A family consists of two grandparents, two parents and three grandchildren. The average age of the grandparents is 67 years, that of the parents is 35 years and that of the grandchildren is 6 years. What is the average age of the whole family?

Introduction / Context:
This problem is a classic example of combined average. Instead of every individual age being listed, you are given average ages for different subgroups within the family. The task is to combine these subgroup averages to find the overall average age for all family members together.

Given Data / Assumptions:
- There are two grandparents with an average age of 67 years.
- There are two parents with an average age of 35 years.
- There are three grandchildren with an average age of 6 years.
- The total number of family members is 2 + 2 + 3 = 7.

Concept / Approach:
To combine averages, we convert each subgroup average back into a total sum for that subgroup by multiplying by the number of people in that subgroup. Then we add all these subgroup sums to get the total sum of ages for the entire family. Finally, we divide this total by the total number of people, which in this case is seven, to get the overall average age.

Step-by-Step Solution:
Step 1: Compute the total age of the grandparents: 2 * 67 = 134 years. Step 2: Compute the total age of the parents: 2 * 35 = 70 years. Step 3: Compute the total age of the grandchildren: 3 * 6 = 18 years. Step 4: Add all these totals to get the family total age: 134 + 70 + 18 = 222 years. Step 5: The total number of family members is 7, so the overall average age is 222 / 7 years. Step 6: Divide 222 by 7 to get 31 with a remainder of 5, so the average age is 31 5/7 years.

Verification / Alternative check:
You can check the mixed fraction form: 7 * 31 = 217 and 217 + 5 = 222, which agrees with the total family age. So 31 5/7 is correctly expressed.

Why Other Options Are Wrong:
- Option 32 6/7: This corresponds to 32 + 6/7 = 32.857..., which is higher than the actual overall average.
- Option 34 4/7: This is even larger, corresponding to 34.571..., which is inconsistent with the given subgroup data.
- Option 35 4/9: This value is around 35.44, far too high given that three of the family members are only 6 years old.

Common Pitfalls:
A common mistake is to simply take the average of the three subgroup averages (67, 35 and 6) without weighting them by the number of people in each group. This ignores the fact that there are different numbers of grandparents, parents and grandchildren. Always convert averages to totals, sum them, and then divide by the overall count.

Final Answer:
The average age of the entire family is 31 5/7 years.