A and B share a rented field. A utilizes 20 horses for 5 months, while B uses 30 cows for 4 months and 45 sheep for 5 months. If 2 horses are equivalent to 5 cows, and 3 cows are equal to 9 sheep, what portion of the rent is A pays?
\(\dfrac{50}{89}\)
Convert all animals into a single unit (sheep) using the equivalence relations.
Step 1 – conversions:
\(3 \text{ cows} = 9 \text{ sheep} \implies 1 \text{ cow} = 3 \text{ sheep}\)
\(2 \text{ horses} = 5 \text{ cows} \implies 1 \text{ horse} = 2.5 \text{ cows} = 2.5 \times 3 = 7.5 \text{ sheep}\)
Step 2 – A's usage in sheep-months:
\(A = 20 \text{ horses} \times 5 \text{ months} = (20 \times 7.5) \times 5 = 150 \times 5 = 750 \text{ sheep-months}\)
Step 3 – B's usage in sheep-months:
\(B_{\text{cows}} = 30 \times 3 \times 4 = 360\)
\(B_{\text{sheep}} = 45 \times 5 = 225\)
\(B_{\text{total}} = 360 + 225 = 585 \text{ sheep-months}\)
Step 4 – A's share of rent:
\(\dfrac{A}{A+B} = \dfrac{750}{750 + 585} = \dfrac{750}{1335} = \dfrac{50}{89}\)
Hence A pays \(\dfrac{50}{89}\) of the rent.
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