The average age of “two boys and their father” exceeds the average age of “the same two boys and their mother” by 3 years. The average age of all four (two boys, father, mother) is 19 years. If the average age of the two boys is 5.5 years, find the ages of the father and the mother.

Introduction:
This problem combines multiple averages over overlapping groups. Using total-sum reasoning with equal group sizes, the difference between the two 3-person averages directly reveals the difference in ages between father and mother. Then the overall 4-person average gives their combined age, from which individual ages are found.

Given Data / Assumptions:

• Avg(father + 2 boys) − Avg(mother + 2 boys) = 3 years.
• Avg(all four) = 19 years → sum(all four) = 76.
• Avg(2 boys) = 5.5 years → sum(2 boys) = 11.

Concept / Approach:
Since both compared groups have 3 people, the average difference equals (sum difference)/3. So father − mother = 3 * 3 = 9. Also, father + mother = 76 − 11 = 65. Solve the simple system: F − M = 9 and F + M = 65.

Step-by-Step Solution:

Verification / Alternative check:
Check averages: Boys’ average 5.5 → 11 total. With F = 37 and M = 28, all-four sum = 76 → average 19. Difference of the two 3-person averages = (F − M)/3 = 9/3 = 3, as required.

Why Other Options Are Wrong:
Other pairs do not satisfy both equations F − M = 9 and F + M = 65 simultaneously.

Common Pitfalls:
Averaging averages without converting to totals, or forgetting both compared groups have equal size, which is key to turning average differences into a clean difference of sums.

Final Answer:
37 yr and 28 yr