If the earth is at one - fourth of its present distance from the sun, the duration of the year will be

Let the present distance between sun & earth = r

Time period = 1 year

If distance between sun & earth reduced to $\frac{\text{r}}{\text{4}}$

The time period = ?                   

${\text{T}}_{\text{1} }=\text{1} \text{year} {\text{T}}_{\text{2}} =?$

${\text{r}}_{\text{1}}=\text{r}$           ${\text{r}}_{\text{2}}=\frac{\text{r}}{4}$

$\frac{{\text{T}}_{\text{1}}^{\text{2}}}{{\text{T}}_{\text{2}}^{\text{2}}}={\left(\frac{{\text{r}}_{\text{1}}}{{\text{r}}_{\text{2}}}\right)}^3$                              

$\Rightarrow {\left(\frac{1}{{\text{T}}_{\text{2}}}\right)}^2={\left(\frac{\text{r}}{\text{r/4}}\right)}^3$

$\Rightarrow \frac{\text{1}}{{\text{T}}_{\text{2}}^{\text{2}}}=64$

${\text{T}}_{\text{2}}=\frac{1}{8}\text{year}$