Short answer (one line)
If you want the pond to stay at the same size next season, restock the same number you sold — that is, restock half the pond. If you only want to make sure there’s at least some fish left next season even if nothing breeds, restock at least 1 fish.
Step-by-step explanation
- Define the starting number: Let P be the number of fish in the pond now.
- You sell half the pond: Selling half means you sell P/2 fish. The pond then has P/2 fish left.
- What 'always some left' could mean:
- Minimum meaning — at least one fish remains next season: then you only need to restock enough so the pond doesn’t become empty. If P/2 is already at least 1, you may not need to add any fish. If you want to be safe if nothing breeds, adding 1 fish guarantees ‘some’ remains.
- Conservation meaning — keep the pond at the same size each year so you can keep selling half every year: if you want to return to P fish after restocking, you must add R fish so that P/2 + R = P. Solve R = P − P/2 = P/2. So restock half the pond (the same number you sold).
- Worked number example:
Suppose the pond has 100 fish (P = 100). You sell half → you sell 50, leaving 50. To keep the pond at 100 fish for next year, restock 50 fish (that is, half the pond). If you only want to be sure there is at least one fish next year and you are worried none will breed, restock 1 fish.
- Adding reproduction (optional):
If the fish reproduce during the year, you can restock fewer than the number you sold and still keep the population. For example, if the remaining fish reproduce enough to make up the shortfall, R can be smaller. To figure that out you need the expected birth rate.
Link to Clause 65, Capitulary de Villis
The Capitulary de Villis (a set of rules from Charlemagne’s court) includes instructions about managing farm resources like ponds. Clause 65 urges careful management of fish so there will be food in future seasons. That historical rule follows the same idea: don’t sell everything now — leave or replace enough fish so the pond continues to supply fish later.
Exam-style exemplary answer (concise)
Let P be the pond’s fish. Selling half leaves P/2. To restore the pond to P for the next season, restock R where P/2 + R = P, so R = P/2. Therefore, restock half the pond (the same number you sold) to ensure the pond is as full next season. If only a single fish is needed to guarantee the pond isn’t empty, restocking 1 fish suffices, but to maintain steady supply year after year, restock half the pond.
Teacher feedback (what makes this an exemplary answer)
- Clear definition of variables (P makes the math simple).
- Two useful interpretations given: minimum survival (at least one fish) and steady-state conservation (keep pond size the same).
- Worked numerical example to show the idea in practice.
- Historical link to Clause 65 to show understanding of the rule’s purpose: long-term provision, not short-term profit.
Tip: if you know the pond’s usual reproduction rate, include that number and adjust the restock amount down — sustainable harvesting depends on both how many you remove and how fast the population grows.