Hari and Murli have 24 cows and 30 cows respectively. Both of them together hired a grazing field for the whole month of November. Initially Hari's cows used for grazing then for the remaining days of the month Murli's cows grazed it. If Hari has paid ₹ 3500 and Murli has paid ₹ 5000 for grazing then for how many days Hari used the grazing field :

Options

A. 14
B. 16
C. 21
D. 20

Answer

To solve the problem, we need to determine how many days Hari used the grazing field based on the payments made by both Hari and Murli and the number of cows they have.

### Step-by-Step Solution:

1. **Identify the number of cows**:
- Hari has 24 cows.
- Murli has 30 cows.

2. **Calculate the total payment made**:
- Hari paid ₹3500.
- Murli paid ₹5000.
- Total payment = ₹3500 + ₹5000 = ₹8500.

3. **Determine the ratio of cows**:
- The ratio of cows between Hari and Murli is 24:30, which simplifies to 4:5.

4. **Calculate the total parts in the ratio**:
- Total parts = 4 (for Hari) + 5 (for Murli) = 9 parts.

5. **Calculate the value of each part**:
- Total payment = ₹8500.
- Value of each part = Total payment / Total parts = ₹8500 / 9 ≈ ₹944.44.

6. **Calculate the payment for each person based on their ratio**:
- Payment for Hari = 4 parts = 4 × ₹944.44 ≈ ₹3777.76.
- Payment for Murli = 5 parts = 5 × ₹944.44 ≈ ₹4722.22.

7. **Set up the equation based on days of grazing**:
- Let \( x \) be the number of days Hari used the grazing field.
- Then, the number of days Murli used the field is \( 30 - x \) (since November has 30 days).
- The cost of grazing per day for each person can be calculated based on the number of cows:
- Cost per day for Hari = 24 cows.
- Cost per day for Murli = 30 cows.
- Therefore, the total cost for Hari for \( x \) days = \( 24x \).
- The total cost for Murli for \( 30 - x \) days = \( 30(30 - x) \).

8. **Set up the equation based on payments**:
- Since the total payments made by both should equal the total cost:
\[ 24x = 3500 \quad \text{(for Hari)} \]
\[ 30(30 - x) = 5000 \quad \text{(for Murli)} \]

9. **Solve for \( x \)**:
- From Hari's equation:
\[ x = \frac{3500}{24} \approx 145.83 \text{ (not feasible, recheck)} \]
- From Murli's equation:
\[ 30(30 - x) = 5000 \implies 900 - 30x = 5000 \implies 30x = 900 - 5000 \implies 30x = -4100 \text{ (not feasible)} \]

10. **Revisiting the grazing cost**:
- The correct approach is to use the ratio of payments:
\[ \frac{3500}{5000} = \frac{4x}{5(30-x)} \]
- Cross-multiplying gives:
\[ 3500 \cdot 5(30 - x) = 5000 \cdot 4x \]
- Simplifying leads to:
\[ 17500(30 - x) = 20000x \]
- Solving this equation will give us the value of \( x \).

11. **Final Calculation**:
- After solving, we find \( x = 14 \).
- Thus, Hari used the grazing field for **14 days**.

### Final Answer:
Hari used the grazing field for **14 days**.