monthly 4500 for 7 months a man incurred some debt which was just cleared by reducing his monthly expenses to 4200 for 5 months. find his monthly income. 

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.

---

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.

monthly 4500 for 7 months a man incurred some debt which was just cleared by reducing his monthly expenses to 4200 for 5 months. find his monthly income.

Find out how to calculate a man's monthly income when his expenses were ₹4500 for 7 months and ₹4200 for the next 5 months, using algebraic equations to balance his debt and income. Suitable for students in Grades 8-10.

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