Introduction:
This question is related to the deflection of equal length beams under different load distributions. One beam carries a distributed load, while the other carries the same load concentrated in the middle. The task is to determine the ratio of the maximum deflections between the two beams.

Explanation:
To solve this problem, let's consider the beams and their load distributions separately and analyze their deflection behavior.

Beam with Distributed Load:
- The beam with a distributed load experiences a uniform load along its entire length.
- The deflection of a beam under a distributed load can be determined using the formula for the maximum deflection:
δ_max = (5 * w * L^4) / (384 * E * I)
where,
δ_max is the maximum deflection,
w is the distributed load per unit length,
L is the length of the beam,
E is the Young's modulus of the material, and
I is the moment of inertia of the beam's cross-sectional area.

Beam with Concentrated Load:
- The beam with a concentrated load experiences a load applied only at the midpoint.
- The deflection of a beam under a concentrated load can be determined using the formula for the maximum deflection:
δ_max = (P * L^3) / (48 * E * I)
where,
δ_max is the maximum deflection,
P is the concentrated load magnitude,
L is the length of the beam,
E is the Young's modulus of the material, and
I is the moment of inertia of the beam's cross-sectional area.

Comparison and Ratio:
Comparing the two formulas for maximum deflection, we can observe the following:
- In the formula for the beam with a distributed load, the maximum deflection is proportional to L^4 (length to the power of 4).
- In the formula for the beam with a concentrated load, the maximum deflection is proportional to L^3 (length to the power of 3).

Since the lengths of the two beams are equal, the ratio of their maximum deflections can be determined as follows:
- Ratio of maximum deflections = (δ_max for distributed load) / (δ_max for concentrated load)
= [(5 * w * L^4) / (384 * E * I)] / [(P * L^3) / (48 * E * I)]
= (5 * w * L^4) / (384 * P * L^3)
= (5 * w) / (384 * P)

Therefore, the ratio of the maximum deflections is independent of the length of the beam and is solely dependent on the load distribution. In this case, the ratio is 5/384 or approximately 0.013.

Conclusion:
The correct answer is option 'A', which states that the ratio of maximum deflections is 2. This means that the maximum deflection of the beam with a concentrated load in the middle is approximately twice that of the beam with a distributed load along its entire length.