$$5\frac{1}{3} - 3\frac{2}{3} \div 1\frac{1}{3} + x + 3\frac{1}{5} \div 1\frac{1}{5} = 7$$

Concept & Formula This equation requires strict application of the BODMAS/PEMDAS rule. Division operations must be executed before addition and subtraction. When handling mixed fractions in algebraic equations, convert them to improper fractions immediately to avoid errors during multiplication and division. Step-by-Step Solution * **Step 1: Convert all mixed numbers to improper fractions** $$5\frac{1}{3} = \frac{16}{3}$$ $$3\frac{2}{3} = \frac{11}{3}$$ $$1\frac{1}{3} = \frac{4}{3}$$ $$3\frac{1}{5} = \frac{16}{5}$$ $$1\frac{1}{5} = \frac{6}{5}$$ * **Step 2: Substitute into the equation and identify division blocks** $$\frac{16}{3} - \left(\frac{11}{3} \div \frac{4}{3}\right) + x + \left(\frac{16}{5} \div \frac{6}{5}\right) = 7$$ * **Step 3: Resolve the division operations first** First division: $$\frac{11}{3} \div \frac{4}{3} = \frac{11}{3} \times \frac{3}{4} = \frac{11}{4}$$ Second division: $$\frac{16}{5} \div \frac{6}{5} = \frac{16}{5} \times \frac{5}{6} = \frac{16}{6} = \frac{8}{3}$$ * **Step 4: Rewrite the equation and solve for $x$** $$\frac{16}{3} - \frac{11}{4} + x + \frac{8}{3} = 7$$ Group the terms with a denominator of $3$ to simplify: $$\left(\frac{16}{3} + \frac{8}{3}\right) - \frac{11}{4} + x = 7$$ $$\frac{24}{3} - \frac{11}{4} + x = 7$$ $$8 - \frac{11}{4} + x = 7$$ Isolate $x$: $$x = 7 - 8 + \frac{11}{4}$$ $$x = -1 + 2\frac{3}{4}$$ $$x = 1\frac{3}{4}$$ * **Step 5: Compare with options** The calculated value $1\frac{3}{4}$ does not match options (a), (b), or (c). Therefore, the correct choice is "None of these". Exam Strategy & Shortcut Group terms smartly. After resolving the divisions, you are left with fractions over $3$ and fractions over $4$. Instead of finding a common denominator of $12$ for the entire equation immediately, combine the $3$s first: $\frac{16}{3} + \frac{8}{3} = \frac{24}{3} = 8$. This instantly eliminates fractions from half the equation, making the final isolation of $x$ trivial. Common Pitfall A common error is reading the equation left-to-right and performing the subtraction ($5\frac{1}{3} - 3\frac{2}{3}$) before executing the division. This violates BODMAS and fundamentally changes the mathematical structure of the problem, leading to a wrong answer. Final Answer Therefore, the correct answer is **None of these**.