The marks obtained by students A and B in 3 examinations are given below Marks of A 30 20 40 Marks of B 70 0…

The marks obtained by students $A$ and $B$ in $3$ examinations are given below:

Mean of $A_{\overline{x}}=\frac{30+20+40}{3}=\frac{90}{3}=30$

Now,

S.D. of $A\left({\sigma }_A\right)=\sqrt{\frac{1}{n}\sum _{}^{}{\left(x_i-\overline{x}\right)}^2}$

    $=\sqrt{\frac{1}{3}\left\{{\left(30-30\right)}^2+{\left(20-30\right)}^2+{\left(40-30\right)}^2\right\}}$

    $=\sqrt{\frac{200}{3}}$

Mean of $B_{\overline{x}}=\frac{70+0+5}{3}=\frac{75}{3}$

S.D. of $B\left({\sigma }_B\right)=\sqrt{\frac{1}{n}\sum _{}^{}{\left(x_i-\overline{x}\right)}^2}$

          $=\sqrt{\frac{1}{3}\left\{{\left(70-25\right)}^2+{\left(6-25\right)}^2+{\left(5-25\right)}^2\right\}}$

          $=\sqrt{\frac{3050}{3}}$

Now, required ratio$=\frac{{\sigma }_A}{{\sigma }_B}\times \frac{B_{\overline{x}}}{A_{\overline{x}}}$

                               $=\left(\frac{\sqrt{{\displaystyle \frac{200}{3}}}}{\sqrt{{\displaystyle \frac{3050}{3}}}}\right)\times \left(\frac{{\displaystyle \frac{75}{3}}}{30}\right)$

                               $=\frac{5}{3\sqrt{61}}$