If the rent for grazing 40 cows for 20 days is Rs. 370, how many cows can graze for 30 days on Rs. 111 ?

Options

A. 6
B. 8
C. 5
D. 12

Answer

To solve the problem step by step, we will use the concept of direct proportionality between the number of cows, the number of days, and the rent.

### Step-by-Step Solution:

1. **Understand the Given Information**:
- Rent for grazing 40 cows for 20 days = Rs. 370.
- We need to find out how many cows (let's denote this as \( x \)) can graze for 30 days on Rs. 111.

2. **Set Up the Proportionality Equation**:
- The relationship can be established using the formula:
\[ \frac{m_1 \cdot d_1 \cdot h_1}{w_1} = \frac{m_2 \cdot d_2 \cdot h_2}{w_2} \]
- Here:
- \( m_1 = 40 \) (number of cows in the first case)
- \( d_1 = 20 \) (number of days in the first case)
- \( w_1 = 370 \) (rent in the first case)
- \( m_2 = x \) (number of cows in the second case)
- \( d_2 = 30 \) (number of days in the second case)
- \( w_2 = 111 \) (rent in the second case)

3. **Substituting the Values**:
- Substitute the known values into the equation:
\[ \frac{40 \cdot 20}{370} = \frac{x \cdot 30}{111} \]

4. **Cross-Multiply to Solve for \( x \)**:
- Cross-multiplying gives:
\[ 40 \cdot 20 \cdot 111 = 370 \cdot x \cdot 30 \]
- Simplifying this:
\[ 88800 = 11100x \]

5. **Isolate \( x \)**:
- Divide both sides by 11100:
\[ x = \frac{88800}{11100} = 8 \]

6. **Conclusion**:
- Therefore, the number of cows that can graze for 30 days on Rs. 111 is \( x = 8 \).

### Final Answer:
The answer is **8 cows** (Option B).