Mr. Joe's family consists of six people—himself, his wife and their four children. It is known that the average age of the family immediately after the birth of the first, second, third and fourth child was 16, 15, 16 and 15 years respectively. Find the age of Mr. Joe's eldest son if the present average age of the entire family is 16 years.

### Concept & Logic To solve this problem, we use the principle of **Total Age = Average Age $\times$ Number of Members**. We must track the total age of the family at each given milestone (birth of each child). Note that the age of a newborn is exactly $0$ years. ### Step-by-step Solution * **At 1st child's birth:** Family size = $3$. Average age = $16$. Total age = $3 \times 16 = 48$ years. (This is simply the sum of the parents' ages, as the newborn is $0$). * **At 2nd child's birth:** Let the gap between the 1st and 2nd child be $x$ years. During this time, the parents and the first child all age by $x$ years. Total age = $48 + 3x$. Given average = $15$, Family size = $4$. $48 + 3x = 4 \times 15 = 60 \Rightarrow 3x = 12 \Rightarrow x = 4$ years. (The first child is now $4$ years old). * **At 3rd child's birth:** Let the gap between the 2nd and 3rd child be $y$ years. Total age = $60 + 4y$. Given average = $16$, Family size = $5$. $60 + 4y = 5 \times 16 = 80 \Rightarrow 4y = 20 \Rightarrow y = 5$ years. * **At 4th child's birth:** Let the gap between the 3rd and 4th child be $z$ years. Total age = $80 + 5z$. Given average = $15$, Family size = $6$. $80 + 5z = 6 \times 15 = 90 \Rightarrow 5z = 10 \Rightarrow z = 2$ years. At this point, the eldest son's age is the sum of the gaps: $x + y + z = 4 + 5 + 2 = 11$ years. * **Present Day:** Let $t$ years have passed since the 4th child's birth. Total present age = $90 + 6t$. Given present average = $16$, Family size = $6$. $90 + 6t = 6 \times 16 = 96 \Rightarrow 6t = 6 \Rightarrow t = 1$ year. The present age of the eldest son is his age at the 4th birth plus $1$ year: $11 + 1 = 12$ years. ### Exam Strategy & Shortcut Instead of complex equations, just track the total age increment mentally. Start: Total age $48$ (3 people). To reach average $15$ for 4 people, total must be $60$. Difference is $12$, shared by 3 existing people $\Rightarrow 4$ years gap. To reach average $16$ for 5 people, total must be $80$. Difference is $20$, shared by 4 existing people $\Rightarrow 5$ years gap. To reach average $15$ for 6 people, total must be $90$. Difference is $10$, shared by 5 existing people $\Rightarrow 2$ years gap. Total gap = $4+5+2 = 11$ years. Present total is $96$, so gap is $96-90 = 6$, shared by 6 people $\Rightarrow 1$ year gap. Final Age $= 11+1 = 12$. This logic is lightning-fast on paper. ### Common Pitfall Students often forget that the parents *and all existing children* age during the gap years. They might divide the age difference incorrectly. Always divide the total age increase by the number of people who were alive *during* that time gap. ### Final Answer **Therefore, the correct answer is 12 years.**