A man spends one-quarter of his income on food, one-fifth of it on rent and remaining, which is ₹ 231, on other commodities. Find his total income.

### Concept & Formula This is a classic basic fractions problem involving expenditure. Since both the food and rent expenses are expressed as fractions of the *total* income (not the remainder), we can simply add these fractions together to find the total fractional portion spent. The remainder can then be equated to the absolute monetary value given to find the total income. ### Step-by-Step Solution * **Given:** Food expense = $\frac{1}{4}$ of total. Rent expense = $\frac{1}{5}$ of total. Remaining balance = ₹ 231. Let total income be $x$. * **Calculation:** Find the total fraction of income spent: $$ \text{Total Spent Fraction} = \frac{1}{4} + \frac{1}{5} $$ $$ \text{Total Spent Fraction} = \frac{5}{20} + \frac{4}{20} = \frac{9}{20} $$ * Find the fraction representing the remaining income: $$ \text{Remaining Fraction} = 1 - \frac{9}{20} = \frac{11}{20} $$ * Equate this remaining fraction to the actual remaining amount to solve for $x$: $$ \frac{11}{20}x = 231 $$ $$ x = 231 \times \frac{20}{11} $$ $$ x = 21 \times 20 = 420 $$ ### Exam Strategy & Shortcut Use the **LCM Value Assumption**. The denominators are 4 and 5, so assume the total income is a multiple of 20 units. Food = $\frac{1}{4}$ of 20 = 5 units. Rent = $\frac{1}{5}$ of 20 = 4 units. Total spent = 9 units. Remaining = $20 - 9 = 11$ units. We are told the remaining amount is 231. Thus, $11 \text{ units} = 231$, meaning $1 \text{ unit} = 21$. Total income is 20 units, so $20 \times 21 = 420$. ### Common Pitfall The main trap is treating the "one-fifth" rent expense as a fraction of the *remaining* income after food, rather than a fraction of the *total* income. The phrasing "one-fifth of **it**" refers back to the original subject, "his income." ### Final Answer Therefore, the correct answer is **₹ 420**.