${{d}_{1}}=\frac{d}{2},{{d}_{2}}+{{d}_{3}}=\frac{d}{2}$ 

$\xrightarrow[{{t}_{2}}=\frac{1}{3}({{t}_{2}}+{{t}_{3}})\Rightarrow {{t}_{2}}-\frac{1}{3}{{t}_{2}}=\frac{1}{3}{{t}_{3}}\Rightarrow \frac{2}{3}{{t}_{2}}=\frac{{{t}_{3}}}{3}\Rightarrow \frac{{{t}_{2}}}{{{t}_{3}}}=\frac{1}{2}]{{{d}_{2}}={{({{v}_{av}})}_{2}}{{t}_{2}},{{d}_{3}}={{({{v}_{av}})}_{3}}{{t}_{3}}}$ 

$({{({{v}_{av}})}_{2}}+2{{({{v}_{av}})}_{3}}){{t}_{2}}=\frac{d}{2}$

$\Rightarrow{{t}_{2}}=\frac{d}{2{{({{v}_{av}})}_{2}}+4{{({{v}_{av}})}_{3}}},{{t}_{3}}=\frac{d}{{{({{v}_{av}})}_{2}}+2{{({{v}_{av}})}_{3}}}$

${{v}_{av}}=\frac{{{d}_{1}}+{{d}_{2}}+{{d}_{3}}}{{{t}_{1}}+{{t}_{2}}+{{t}_{3}}}$

$=\frac{d}{\frac{d}{2{{({{v}_{av}})}_{1}}}+\frac{d}{2{{({{v}_{av}})}_{2}}+4{{({{v}_{av}})}_{3}}}+\frac{d}{{{({{v}_{av}})}_{2}}+2{{({{v}_{av}})}_{3}}}}$

$\Rightarrow {{v}_{av}}=\frac{1}{\frac{1}{2{{({{v}_{av}})}_{1}}}+\frac{1}{2{{({{v}_{av}})}_{2}}+4{{({{v}_{av}})}_{3}}}+\frac{1}{{{({{v}_{av}})}_{2}}+2{{({{v}_{av}})}_{3}}}}$

$\xrightarrow{{{({{v}_{av}})}_{1}}=10\frac{m}{s},{{({{v}_{av}})}_{2}}=4\frac{m}{s},{{({{v}_{av}})}_{3}}=3\frac{m}{s}}$

${{v}_{av}}=\frac{1}{\frac{1}{20}+\frac{1}{20}+\frac{1}{10}}=\frac{20}{4}=5\frac{m}{s}$