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

The correct option is 129.

To find the new number of days in a year, we can use Kepler's Third Law of Planetary Motion, which states that the square of the orbital period (T) of a planet is directly proportional to the cube of the semi-major axis of its orbit (r, the average distance between the planet and the Sun):

$T^2\propto r^3$

This relationship can also be written in ratio form for two different states as:

${\left(\frac{T_2}{T_1}\right)}^2={\left(\frac{r_2}{r_1}\right)}^3$

Or, by taking the square root on both sides:

$\frac{T_2}{T_1}={\left(\frac{r_2}{r_1}\right)}^{3/2}$

Let the initial average distance between the Earth and the Sun be $r_1=r$ and the initial orbital period (number of days in a year) be $T_1=365\text{days}$.
According to the question, the new distance is half of its present value, so $r_2=\frac{r}{2}$.

Substituting these values into our ratio formula:

$\frac{T_2}{365}={\left(\frac{r/2}{r}\right)}^{3/2}$

$\frac{T_2}{365}={\left(\frac{1}{2}\right)}^{3/2}$

$\frac{T_2}{365}=\frac{1}{2\sqrt{2}}$

Now, we solve for $T_2$:

$T_2=\frac{365}{2\sqrt{2}}$

Using the approximation $\sqrt{2}\approx 1.414$:

$T_2=\frac{365}{2\times 1.414}=\frac{365}{2.828}\approx 129.07\text{days}$

Therefore, if the distance between the Earth and the Sun becomes half its present value, the number of days in a year would be approximately 129 days.