$$\frac{4335}{4(x)24} \div 1\frac{7}{8} = \frac{289}{528}$$

Concept & Strategy This question involves identifying a missing digit in a 4-digit number (represented as $4(x)24$) within an equation. The optimal strategy is not cross-multiplication immediately, but rather algebraic simplification. Isolate the term containing the unknown digit and use cross-cancellation to reveal the hidden value. Step-by-Step Solution * **Step 1: Simplify the known elements** Convert the mixed fraction to an improper fraction: $$1\frac{7}{8} = \frac{15}{8}$$ Rewrite the equation: $$\frac{4335}{4(x)24} \div \frac{15}{8} = \frac{289}{528}$$ * **Step 2: Convert division to multiplication** Multiply by the reciprocal of the divisor: $$\frac{4335}{4(x)24} \times \frac{8}{15} = \frac{289}{528}$$ * **Step 3: Isolate the unknown term** Multiply both sides by $\frac{15}{8}$ to isolate the fraction containing the unknown: $$\frac{4335}{4(x)24} = \frac{289}{528} \times \frac{15}{8}$$ $$\frac{4335}{4(x)24} = \frac{289 \times 15}{4224}$$ * **Step 4: Cancel common factors** Check if the numerators relate to each other to simplify. Let's divide $4335$ by $15$: $$4335 \div 15 = 289$$ Therefore, the equation simplifies perfectly: $$\frac{289 \times 15}{4(x)24} = \frac{289 \times 15}{4224}$$ * **Step 5: Compare denominators** Since the numerators are identical ($289 \times 15$), the denominators must also be identical: $$4(x)24 = 4224$$ By direct comparison, the missing digit $x$ is $2$. Exam Strategy & Shortcut In competitive exams, numbers are chosen to cancel out. Notice the $289$ on the right side. Before multiplying large numbers, test if the large numerator on the left ($4335$) is a multiple of $289$. $289 \times 15 = 4335$. This immediate recognition allows you to cancel the numerators almost instantly, turning a complex arithmetic problem into a simple visual comparison. Common Pitfall A major time-trap is attempting to cross-multiply everything immediately: $4335 \times 8 \times 528 = 289 \times 15 \times 4(x)24$. This results in massive numbers that are highly prone to calculation errors. Always simplify and cancel horizontally before cross-multiplying. Final Answer Therefore, the correct answer is **2**.