A man spends 80% of his income. With an increase in the cost of living, his expenditure increases by $37 \frac{1}{2}\%$ and his income increases by $16 \frac{2}{3}\%$. His present percent savings are

### Concept & Formula The core relationship in these problems is always: $$Income = Expenditure + Savings$$ When dealing with mixed percentages like $37 \frac{1}{2}\%$ and $16 \frac{2}{3}\%$, it is highly efficient to convert them to their fractional equivalents. * $37 \frac{1}{2}\% = \frac{3}{8}$ * $16 \frac{2}{3}\% = \frac{1}{6}$ ### Step-by-Step Solution * **Initial Scenario:** Let the initial income be $100$ units. Expenditure = $80\%$ of $100 = 80$ units. Savings = $100 - 80 = 20$ units. * **New Income Calculation:** Income increases by $16 \frac{2}{3}\%$, which is exactly $\frac{1}{6}$ of the original. Increase = $100 \times \frac{1}{6} = \frac{100}{6} = \frac{50}{3}$. New Income = $100 + \frac{50}{3} = \frac{350}{3}$. * **New Expenditure Calculation:** Expenditure increases by $37 \frac{1}{2}\%$, which is $\frac{3}{8}$. Increase = $80 \times \frac{3}{8} = 30$. New Expenditure = $80 + 30 = 110$. * **New Savings Calculation:** New Savings = New Income - New Expenditure New Savings = $\frac{350}{3} - 110 = \frac{350 - 330}{3} = \frac{20}{3}$. * **Present Percent Savings:** Percentage = $(\frac{\text{New Savings}}{\text{New Income}}) \times 100$ Percentage = $(\frac{\frac{20}{3}}{\frac{350}{3}}) \times 100$ Percentage = $(\frac{20}{350}) \times 100 = (\frac{2}{35}) \times 100 = \frac{40}{7}\% = 5 \frac{5}{7}\%$. ### Exam Strategy & Shortcut Instead of starting with 100, look at the fractional increases: $\frac{1}{6}$ for income and $\frac{3}{8}$ for expenditure. Assume an initial income that is a multiple of both 6 and the denominator of the expenditure fraction. Since expenditure is $\frac{4}{5}$ of income (80%), assume Income = 300. Income = 300. Spend = 240. Save = 60. New Income = $300 + (300 \times \frac{1}{6}) = 350$. New Spend = $240 + (240 \times \frac{3}{8}) = 240 + 90 = 330$. New Save = $350 - 330 = 20$. % Save = $\frac{20}{350} \times 100 = \frac{40}{7}\% = 5 \frac{5}{7}\%$. This completely avoids messy fractional addition! ### Common Pitfall Students often calculate the new savings as a percentage of the *original* income instead of the *present* income. Always read carefully: "His present percent savings" implies (Present Savings / Present Income). ### Final Answer **Therefore, the correct answer is $5 \frac{5}{7}\%$.**