Two individuals, A and B rent a pasture. A uses 16 cows for a duration of 3 months and 20 sheep for 4 months and B uses 30 sheep for 6 months. If 4 cows are equivalent to 8 sheep, determine A's portion of the rent.

Correct — A. Convert the cows to sheep, then weigh by months.

4 cows = 8 sheep → 1 cow = 2 sheep
16 cows = 16 × 2 = 32 sheep

A: 32 × 3 + 20 × 4 = 96 + 80 = 176 sheep-months
B: 30 × 6 = 180 sheep-months

A's share = 176 ÷ (176 + 180) = 176⁄356 = 44⁄89 → option (a)

Why the others are wrong

• (b)
  54⁄89 — 54⁄89 gives A more than half (half is 44.5⁄89), but A's 176 sheep-months are fewer than B's 180, so A pays a little under half.
• (c)
  64⁄89 — 64⁄89 would mean A grazed far more than B. In fact B grazes more, 180 sheep-months against 176, so A's share is below 1⁄2.
• (d)
  74⁄89 — 74⁄89 leaves B only 15⁄89, as if B grazed about a fifth of what A did. The totals are 176 and 180, nearly equal, so the shares are too.

Grazing rent is shared in the ratio of animal-months. Convert to the smaller animal, here the sheep, and every count stays a whole number: 1 cow = 2 sheep.

Then multiply each group by its months and compare the two totals. When the totals are close, as 176 and 180 are, each share sits close to one half.

Key facts

• 4 cows = 8 sheep gives 1 cow = 2 sheep.
• A grazes 16 cows × 3 months = 96 sheep-months plus 20 sheep × 4 months = 80.
• 176⁄356 reduces by 4 to 44⁄89.

Common traps

• Halving the cows instead of doubling them: 16 cows are 32 sheep, not 8.
• Leaving out the months: A's cows graze 3 months and A's sheep 4, so each group needs its own multiplier.

No directly related past PYQ was found.