In a farm there are cows and hens. If heads are counted they are 180, if legs are counted they are 420. The number of cows in the farm is

Options

A. 130
B. 150
C. 50
D. 30

Answer

To solve the problem of determining the number of cows on the farm, we can set up a system of equations based on the information provided.

### Step-by-Step Solution:

1. **Define Variables**:
Let \( x \) be the number of cows and \( y \) be the number of hens.

2. **Set Up the Equations**:
- From the problem, we know that the total number of heads (cows and hens) is 180. Since each animal has one head, we can write the first equation as:
\[ x + y = 180 \]
- We also know that the total number of legs is 420. Cows have 4 legs each and hens have 2 legs each, so we can write the second equation as:
\[ 4x + 2y = 420 \]

3. **Simplify the Second Equation**:
We can simplify the second equation by dividing all terms by 2:
\[ 2x + y = 210 \]

4. **Solve the System of Equations**:
Now we have the following system of equations:
\[ \begin{align*} (1) & \quad x + y = 180 \\ (2) & \quad 2x + y = 210 \end{align*} \]
We can subtract equation (1) from equation (2):
\[ (2x + y) - (x + y) = 210 - 180 \]
This simplifies to:
\[ x = 30 \]

5. **Find the Number of Hens**:
Now that we have \( x = 30 \), we can substitute this value back into equation (1) to find \( y \):
\[ 30 + y = 180 \]
Solving for \( y \):
\[ y = 180 - 30 = 150 \]

6. **Conclusion**:
Therefore, the number of cows in the farm is \( \boxed{30} \).