Let's solve this step by step.

Given:

• For the first 7 months, a man incurred debt by spending ₹4500 monthly.
• For the next 5 months, he reduced his expenses to ₹4200 to clear the debt.
• We need to find his monthly income.

Let:

• $I$ represent his monthly income.

Step 1: Calculate Total Expenses for Each Period

• For the first 7 months: $\text{Total expenses} = 4500 \times 7 = ₹31500$

• For the next 5 months: $\text{Total expenses} = 4200 \times 5 = ₹21000$

Step 2: Express Income and Debt Conditions

1. For the first 7 months: His total income in this period is $7I$, but he incurred some debt due to higher expenses. The debt after 7 months is: $\text{Debt} = 4500 \times 7 - 7I = ₹31500 - 7I$

2. For the next 5 months: He reduced his monthly expenses to ₹4200 and used the extra savings to clear the debt. His savings per month are $I - 4200$, and the total savings over 5 months is: $\text{Total savings} = 5(I - 4200) = 5I - 21000$

Step 3: Set up the equation to clear the debt

Since his savings over the 5 months are used to clear the debt: $\text{Debt} = \text{Total savings}$ $31500 - 7I = 5I - 21000$

Step 4: Solve for $I$

Simplifying the equation: $31500 + 21000 = 7I + 5I$ $52500 = 12I$ $I = \frac{52500}{12} = ₹4375$

Conclusion:

The man's monthly income is ₹4375.

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Would you like more details or further explanation? Here are five related questions to expand on this:

1. What would his savings be per month after clearing the debt?
2. How would the debt change if his initial expenses were ₹5000 instead of ₹4500?
3. What if he had reduced his expenses to ₹4000 for 6 months instead of 5?
4. How much total debt would he have incurred after 7 months if his income was ₹4200?
5. What would happen if his monthly income was unknown but his total savings over the last 5 months was ₹20000?

Tip: To manage debt effectively, it's crucial to compare your monthly expenses against your income regularly.