The monthly incomes of two persons are in the ratio 4 : 7 and their expenses are in the ratio 11 : 20. If each saves Rs 400 per month, then their monthly incomes must be respectively

Options

A. 3600 and 4200
B. 4000 and 7000
C. 4200 and 7350
D. 4800 and 8400

Answer

To solve the problem step by step, let's denote the monthly incomes and expenses of the two persons as follows:

1. **Let the incomes of the two persons be:**
- Person A: \( 4X \)
- Person B: \( 7X \)

2. **Let the expenses of the two persons be:**
- Person A: \( 11Y \)
- Person B: \( 20Y \)

3. **Given that both persons save Rs. 400 each month, we can set up the following equations based on the relationship between income, expenses, and savings:**
- For Person A:
\[ 4X - 11Y = 400 \quad \text{(1)} \]
- For Person B:
\[ 7X - 20Y = 400 \quad \text{(2)} \]

4. **Now, we will solve these equations simultaneously. First, we can multiply equation (1) by 7 and equation (2) by 4 to eliminate \( X \):**
- Multiplying equation (1) by 7:
\[ 28X - 77Y = 2800 \quad \text{(3)} \]
- Multiplying equation (2) by 4:
\[ 28X - 80Y = 1600 \quad \text{(4)} \]

5. **Next, we will subtract equation (4) from equation (3):**
\[ (28X - 77Y) - (28X - 80Y) = 2800 - 1600 \]
This simplifies to:
\[ 3Y = 1200 \]

6. **Now, we can solve for \( Y \):**
\[ Y = \frac{1200}{3} = 400 \]

7. **Substituting \( Y = 400 \) back into equation (1) to find \( X \):**
\[ 4X - 11(400) = 400 \]
Simplifying gives:
\[ 4X - 4400 = 400 \]
\[ 4X = 4800 \]
\[ X = \frac{4800}{4} = 1200 \]

8. **Finally, we can find the monthly incomes of both persons:**
- Person A's income:
\[ 4X = 4 \times 1200 = 4800 \]
- Person B's income:
\[ 7X = 7 \times 1200 = 8400 \]

Thus, the monthly incomes of the two persons are Rs. 4800 and Rs. 8400 respectively.

### Summary of the Solution:
- Monthly Income of Person A: Rs. 4800
- Monthly Income of Person B: Rs. 8400