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

### Concept & Strategy This problem involves addition and subtraction of mixed fractions. The fastest and most error-free method is to completely separate the integer (whole number) parts from the fractional parts. Calculate them independently, and then combine the final results. ### Step-by-Step Solution * **Step 1:** Separate the expression into whole numbers and fractions. Whole numbers: $(2 + 1 - 4)$ Fractions: $(\frac{1}{4} + \frac{1}{3} - \frac{1}{2})$ * **Step 2:** Solve the whole number portion. $2 + 1 - 4 = -1$ * **Step 3:** Solve the fraction portion by finding a common denominator. The Lowest Common Multiple (LCM) of 4, 3, and 2 is 12. $\frac{1 \times 3}{12} + \frac{1 \times 4}{12} - \frac{1 \times 6}{12}$ $\frac{3 + 4 - 6}{12} = \frac{1}{12}$ * **Step 4:** Combine the whole number and fraction results. Result $= -1 + \frac{1}{12}$ * **Step 5:** Simplify into a final single fraction. $-1 + \frac{1}{12} = \frac{-12 + 1}{12} = -\frac{11}{12}$ ### Exam Strategy & Shortcut Avoid converting mixed fractions into improper fractions (e.g., $2\frac{1}{4} \rightarrow \frac{9}{4}$) when the whole numbers are small. Converting to improper fractions inflates the numerators and makes finding common denominators mathematically heavier and slower. The split method (Whole + Fraction) takes seconds and keeps numbers small. ### Common Pitfall A frequent mistake is mishandling the negative sign in the final combination step. A student might calculate $-1$ and $\frac{1}{12}$, and incorrectly write the answer as $-1\frac{1}{12}$. Remember that $-1 + \frac{1}{12}$ means you are moving from -1 back towards zero on the number line, resulting in $-\frac{11}{12}$. ### Final Answer **Therefore, the correct answer is -\frac{11}{12}.**