A man's income is increased by ₹ 1200 and at the same time, the rate of tax to be paid is reduced from 12% to 10%. He now pays the same amount of tax as before. What is his increased income if 20% of his income is exempted from tax in both cases?

### Concept & Logic This problem revolves around **Equating Tax Amounts**. The core logic is that the actual tax amount paid in both scenarios is perfectly identical, even though the base income increased and the applied tax rate decreased. We must express the tax paid algebraically for both scenarios and set them equal. $$\text{Tax Paid} = \text{Tax Rate} \times \text{Taxable Income}$$ ### Step-by-step Solution * **Given:** * Let initial income = $x$ * New increased income = $x + 1200$ * Taxable income in both cases = $(100\% - 20\%) = 80\%$ of total income. * Initial tax rate = $12\%$ * New tax rate = $10\%$ * **Calculation:** * Initial tax paid: $12\%$ of $(80\% \text{ of } x) = 0.12 \times 0.8x$ * New tax paid: $10\%$ of $(80\% \text{ of } (x + 1200)) = 0.10 \times 0.8(x + 1200)$ * Since the tax paid is identical: $$0.12 \times 0.8x = 0.10 \times 0.8(x + 1200)$$ * Notice that $0.8$ is on both sides. Divide both sides by $0.8$ to simplify: $$0.12x = 0.10(x + 1200)$$ $$0.12x = 0.10x + 120$$ * Subtract $0.10x$ from both sides: $$0.02x = 120$$ * Solve for $x$: $$x = \frac{120}{0.02} = 6000$$ * The question asks for his *increased* income, not his original income: $$\text{Increased Income} = x + 1200 = 6000 + 1200 = 7200$$ ### Exam Strategy & Shortcut Notice that the 20% exemption applies equally to *both* cases. Because we are equating the taxes, this constant multiplier ($80\%$) cancels out entirely. You can directly jump to equating the unexempted tax equations: $$12\% \text{ of } x = 10\% \text{ of } (x + 1200)$$ $$12x = 10x + 12000$$ $$2x = 12000 \implies x = 6000$$ Increased income $= 6000 + 1200 = 7200$. This mentally bypasses the unnecessary decimals. ### Common Pitfall The most dangerous pitfall is solving for $x$ (the initial income) and stopping there. In many exams, ₹ 6000 would be given as option (a) as bait. Always re-read the last line of the question to see exactly what is being asked. Here, it specifically requests the *increased* income ($x + 1200$). ### Final Answer **Therefore, the correct answer is ₹ 7200.**