Tag: composition of ratios

The triplicate ratio of $a : b$ is $a^{3} : b^{3}$
$\therefore$ The triplicate ratio of $a^{2} : b^{2}$ is $(a^{2})^{3} : (b^{2})^{3} = a^{6} : b^{6}$.

The triplicate ratio of $ab^{\frac {2}{3}} : a^{\frac {2}{3}} b$ is $(ab^{\frac {2}{3}})^{3} : (a^{\frac {2}{3}}b)^{3}$
$= a^{3} \cdot b^{2} : a^{2}b^{3}$
$a^{3}b^{2} : a^{2}b^{3} = \dfrac {a^{3}b^{2}}{a^{2}b^{3}} = \dfrac {a}{b}$

The triplicate ratio of $a : b$ is $a^{3} : b^{3}$
$\therefore$ The triplicate ratio of $\dfrac {x}{2} : \sqrt [3]{x}$ is $\left (\dfrac {x}{2}\right )^{3} : \left (\sqrt [3]{x}\right )^{3} = \dfrac {x^{3}}{8} : x = \dfrac {x^{2}}{8}$
$\therefore x^{2} : 8$.

The subduplicate ratio of $a : b$ is $\sqrt {a} : \sqrt {b}$
$\therefore$ The subduplicate ratio of $49 : 169$ is $\sqrt {49} : \sqrt {169} = 7 : 13$

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b}$.
$\therefore$ The subtriplicate ratio of $125y^{3} : 1000z^{6}$ is $\sqrt [3]{125y^{3}} : \sqrt [3]{1000z^{6}} = 5y : 10z^{2}$.

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b}$.
$\therefore$ the subtriplicate ratio of $8x^{3} : y^{3}$ is $\sqrt [3]{8x^{3}} : \sqrt [3]{y^{3}} = 2x : y$

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b}$
$\therefore$ The subtriplicate ratio of $64 m^{6} : 27 n^{3}$ is $\sqrt [3]{64m^{6}} : \sqrt [3]{27n^{3}}$
$= 4m^{2} : 3n$

The subduplicate ratio of $a : b$ is $\sqrt {a} : \sqrt {b}$
$\therefore$ The subduplicate ratio of $121 : 100$ is $\sqrt {121} : \sqrt {100} = 11 : 10$

The subduplicate ratio of $\dfrac {1}{4} : \dfrac {1}{36}$ is $\sqrt {\dfrac {1}{4}} : \sqrt {\dfrac {1}{36}} = \dfrac {1}{2} : \dfrac {1}{6}$

The subduplicate ratio of $a : b$ is $\sqrt {a} : \sqrt {b}$
$\therefore$ The subduplicate ratio of $x^{4} : y^{4}$ is $\sqrt {x^{4}} : \sqrt {y^{4}} = x^{2} : y^{2}$.