A man has some hens and cows. If the number of heads : number of feet = 12 : 35, find out the number of hens, if the number of heads alone is 48.

Options

A. 28
B. 26
C. 24
D. 22

Answer

To solve the problem step by step, we will follow the given information and use algebra to find the number of hens.

### Step 1: Understand the Ratio of Heads to Feet
The ratio of the number of heads to the number of feet is given as 12:35. This means for every 12 heads, there are 35 feet.

### Step 2: Set Up the Variables
Let:
- \( H \) = number of hens
- \( C \) = number of cows

From the problem, we know that:
- Each hen has 1 head and 2 feet.
- Each cow has 1 head and 4 feet.

### Step 3: Express Total Heads
The total number of heads (hens + cows) is given as 48. Therefore, we can write the equation:
\[ H + C = 48 \]

### Step 4: Express Total Feet
Using the information about the number of feet:
- Total feet from hens = \( 2H \)
- Total feet from cows = \( 4C \)

Thus, the total number of feet can be expressed as:
\[ 2H + 4C \]

### Step 5: Relate Heads and Feet Using the Ratio
From the ratio of heads to feet (12:35), we can set up the equation:
\[ \frac{H + C}{2H + 4C} = \frac{12}{35} \]

### Step 6: Substitute Total Heads into the Ratio Equation
Since \( H + C = 48 \), we can substitute this into the equation:
\[ \frac{48}{2H + 4C} = \frac{12}{35} \]

### Step 7: Cross-Multiply to Solve for Feet
Cross-multiplying gives:
\[ 48 \cdot 35 = 12 \cdot (2H + 4C) \]
Calculating the left side:
\[ 1680 = 12(2H + 4C) \]

### Step 8: Simplify the Equation
Dividing both sides by 12:
\[ 140 = 2H + 4C \]

### Step 9: Solve the System of Equations
Now we have a system of equations:
1. \( H + C = 48 \)
2. \( 2H + 4C = 140 \)

From the first equation, we can express \( C \) in terms of \( H \):
\[ C = 48 - H \]

Substituting this into the second equation:
\[ 2H + 4(48 - H) = 140 \]
Expanding this gives:
\[ 2H + 192 - 4H = 140 \]
Combining like terms:
\[ -2H + 192 = 140 \]
Subtracting 192 from both sides:
\[ -2H = 140 - 192 \]
\[ -2H = -52 \]
Dividing by -2:
\[ H = 26 \]

### Conclusion
The number of hens is \( \boxed{26} \).