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

by : 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:

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 women in the household, so the solution:
 * $m = 4, w = 0, c = 16$

is usually ruled out.

Hence we have:
 * $m = 1, w = 5, c = 14$