Tag: composition of ratios

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 $[(x + y)^{\frac {2}{3}}]^{3} : [(x - y)^{\frac {2}{3}}]^{3}$
$= (x + y)^{2} : (x - y)^{2}$

The subduplicate ratio of $a : b$ is $\sqrt {a} : \sqrt {b}$
$\therefore$ The subduplicate ratio of $25x^{2} : 196y^{2}$ is $\sqrt {25x^{2}} : \sqrt {196y^{2}}$
$= 5x : 14y$.

The given ratio can be simplified as $\sqrt {16} : \sqrt {64} = 4 : 8 = 1 : 2$
Now, the duplicate ratio of $a : b$ is $a^{2} : b^{2}$ and so that of $1 : 2$ is $1^{2} : 2^{2} = 1 : 4$.

$\dfrac {(x^{2} - y^{2})^{2}}{(x + y)^{2}} = \dfrac {((x - y)(x + y))^{2}}{(x + y)^{2}} = \dfrac {(x - y)^{2}}{1}$
$\therefore$ The subduplicate ratio of $(x - y)^{2} : 1$ is $\sqrt {(x - y)^{2}} : \sqrt {1} = (x - y) : 1$.

$x : y = 3 : 8$ and $y : z = 4 : 9$
$\therefore \dfrac {x}{y}\times \dfrac {y}{z} = \dfrac {3}{8} \times \dfrac {4}{9} = \dfrac {1}{6}$
$\therefore x : z = 1 : 6$
$\therefore$ The triplicate ratio of $x : z$ is $x^{3} : z^{3} = (1)^{3} : (6)^{3} = 1 : 216$

$x : y = 30 : 20$ and $y : z = 24 : 25$
$\therefore \dfrac {x}{y}\times \dfrac {y}{z} = \dfrac {30}{20} \times \dfrac {24}{25} = \dfrac {36}{25}$
$\therefore x : z = 36 : 25$
$\therefore$ The subduplicate ratio of $x : z$ is $\sqrt {36} : \sqrt {25} = 6 : 5$