If the distance between the earth and the sun were half its present value, the number of days in a year would have been-

Solution :
According to Kepler's law: $$\frac{{T_1^2}}{{T_2^2}} = \frac{{R_1^3}}{{R_2^3}}$$
Here,
$$\eqalign{ & {T_1} = 365{\text{ days}}\,{\text{; }}\,\,{T_2} = ?{\text{;}}\,\,\,\,{R_1} = R\,{\text{; }}\,\,{R_2} = \frac{R}{2} \cr & \Rightarrow {T_2} = {T_1}{\left( {\frac{{{R_2}}}{{{R_1}}}} \right)^{\frac{3}{2}}} = 365{\left[ {\frac{{\frac{R}{2}}}{R}} \right]^{\frac{3}{2}}} = 129{\text{ days}} \cr} $$