Consider equivalent ratios[br][math]\frac{4}{5}=\frac{12}{15}[/math][br][math]\therefore\frac{5}{4}=\frac{15}{12}[/math]            [math]\therefore If,\frac{a}{b}=\frac{c}{d}[/math] then [math]\frac{b}{a}=\frac{d}{c}[/math][br]The Property is known as [b][i]INVERTENDO.[br][br][/i][/b]Now take ratios of first terms of both ratios to the second terms.[br][math]\frac{4}{12}=\frac{1}{3}[/math] and [math]\frac{5}{15}=\frac{1}{3}[/math][br][math]\therefore If,\frac{a}{b}=\frac{c}{d}[/math] then [math]\frac{a}{c}=\frac{b}{d}[/math][br]The Property is known as [b][i]ALTERNENDO[/i][/b]

Now take the ratios of addition both the terms of one ratio to the second term.[br][math]\frac{4+5}{5}=\frac{9}{5}[/math]  and  [math]\frac{12+15}{15}=\frac{27}{15}=\frac{9}{5}[/math][br][math]\therefore If,\frac{a}{b}=\frac{c}{d}[/math]  then [math]\frac{a+b}{b}=\frac{c+d}{d}[/math] [br]The property is known as [i][b]COMPONENDO.[br][br][/b][/i]Now take the ratios of subtraction of both the terms of one ratio to the second term.[br][math]\frac{4-5}{5}=\frac{-1}{5}[/math][b][i] [/i][/b]and [math]\frac{12-15}{15}=\frac{-3}{15}=\frac{-1}{5}[/math][br][math]\therefore If,\frac{a}{b}=\frac{c}{d}[/math] then [math]\frac{a+b}{b}=\frac{c+d}{d}[/math][br]The property is known as[b][i] DIVIDENDO.[br][br][/i][/b]Now take the ratios of addition of both the terms of one ratio to the subtraction of both term.[br][math]\frac{4+5}{4-5}=\frac{9}{-1}=-\frac{9}{1}[/math]  and  [math]\frac{12+15}{12-15}=\frac{27}{-3}=\frac{9}{-1}=-\frac{9}{1}[/math][br][math]\therefore If,\frac{a}{b}=\frac{c}{d}[/math]  then  [math]\frac{a+b}{a-b}=\frac{c+d}{c-d}[/math][br]The property is known as[i][b] COMPONENDO-DIVIDENDO.[/b][/i][br][i][b][br][/b][/i]