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?

Correct — A. Convert every animal to cows, then weigh by months.

3 horses = 5 cows → 21 horses = 21 × 5⁄3 = 35 cows
6 cows = 10 sheep → 40 sheep = 40 × 6⁄10 = 24 cows

A: 35 × 6 + 15 × 2 = 210 + 30 = 240 cow-months
B: 15 × 6 + 24 × 7.5 = 90 + 180 = 270 cow-months

A's share = 240 ÷ (240 + 270) = 240⁄510 = 8⁄17 → option (a)

Why the others are wrong

• (b)
  19⁄17 — 19⁄17 is more than 1, so A would pay more than the whole rent. A share of a shared rent must lie between 0 and 1.
• (c)
  16⁄17 — 16⁄17 would leave B just 1⁄17, yet B grazes more: 270 cow-months against A's 240. A's share must be under half, and 8⁄17 is.
• (d)
  20⁄17 — 20⁄17 exceeds 1 as well: A would pay the whole rent plus 3⁄17 more. No partner's share can be bigger than the total.

Shared pasture rent is split in the ratio of grazing, counted in animal-months. Different animals eat different amounts, so first turn every animal into one unit using the given equivalences.

Then multiply each group by its months, add each person's groups, and divide by the grand total. A share is a part of the whole, so it always lies between 0 and 1.

Key facts

• 3 horses = 5 cows gives 1 horse = 5⁄3 cows.
• 6 cows = 10 sheep gives 1 sheep = 3⁄5 cow.
• Rent share = own animal-months ÷ total animal-months.

Common traps

• Converting the wrong way: 3 horses = 5 cows makes 21 horses 35 cows, not 21 × 3⁄5 = 12.6 cows.
• Using one period for every group: A's cows graze 2 months while B's cows graze 6.
• Giving B's share, 270⁄510 = 9⁄17, when the question asks what A should cover.

No directly related past PYQ was found.