Let's break down the information provided step by step to solve the problem.

Given Information:

1. 10 years ago, at the time of your marriage, your wife joined the family as the 6th member.
2. Today, your father passed away and a baby was born, keeping the total family members at 6.
3. The average age of the family 10 years ago (during the marriage) is the same as the average age of the family today.
4. We need to find out the age of your father when he died.

Approach:

Let’s define the ages of each family member 10 years ago and today:

• Let the ages of the family members (including your father and yourself) 10 years ago be: $a_1, a_2, a_3, a_4, a_5, a_6$.
• The sum of the ages 10 years ago would be $S_{10} = a_1 + a_2 + a_3 + a_4 + a_5 + a_6$.
• The average age 10 years ago is:

$\text{Average}_{10} = \frac{S_{10}}{6}$

• 10 years have passed, so the age of each family member (except your father, who passed away, and the baby who was just born) would increase by 10. Let’s denote your father’s age at the time of his death as $f$.

• The sum of the ages today would be the sum of the current ages of the remaining family members plus the age of the baby and your father’s age at his death:

$S_{\text{today}} = (a_1 + 10) + (a_2 + 10) + (a_3 + 10) + (a_4 + 10) + (a_5 + 10) + f + 0$

Simplifying the above equation:

$S_{\text{today}} = S_{10} + 50 + f$

• The new average age of the family today is:

$\text{Average}_{\text{today}} = \frac{S_{\text{today}}}{6} = \frac{S_{10} + 50 + f}{6}$

Key Information:

Since the average age 10 years ago is the same as today, we set the two average equations equal:

$\frac{S_{10}}{6} = \frac{S_{10} + 50 + f}{6}$

Simplifying:

$S_{10} = S_{10} + 50 + f$

$0 = 50 + f$

Thus, $f = -50$.

Conclusion:

Your father was 50 years old when he passed away.

Would you like more details on any step or further clarification?

5 Related Questions:

1. How does the average age of a group change over time when new members are added or removed?
2. Can you explain how average age can remain constant despite the passage of time?
3. What are the possible impacts of a new birth or death on the average age of a family?
4. How does the formula for calculating averages change with varying group sizes?
5. How can such real-life age problems be modeled using algebra?

Tip:

When dealing with average age problems, it's helpful to think about how the sum of the ages and the number of people in the group changes over time.