Diophantus's youth lasted 1/6 of his life. He had his first beard in the next 1/12 of his life. At the end of the following 1/7 of his life Diophantus got married. Five years from then his son was born. His son lived exactly 1/2 of Diophantus's life. Diophantus died 4 years after the death of his son. How long did Diophantus live?

Set Diophantus's total life as x years. Youth: x/6. First beard: x/12. Marriage: x/7. Son born: 5 years later. Son's life: x/2. Diophantus lived 4 more years after son's death. So x = x/6 + x/12 + x/7 + 5 + x/2 + 4. Solving: x = x(1/6+1/12+1/7+1/2) + 9 = x(14/84+7/84+12/84+42/84) + 9 = x(75/84) + 9. Then x - 75x/84 = 9, so 9x/84 = 9, giving x = 84.