Mr. X spends $\frac{1}{4}$ of his income on food and $\frac{1}{3}$ less than what he spent on food, on the education of his children. What fraction of his income did he spend on food and education?

### Concept & Formula The problem requires computing the total fraction spent by summing up the relative component allocations. The key structural nuance is parsing "$\frac{1}{3}$ less than what he spent on food". $$ \text{Total Fraction} = \text{Fraction}_{\text{food}} + \text{Fraction}_{\text{education}} $$ ### Step-by-Step Solution * **Given:** * Fraction spent on food = $\frac{1}{4}$ * Fraction spent on education = $\frac{1}{4} - \left(\frac{1}{3} \times \frac{1}{4}\right)$ * **Calculation:** * Find the education expenditure fraction: $$ \text{Education Expenditure} = \frac{1}{4} \times \left(1 - \frac{1}{3}\right) = \frac{1}{4} \times \frac{2}{3} = \frac{1}{6} $$ * Sum the expenditure fractions for food and education: $$ \text{Total Spent} = \frac{1}{4} + \frac{1}{6} $$ * The lowest common multiple of 4 and 6 is 12: $$ \text{Total Spent} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12} $$ ### Exam Strategy & Shortcut **Variable Multiplier Option:** Let Mr. X's total income be $12$ units (LCM of 4 and 3). * Amount spent on food = $\frac{1}{4} \times 12 = 3$ units. * Amount spent on education is $\frac{1}{3}$ less than food = $3 - \left(\frac{1}{3} \times 3\right) = 3 - 1 = 2$ units. * Combined spending = $3 + 2 = 5$ units. * Required fraction = $\frac{5}{12}$. ### Common Pitfall Misinterpreting "$\frac{1}{3}$ less than what he spent on food" as a absolute arithmetic deduction from overall income (i.e., doing $\frac{1}{4} - \frac{1}{3}$), which produces negative results, or subtracting it directly from the whole. It means a reduction relative to the value of the food allocation itself. ### Final Answer **Therefore, the correct answer is 5/12.**