Propositiones ad Acuendos Juvenes/Problems/47 - De Episcopo qui Jussit XII Panes in Clero Dividi

by : Problem $47$

 * De Episcopo qui Jussit $\text {XII}$ Panes in Clero Dividi: A Bishop Dividing $12$ Loaves Among his Clergy
 * A certain bishop ordered that $12$ loaves be divided among his clergy.
 * He ordered that:
 * each priest should receive $2$ loaves,
 * each deacon one half
 * and each reader one quarter.
 * There were as many loaves as clergy.


 * How many priests, deacons and readers must there be?

Solution

 * $5$ priests
 * $1$ deacon
 * $6$ readers.

Proof
Let $p$, $d$ and $r$ denote the number of priests, deacons and readers respectively.

We are to solve for $p, d, r\in \N$:

Thus:
 * $d = 36 - 7 p$

Inspecting possible contenders for $p$ and $d$ individually, and calculating $r$:

Only $2$ of these satisfie the condition that $12 - \paren {p + 2} \ge 0$.
 * $p = 4$, $d = 8$, $r = 0$
 * $p = 5$, $d = 1$, $r = 6$

It is understood that there is at least one reader.

Hence the result:
 * $p = 5$, $d = 1$, $r = 6$

Thus:
 * the priests get $10$ loaves between them;
 * the deacon gets half a loaf;
 * the readers get $1 \frac 1 2$ loaves divided between them.