A family has two grandparents, two parents and three grandchildren. The average age of the two grandparents is 67 years, the average age of the two parents is 35 years and the average age of the three grandchildren is 6 years. What is the overall average age (in years) of all members of the family together?

Introduction / Context:
This problem deals with averages over different subgroups within a family and then combines those subgroups to find an overall average. We are given the average ages of grandparents, parents and grandchildren separately and need to determine the average age of the entire family including all members together. It is a classic weighted average situation based on group sizes.

Given Data / Assumptions:

• Number of grandparents = 2, average age = 67 years.
• Number of parents = 2, average age = 35 years.
• Number of grandchildren = 3, average age = 6 years.
• Total number of family members = 2 + 2 + 3 = 7.
• We must find overall average age of these 7 people.

Concept / Approach:
To find the overall average, we first compute the total ages of each subgroup by multiplying the subgroup average by its size. Then, we sum these totals to get the total age of the entire family. Finally, we divide this total by the total number of family members. This is equivalent to a weighted average where each subgroup contributes according to its size.

Step-by-Step Solution:
Step 1: Total age of grandparents = 2 * 67 = 134 years. Step 2: Total age of parents = 2 * 35 = 70 years. Step 3: Total age of grandchildren = 3 * 6 = 18 years. Step 4: Total age of all family members = 134 + 70 + 18. 134 + 70 = 204; 204 + 18 = 222 years. Step 5: Total number of members = 7. Overall average age = 222 / 7 years. Step 6: Divide 222 by 7: 7 * 31 = 217, remainder = 5. So, 222 / 7 = 31 5/7 years.
Verification / Alternative check:
We can approximate the result to check plausibility: 31 5/7 ≈ 31.714 years, which is reasonable because grandparents are quite old, parents are middle-aged and children are very young, pulling the overall average down from the grandparents but above the parents average.
Why Other Options Are Wrong:
32 6/7 and 34 4/7 years are too high and would imply a higher total age than 222 when multiplied by 7. 35 4/9 is even larger and clearly inconsistent. 30 6/7 is slightly smaller but does not match the exact division of 222 by 7.
Common Pitfalls:
Some students mistakenly average the three subgroup averages directly (67, 35 and 6) without weighting them by how many people are in each group. Another common error is miscalculating the total number of members or forgetting to multiply each group average by the number of people in that group.
Final Answer:
The average age of the whole family is 31 5/7 years.