Propositiones ad Acuendos Juvenes/Problems/49 - De Carpentariis

Propositiones ad Acuendos Juvenes by Alcuin of York: Problem $49$

$7$ carpenters made $7$ wheels each.

How many carts were fitted?

$12$, with one wheel left over.

Let $x$ be the number of carts fitted.

Let $y$ be the number of wheels made.

It is presumed that each cart has $4$ wheels.

Thus we have:

$\ds x$

$=$

$\ds \floor {\dfrac y 4}$

where $\floor {\, \cdot \,}$ denotes the floor function

$\ds $

$=$

$\ds \floor {\dfrac {7 \times 7} 4}$

$\ds $

$=$

$\ds \floor {\dfrac {49} 4}$

$\ds $

$=$

$\ds 12$

That is, we have that:

$49 = 4 \times 12 + 1$

The result follows.

$\blacksquare$