In a zoo, each pigeon has 2 legs, and each rabbit has 4 legs. The head count of the two species together is 12, and the leg count is 32. How many pigeons and how many rabbits are there in the zoo?

Options

A. 4,8
B. 6,6
C. 6,8
D. 8,4

Answer

To solve the problem, we will use a system of equations based on the information given about the pigeons and rabbits in the zoo.

### Step 1: Define the Variables
Let:
- \( P \) = number of pigeons
- \( R \) = number of rabbits

### Step 2: Set Up the Equations
From the problem, we know:
1. The total number of heads (pigeons + rabbits) is 12:
\[ P + R = 12 \quad \text{(Equation 1)} \]
2. The total number of legs (pigeons have 2 legs and rabbits have 4 legs) is 32:
\[ 2P + 4R = 32 \quad \text{(Equation 2)} \]

### Step 3: Simplify Equation 2
We can simplify Equation 2 by dividing everything by 2:
\[ P + 2R = 16 \quad \text{(Equation 3)} \]

### Step 4: Solve the System of Equations
Now, we have two equations:
1. \( P + R = 12 \) (Equation 1)
2. \( P + 2R = 16 \) (Equation 3)

Next, we can subtract Equation 1 from Equation 3:
\[ (P + 2R) - (P + R) = 16 - 12 \]
This simplifies to:
\[ R = 4 \]

### Step 5: Substitute to Find P
Now that we have \( R = 4 \), we can substitute this value back into Equation 1 to find \( P \):
\[ P + 4 = 12 \]
Subtracting 4 from both sides gives:
\[ P = 8 \]

### Conclusion
Thus, the number of pigeons is \( P = 8 \) and the number of rabbits is \( R = 4 \).

### Final Answer
- Number of Pigeons: 8
- Number of Rabbits: 4

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