Question 158504
About all we know of Diophantus personal life is that contained in the following summary of ipitaph given in the greek Anthology. Diophantus passed one sixth of his life in childhood, one twelfth in youth and one seventh more as a bachelor. Five years after his marriage was born a son who died four years before his father, at half his father, at half his father's {final)age. How old was Diophantus when he died.

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Suppose Diophantus was N years old when he died.

>>...Diophantus passed one sixth of his life in childhood,...<<

His childhood lasted {{{N/6}}} years.  Therefore,
he was {{{N/6}}} years old when his childhood ended.

>>...one twelfth in youth...

His youth lasted {{{N/12}}} years so he was {{{N/6+N/12}}} years 
old when his youth ended.

>>...and one seventh more as a bachelor...<

His bachelorhood lasted {{{N/7}}} years so he was {{{N/6+N/12+N/7}}} 
years old when his bachelorhood ended.

>>...Five years after his marriage was born a son...<< 

His childless marriage lasted 5 years, so he was {{{N/6+N/12+N/7+5}}} 
years old when his childless marriage ended.

>>...who died...at half his father's {final)age...<<

His life with a live son lasted {{{N/2}}} years, so 
he was {{{N/6+N/12+N/7+5+N/2}}} years old when his son died.

>>...who died four years before his father,...<<

So he was {{{N/6+N/12+N/7+5+N/2+4}}} years old when he died.

And he was also {{{N}}} years old when he died. So we set 
these equal:

{{{N=N/6+N/12+N/7+5+N/2+4}}}

Combine 5 and 4:

{{{N=N/6+N/12+N/7+9+N/2}}}

Clear of fractions by multiplying through by {{{LCD=84}}}

{{{(84)(N)=(84)(N/6)+(84)(N/12)+(84)(N/7)+84(9)+(84)(N/2)}}}

{{{84N=14N+7N+12N+756+42N}}}

Combine like terms:

{{{84N=75N+756}}}

Add -75N to both sides:

{{{9N=756}}}

Divide both sides by 9

{{{N=84}}}

Diophantus was 84 years old when he died.

Edwin</pre></font></b>
AnlytcPhil@aol.com