Two individuals, A and B, have rented a field. A places 21 horses in the field for 6 months and 15 cows for 2 months. B adds 15 cows for 6 months and 40 sheep for 7.5 months. If 3 horses consume the same amount of food as 5 cows, and 6 cows eat as much as 10 sheep, what portion of the rent should A cover?
\(\dfrac{8}{17}\)
Step 1 — Convert everything to cow-months. Given \(3\text{ horses} = 5\text{ cows}\), so 1 horse = \(\tfrac{5}{3}\) cow. Given \(6\text{ cows} = 10\text{ sheep}\), so 1 sheep = \(\tfrac{3}{5}\) cow.
Step 2 — A's contribution (in cow-months):
\(21\text{ horses} \times 6 = 21 \times \tfrac{5}{3} \times 6 = 210\)
\(15\text{ cows} \times 2 = 30\)
\(\text{A total} = 210 + 30 = 240\)
Step 3 — B's contribution (in cow-months):
\(15\text{ cows} \times 6 = 90\)
\(40\text{ sheep} \times 7.5 = 40 \times \tfrac{3}{5} \times 7.5 = 180\)
\(\text{B total} = 90 + 180 = 270\)
Step 4 — A's rent share:
\(\dfrac{240}{240 + 270} = \dfrac{240}{510} = \dfrac{8}{17}\)
Hence, A pays \(\tfrac{8}{17}\) of the rent.
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