Comparision of two numbers in the form of division is called ratio.

**Example:** Comparision of \(x\) and \(y = x : y = \frac{x}{y}\)

**(1). Compound Ratio:** The multiplication of two or more ratios is called compounded ratio.

**Example:** Find the compounded ratio of \(2 : 3\) and \(4 : 5\)?

**Solution:** compound ratio of \(2 : 3\) and \(4 : 5\) is $$ 2 : 3 \times 4 : 5 $$ $$ = \frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15} $$

**(2). Duplicate Ratio:** Square of any ratio is called duplicate ratio and square-root of any ratio is called sub-duplicate ratio.

**Example (1):** Find the duplicate ratio of \(3 : 4\)?

**Solution:** duplicate ratio of \(3 : 4 = 3^2 : 4^2 = 9 : 16\)

**Example (2):** Find the sub-duplicate ratio of \(4 : 9\)?

**Solution:** sub-duplicate ratio of \(4 : 9 = \sqrt 4 : \sqrt 9 = 2 : 3\)

**(3). Triplicate Ratio:** Cube of any ratio is called triplicate ratio and cube-root of any ratio is called sub-triplicate ratio.

**Example (1):** Find the triplicate ratio of \(2 : 3\)?

**Solution:** triplicate ratio of \(2 : 3 = 2^3 : 3^3 = 8 : 27\)

**Example (2):** Find the sub-triplicate ratio of \(27 : 64\)?

**Solution:** sub-triplicate ratio of \(27 : 64 = \sqrt[3] {27} : \sqrt[3] {64} = 3 : 4\)

Proportion is generated from the word proportional which means equivalent. When two ratios are equivalent to each other then it is called proportion. The sign of proportion is (::).

**Example:** \(2 : 3 :: 4 : 9 \Rightarrow \frac{2}{3} = \frac{4}{9}\), Here \(\frac{2}{3}\) is equivalent to \(\frac{4}{9}\)

**(1). Third Proportion:** Let proportion \(A : B :: B : K\), then \(K\) is called third proportion of \(A\) and \(B\), and mathematically will be expressed as-$$ Proportion = A : B :: B : K $$ $$ \frac{A}{B} = \frac{B}{K} \Rightarrow \bbox[5px,border:2px solid #800000] {K = \frac{B^2}{A}} $$

**Example:** Find the third proportion of \(2\) and \(3\)?

**Solution:** Given values, \(A = 2\) and \(B = 3\) then, Third proportion of \(2\) and \(3\) will be $$ K = \frac{B^2}{A} $$ $$ K = \frac{3^2}{2} = \frac{9}{2} = 4.5 $$

**(2). Fourth Proportion:** Let proportion \(A : B :: C : K\), then \(K\) is called fourth proportion of \(A\), \(B\) and \(C\), and mathematically will be expressed as-$$ Proportion = A : B :: C : K $$ $$ \frac{A}{B} = \frac{C}{K} \Rightarrow \bbox[5px,border:2px solid #800000] {K = \frac{B \times C}{A}} $$

**Example:** Find the fourth proportion of \(2\), \(3\) and \(4\)?

**Solution:** Given values, \(A = 2\), \(B = 3\) and \(C = 4\) then, Fourth proportion of \(2\), \(3\) and \(4\) will be $$ K = \frac{B \times C}{A} $$ $$ K = \frac{3 \times 4}{2} = \frac{12}{2} = 6 $$

**(3). Mean Proportion:** Let proportion \(A : K :: K : B\), then \(K\) is called mean proportion of \(A\) and \(B\), and mathematically will be expressed as-$$ Proportion = A : K :: K : B $$ $$ \frac{A}{K} = \frac{K}{B} $$ $$ K^2 = A \times B \Rightarrow \bbox[5px,border:2px solid #800000] {K = \sqrt {AB}} $$

**Example:** Find the mean proportion between \(4\) and \(5\)?

**Solution:** Given values, \(A = 4\) and \(B = 5\) then, mean proportion of \(4\) and \(5\) will be $$ K = \sqrt {AB} $$ $$ K = \sqrt {4 \times 5} = \sqrt {20} $$

Lec 1: Introduction
Exercise-1
Lec 2: Rules of Componendo and Dividendo
Exercise-2
Lec 3: Rules of Partnership
Exercise-3
Exercise-4
Exercise-5