Propositiones ad Acuendos Juvenes/Problems/35 - De Obitu Cuiusdam Patrisfamilias

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Propositiones ad Acuendos Juvenes by Alcuin of York: Problem $35$

De Obitu Cuiusdam Patrisfamilias
A Dying Man's Will
A dying man left $960$ shillings and a pregnant wife.
He directed that:
if a boy is born he should receive $\frac 9 {12}$ of the estate and the mother $\frac 3 {12}$.
However if a daughter is born she would receive $\frac 7 {12}$ of the estate and the mother $\frac 5 {12}$.
It happened however that twins were born -- a boy and a girl.
How much should the mother receive, how much the son, how much the daughter?


Solution

$9$ and $3$ make $12$, and there are $12$ troy ounces to the troy pound.

Similarly, $5$ and $7$ make $12$.

Twice $12$ is $24$.

$24$ (troy) ounces makes $2$ (troy) pounds, that is, $40$ shillings.

Divide $960$ into $24$ parts.

Each part will be $40$ shillings.

Then take $9$ parts of $40$ shillings.

The son receives these $9$ $40$s, which is $18$ pounds, making $360$ shillings.

The mother takes $3$ parts compared to the son and $5$ parts compared to the daughter, and $3$ and $5$ make $8$.

So, take $8$ parts of $40$.

The mother receives $8$ of those $40$s, making $16$ pounds, or $320$ shillings.

Then take what remains, which is $7$ parts, that is $7$ parts of $40$, which is $14$ pounds or $280$ shillings.

This is what the daughter receives.

Adding $360$ and $320$ and $280$ gives $960$ shillings, or $48$ pounds.

$\blacksquare$


Historical Note

David Eugene Smith discusses the general testament problem in his History of Mathematics, Volume 2 of $1925$.

He explains this type of problem and its legal origins in the law of the Roman Empire.

Commentators have argued that Alcuin's solution shows that he does not understand the law.

David Wells's take on this refers back to the "original translator", who suggests adding the original fractions they expected.

Thus we have:

$\dfrac 3 4 + \dfrac 7 {12} + \dfrac 1 3$ (which was the mother's average expectation) for a total of $\dfrac 5 3$.

Multiplying their original expectations by $\dfrac 3 5$, this leaves:

the mother with $432$ shillings
the boy with $336$ shillings
the girl with $192$ shillings.


Sources

although Wells cites this as being Problem $\text {XXV}$ (which is $25$)
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