Propositiones ad Acuendos Juvenes/Problems/9 - De Sago

Propositiones ad Acuendos Juvenes by Alcuin of York: Problem $9$

De Sago

A Cloak

I have a cloak, $100$ cubits long and $80$ cubits wide.

I wish to make small cloaks with it, each small cloak $5$ cubits long and $4$ cubits wide.

How many small cloaks can I make?

Solution

$400$.

Proof

It is assumed there is no cloth spent on hems, and the making of a small cloak consists of little more than cutting a piece of cloth to the required size.

We have that:

$5$ goes into $100$ a total of $20$ times

$4$ goes into $80$ also a total of $20$ times.

Thus the cloth can be cut into $20 \times 20 = 400$ pieces that are $5$ cubits long and $4$ cubits wide.

$\blacksquare$

Historical Note

According to the translator John Hadley, the explanation for the answer given in the original manuscript was "not enlightening".

It is supposed that a scribe may have skipped a line when transcribing it.

The text as given by Bede makes little sense:

The eightieth part of $400$ is $5$ and the hundredth part of $400$ is $4$.

Both eighty $5$s and one hundred $4$s give the same result, $400$.

There are that many cloaks.

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