Answer for a 13-year-old: How many fish to restock if you can only sell half the pond this year (Clause 65, Capitulary de Villis)

Short idea: If you can only sell half the pond, selling half leaves you with the other half. To make sure there are always fish left next season, you should restock at least as many fish as you sold (that is, restock half the pond). For safety, add a small extra number so the pond can recover from losses.

Step-by-step (with math):

1. Call the number of fish in the pond now P.
2. You may sell only half the pond, so you sell P/2 fish this year. That leaves P/2 fish in the pond after the sale.
3. Let R be the number of fish you restock right after the sale.
4. After restocking, the pond will have (P/2) + R fish for next season.
5. If your goal is simply to "always have some left" next season (i.e., at least 1 fish), in most realistic cases you don’t need to restock anything because P/2 will usually be at least 1. Formally, you need R ≥ 1 − (P/2). If P/2 ≥ 1, then R ≥ 0.
6. If your goal is to restore the pond to the original size P (a sustainable approach consistent with rules like Clause 65 that encourage keeping resources for the future), set (P/2) + R ≥ P. That gives R ≥ P/2. So you should restock at least P/2 fish — the same number you sold.
7. Best practice: restock a little more than P/2 (for example, P/2 + 10%), to allow for natural deaths or predators and to make the pond healthier for the next season.

Numeric example (easy numbers):

• Suppose P = 100 fish.
• You sell half: 100/2 = 50 fish sold, leaving 50 in the pond.
• To return to 100 for next season, restock R = 50 fish (that is, P/2).
• If you want a safety buffer, restock 50 + 10% of 100 = 60 fish. Then the pond would have 50 + 60 = 110 fish next season.

Why this matches the spirit of Clause 65: Clause 65 of the Capitulary de Villis and similar medieval rules asked stewards to care for estate resources so they weren’t exhausted. Restocking at least what you sold (P/2) is a simple way to keep the pond sustainable for future seasons.

Teacher feedback — Ally McBeal cadence (short, punchy, encouraging):

Nice start — clear claim. Nice math — you set P, used fractions, solved cleanly. Rhythm is good: quick setup, clear equation, real example. To be even stronger: name your goal (minimum one fish vs. restoring full stock), show both answers, and explain why a buffer helps in real life. Great use of numbers — tidy, accurate, thoughtful.

Bottom line: Restock at least the number you sold (P/2) to keep the pond at its original size; add a small extra amount if you want a safety buffer.