Ten years ago, the average age of a family of 4 members was 24 years. Since then, two children have been born in the family, and the age difference between the two children is 2 years. If the present average age of the family (all 6 members) is still 24 years, what is the present age of the younger child?

Introduction / Context:
This problem is about average age in a family over time. Ten years ago there were 4 members, and their average age was 24 years. Over the next 10 years, two children were born, with a 2-year age difference between them. Interestingly, the current average age remains the same. Using this information, we determine the current age of the youngest child by focusing on total age changes over time.

Given Data / Assumptions:

• Number of family members 10 years ago = 4.
• Average age 10 years ago = 24 years.
• Present number of family members = 6 (original 4 plus 2 children).
• Present average age of all 6 members = 24 years.
• Age difference between the two children = 2 years.
• We need to find the present age of the younger child.

Concept / Approach:
Average age times number of persons gives total age. Over 10 years, each of the original 4 members ages by 10 years, adding a fixed total of 4 * 10 years to their combined age. The two children contribute additional age. Because the average stays the same, we can equate the total present age to a fixed value and then compute how much age belongs to the children. From there, we use the age difference to find each child's age.

Step-by-Step Solution:
Step 1: Ten years ago, total age of 4 members = 4 * 24 = 96 years. Step 2: After 10 years, each of the original 4 members gains 10 years. So increase for these 4 = 4 * 10 = 40 years. Total present age of original 4 members = 96 + 40 = 136 years. Step 3: Currently there are 6 family members, and the present average age is still 24. Therefore, total present age of all 6 members = 6 * 24 = 144 years. Step 4: Combined age of the two children now = total present age - present age of original 4. Children's total age = 144 - 136 = 8 years. Step 5: Let the present age of the younger child be x years. Then the older child is x + 2 years (2-year age difference). Step 6: Sum of their ages = x + (x + 2) = 2x + 2. We know 2x + 2 = 8, so 2x = 6, x = 3.
Verification / Alternative check:
If the younger child is 3, the older is 5. Their total age is 3 + 5 = 8, which matches the previous calculation. Thus, the total present age of the family is 136 + 8 = 144, and the average is 144 / 6 = 24, as required.
Why Other Options Are Wrong:
If the younger child were 2, the older would be 4, total = 6, which is too small compared to the required 8. If the younger child were 1 or 4 or 5, the sums would be 1 + 3 = 4, 4 + 6 = 10, 5 + 7 = 12 respectively, none of which equal 8.
Common Pitfalls:
Forgetting that the average remained constant and not using it to compute current total age can lead to unnecessary algebra. Another common mistake is mixing up years ago with present ages, which can shift totals incorrectly.
Final Answer:
The present age of the youngest child is 3 years.