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
- c. 800: Alcuin of York: Propositiones ad Acuendos Juvenes ... (previous) ... (next)
- 1992: John Hadley/2 and David Singmaster: Problems to Sharpen the Young (Math. Gazette Vol. 76, no. 475: pp. 102 – 126) www.jstor.org/stable/3620384