Given:
- The average age of 5 members of a family is 24 years.
- The youngest member is 8 years old.

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

Solution:
Let's assume the sum of ages of all 5 family members at present is S.
So, according to the given information:
S/5 = 24 ... (1)

We also know that the age of the youngest member is 8 years.
So, the sum of ages of the other 4 family members at present is S - 8.

Let's assume the sum of ages of the other 4 family members at the time of the birth of the youngest member (x years ago) is S'.
So, the sum of ages of all 5 family members at that time would be S' + 8.

Now, we need to find the average age of the family at the time of the birth of the youngest member, which is (S' + 8)/5.

Calculation:
To find the average age at the time of the birth of the youngest member, we need to find S'.

We can use the concept of proportions to find the value of S'.

According to (1):
S/5 = 24

Cross-multiplying:
S = 5 * 24
S = 120

So, the sum of ages of all 5 family members at present is 120 years.

Now, let's consider the sum of ages of the other 4 family members at the time of the birth of the youngest member as S'.

Using the concept of proportions again:
(S' + 8)/(S - 8) = 5/4 ... (2) (As there are 4 family members other than the youngest member)

Cross-multiplying:
4(S' + 8) = 5(S - 8)
4S' + 32 = 5S - 40

Simplifying the equation:
S' - 5S = -72 ... (3)

Now, substituting the value of S from (1):
S' - 5(120) = -72
S' - 600 = -72
S' = -72 + 600
S' = 528

So, the sum of ages of the other 4 family members at the time of the birth of the youngest member was 528 years.

Now, substituting the value of S' in (2):
(S' + 8)/(S - 8) = 5/4
(528 + 8)/(120 - 8) = 5/4
536/112 = 5/4

Cross-multiplying:
4 * 536 = 5 * 112
2144 = 560

This is not true. Therefore, our initial assumption is incorrect.

Hence, we cannot determine the average age of the family at the time of the birth of the youngest member with the given information.