The ratio of incomes of two persons is 9 : 7 and the ratio of their expenditures is 4 : 3. If each of them manages to save Rs. 2,000 per month, then one person monthly income is :

Question

Accepted Answer

To determine why Option 1 (Rs. 18,000) is the correct answer, let's analyze the problem using the given ratios and savings information.

1. **Understanding the Ratios:**
   - The ratio of incomes of two persons is 9:7. Let's denote the incomes of the two persons as 9x and 7x, respectively.
   - The ratio of their expenditures is 4:3. Let's denote their expenditures as 4y and 3y, respectively.

2. **Savings Information:**
   - Each person saves Rs. 2,000 per month. Therefore, the savings for each person can be expressed as:
     - For the first person: Income - Expenditure = Savings
       $$
       9x - 4y = 2000
       $$
     - For the second person: Income - Expenditure = Savings
       $$
       7x - 3y = 2000
       $$

3. **Solving the Equations:**
   - We have two equations:
     1. $9x - 4y = 2000$
     2. $7x - 3y = 2000$

- To solve these equations, we can use the method of elimination or substitution. Let's use elimination:
     - Multiply the first equation by 3:
       $$
       27x - 12y = 6000
       $$
     - Multiply the second equation by 4:
       $$
       28x - 12y = 8000
       $$

- Subtract the first modified equation from the second:
     $$
     (28x - 12y) - (27x - 12y) = 8000 - 6000
     $$
     $$
     x = 2000
     $$

- Substitute $x = 2000$ back into one of the original equations to find $y$:
     - Using $9x - 4y = 2000$:
       $$
       9(2000) - 4y = 2000
       $$
       $$
       18000 - 4y = 2000
       $$
       $$
       4y = 16000
       $$
       $$
       y = 4000
       $$

4. **Calculating the Monthly Income:**
   - The monthly income of the first person is $9x = 9 \times 2000 = 18000$.

5. **Conclusion:**
   - Therefore, the monthly income of one person is Rs. 18,000, which corresponds to Option 1.

**Explanation for Incorrect Options:**
- **Option 2 (Rs. 16,000):** This option is incorrect because, based on the calculations, the income derived from the given ratios and savings does not match Rs. 16,000.
- **Option 3 (Rs. 9,000):** This option is incorrect as it does not align with the calculated income of Rs. 18,000.
- **Option 4 (Rs. 20,000):** This option is incorrect because the calculations show that the income is Rs. 18,000, not Rs. 20,000.

Thus, the correct answer is indeed Option 1, Rs. 18,000.