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 can follow these instructions:

### Step 1: Understand the problem
We know that the ratio of the number of heads (hens + cows) to the number of feet (hens' feet + cows' feet) is given as 12:35. We also know that the total number of heads is 48.

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

From the problem, we can establish the following relationships:
1. Total heads (H + C) = 48
2. Each hen has 1 head and 2 feet, and each cow has 1 head and 4 feet.

### Step 3: Express the total number of feet
The total number of feet can be expressed as:
\[ \text{Total feet} = 2H + 4C \]

### Step 4: Use the ratio of heads to feet
According to the given ratio:
\[ \frac{\text{Total heads}}{\text{Total feet}} = \frac{12}{35} \]
Substituting the expressions for heads and feet, we get:
\[ \frac{H + C}{2H + 4C} = \frac{12}{35} \]

### Step 5: Substitute the total heads
Since we know that \( H + C = 48 \), we can substitute this into the equation:
\[ \frac{48}{2H + 4C} = \frac{12}{35} \]

### Step 6: Cross-multiply to solve for feet
Cross-multiplying gives us:
\[ 48 \cdot 35 = 12 \cdot (2H + 4C) \]
Calculating the left side:
\[ 1680 = 12(2H + 4C) \]

### Step 7: Simplify the equation
Dividing both sides by 12:
\[ 140 = 2H + 4C \]

### Step 8: Solve the system of equations
Now we have two 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 \( C \) into the second equation:
\[ 2H + 4(48 - H) = 140 \]
Expanding this:
\[ 2H + 192 - 4H = 140 \]
Combining like terms:
\[ -2H + 192 = 140 \]
Subtracting 192 from both sides:
\[ -2H = 140 - 192 \]
\[ -2H = -52 \]
Dividing by -2:
\[ H = 26 \]

### Step 9: Find the number of cows
Now substitute \( H \) back to find \( C \):
\[ C = 48 - H = 48 - 26 = 22 \]

### Final Answer
The number of hens is \( H = 26 \).

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