Tag: ratio and proportions

The duplicate ratio of $a : b$ is $a^{2} : b^{2}$.
$\therefore$ The duplicate ratio of $5 : 7$ is $5^{2} : 7^{2} = 25 : 49$

The duplicate ratio of $a : b$ is $b : a$
$\therefore$ The duplicate ratio of $\dfrac {x}{2} : \dfrac {y}{3}$ is $\left (\dfrac {x}{2}\right )^{2} : \left (\dfrac {y}{3}\right )^{2} = \dfrac {x^{2}}{4} : \dfrac {y^{2}}{9}$

The triplicate ratio of $a : b$ is $a^{3} : b^{3}$
$\therefore$ The triplicate ratio of $2 : 5$ is $2^{3} : 5^{3} = 8 : 125$.

The duplicate ratio of $a : b$ is $a^{2} : b^{2}$
$\therefore$ The duplicate ratio of $2\sqrt {2} : 3\sqrt {5}$ is $(2\sqrt {2})^{2} : (3\sqrt {5})^{2} = 8 : 45$

The triplicate ratio of $a : b$ is $a^{3} : b^{3}$
$\therefore$ The triplicate ratio of $a^{3} : b^{3}$ is $(a^{3})^{3} : (b^{3})^{3} = a^{9} : b^{9}$

The triplicate ratio of $a : b$ is $a^{3} : b^{3}$
$\therefore$ The triplicate ratio of $4 : 7$ is $4^{3} : 7^{3} = 64 : 243$

By the defination of compound ratio these ratio can be expressed as
$\dfrac {a - b}{a + b}\times \dfrac {(a - b)^{2}}{(a + b)^{2}} \times \dfrac {(a + b)^{3}}{(a - b)^{3}} = \dfrac {1}{1}$
Hence, $1 : 1$.

$\sqrt [3]{9} : \sqrt {8} = (9)^{\dfrac {1}{3}} : (8)^{\dfrac {1}{2}}$
The triplicate ratio of $a : b$ is $a^{3} : b^{3}$
$\therefore$ The triplicate ratio of $\sqrt [3]{9} : \sqrt {8}$ is $[(9)^{\dfrac {1}{3}}]^{3} : [(8)^{\dfrac {1}{2}}]^{3}$
$= 9 : (2\sqrt {2})^{3}$
$= 9 : 8(\sqrt {2})^{3}$
$= 9 : 16\sqrt {2}$.

The subduplicate ratio of a:ba:b