Question 163757
This problem came up a few weeks ago, this is what I submitted then
:
Diophantus passed one-sixth of his life in childhood, one-twelfth in youth, one-seventh more as a bachelor. Five years after his marriage, a son was born who died four years before his father, at half his father's [final] age. How old was Diophantus when he died.
:
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 = 84 is his final age
:
Check:
14 + 7 + 12 + 5 + 42 + 4 = 84