Propositiones ad Acuendos Juvenes/Problems/33 - De Alio Patrefamilias Erogante Suae Familiae Annonam

by : Problem $33$

 * De Alio Patrefamilias Erogante Suae Familiae Annonam: Another Landlord Apportioning Grain
 * A gentleman has a household of $30$ persons and directs that they be given $30$ 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

 * $3$ men, $5$ women and $22$ children.

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

We have:

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 $w$:

It is implicit that there are at least some men and some women in the household, so the solutions:
 * $m = 6, w = 0, c = 24$
 * $m = 0, w = 10, c = 20$

are usually ruled out.

Hence we have:
 * $m = 3, w = 5, c = 22$