Tag: composition of ratios

By the definition of compounded ratio these ratio can be expressed as
$\dfrac {(x^{2} - y^{2})}{(x^{2} + y^{2})} \times \dfrac {(x^{4} - y^{4})}{(x + y)^{4}}$
$\dfrac {(x^{2} - y^{2})}{(x^{2} + y^{2})} \times \dfrac {(x^{2} - y^{2})(x^{2} + y^{2})}{(x + y)^{4}}$
$= \dfrac {(x^{2} - y^{2})^{2}}{(x + y)^{4}}$
$= \dfrac {[(x - y) (x + y)]^{2}}{(x + y)^{4}}$
$= \dfrac {(x - y)^{2}}{(x + y)^{2}}$
Hence $(x - y)^{2} : (x + y)^{2}$

Given, 

Given, $\displaystyle \frac{1}{a} : \frac{1}{b} : \frac{1}{c} = 3 : 4 : 5$

By the defination of compound ratio $4 : 5, 10 : 6$ and $3 : 5$ can be expressed as
$\dfrac {4}{5}\times \dfrac {10}{6} \times \dfrac {3}{5} = \dfrac {4}{5}$
Hence $4 : 5$

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 $1 : 9$

let be $ a=\dfrac{3}{5}, b=\dfrac{7}{8}, c=\dfrac{5}{10}$
therefore the $ratio=a\times b\times c=\dfrac{3\times 7\times 5}{5\times 8\times 10}=\dfrac{21}{80}$

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 $l : o$