Tag: ratio and proportions

The duplicate ratio of $a : b$ is $a^{2} : b^{2}$
$\therefore$ The duplicate ratio of $\dfrac {1}{6} : \dfrac {1}{5}$ is $\left (\dfrac {1}{6}\right )^{2} : \left (\dfrac {1}{5}\right )^{2} = \left (\dfrac {1}{36}\right ) : \left (\dfrac {1}{25}\right )$
$= \dfrac {\dfrac {1}{36}}{\dfrac {1}{25}} = \dfrac {25}{36}$.

$x : y = 4 : 9$
$y : z = 3 : 8$
$\therefore \dfrac {x}{y} \times \dfrac {y}{z} = \dfrac {4}{9}\times \dfrac {3}{8}$
$\therefore \dfrac {x}{z} = \dfrac {1}{6}\Rightarrow x : z = 1 : 6$
$\therefore$ The duplicate ratio of $x : z$ is $(1)^{2} : (6)^{2} = 1 : 36$.

$a : b = 2 : 3$
$b : c = 9 : 8$
$\therefore \dfrac {a}{b} \times \dfrac {b}{c} = \dfrac {2}{3} \times \dfrac {9}{8}$
$\therefore \dfrac {a}{c} = \dfrac {3}{4} \Rightarrow a : c = 3 : 4$.
$\therefore$ The duplicate ratio of $3 : 4$ is $3^{2} : 4^{2} = 9 : 16$

Let the salaries of Laxman and Gopal one year before be ${L} _{1} \; & \; {G} _{1}$ respectively and now be ${L} _{2} \; & \;  {G} _{2}$ respectively.

The triplicate ratio of $3 : 4$ is $3^{3} : 4^{3} = 27 : 64$
$\therefore \dfrac {4x + 3}{9x + 10} = \dfrac {27}{64}$
$\Rightarrow 256x + 192 = 243x + 270$
$\Rightarrow 256x - 243x = 270 - 192$
$\Rightarrow 13x = 78$
$\Rightarrow x = \dfrac {78}{13} = 6$
$\Rightarrow x = 6$

The triplicate ratio of $4 : 3$ is $4^{3} : 3^{3}$
$\therefore \dfrac {5x + 3}{3x + 1} = \dfrac {64}{27}$
$\Rightarrow 135x + 81 = 192x + 64$
$\Rightarrow 192x - 135x = 81 - 64$
$\Rightarrow 57x = 17$
$\therefore x = \dfrac {17}{57}$

$x : y = 2 : 5, y : z = 15 : 8, z : w = 3 : 2$
$\therefore \dfrac {x}{y}\times \dfrac {y}{z}\times \dfrac {z}{w} = \dfrac {2}{5} \times \dfrac {15}{8}\times \dfrac {3}{2} = \dfrac {9}{8}$
$\therefore x : w = 9 : 8$
$\therefore$ The triplicate ratio of $9 : 8$ is $9^{3} : 8^{3} = 729 : 512$

The subduplicate ratio of $9 : 1$ is $\sqrt {9} : \sqrt {1} = 3 : 1$
$\therefore \dfrac {x + y}{x - y} = \dfrac {3}{1} \Rightarrow x + y = 3x - 3y$
$\Rightarrow 2x = 4y \Rightarrow \dfrac {x}{y} = \dfrac {4}{2} = \dfrac {2}{1}$

The triplicate ratio of $1 : 2$ is $1^{3} : 2^{3} = 1 : 8$
$\therefore \dfrac {x - 4}{x + 2} = \dfrac {1}{8}\Rightarrow 8x - 32 = x + 2$
$\Rightarrow 8x - x = 2 + 32$
$\Rightarrow 7x = 34$
$\Rightarrow x = \dfrac {34}{7}$

The triplicate ratio of $a : b$ is $a^{3} : b^{3}$
$\therefore$ The triplicate ratio of $(x + y)^{\frac {2}{3}} : (x - y)^{\frac {2}{3}}$ is $[(x + y)^{\frac {2}{3}}]^{3} : [(x - y)^{\frac {2}{3}}]^{3}$
$= (x + y)^{2} : (x - y)^{2}$