In a joint family , the average age of grandparents is 67 years , the average age of parents is 35 years and that of three grand children is 6 years .What is the average age of the family ?

Options

A. `28(4)/(7)` years
B. `31(5)/(7)` years
C. `32 (1)/(2)` years
D. `29(4)/(7)` years

Answer

To find the average age of a joint family consisting of grandparents, parents, and grandchildren, we can follow these steps:

### Step 1: Identify the number of individuals in each group
- **Grandparents**: 2 (1 grandfather and 1 grandmother)
- **Parents**: 2 (1 father and 1 mother)
- **Grandchildren**: 3

**Total number of individuals** = 2 (grandparents) + 2 (parents) + 3 (grandchildren) = **7 individuals**

### Step 2: Calculate the total age of each group
- **Average age of grandparents** = 67 years
- Total age of grandparents = Average age × Number of grandparents = 67 × 2 = **134 years**

- **Average age of parents** = 35 years
- Total age of parents = Average age × Number of parents = 35 × 2 = **70 years**

- **Average age of grandchildren** = 6 years
- Total age of grandchildren = Average age × Number of grandchildren = 6 × 3 = **18 years**

### Step 3: Calculate the total age of the family
- Total age of the family = Total age of grandparents + Total age of parents + Total age of grandchildren
- Total age of the family = 134 + 70 + 18 = **222 years**

### Step 4: Calculate the average age of the family
- Average age of the family = Total age of the family / Total number of individuals
- Average age of the family = 222 / 7 = **31.7143 years** (approximately)

### Step 5: Convert the average age into a mixed fraction
- 31.7143 can be expressed as 31 years and a fraction.
- To find the fraction, take the decimal part (0.7143) and convert it to a fraction:
- 0.7143 is approximately 5/7 when expressed as a fraction.

Thus, the average age of the family can be expressed as **31 years and 5/7 years**.

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

---