1. The duration of a day is determined by the Earth's angular speed of rotation.

2. If the Earth's radius changes but its mass remains the same, and no external torque acts, then angular momentum is conserved.

3. Let the initial radius be $R$ and final radius be $R' = \frac{3R}{4}.$

4. For a spherical body, moment of inertia is proportional to $MR^2$: $I \propto MR^2.$ Since mass remains constant, $\frac{I'}{I} = \left(\frac{R'}{R}\right)^2 = \left(\frac{3}{4}\right)^2 = \frac{9}{16}.$

5. By conservation of angular momentum, $I\omega = I'\omega'.$ Therefore, $\omega' = \omega \cdot \frac{I}{I'} = \omega \cdot \frac{16}{9}.$

6. Time period $T$ is inversely proportional to angular speed: $T = \frac{2\pi}{\omega}.$ Hence, $\frac{T'}{T} = \frac{\omega}{\omega'} = \frac{9}{16}.$

7. The present duration of a day is $24$ hours, so $T' = 24 \times \frac{9}{16} = 13.5 \text{ hours}.$

8. Converting $0.5$ hour into minutes: $0.5 \times 60 = 30 \text{ minutes}.$

So the new duration of the day is $13 \text{ hours } 30 \text{ minutes}.$

Therefore, the blank should be filled with 13.