The ratio is used for comparing two quantities of the sane kind.
The ratio formula for two numbers says a and b is given by a:b or a/b. When two or more such ratios are equal, they are said to be in proportion.
The concept of ratio and proportion is majorly based on ratios and fractions.
Solution:
Given,$125{{a}^{3}}:343{{b}^{6}}$
$={}^{3}\sqrt{\left( 125{{a}^{3}} \right)}:{}^{3}\sqrt{\left( 343{{b}^{6}}
\right)}$
$={{\left( 125{{a}^{3}} \right)}^{{1}/{3}\;}}:{{\left( 343{{b}^{6}}
\right)}^{{1}/{3}\;}}$
$=5a:7{{b}^{2}}$
Therefore, sub – triplicate ratio is $5a:7{{b}^{2}}$
Solution:
Given,${64{{m}^{3}}}/{729n{}^{3}}\;:{216{{m}^{3}}}/{27{{n}^{3}}}\;$
${{=}^{3}}\sqrt{\left( 64{{m}^{3}}/729{{n}^{3}}
\right)}{{:}^{3}}\sqrt{\left( 216{{m}^{3}}/27{{n}^{3}} \right)\overset{\grave{\
}}{\mathop{\ }}\,}$
$={{\left( 64{{m}^{3}}/729{{n}^{3}} \right)}^{1/3}}:{{\left(
216{{m}^{3}}/27{{n}^{3}} \right)}^{1/3}}$
By simplification we get,
$=4m/9n:6m/3n$
$=\left( 4m/9n \right)\times \left( 3n/6m \right)$
$=2/9$
Therefore, sub – triplicate ratio is $2:9.$