Propositiones ad Acuendos Juvenes/Problems/36 - De Salutatione Cuiusdam Senis ad Puerum

Propositiones ad Acuendos Juvenes by Alcuin of York: Problem $36$

An old man greeted a boy as follows:

"May you live long --

as long as you have lived so far,

and as long again as your age would be then,

and then to three times that age,

and let God add one year more

and you will be $100$.

How old was the boy at the time?

$8$ years and $3$ months.

Let $x$ years be the age of the boy at the time.

Then we have:

$\ds 3 \paren {x + x + \paren {x + x} } + 1$

$=$

$\ds 100$

$3$ times his age plus his age plus that age again plus $1$

$\ds \leadsto \ \ $

$\ds 12 x$

$=$

$\ds 99$

simplifying

$\ds \leadsto \ \ $

$\ds x$

$=$

$\ds 8 \tfrac 1 4$

That is, $8$ years and $3$ months.

$\blacksquare$