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Let's address the first question from Section C (Question 26):

**Q26.** 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. 2000 per month, find their monthly incomes.

### Solution:
Let the incomes of the two persons be \(9x\) and \(7x\), and their expenditures be \(4y\) and \(3y\), respectively.

We are given that each person saves Rs. 2000 per month. Thus, we can form two equations based on the savings:

\[
9x - 4y = 2000 \quad \text{(for the first person)}
\]
\[
7x - 3y = 2000 \quad \text{(for the second person)}
\]

Now we solve these two equations simultaneously.

1. Multiply the first equation by 3 and the second equation by 4 to eliminate \(y\):
   \[
   3(9x - 4y) = 3(2000) \implies 27x - 12y = 6000
   \]
   \[
   4(7x - 3y) = 4(2000) \implies 28x - 12y = 8000
   \]

2. Subtract the first equation from the second:
   \[
   (28x - 12y) - (27x - 12y) = 8000 - 6000
   \]
   \[
   x = 2000
   \]

Thus, \(x = 2000\).

Now, substitute \(x = 2000\) into one of the original equations to find \(y\). Using the first equation:
\[
9(2000) - 4y = 2000
\]
\[
18000 - 4y = 2000
\]
\[
4y = 16000 \implies y = 4000
\]

Thus, the incomes of the two persons are:
\[
\text{First person's income} = 9x = 9(2000) = Rs. 18,000
\]
\[
\text{Second person's income} = 7x = 7(2000) = Rs. 14,000
\]

### Final Answer:
The monthly incomes of the two persons are Rs. 18,000 and Rs. 14,000, respectively.

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Would you like to explore another question or dive deeper into any part of the solution?

Here are some related questions you might want to consider:
1. How would the solution change if the savings were different for each person?
2. What happens if the ratio of incomes is changed to 5:3 but expenditures remain the same?
3. Can you derive a general formula for income given ratios of incomes and expenditures?
4. What if one of the persons has no savings? How would that affect their income?
5. How do you solve similar problems where expenses depend on other variables?

### Tip:
When solving systems of linear equations, multiplying both equations to align coefficients of one variable is a powerful method to eliminate that variable efficiently.

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. 2000 per month, find their monthly incomes.

Learn how to solve a real-life problem involving income and expenditure ratios using systems of linear equations. This example provides a detailed step-by-step solution for finding the monthly incomes of two persons. Suitable for students in Grades 8-10.

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