Question 781909
Diophantus's youth lasted one sixth of his life.
 He grew a beard after one twelfth more.
 After one seventh more of his life, he married.
 5 years later, he and his wife had a son.
 The son lived exactly one half as long as his father, and Diophantus died four years after his son.

How many years did Diophantus live?


Let x = his final age
:
a lengthy but simple equation
Final age = child + youth + bach + marriage + son's age + 4 
x = {{{1/6}}}x + {{{1/12}}}x + {{{1/7}}}x + 5 + {{{1/2}}}x + 4 
:
Common denominator of 12*7 = 84
x = {{{14/84}}}x + {{{7/84}}}x + {{{12/84}}}x + 5 + {{{42/84}}}x + 4
:
x = {{{75/84}}}x + 9
:
{{{84/84}}}x - {{{75/84}}}x = 9
:
{{{9/84}}}x = 9
:
x = 9 * {{{84/9}}}
:
x = 84 is his final age
:
Check:
14 + 7 + 12 + 5 + 42 + 4 = 84