Nagaraj could save 10% of his income. But 2 years later, when his income increased by 20%, he could save the same amount only as before. By how much percentage has his expenditure increased?

### Concept & Logic The problem involves the core financial equation: $\text{Income} = \text{Expenditure} + \text{Savings}$. By assuming an initial base income of $100$, we can easily track the absolute changes in income, savings, and expenditure to find the final percentage increase. ### Step-by-Step Solution * **Given**: Initial savings = $10\%$ of income. Later, income increases by $20\%$, but the absolute savings amount remains constant. * **Calculation**: 1. Assume Nagaraj's initial income is $100$. 2. Initial Savings = $10\%$ of $100 = 10$. 3. Initial Expenditure = $\text{Income} - \text{Savings} = 100 - 10 = 90$. 4. New Income = Initial income increased by $20\% = 100 + 20 = 120$. 5. New Savings = Same amount as before = $10$. 6. New Expenditure = $\text{New Income} - \text{New Savings} = 120 - 10 = 110$. 7. Increase in Expenditure = $\text{New Expenditure} - \text{Initial Expenditure} = 110 - 90 = 20$. 8. Percentage increase in expenditure: $$\text{Percentage Increase} = \left( \frac{\text{Increase}}{\text{Initial Expenditure}} \right) \times 100$$ $$= \left( \frac{20}{90} \right) \times 100$$ $$= \frac{200}{9}\%$$ $$= 22\frac{2}{9}\%$$ ### Exam Strategy & Shortcut Use the base-$100$ method mentally. Income $100 \rightarrow$ Save $10 \rightarrow$ Spend $90$. Income $120 \rightarrow$ Save $10 \rightarrow$ Spend $110$. Increase is $20$ on a base of $90$. Fraction is $\frac{2}{9}$. Since $\frac{1}{9} = 11.11\%$ (or $11\frac{1}{9}\%$), then $\frac{2}{9} = 22.22\%$ or $22\frac{2}{9}\%$. This entire process can be executed in under 15 seconds without writing full equations. ### Common Pitfall A common mistake is calculating the percentage increase in expenditure against the *new* expenditure ($110$) or the *income* ($100$ or $120$) instead of the *original* expenditure ($90$). Percentage change must always be calculated relative to the starting value of the specific metric being tracked. ### Final Answer **Therefore, the correct answer is $22\frac{2}{9}\%$.**