The average age of four children is 12 years. The average age of these four children together with their father is 20 years. What is the father's present age?

Introduction / Context:
This question checks your understanding of average age and how to move between totals and averages. We know the average age of four children and the average age when their father is included. From this, we must determine the father's present age.

Given Data / Assumptions:

- There are four children, and their average age is 12 years.
- When these four children and their father are taken together (five persons), the average age is 20 years.
- All ages are in years and positive.

Concept / Approach:
Average age is equal to the total of all ages divided by the number of people. Using the average of just the children, we can find the total of their ages. Using the average of the children plus the father, we can find the total age of all five people. The father's age is then the difference between these two totals.

Step-by-Step Solution:
Step 1: Number of children = 4 and their average age = 12 years. Step 2: Total age of the four children = 4 × 12 = 48 years. Step 3: Number of persons including the father = 4 children + 1 father = 5 persons. Step 4: Average age of these five persons = 20 years. Step 5: Total age of the four children and the father together = 5 × 20 = 100 years. Step 6: Father's present age = total age of group − total age of children = 100 − 48 = 52 years.

Verification / Alternative check:
We can verify by reconstructing the averages. If the children together sum to 48 years and the father is 52 years old, then the total for all five is 48 + 52 = 100 years. Dividing by 5 gives an average of 20 years, which matches the given information. The children's average remains 48 / 4 = 12 years, also consistent. Hence, the father's age must be 52 years.

Why Other Options Are Wrong:
If the father were 48, 54, 56 or 62 years old, the total ages and resulting averages would not match both the given average of 12 years for the children and 20 years for the combined group. Only 52 years satisfies both averages simultaneously.

Common Pitfalls:
A common mistake is to average the averages directly, which is not valid because the number of people differs in each group. Another error is to forget to multiply the averages by the number of people when converting to total age. Always use the relationship total age = average age × number of persons.

Final Answer:
The father's present age is 52 years.