Eight years ago there were 5 members in the family of Arthur and at that time the average age of the family members was 36 years. Since then Arthur has married and has had a child, and even now the average age of the family is the same as before. If the difference between the present age of the wife of Arthur and the present age of the child is 26 years, what was the age of his wife at the time of the child birth?

Introduction:
This problem tests understanding of average age over time and how adding new members affects the average. It also involves a typical age difference relation between a mother and her child. By using the fact that the average age of the family remains unchanged after some years and new additions, we can find the current ages and then the age of the wife at the time of the child birth.

Given Data / Assumptions:
- Eight years ago there were 5 members in Arthur family. - Eight years ago the average age of the family was 36 years. - Since then, Arthur has married and has had a child, so now there are 7 members. - The present average age of the family is still 36 years. - The present age of the wife of Arthur is 26 years more than the present age of the child. - We need the age of the wife at the time of the child birth.

Concept / Approach:
Eight years ago, we can find the total age of the original 5 members from the average. After 8 years, these original members are all 8 years older. The wife and child were not counted then, but they are now. The total current age of all 7 members can be found using the current average. The difference between this current total and the total age of the original members now gives the combined present age of the wife and child. Using the difference between their ages, we can solve for both current ages and then subtract the child current age from the wife current age to find her age at birth of the child.

Step-by-Step Solution:
Step 1: Eight years ago, total age of the 5 members = 5 * 36 = 180 years. Step 2: Eight years later, each of these 5 members is 8 years older, so their total present age = 180 + 5 * 8 = 180 + 40 = 220 years. Step 3: At present, the family has 7 members (the original 5 members, the wife and the child). Step 4: Present average age of the family is 36 years, so total present age of all 7 members = 7 * 36 = 252 years. Step 5: Combined present age of the wife and the child = total present age of 7 members minus present age of original 5 members = 252 - 220 = 32 years. Step 6: Let the present age of the wife be W years and the present age of the child be C years. Step 7: We have two equations: W + C = 32 and W - C = 26. Step 8: Add the equations: (W + C) + (W - C) = 32 + 26 gives 2W = 58. Step 9: Therefore, W = 58 / 2 = 29 years. Step 10: Substitute W = 29 into W + C = 32 to get C = 3 years. Step 11: The age of the wife at the time of the child birth = present age of wife minus present age of child = 29 - 3 = 26 years.

Verification / Alternative Check:
If the wife is now 29 and the child is 3, their combined age is 32, matching the value found from the total averages. The age difference is 29 - 3 = 26 years, which matches the condition. The original 5 members have a combined present age of 220 years. Adding 32 for the wife and child gives 252 years. Dividing 252 by 7 gives 36, which confirms the present average. Thus all conditions are satisfied.

Why Other Options Are Wrong:
Options 25, 20 and 27 years do not match the age difference or the combined age constraint of 32 years. Option 29 years is the present age of the wife, not her age at the time of the child birth.

Common Pitfalls:
Some students may forget that the original members age increases over time and incorrectly keep their total age fixed. Others may misinterpret the age difference condition or confuse present ages with ages at the time of birth. Carefully setting up equations for the sums and differences of ages helps avoid these errors.

Final Answer:
The age of the wife of Arthur at the time of the child birth was 26 years.