In a park, there are some cows and some ducks. If total number of heads in the park are 68 and number of their legs together is 198, then find the number of ducks in the park.

Answer

To solve the problem step by step, we need to determine the number of cows and ducks in the park based on the given information about heads and legs.

### Step 1: Define Variables
Let:
- \( C \) = number of cows
- \( D \) = number of ducks

### Step 2: Set Up Equations
From the problem, we know:
1. The total number of heads (cows + ducks) is 68:
\[ C + D = 68 \quad \text{(Equation 1)} \]
2. The total number of legs (cows have 4 legs each and ducks have 2 legs each) is 198:
\[ 4C + 2D = 198 \quad \text{(Equation 2)} \]

### Step 3: Simplify Equation 2
We can simplify Equation 2 by dividing all terms by 2:
\[ 2C + D = 99 \quad \text{(Equation 3)} \]

### Step 4: Solve the System of Equations
Now we have two equations:
1. \( C + D = 68 \) (Equation 1)
2. \( 2C + D = 99 \) (Equation 3)

We can subtract Equation 1 from Equation 3:
\[ (2C + D) - (C + D) = 99 - 68 \]
This simplifies to:
\[ C = 31 \]

### Step 5: Substitute Back to Find D
Now that we know \( C \), we can substitute it back into Equation 1 to find \( D \):
\[ 31 + D = 68 \]
Subtracting 31 from both sides gives:
\[ D = 68 - 31 = 37 \]

### Conclusion
The number of ducks in the park is \( D = 37 \).

### Summary of Steps:
1. Define variables for cows and ducks.
2. Set up equations based on the total number of heads and legs.
3. Simplify the equation for legs.
4. Solve the system of equations to find the number of cows.
5. Substitute back to find the number of ducks.