Eight years ago, there were 5 members in Arthur's family and the average age of the family at that time was 36 years. Since then Arthur got married and a child was born. Even now, the average age of the family (now 7 members) is still 36 years. If the difference between the present age of Arthur's wife and their child is 26 years, what was the age of his wife at the time of the child's birth?

Introduction / Context:
This question is a multi step age problem involving changes in family size and average age over time. Eight years ago, Arthur's family had 5 members. Since then, his wife and child have joined, and yet the average age has remained the same. We are given the present age difference between the wife and child and asked to find the wife's age at the time of the child's birth. It requires careful bookkeeping of total ages and age gaps.

Given Data / Assumptions:

• 8 years ago: 5 family members, average age = 36 years.
• Present: 7 family members (the original 5 plus wife plus child).
• Even now, the average age is 36 years.
• The present difference between the wife's age and the child's age is 26 years.
• We must find the wife's age at the time of the child's birth.
• All ages increase linearly with time and there are no deaths or other new members besides the wife and child in this period.

Concept / Approach:
We start by computing total ages at different times using the average information. The sum of ages of the original 5 members 8 years ago is 5 * 36. Eight years later, these 5 have each aged by 8 years, so we can find their current total. We also compute the current total age of all 7 members from the current average. The difference between these totals gives the combined present ages of the wife and child. Using the stated age difference between wife and child, we solve for their individual present ages, and finally subtract the child's current age from the wife's current age to get her age at the child's birth.

Step-by-Step Solution:
Step 1: Eight years ago, sum of ages of 5 members = 5 * 36 = 180 years. Step 2: In the present, each of those 5 members is 8 years older, so total increase in their ages = 5 * 8 = 40 years. Step 3: Present total age of the original 5 members = 180 + 40 = 220 years. Step 4: Currently, there are 7 members and the average age is still 36 years. Step 5: Present total age of all 7 members = 7 * 36 = 252 years. Step 6: Combined present age of wife (W) and child (C) = 252 - 220 = 32 years. Step 7: Given that the present difference between wife's and child's ages is 26 years, we have W - C = 26. Step 8: Also W + C = 32 from the total just found. Step 9: Solve these two equations: adding them gives 2W = 32 + 26 = 58, so W = 29. Step 10: Then C = 32 - 29 = 3 years (present age of the child). Step 11: The child is 3 years old now, so the child was born 3 years ago. Step 12: Wife's present age is 29, so at the time of the child's birth her age was 29 - 3 = 26 years.

Verification / Alternative check:
We can verify the average information. Eight years ago, the sum of ages of the original 5 was 180. Now, those 5 contribute 220 years. Adding the present wife and child ages, 29 and 3, the total is 220 + 29 + 3 = 252. Dividing 252 by 7 gives an average of 36, matching the present average stated. The present age difference between wife and child is 29 - 3 = 26, which aligns with the given difference. This confirms all conditions are satisfied by these ages.

Why Other Options Are Wrong:
If the wife had been 25 or 20 at the time of the child's birth, her present age would be lower, which would change the combined ages and break the given average conditions. The option 'cannot be determined' is incorrect because the data lead to a unique solution for the wife and child's ages. The value 29 years is the wife's present age, not her age at the child's birth; the question specifically asks for her age at birth, which is 26.

Common Pitfalls:
A common mistake is to misinterpret the age difference as referring to some earlier time instead of the present. Others forget to account for the fact that the original 5 members have aged 8 years each, leading to an incorrect present total. Some students mix up the number of family members when computing totals. Keeping track of time shifts and totals systematically is essential in multi step age problems like this one.

Final Answer:
Arthur's wife was 26 years old at the time of the child's birth.