P, Q, and R hire a meadow for ₹ 2,920. P grazes 10 cows for 20 days, Q grazes 30 cows for 8 days, and R grazes 16 cows for 9 days. If rent is proportional to (cows × days), how much rent should R pay?

Introduction / Context:
Shared rentals in grazing problems are allocated proportional to the product of the number of animals and the duration. Compute each party's usage weight and distribute the total cost accordingly.

Given Data / Assumptions:

• P: 10 cows × 20 days = 200 cow-days.
• Q: 30 cows × 8 days = 240 cow-days.
• R: 16 cows × 9 days = 144 cow-days.
• Total rent = ₹ 2,920.

Concept / Approach:
R's share = (R's cow-days / total cow-days) * total rent.

Step-by-Step Solution:
Total cow-days = 200 + 240 + 144 = 584. R's fraction = 144/584 = 18/73 after dividing by 8. R's rent = (18/73) * 2,920 = 40 * 18 = ₹ 720 (since 2,920/73 = 40).

Verification / Alternative check:
P's and Q's portions similarly sum with R's to ₹ 2,920, reinforcing proportional correctness.

Why Other Options Are Wrong:
₹ 735, ₹ 820, ₹ 575 deviate from the precise 144/584 fraction; only ₹ 720 equals (18/73)*2,920.

Common Pitfalls:
Using cows + days instead of cows × days or failing to reduce the fraction cleanly.

Final Answer:
₹ 720