Tag: ratio and proportions

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 $\sqrt {36} : \sqrt {25} = 6 : 5$

$x : y = 1 : 2$ and $y : z = 8 : 3$
$\therefore x : z = \dfrac {x}{y} \times \dfrac {y}{z} = \dfrac {1}{2} \times \dfrac {8}{3} = \dfrac {4}{3} = 4 : 3$
Now, the reciprocal ratio of $z : x$ is $x : z$
$\therefore x : z = 4 : 3$

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b}$
$\therefore$ The subtriplicate ratio of $(x^{4} - y^{4})^{3} : (x^{2} + y^{2})^{6}$ is $\sqrt [3]{(x^{4} - y^{4})^{3}} : \sqrt [3]{(x^{2} + y^{2})^{6}} = (x^{4} - y^{4}) : (x^{2} + y^{2})^{2}$
$= \dfrac {x^{4} - y^{4}}{(x^{2} + y^{2})^{2}} = \dfrac {(x^{2} - y^{2})(x^{2} + y^{2})}{(x^{2} + y^{2})^{2}}$
$= \dfrac {x^{2} - y^{2}}{x^{2} + y^{2}} = x^{2} - y^{2} : x^{2} + y^{2}$

The reciprocal ratio of $\dfrac {1}{a} : \dfrac {1}{b}$ is $a : b$
$\therefore$ The reciprocal ratio of $\dfrac {1}{7} : \dfrac {1}{8}$ is $7 : 8$.

$a : b = 2 : 3$ and $b : c = 4 : 7$
$\therefore a : c = \dfrac {a}{b} \times \dfrac {b}{c} = \dfrac {2}{3} \times \dfrac {4}{7}$
$= \dfrac {8}{21}$
$\therefore a : c = 8 : 21$
$\therefore$ The reciprocal ratio of $a : c$ is $21 : 8$.

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b}$
$\therefore$ The subtriplicate ratio of $(a + b)^{3} : (a^{2} - b^{2})^{3}$ is $\sqrt [3]{(a + b)^{3}} : \sqrt [3]{(a^{2} - b^{2})^{3}} = (a + b) : (a^{2} - b^{2})$
$= \dfrac {a + b}{a^{2} - b^{2}} = \dfrac {a + b}{(a - b)(a + b)} = \dfrac {1}{a - b} = 1 : (a - b)$.