$\frac{7\frac{1}{2} - 5\frac{3}{4}}{3\frac{1}{2} + x} \div \frac{\frac{1}{2} + 1\frac{1}{4}}{1\frac{1}{5} + 3\frac{1}{2}} = 0.6$

### Concept & Strategy The problem requires resolving a complex equation involving mixed fractions and division. The core strategy is to convert all mixed fractions and decimals into improper fractions, simplify the numerators and denominators independently, and then solve for the unknown variable $x$. The key rule to remember is that dividing by a fraction is the same as multiplying by its reciprocal: $$\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \times \frac{D}{C}$$ ### Step-by-Step Solution * **Step 1: Simplify the First Complex Fraction** Numerator: $7\frac{1}{2} - 5\frac{3}{4} = \frac{15}{2} - \frac{23}{4} = \frac{30 - 23}{4} = \frac{7}{4}$ Denominator: $3\frac{1}{2} + x = \frac{7}{2} + x$ First Term = $\frac{\frac{7}{4}}{\frac{7}{2} + x}$ * **Step 2: Simplify the Second Complex Fraction** Numerator: $\frac{1}{2} + 1\frac{1}{4} = \frac{1}{2} + \frac{5}{4} = \frac{2 + 5}{4} = \frac{7}{4}$ Denominator: $1\frac{1}{5} + 3\frac{1}{2} = \frac{6}{5} + \frac{7}{2} = \frac{12 + 35}{10} = \frac{47}{10}$ Second Term = $\frac{\frac{7}{4}}{\frac{47}{10}}$ * **Step 3: Setup and Solve the Equation** The original equation is now: $$\frac{\frac{7}{4}}{\frac{7}{2} + x} \div \frac{\frac{7}{4}}{\frac{47}{10}} = 0.6$$ Convert $0.6$ to a fraction ($\frac{3}{5}$) and change division to multiplication by the reciprocal: $$\frac{\frac{7}{4}}{\frac{7}{2} + x} \times \frac{\frac{47}{10}}{\frac{7}{4}} = \frac{3}{5}$$ Notice that the $\frac{7}{4}$ in the numerator and denominator cancel out perfectly: $$\frac{\frac{47}{10}}{\frac{7}{2} + x} = \frac{3}{5}$$ * **Step 4: Isolate $x$** Cross-multiply to solve for $x$: $$5 \times \frac{47}{10} = 3 \times \left(\frac{7}{2} + x\right)$$ $$\frac{47}{2} = \frac{21}{2} + 3x$$ $$3x = \frac{47}{2} - \frac{21}{2}$$ $$3x = \frac{26}{2} = 13$$ $$x = \frac{13}{3} = 4\frac{1}{3}$$ ### Exam Strategy & Shortcut Instead of expanding every fraction immediately, leave identical terms (like $\frac{7}{4}$) unsimplified when they appear on opposite sides of a division operation. Scanning the structure first reveals that the numerators will cancel out, saving you from tedious cross-multiplication. ### Common Pitfall A major trap is attempting to distribute the division across the addition in the denominator. Remember that fractions act as solid structural blocks; you must fully resolve the numerator and denominator before applying operations between two complex fractions. ### Final Answer **Therefore, the correct answer is 4(1/3).**