Tag: composition of ratios

By the defination of compound ratio $18 : 9, 16 : 13$ and $6 : 9$ can be expressed as
$\dfrac {18}{9}\times \dfrac {16}{13}\times \dfrac {6}{9} = \dfrac {64}{39}$
Hence $64 : 39$

By the defination of compound ratio $2 : 3$ and $5 : 7$ can be expressed as
$\dfrac {2}{3}\times \dfrac {5}{7} = \dfrac {10}{21}$
Hence $10 : 21$

By the defination of compound ratio $10 : 30$ and $60 : 80$
$\dfrac {10}{30}\times \dfrac {60}{80} = \dfrac {1}{4}$
Hence $1 : 4$.

The duplicate ratio of $a : b$ is $b : a$
$\therefore$ The duplicate ratio of $3a : 4b$ is $(3a)^{2} : (4b)^{2} = 9a^{2} : 16b^{2}$

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

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 $(2\sqrt {2})^{2} : (3\sqrt {5})^{2} = 8 : 45$