Propositiones ad Acuendos Juvenes/Problems/32 - De Quodam Patrefamilias Distribuente Annonam

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Propositiones ad Acuendos Juvenes by Alcuin of York: Problem $32$

De Quodam Patrefamilias Distribuente Annonam
A Lord of the Manor Distributing Grain
A gentleman has a household of $20$ persons and orders that they be given $20$ measures of grain.
He directs that:
each man should receive $3$ measures,
each woman $2$ measures,
and each child $\frac 1 2$ a measure.
How many men, women and children must there be?


Solution

$1$ man, $5$ women and $14$ children.


Proof

Let $m$, $w$ and $c$ denote the number of men, women and children respectively.

We have:

\(\ds 3 m + 2 w + \dfrac c 2\) \(=\) \(\ds 20\) apportioning the measures of grain
\(\text {(1)}: \quad\) \(\ds \leadsto \ \ \) \(\ds 6 m + 4 w + c\) \(=\) \(\ds 40\)
\(\text {(2)}: \quad\) \(\ds m + w + c\) \(=\) \(\ds 20\) the total number of people
\(\text {(1)}: \quad\) \(\ds \leadsto \ \ \) \(\ds 5 m + 3 w\) \(=\) \(\ds 20\) $(1) - (2)$

We note that $5 m$ is a multiple of $5$.

Hence $3 w$ also has to be a multiple of $5$.

Thus $w$ has to be a multiple of $5$.

Hence the following possible solutions for $m$ and $w$:

\(\ds w = 0\) \(,\) \(\ds m = 4\)
\(\ds w = 5\) \(,\) \(\ds m = 1\)

It is implicit that there are at least some women in the household, so the solution:

$m = 4, w = 0, c = 16$

is usually ruled out.

Hence we have:

$m = 1, w = 5, c = 14$

$\blacksquare$


Sources

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