To understand why the correct answer is option 'C', let's first discuss the concept of consolidation and the difference between double drainage and single drainage.

Consolidation is a process by which the excess pore water pressure in a saturated soil is dissipated over time. When a load is applied to a clay layer, it causes the water in the pores to be squeezed out, resulting in an increase in effective stress. As the excess pore water pressure dissipates, the soil gradually consolidates and undergoes settlement.

Double drainage refers to a condition where the soil is allowed to drain from both the top and the bottom, enabling the dissipation of excess pore water pressure in two directions. On the other hand, single drainage only allows drainage from one direction, typically from the top.

Now, let's analyze the given scenario. The saturated clay layer takes 5 years to attain a 50% degree of consolidation under double drainage conditions. This means that after 5 years, the excess pore water pressure is reduced by 50%.

Since single drainage only allows drainage from one direction, it hinders the dissipation of excess pore water pressure compared to double drainage. As a result, the consolidation process is slower under single drainage conditions.

Based on this information, we can conclude that the time required to attain the same conditions (50% degree of consolidation) under single drainage will be longer than 5 years.

Let's consider the options:

a) 10 years: This option suggests that the time required for consolidation under single drainage is double that of double drainage. However, this assumption is not accurate because the consolidation process is not linear. It is a time-dependent process, and the rate of consolidation decreases over time.

b) 15 years: This option suggests that the time required for consolidation under single drainage is 3 times that of double drainage. While this is a possibility, it is not the correct answer.

c) 20 years: This option suggests that the time required for consolidation under single drainage is 4 times that of double drainage. It is a reasonable assumption considering that the consolidation process is slower under single drainage conditions.

d) 25 years: This option suggests that the time required for consolidation under single drainage is 5 times that of double drainage. While this is a possibility, it is not the correct answer.

Therefore, based on the given information and the analysis above, the correct answer is option 'C' - 20 years.

Tv = Cv*t/D^2
t1/t2 =d1^2/d2^2
5/t2 = (d1/2)^2/d1^2
on Solving t2 = 20years
d1/2 because two way drainage