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

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Propositiones ad Acuendos Juvenes by Alcuin of York: 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 $p, d, r\in \N$:

\(\ds 2 p + \dfrac d 2 + \dfrac r 4\) \(=\) \(\ds 12\)
\(\text {(1)}: \quad\) \(\ds \leadsto \ \ \) \(\ds 8 p + 2 d + r\) \(=\) \(\ds 48\)
\(\text {(2)}: \quad\) \(\ds p + d + r\) \(=\) \(\ds 12\)
\(\text {(3)}: \quad\) \(\ds \leadsto \ \ \) \(\ds 7 p + d\) \(=\) \(\ds 36\) $(1) - (2)$

Thus:

$d = 36 - 7 p$


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

\(\, \ds p = 1: \, \) \(\ds d\) \(=\) \(\ds 36 - 1 \times 7\) \(\ds = 29\)
\(\, \ds p = 2: \, \) \(\ds d\) \(=\) \(\ds 36 - 2 \times 7\) \(\ds = 22\)
\(\, \ds p = 3: \, \) \(\ds d\) \(=\) \(\ds 36 - 3 \times 7\) \(\ds = 15\)
\(\, \ds p = 4: \, \) \(\ds d\) \(=\) \(\ds 36 - 4 \times 7\) \(\ds = 8\)
\(\, \ds p = 5: \, \) \(\ds d\) \(=\) \(\ds 36 - 5 \times 7\) \(\ds = 1\)

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.

$\blacksquare$


Sources

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