Solve $4\frac{2}{3} + 3\frac{1}{2} - 1\frac{2}{3} = x$

Concept & Strategy When adding or subtracting mixed fractions, process the integer components and the fractional components independently. Always look for canceling terms before calculating common denominators. $$A\frac{b}{c} + X\frac{y}{z} = (A + X) + \left(\frac{b}{c} + \frac{y}{z}\right)$$ Step-by-Step Solution * Separate the whole numbers and fractions from the equation: $$(4 + 3 - 1) + \left(\frac{2}{3} + \frac{1}{2} - \frac{2}{3}\right)$$ * Calculate the integer sum: $$4 + 3 - 1 = 6$$ * Evaluate the fractions. Notice that $+\frac{2}{3}$ and $-\frac{2}{3}$ perfectly cancel each other out: $$\frac{2}{3} - \frac{2}{3} + \frac{1}{2} = \frac{1}{2}$$ * Recombine the integer and fractional parts: $$6 + \frac{1}{2} = 6\frac{1}{2}$$ Exam Strategy & Shortcut Do not convert to improper fractions! By visually scanning the equation $4\frac{2}{3} + 3\frac{1}{2} - 1\frac{2}{3}$, you can instantly see the $\frac{2}{3}$ fractions cancel out. The math reduces to $4 + 3\frac{1}{2} - 1$, which can be done mentally in under 5 seconds to get $6\frac{1}{2}$. Common Pitfall The most common mistake is rigidly converting all mixed numbers to improper fractions (e.g., $\frac{14}{3} + \frac{7}{2} - \frac{5}{3}$). This forces you to find an LCM of 6, creating large numerators and wasting valuable exam time on unnecessary arithmetic. Final Answer Therefore, the correct answer is **$6\frac{1}{2}$**.