Diophantus of Alexandria

Let $x$ = Diophantus' Final Age, so we can say that: $\frac{x}{6}$ = Diophantus' Childhood , $\frac{x}{12}$ = Diophantus' Adolescence , $\frac{x}{7}$ = Diophantus' as a Bachelor. Then, 5 years passed and his son was born ( +5 ). Since the fact that Diophantus' son died at "one-half his father's age", we can say that $\frac{x}{2}$ = Diophantus' son final age, And finally we know that Diophantus died four years after that ( +4 ). All of that gives us the equation: $\frac{x}{6}$ + $\frac{x}{12}$ + $\frac{x}{7}$ + $5$ + $\frac{x}{2}$ + $4$ = $x$ , then of a little algebra, we get that $x$ = $84$ . To calculate how old was Diophantus when his son born, we have two paths, substitute the value of $x$ = $84$ on the following expression (that eliminates the terms that include the Diophantus' son lifetime) : $\frac{x}{6}$ + $\frac{x}{12}$ + $\frac{x}{7}$ + $5$ . Or substract $4+\frac{x}{2}$ (son's age and last years of Diophantus' life). For the second way, we get that $x-(4+\frac{x}{2})$ = $84-4-42$ . For that, the answer is $\boxed{38}$

let his age at his death be x so we know that his son was bron when he was x/12+x/6+x/7+5=y years old. and we know that his son died 4 years before he died namely when Diophantus was x-4 years old. The age of the sons death is the difference between the day of his birth to his death: (x-4)-y=x/2. solving we get 84 so the age Diophantus when his son died 38