There are deers and ducks in a zoo. If heads are counted then there are 180 total and if legs are counted then there are 448 total. Find the number of deers and ducks .

Answer

To solve the problem of finding the number of deers and ducks in the zoo, we can set up a system of equations based on the information provided.

### Step-by-step Solution:

1. **Define Variables**:
Let:
- \( d \) = number of ducks
- \( r \) = number of deers

2. **Set Up the First Equation**:
Since each animal has one head, the total number of heads (which is 180) gives us our first equation:
\[ d + r = 180 \quad \text{(Equation 1)} \]

3. **Set Up the Second Equation**:
Ducks have 2 legs each and deers have 4 legs each. The total number of legs counted is 448, which gives us our second equation:
\[ 2d + 4r = 448 \]
We can simplify this equation by dividing everything by 2:
\[ d + 2r = 224 \quad \text{(Equation 2)} \]

4. **Solve the System of Equations**:
Now we have two equations:
- \( d + r = 180 \) (Equation 1)
- \( d + 2r = 224 \) (Equation 2)

We can subtract Equation 1 from Equation 2:
\[ (d + 2r) - (d + r) = 224 - 180 \]
Simplifying this, we get:
\[ r = 44 \]

5. **Substitute Back to Find \( d \)**:
Now that we have \( r \), we can substitute it back into Equation 1 to find \( d \):
\[ d + 44 = 180 \]
Subtracting 44 from both sides gives:
\[ d = 180 - 44 = 136 \]

6. **Final Answer**:
Therefore, the number of deers is \( r = 44 \) and the number of ducks is \( d = 136 \).

### Summary:
- Number of deers = 44
- Number of ducks = 136