The average monthly income of X and Y is Rs 5050. The average monthly income of Y and Z is Rs 6250 and the average monthly income of X and Z is Rs 5200. The monthly Income of X is :

Options

A. Rs 4050
B. Rs 3500
C. Rs 4000
D. Rs 5000

Answer

To find the monthly income of X, we can use the information given in the problem about the averages of the incomes of X, Y, and Z. Let's break it down step by step.

### Step-by-Step Solution:

1. **Set up the equations based on averages**:
- The average monthly income of X and Y is Rs 5050:
\[ \frac{X + Y}{2} = 5050 \]
Multiplying both sides by 2:
\[ X + Y = 10100 \quad \text{(Equation 1)} \]

- The average monthly income of Y and Z is Rs 6250:
\[ \frac{Y + Z}{2} = 6250 \]
Multiplying both sides by 2:
\[ Y + Z = 12500 \quad \text{(Equation 2)} \]

- The average monthly income of X and Z is Rs 5200:
\[ \frac{X + Z}{2} = 5200 \]
Multiplying both sides by 2:
\[ X + Z = 10400 \quad \text{(Equation 3)} \]

2. **Add all three equations**:
\[ (X + Y) + (Y + Z) + (X + Z) = 10100 + 12500 + 10400 \]
This simplifies to:
\[ 2X + 2Y + 2Z = 33000 \]
Dividing the entire equation by 2:
\[ X + Y + Z = 16500 \quad \text{(Equation 4)} \]

3. **Now, we can find the individual incomes**:
- From Equation 1: \(X + Y = 10100\)
- From Equation 4: \(X + Y + Z = 16500\)
- To find Z, we can subtract Equation 1 from Equation 4:
\[ Z = 16500 - 10100 = 6400 \]

4. **Substituting Z back to find Y**:
- From Equation 2: \(Y + Z = 12500\)
- Substitute Z:
\[ Y + 6400 = 12500 \]
\[ Y = 12500 - 6400 = 6100 \]

5. **Finally, substitute Y back to find X**:
- From Equation 1: \(X + Y = 10100\)
- Substitute Y:
\[ X + 6100 = 10100 \]
\[ X = 10100 - 6100 = 4000 \]

### Conclusion:
The monthly income of X is Rs 4000.