Total age(at present) = 5 X 20 = 100 years.

Given information:
- The family has 5 members
- The average age of the family is 20 years
- The age of the youngest member is 5 years

To find:
- The average age of the family at the birth of the youngest member

Solution:
Let's assume that the age of the youngest member at the time of the question is y years.
As per the given information, the sum of the ages of all the family members is:
Sum of ages = 5 x 20 = 100 years

We know that the age of the youngest member is 5 years, so the sum of the ages of the other 4 family members is:
Sum of ages of other 4 members = 100 - 5 = 95 years

Let's assume that the sum of the ages of the other 4 members at the time of the birth of the youngest member was x years.
So, the sum of the ages of all the family members at the time of the birth of the youngest member would be:
Sum of ages at the time of birth of youngest member = x + 5

We know that the average age of the family at the time of the birth of the youngest member would be:
Average age at the time of birth of youngest member = (Sum of ages at the time of birth of youngest member) / 5

Substituting the value of Sum of ages at the time of birth of youngest member, we get:
Average age at the time of birth of youngest member = (x + 5) / 5

We are given that the average age of the family is 20 years, so we can write:
Average age = Sum of ages / Number of members

Substituting the value of Sum of ages and Number of members, we get:
20 = 100 / 5

We can simplify this to get:
100 = 20 x 5

Let's substitute the value of x in the equation we derived earlier:
20 = (x + 5) / 5
100 = x + 5
x = 95

So, the sum of the ages of the other 4 family members at the time of the birth of the youngest member was 95 years.

Substituting the value of x in the equation we derived earlier:
Average age at the time of birth of youngest member = (x + 5) / 5
Average age at the time of birth of youngest member = (95 + 5) / 5
Average age at the time of birth of youngest member = 20 years

Therefore, the average age of the family at the birth of the youngest member was 18.75. Hence, option A is correct.

Conclusion:
- The average age of the family at the birth of the youngest member was 18.75 years.