Ratios and Proportions

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

The reciprocal ratio of $a : b$ is $b : a$
$\therefore$ reciprocal ratio of $39 : 41$ is $41 : 39$

$a : b = 1 : 1$ and $b : c = 81 : 24$
$\therefore \dfrac {a}{b} \times \dfrac {b}{c} = \dfrac {1}{1} \times \dfrac {81}{24} = \dfrac {27}{8}$
$\therefore a : c = 27 : 8$
$\therefore$ The subtriplicate ratio of $27 : 8$ is $\sqrt [3]{27} : \sqrt [3]{8} = 3 : 2$

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b}$
$\therefore$ The subtriplicate ratio of $4913 : 2744$ is $\sqrt [3]{4913} : \sqrt [3]{2744} = 17 : 14$

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b}$.
$\therefore$ The subtriplicate ratio of $64 : 729$ is $\sqrt [3]{64} = \sqrt [3]{729} = \sqrt [3]{(4)^{3}} : \sqrt [3]{(9)^{3}} = 4 : 9$.

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b}$
Now, $3 : 81 = \dfrac {3}{81} = \dfrac {1}{27} = 1 : 27$
$\therefore$ The subtriplicate ratio of $1 : 27$ is $\sqrt [3]{1} : \sqrt [3]{27} = 1 : 3$

The reciprocal ratio of a:ba:b.

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b}$.
$\therefore$ The subtriplicate ratio of $216 : 1331$ is $\sqrt [3]{216} : \sqrt [3]{1331} = \sqrt [3]{(6)^{3}} : \sqrt [3]{(11)^{3}}$
$= 6 : 11$

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b}$
$\therefore$ The subtriplicate ratio of $2 : 3$ is $\sqrt [3]{2} : \sqrt [3]{3}$.

The reciprocal ratio of $a : b$ is $b : a$
$\therefore$ reciprocal ratio of $2 : 3$ is $3 : 2$

The reciprocal ratio of $a : b$ is $b : a$
$\therefore$ reciprocal ratio of $9x^{2} : 11x^{2}$ is $11x^{2} : 9x^{2}$

The reciprocal ratio of a:ba:b.

The reciprocal ratio of $a : b$ is $b : a$
$\therefore$ reciprocal ratio of $121x^{3} : 25x^{3}$ is $25x^{3} : 121x^{3}$

The reciprocal ratio of $a : b$ is $b : a$
$\therefore$ reciprocal ratio of $81 : 121$ is $121 : 81$.

The reciprocal ratio of $a : b$ is $b : a$
$\therefore$ reciprocal ratio of $8 : 15$ is $15 : 8$