A, B, and C rent a pasture jointly. A puts 10 oxen for 7 months, B puts 12 oxen for 5 months, and C puts 15 oxen for 3 months for grazing. If the total rent for the pasture is Rs. 175, how much should C pay as his share of the rent?

Introduction / Context:
This problem is an example of cost sharing based on usage. The cost of renting the pasture must be divided according to how much each person uses it. Usage here is measured in ox months, which is similar to the capital time idea in partnership questions. The more oxen a person keeps and the longer they graze, the bigger their share of the rent.

Given Data / Assumptions:
- A keeps 10 oxen for 7 months.
- B keeps 12 oxen for 5 months.
- C keeps 15 oxen for 3 months.
- Total rent for the pasture is Rs. 175.
- Rent is shared in proportion to the product of number of oxen and number of months.

Concept / Approach:
We compute ox months for each person by multiplying the number of oxen by the number of months they graze. These ox months represent how much grazing each person used. The total ox months give the overall usage. Each person's fraction of total ox months equals their fraction of the rent. Therefore C's share of the rent equals total rent multiplied by C's ox months divided by total ox months.

Step-by-Step Solution:
Step 1: Compute A's ox months: 10 oxen * 7 months = 70 ox months.Step 2: Compute B's ox months: 12 oxen * 5 months = 60 ox months.Step 3: Compute C's ox months: 15 oxen * 3 months = 45 ox months.Step 4: Total ox months = 70 + 60 + 45 = 175 ox months.Step 5: Total rent is Rs. 175. Therefore, rent per ox month = 175 / 175 = Rs. 1 per ox month.Step 6: C used 45 ox months, so C's share of the rent = 45 * 1 = Rs. 45.

Verification / Alternative check:
We can also compute shares of A and B to verify the total. A's share = 70 * 1 = Rs. 70. B's share = 60 * 1 = Rs. 60. C's share = 45 * 1 = Rs. 45. Adding 70 + 60 + 45 gives Rs. 175, which matches the total rent. This confirms that the rate of Rs. 1 per ox month and C's share of Rs. 45 are correct.

Why Other Options Are Wrong:
If C paid Rs. 50, 55, or 60, his share would exceed the fair amount determined by his 45 ox months. This would also cause the sum of all three contributions either to exceed or not match the total rent of Rs. 175. Only Rs. 45 leads to a consistent division of the rent according to usage.

Common Pitfalls:
Some students add the numbers of oxen directly without multiplying by months, which ignores the time factor. Others mistakenly divide the rent in proportion to months only or in proportion to oxen only. Another error is to treat 10 : 12 : 15 as the sharing ratio, which is incorrect because the grazing periods are different. Always multiply number of animals by time to get correct ox months before sharing the cost.

Final Answer:
C must pay Rs. 45 as his share of the rent.