The riddle of Diophantus

Statement

Diophantus lived thus: $\frac{1}{6}$ of his life he was a boy, $\frac{1}{12}$ a youth, $\frac{1}{7}$ a bachelor; he married and after $5$ years had a son who lived $\frac{1}{2}$ of his father’s life; Diophantus died $4$ years after his son. How many years did he live?

Solution

Let $v$ be the length of his life. Adding up all the phases we obtain the equation: $\frac{v}{6}+\frac{v}{12}+\frac{v}{7}+5+\frac{v}{2}+4 = v$ We reduce to the same denominator $84$: $\frac{14v+7v+12v+42v}{84}+9 = v \implies \frac{75v}{84}+9=v \implies 9 = \frac{9v}{84} \implies \boxed{v=84}$ Diophantus lived $84$ years.

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Topics: First-degree equations
Concepts: Equation
Skills: Modelling · Solving equations
People: Diophantus of Alexandria
Exercise type: Word problem