Krishna has some hens and some goats. If the total number animal heads are 81 and the total number of animnal legs are 234, how many goats does Krishna have?

Options

A. 45
B. 24
C. 36
D. Cannot be determined

Answer

To solve the problem, we need to set up a system of equations based on the information given about the heads and legs of the animals.

### Step 1: Define the variables
Let:
- \( x \) = number of goats
- \( y \) = number of hens

### Step 2: Set up the equations
From the problem, we know:
1. The total number of heads (goats + hens) is 81:
\[ x + y = 81 \quad \text{(Equation 1)} \]
2. The total number of legs (goats have 4 legs and hens have 2 legs) is 234:
\[ 4x + 2y = 234 \quad \text{(Equation 2)} \]

### Step 3: Simplify Equation 2
We can simplify Equation 2 by dividing every term by 2:
\[ 2x + y = 117 \quad \text{(Equation 3)} \]

### Step 4: Solve the system of equations
Now we have two equations:
1. \( x + y = 81 \) (Equation 1)
2. \( 2x + y = 117 \) (Equation 3)

We can subtract Equation 1 from Equation 3 to eliminate \( y \):
\[ (2x + y) - (x + y) = 117 - 81 \]
This simplifies to:
\[ x = 36 \]

### Step 5: Substitute back to find \( y \)
Now that we have \( x \), we can substitute it back into Equation 1 to find \( y \):
\[ 36 + y = 81 \]
Subtracting 36 from both sides gives:
\[ y = 45 \]

### Step 6: Conclusion
Thus, Krishna has:
- Number of goats \( x = 36 \)
- Number of hens \( y = 45 \)

So, the number of goats Krishna has is **36**.

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