Ratio and Proprotion

Ratio :

Ratio is the comparison of two quantities of same kind. If A and B are two quantities then,

• The ratio of A to B is represented by A : B and its value is $\frac{A}{B}$
• The first term A is called the Antecedent and the second term B is called the consequent
• A : B has no unit
• If we multiply or divide the terms of the ratio by the same non-zero number, the resulting value will still be the same.

Proportion :

It is a statement that equates two ratios.

Example : $\frac12 =\frac24$

1. A proportion is represented using the symbol “::”.

i.e A : B = C : D is represented as A : B :: C : D.

In the above statement,

• A and D are called Extremes, B and C are called Means.
• D is the fourth proportional to A, B and C

Note:

Product of the means = Product if the extremes.

=> A x D = B x C

2. If $\frac AB = \frac BC$

Here, B is called the mean proportional and C is called the third proportional

3. If $\frac AB =\frac CD$, then $\frac AB =\frac CD = \frac{A+C}{B+D}$

• Invertendo:

If $\frac AB = \frac CD$, then $\frac BA = D/C$

• Altrendo:

If $\frac AB = \frac CD$, then $\frac AC = \frac BD$

• Componendo:

If $\frac AB = \frac CD$, then $\frac{A+B}{B} = \frac{C+D}{D}$

• Dividendo:

If $\frac{A}{B} = \frac{C}{D}$, then $\frac{A-B}{B} = \frac{C-D}{D}$

• Componendo and Dividendo:

If $\frac AB =  \frac CD$, then $\frac{A+B}{A-B} = \frac{C+D}{C-D}$

4. Types of Ratio

• Duplicate Ratio:

Duplicate ratio of two numbers A and B is the ratio of their squares, i.e $\frac{A^2}{B^2}$

• Triplicate Ratio

Triplicate ratio of two numbers A and B is the ratio of their cubes, i.e $\frac{A^3}{B^3}$

• Sub-duplicate Ratio

Subduplicate ratio of two numbers A and B is the ratio of their square roots, i.e $\frac{\sqrt{a}}{\sqrt{b}}$

• Sub-triplicate Ratio

Sub-triplicate ratio of two numbers A and B is the ratio of their cube roots, i.e $\frac{\sqrt[3]{A}}{\sqrt[3]{B}}$

• Inverse or Reciprocal Ratio

Inverse ratio of A and B is $\frac{1}{A} : \frac{1}{B}$