The average age of a couple and their son at the time of the son's marriage was 40 years. The son got married, and a child was born 2 years after the marriage. When the child turned 10 years old, the average age of the family of five became 38 years. What was the daughter in law's age at the time of marriage?

Introduction / Context:
This is a multi step family age and average problem. The situation starts with three people (a couple and their son), then expands to five members after marriage and childbirth. We are given averages at two different times and the timing of the childbirth relative to marriage. The objective is to find the daughter in law's age at marriage by tracking totals and time intervals carefully.

Given Data / Assumptions:

• At the time of the son's marriage, the average age of the couple and their son was 40 years.
• So at that time, the total age of the couple and their son was 3 * 40 = 120 years.
• The son got married then, and a child was born 2 years after the marriage.
• When the child turned 10 years old, the average age of the family of 5 (couple, son, daughter in law, child) became 38 years.
• We must find the daughter in law's age at the time of marriage.

Concept / Approach:
Take the time of marriage as the starting point. At that moment, three people have a known total age of 120 years, and the daughter in law has some unknown age D. Twelve years later (2 years until birth plus 10 years until the child is 10), everyone except the unborn child becomes 12 years older, and the child is 10 years old. We express the total family age at that later time in terms of D and equate it to the given average based total of five people. This provides a simple equation in D.

Step-by-Step Solution:
At the time of marriage, total age of couple and son = 3 * 40 = 120 years. Let the daughter in law's age at marriage be D years. Two years after marriage, a child is born. Twelve years after marriage (2 years + 10 years), the child is 10 years old. In those 12 years, the couple, the son and the daughter in law each age by 12 years. So, at the time when the child is 10, total age of the couple and son = 120 + 3 * 12 = 120 + 36 = 156 years. The daughter in law's age at that time is D + 12 years. The child's age at that time is 10 years. Hence, total family age when the child is 10 = 156 + (D + 12) + 10 = 178 + D. Given that the average age of the 5 family members is 38 years then, total age = 5 * 38 = 190 years. So 178 + D = 190 which gives D = 190 - 178 = 12 years.

Verification / Alternative check:
At marriage, the couple and son total 120 and the daughter in law is 12, so the four adults total 132. Twelve years later, their ages increase by 12 each: 4 * 12 = 48, giving a total of 132 + 48 = 180. Adding the 10 year old child gives 190. The average at that time is 190 / 5 = 38, as given. This confirms that the daughter in law's age at marriage must be 12 years.

Why Other Options Are Wrong:
If the daughter in law had been 10, 13, 14 or 16 at marriage, the total family age when the child is 10 would not be 190 and the resulting average would differ from 38. Each alternative value changes the total by more or less than 190, so only D = 12 is consistent with the given data.

Common Pitfalls:
A common mistake is to assume the first mentioned average is at present instead of at the time of marriage. Another pitfall is forgetting that the child is born 2 years after marriage, so the time gap from marriage to the child being 10 is 12 years, not just 10 years. Tracking time carefully and working with totals instead of partial averages makes the solution more reliable.

Final Answer:
Therefore, the daughter in law was 12 years old at the time of marriage.