There are pigeons and rabbits in a zoo. If heads are counted then there are 90 total and if legs are counted then there are 224 total. Find the number of pigeon and rabbits.

Answer

To solve the problem of finding the number of pigeons and rabbits in the zoo, we can follow these steps:

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

### Step 2: Set Up Equations
From the problem, we know:
1. Each animal has one head, and the total number of heads is 90:
\[ P + R = 90 \quad \text{(Equation 1)} \]

2. Pigeons have 2 legs and rabbits have 4 legs. The total number of legs is 224:
\[ 2P + 4R = 224 \quad \text{(Equation 2)} \]

### Step 3: Simplify Equation 2
We can simplify Equation 2 by dividing all terms by 2:
\[ P + 2R = 112 \quad \text{(Equation 3)} \]

### Step 4: Solve the Equations
Now we have two equations:
1. \( P + R = 90 \) (Equation 1)
2. \( P + 2R = 112 \) (Equation 3)

We can subtract Equation 1 from Equation 3:
\[ (P + 2R) - (P + R) = 112 - 90 \]
This simplifies to:
\[ R = 22 \]

### Step 5: Substitute to Find P
Now that we have \( R \), we can substitute it back into Equation 1 to find \( P \):
\[ P + 22 = 90 \]
\[ P = 90 - 22 \]
\[ P = 68 \]

### Conclusion
Thus, the number of pigeons is \( P = 68 \) and the number of rabbits is \( R = 22 \).

### Summary of Results
- Number of Pigeons: 68
- Number of Rabbits: 22

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