Three milkmen rent a pasture. A grazes 24 cows for 4 months, B grazes 10 cows for 6 months, and C grazes 56 cows for 5 months. If A's share of the rent is Rs. 960, what is the total rent of the field?

Introduction / Context:
This is a typical cows and pasture problem where rent is shared in proportion to the total grazing done by each person. Grazing is measured in terms of the product of the number of animals and the time they graze. From one person's known share of rent, we are asked to find the total rent of the pasture.

Given Data / Assumptions:

• A grazes 24 cows for 4 months.
• B grazes 10 cows for 6 months.
• C grazes 56 cows for 5 months.
• A's share of rent is Rs. 960.
• Rent is shared in proportion to cows * months for each milkman.
• We must find the total rent of the field.

Concept / Approach:
The cost of using a common pasture is naturally proportional to the total grazing pressure applied, which is number of cows multiplied by the number of months. First we compute this product for each milkman, then form a ratio. Since we know A's rupee share and his part in the ratio, we can find the value of one ratio unit and from there compute the full rent.

Step-by-Step Solution:
Step 1: Compute A's grazing units: 24 cows * 4 months = 96 cow months. Step 2: Compute B's grazing units: 10 cows * 6 months = 60 cow months. Step 3: Compute C's grazing units: 56 cows * 5 months = 280 cow months. Step 4: Total grazing units = 96 + 60 + 280 = 436 cow months. Step 5: The ratio A : B : C is 96 : 60 : 280. Step 6: Sum of ratio parts is 436, matching the total cow months. Step 7: A's share of rent is Rs. 960, which corresponds to 96 parts. Step 8: Value of one part = 960 / 96 = Rs. 10. Step 9: Total rent = 436 parts * Rs. 10 per part = Rs. 4360.

Verification / Alternative check:
If the total rent is Rs. 4360 and each cow month is Rs. 10, then A pays 96 * 10 = Rs. 960, which matches the given information. B pays 60 * 10 = Rs. 600 and C pays 280 * 10 = Rs. 2800. Adding these gives 960 + 600 + 2800 = Rs. 4360, which is consistent with the total rent and the proportional distribution based on grazing units.

Why Other Options Are Wrong:
Rs. 4530, Rs. 4440 and Rs. 4280 do not give a per unit rent of exactly Rs. 10 when divided by 436, and they do not reproduce A's share as Rs. 960 when multiplied by A's 96 cow months. Therefore they are inconsistent with the given information and the proportional method and must be rejected.

Common Pitfalls:
A common mistake is to compare only the number of cows and ignore the different grazing durations. Another error is to attempt to use percentages instead of direct proportionality based on cow months. Always remember that in such problems, both the number of animals and the length of time they graze contribute equally to the cost share.

Final Answer:
The total rent of the field is Rs. 4360.