Ratios and Proportions

By the defination of compound ratio these ratio can be expressed as
$\dfrac {3}{5}\times \dfrac {7}{9}\times \dfrac {15}{28} = \dfrac {1}{4}$
Hence $1 : 4$

By the defination of compound ratio $5 : 7, 12 : 8$ and $13 : 6$ can be expressed as
$\dfrac {5}{7}\times \dfrac {12}{8}\times \dfrac {13}{6} = \dfrac {65}{28}$
Hence $65 : 28$

By the defination of compound ratio $9 : 27$ and $4 : 12$ can be expressed as
$\dfrac {9}{27} \times \dfrac {4}{12} = \dfrac {1}{9}$
Hence $1 : 9$

By the defination of compound ratio $5 : 7, 14 : 10$ and $2 : 3$ can be expressed as
$\dfrac {5}{7} \times \dfrac {14}{10}\times \dfrac {2}{3} = \dfrac {2}{3}$
Hence $2 : 3$

By the defination of compound ratio $l : m, m : n$ and $n : o$ can be expressed as
$\dfrac {l}{m}\times \dfrac {m}{n} \times \dfrac {n}{o} = \dfrac {l}{o}$
Hence $l : o$

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 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 $4^{3} : 7^{3} = 64 : 243$