A and B have their annual average income Rs. 80,000. B and C have their annual average Income Rs. 75,000. C and A have their annual average income Rs. 78,000. The annual income of A is

Options

A. Rs. 81000
B. Rs. 82000
C. Rs. 83000
D. Rs. 84000

Answer

To find the annual income of A, we can set up equations based on the information given in the question.

### Step-by-Step Solution:

1. **Set up the equations based on averages:**
- A and B have an average income of Rs. 80,000.
\[ \frac{A + B}{2} = 80,000 \implies A + B = 160,000 \quad \text{(Equation 1)} \]
- B and C have an average income of Rs. 75,000.
\[ \frac{B + C}{2} = 75,000 \implies B + C = 150,000 \quad \text{(Equation 2)} \]
- C and A have an average income of Rs. 78,000.
\[ \frac{C + A}{2} = 78,000 \implies C + A = 156,000 \quad \text{(Equation 3)} \]

2. **Add all three equations:**
\[ (A + B) + (B + C) + (C + A) = 160,000 + 150,000 + 156,000 \]
This simplifies to:
\[ 2A + 2B + 2C = 466,000 \]
Dividing the entire equation by 2 gives:
\[ A + B + C = 233,000 \quad \text{(Equation 4)} \]

3. **Substitute Equation 1 into Equation 4 to find C:**
From Equation 1, we have:
\[ B = 160,000 - A \]
Substitute B in Equation 2:
\[ (160,000 - A) + C = 150,000 \]
Rearranging gives:
\[ C = 150,000 - 160,000 + A \implies C = A - 10,000 \quad \text{(Equation 5)} \]

4. **Substitute Equation 5 into Equation 3 to find A:**
Substitute C in Equation 3:
\[ (A - 10,000) + A = 156,000 \]
Simplifying gives:
\[ 2A - 10,000 = 156,000 \]
Adding 10,000 to both sides:
\[ 2A = 166,000 \]
Dividing by 2 gives:
\[ A = 83,000 \]

### Final Answer:
The annual income of A is **Rs. 83,000**.