Tag: ratio and proportions

The sub-duplicate ratio of $x : y$ is $\sqrt {x} : \sqrt {y}$
$\therefore$ The subduplicate ratio of $9a^{2}b^{2} : 16r^{2}s^{2}$ is $\sqrt {9a^{2}b^{2}} : \sqrt {16r^{2}s^{2}}$
$= 3ab : 4rs$

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

The subduplicate ratio of $a : b$ is $\sqrt {a} : \sqrt {b}$
$\therefore$ The subduplicate ratio of $\dfrac {1}{25} : \dfrac {1}{64}$ is $\sqrt {\dfrac {1}{25}} : \sqrt {\dfrac {1}{64}} = \dfrac {1}{5} : \dfrac {1}{8}$

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

The sub-duplicate ratio of $a : b$ is $\sqrt {a} : \sqrt {b}$.
$\therefore$ The subduplicate ratio of $9 : 16$ is $\sqrt {9} : \sqrt {16} = 3 : 4$

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 $\sqrt [3]{125y^{3}} : \sqrt [3]{1000z^{6}} = 5y : 10z^{2}$.