio is a1/3: b1/3

Example:

If 8: 27 is a ratio, then its sub-triplicate ratio is 81/3: 271/3

81/3: 271/3 = (23)1/3: (33)1/3 = 2: 3

Inverse or Reciprocal Ratio:

The inverse ratio of a: b is 1/a: 1/b

Example:

If 2: 3 is a ratio, then its inverse ratio is ½: 1/3