There are some shepherds and their sheep in a grazing field. The no. of total heads are 60 and total legs are 168 including both men and sheep. The no. of sheep is :

Options

A. 18
B. 26
C. 24
D. 36

Answer

To solve the problem step by step, we will define the variables, set up equations based on the information given, and then solve for the unknowns.

### Step 1: Define the Variables
Let:
- \( x \) = number of sheep
- \( y \) = number of shepherds

### Step 2: Set Up the Equations
From the problem, we know:
1. The total number of heads (shepherds + sheep) is 60:
\[ x + y = 60 \quad \text{(Equation 1)} \]

2. The total number of legs (shepherds have 2 legs each and sheep have 4 legs each) is 168:
\[ 2y + 4x = 168 \quad \text{(Equation 2)} \]

### Step 3: Simplify Equation 2
We can simplify Equation 2 by dividing everything by 2:
\[ y + 2x = 84 \quad \text{(Equation 3)} \]

### Step 4: Solve the System of Equations
Now we have two equations:
1. \( x + y = 60 \) (Equation 1)
2. \( y + 2x = 84 \) (Equation 3)

We can solve for \( y \) in terms of \( x \) using Equation 1:
\[ y = 60 - x \]

Now, substitute \( y \) in Equation 3:
\[ (60 - x) + 2x = 84 \]

### Step 5: Simplify and Solve for \( x \)
Combine like terms:
\[ 60 + x = 84 \]
Subtract 60 from both sides:
\[ x = 84 - 60 \]
\[ x = 24 \]

### Step 6: Find the Number of Sheep
Since \( x \) represents the number of sheep, we conclude that:
\[ \text{Number of sheep} = 24 \]

### Final Answer
The number of sheep is **24**.

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