Propositiones ad Acuendos Juvenes/Problems/40 - De Homine et Ovibus in Monte Pascentibus
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Propositiones ad Acuendos Juvenes by Alcuin of York: Problem $40$
- De Homine et Ovibus in Monte Pascentibus
- A Man and some Sheep Grazing on the Hillside
- A man looked at some sheep grazing on the hillside, and said,
- "I wish I had these,
- and as many again,
- plus half of half that total,
- plus half of that last amount;
- then, counting myself, I would take $100$ to my home.
- How many sheep did he see?
Solution
- $36$.
Proof
Let $x$ be the number of sheep seen.
Then we have:
- $x + x + \dfrac {x + x} 4 + \dfrac 1 2 \paren {\dfrac {x + x} 4} + 1 = 100$
That is, after algebra:
- $\dfrac {11 x} 4 = 99$
Hence the result.
$\blacksquare$
Historical Note
The alert reader will note that this is numerically identical to Problems $2$: De Viro Ambulante in Via.
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