The epitaph on Diophantus tomb reads:
This tomb holds Diophantus. Ah, what a marvel! And the tomb tells scientifically the measure of his life. God vouchsafed that he should be a boy for the sixth part part of his life; when a twelfth was added., his cheeks acquired a beard; he kindled for him the light of marriage after the seventh, and in the fifth year of his marriage He granted him a son. Alas! late-begotten and miserable child, when he had reached the measure of half his father's life, the chill grave took him. After consoling his grief by this science of numbers for four years, he reached the end of his life. Greek Mathematical Works II: Aristarchus to Pappus of Alexandria, Loeb Classical Library, Translated by Ivor Thomas, Harvard University Press, 1941.2
How old was Diophantus when he died and how old was his son? Let's assume the phrase half his father's life to mean half the total life, not half the life at the time of the sons death.
Assume Diophantus lived to x years.
The least common denominator of the denominators is 84. Multiplying out all the terms with this common denominator gives
Grouping multipliers of x on one side and constants on the other gives.
Diophantus was X = 84 years old. He was a boy for 14 years and grew a beard after 7 more (21). Twelve year later he marries at age 33 and had a son 5 years after that (38). The son died at the age of 42, when Diophantus was 80. Diophantus dies at the age of 84.
