how old was he?

Question

Diophantus spent 1/6 of his life in childhood, 1/12 as a youth, and 1/7 more as a  bachelor. Five years after his marriage was born a son who died four years before his father at half the age at which his father died. How old was Diophantus when he died?

Answer 1

Let x be the Diophantus' lifespan, and set t (for time, in years)=0 at his birth  Before he was married, he was

(1/6+1/12+1/7)x years old.

 

At t=(1/6+1/12+1/7)x + 5, his child was born.  Let y = his son's lifespan. 

 

The son dies 4 years before his father, or at t=x-4.  And he was born at t=(1/6 + 1/12 + 1/7)x +5, giving a lifespan of

y = (x-4) - [ (1/6 + 1/12 + 1/7)x + 5]. 

 

We also know that x/2 = y, since the son's lifespan is half the father's.  Substituting this into the first equation yields

 

x/2 = (x-4) - [(1/6 + 1/12 + 1/7)x + 5].

 

Now solve for x.