When Raj Tarun was born, his father was 32 years older than his brother and his mother was 25 years older than his sister. At present, Raj Tarun's brother is 6 years older than him and his mother is 3 years younger than his father. How old was Raj Tarun's sister when he was born?

Introduction / Context:
This family age puzzle links the ages of Raj Tarun, his brother, sister, father and mother at the time of his birth using age differences that remain constant over time. We are asked to find the sister's age when Raj Tarun was born, using the current difference between Raj and his brother and the relationships given at birth time.

Given Data / Assumptions:

• At the time of Raj Tarun's birth, his father was 32 years older than his brother.
• At the same time, his mother was 25 years older than his sister.
• At present, Raj Tarun's brother is 6 years older than him.
• At present, his mother is 3 years younger than his father.
• We must find his sister's age at the time of his birth.

Concept / Approach:
Age differences between people remain constant over time. At Raj Tarun's birth, his brother was already 6 years old, because he is 6 years older now. So we can compute the father's age then as brother's age plus 32, and the mother's age as father's age minus 3, which also equals sister's age plus 25. From this, we solve for the sister's age at that time.

Step-by-Step Solution:
At Raj Tarun's birth, his own age was 0 years. His brother is 6 years older than him now, so at the time of his birth the brother was 6 years old. At that time, father's age = brother's age + 32 = 6 + 32 = 38 years. Mother is 3 years younger than father at all times, so at that moment mother's age = 38 - 3 = 35 years. We are told that at that same time mother was 25 years older than the sister. So sister's age at Raj Tarun's birth = mother's age - 25 = 35 - 25 = 10 years.

Verification / Alternative check:
Confirm the relationships at birth: Brother was 6, father was 38, which is indeed 32 years older than the brother (38 - 6 = 32). Mother was 35, which is 3 years younger than father (38 - 35 = 3). Sister was 10, and mother was 25 years older (35 - 10 = 25). All conditions hold perfectly, verifying our calculation.

Why Other Options Are Wrong:
If the sister had been 5 years old, mother would be only 30 at that time, which breaks the 3 year age gap with father at 38. Values like 8, 12 or 15 also fail to maintain the stated 25 year difference with the mother once constraints are propagated. Only 10 years for the sister keeps every difference consistent with the data in the question.

Common Pitfalls:
The main difficulty is keeping track of which age difference applies at which moment and avoiding confusing present ages with ages at birth. Some learners incorrectly try to introduce the present age of Raj Tarun, which is not needed. Focusing on fixed differences and using a timeline from the birth moment makes the logic clear.

Final Answer:
Therefore, Raj Tarun's sister was 10 years old when he was born.