Madan pays income tax at the rate of 10%. If his income increased by 10% and his tax rate increases to 15%, his net income would increase by ₹ 350. What is Madan's income?

### Concept & Formula Net Income is what remains after deducting tax from the gross income. $$Net Income = Gross Income - Tax$$ By setting up an algebraic equation for the initial and final net incomes, we can solve for the unknown base income. ### Step-by-Step Solution * **Initial Scenario:** Let Madan's original gross income be $100x$. Original Tax = $10\%$ of $100x = 10x$. Original Net Income = $100x - 10x = 90x$. * **New Scenario:** Gross Income increases by $10\%$. New Gross Income = $110x$. New Tax Rate is $15\%$. New Tax = $15\%$ of $110x = 110x \times 0.15 = 16.5x$. New Net Income = $110x - 16.5x = 93.5x$. * **Formulating the Equation:** The problem states that net income increases by ₹ $350$. New Net Income - Original Net Income = $350$ $93.5x - 90x = 350$ $3.5x = 350$ $x = \frac{350}{3.5} = 100$. * **Finding the Original Income:** Original Income = $100x = 100 \times 100 = 10,000$. ### Exam Strategy & Shortcut Assume initial income is 100. Initial tax = 10. Initial Net = 90. New income = 110. New tax = 15% of 110 = 16.5. New Net = 110 - 16.5 = 93.5. Difference in Net Income = $93.5 - 90 = 3.5$ units. We are given that 3.5 units correspond to ₹ 350. So, 1 unit = ₹ 100. The total original income is 100 units = $100 \times 100 = 10,000$. Using the "Assume 100" method makes the arithmetic entirely mental. ### Common Pitfall Students sometimes calculate the new 15% tax on the *original* income instead of the *new* increased income. Always apply the new tax rate to the updated income figure. ### Final Answer **Therefore, the correct answer is ₹ 10,000.**