Propositiones ad Acuendos Juvenes/Problems/8 - De Cupa

Propositiones ad Acuendos Juvenes by Alcuin of York: Problem $8$

De Cupa

A Cask and Three Pipes

A cask is filled with $100$ metretae through $3$ pipes.

One third plus one sixth of the capacity flows through one pipe,

one third of the capacity flows through another,

but only a sixth of the capacity flows through the third.

How many sextarii flow in through each pipe?

Solution

$3600$ sextarii flow in through the first pipe;

$2400$ sextarii flow in through the second pipe;

$1200$ sextarii flow in through the third pipe.

Proof

First note that there are $72$ sextarii to the metreta.

Hence there are $7200$ sextarii in total in the cask.

Through the first pipe we have:

$\dfrac 1 3 \times 7200 + \dfrac 1 6 \times 7200 = 3600$

Through the second pipe we have:

$\dfrac 1 3 \times 7200 = 2400$

Through the third pipe we have:

$\dfrac 1 6 \times 7200 = 1200$

$\blacksquare$

Historical Note

While some authors classify this as a cistern problem, David Singmaster argues that it lacks the characteristic usage of flow rates that normally occur in such.

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