**Introduction:**
In this scenario, we have a cantilever beam carrying a uniformly distributed load. We are given two cases:
1. The load is applied from the fixed end to the center of the beam.
2. The load is applied from the center of the beam to the free end.

We need to determine the ratio of deflections between these two cases.

**Analysis:**
To analyze the deflections in the two cases, we can use the principles of structural mechanics and beam theory. Let's consider each case separately.

**Case 1: Load applied from fixed end to center**
In this case, the cantilever beam is subjected to a uniformly distributed load from the fixed end to the center. The beam is fixed at one end and free at the other end.

**Case 2: Load applied from center to free end**
In this case, the cantilever beam is subjected to a uniformly distributed load from the center to the free end. The beam is fixed at one end and free at the other end.

**Calculating the deflections:**
To determine the deflections in each case, we can use the equation for the deflection of a cantilever beam under uniform load:

δ = (w * L^4) / (8 * E * I)

where:
- δ is the deflection at the free end of the beam
- w is the intensity of the uniformly distributed load
- L is the length of the beam
- E is the modulus of elasticity of the beam material
- I is the moment of inertia of the beam cross-section

**Ratio of deflections:**
To find the ratio of deflections between the two cases, we need to calculate the deflections for each case and then compare them.

Let's assume that the length of the beam is L, the intensity of the uniformly distributed load is w, the modulus of elasticity of the beam material is E, and the moment of inertia of the beam cross-section is I.

For case 1, the load is applied from the fixed end to the center. The length of the loaded portion is L/2. Therefore, the deflection at the free end can be calculated using the equation mentioned above.

For case 2, the load is applied from the center to the free end. The length of the loaded portion is also L/2. Therefore, the deflection at the free end can be calculated using the same equation.

Finally, we can find the ratio of deflections by dividing the deflection in case 2 by the deflection in case 1.

**Conclusion:**
In conclusion, the ratio of deflections between the two cases can be calculated using the equation for the deflection of a cantilever beam under uniform load. By comparing the deflections in each case and taking the ratio, we can determine the relative deflections. This analysis is important in understanding the behavior of cantilever beams under different loading conditions.