Roman distances
Marcus' atrium was a square with each side 50 pedes long. How many times did Marcus have to walk round his atrium to complete his daily exercise of 8 stadia?
Age
11 to 14
Challenge level
Problem
In ancient Rome they used the following units of length:
5 pedes = 1 passus,
125 passus = 1 stadium.
The atrium (courtyard) of Marcus' home was a square with each side 50 pedes long.
How many times did Marcus have to walk round his atrium to complete his daily exercise of 8 stadia?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: 25
\begin{eqnarray} 8 \; \text{stadia} &=& 8 \times 125 \; \text{passus} \\ &=& 8 \times 125 \times 5 \; \text{pedes}\\ &=& 5000 \; \text{pedes} \end{eqnarray}
Marcus has to walk $5000 \; \text{pedes}$.
$1$ circuit of his atrium is $50 \times 4 = 200 \; \text{pedes}$ long, so he has to walk round his atrium $\frac{5000}{200}=25$ times.
In Roman numerals, $25$ is XXV.