Solve $2\frac{2}{9} + 4\frac{1}{18} - 1\frac{1}{2} = x$

### Concept & Formula When adding or subtracting mixed fractions, the fastest method is to separate the whole numbers from the fractional parts. $$A\frac{b}{c} + D\frac{e}{f} = (A + D) + \left(\frac{b}{c} + \frac{e}{f}\right)$$ ### Step-by-Step Solution * **Given Equation:** $$2\frac{2}{9} + 4\frac{1}{18} - 1\frac{1}{2} = x$$ * **Step 1: Separate Whole Numbers and Fractions** Group the integers and the proper fractions separately: $$x = (2 + 4 - 1) + \left(\frac{2}{9} + \frac{1}{18} - \frac{1}{2}\right)$$ * **Step 2: Simplify the Whole Numbers** $$2 + 4 - 1 = 5$$ * **Step 3: Simplify the Fractions** Find the Lowest Common Multiple (LCM) of the denominators $9, 18,$ and $2$, which is $18$. Convert all fractions to have the denominator $18$: $$\frac{2}{9} = \frac{4}{18}$$ $$\frac{1}{18} = \frac{1}{18}$$ $$\frac{1}{2} = \frac{9}{18}$$ Now, evaluate the fraction part: $$\frac{4}{18} + \frac{1}{18} - \frac{9}{18} = \frac{4 + 1 - 9}{18} = \frac{-4}{18} = -\frac{2}{9}$$ * **Step 4: Combine Both Parts** $$x = 5 - \frac{2}{9}$$ To subtract the fraction, borrow $1$ from $5$: $$x = 4 + 1 - \frac{2}{9} = 4 + \frac{9 - 2}{9} = 4\frac{7}{9}$$ ### Exam Strategy & Shortcut Never convert mixed fractions into improper fractions for addition/subtraction (e.g., $20/9 + 73/18 - 3/2$). That balloons the numbers and invites calculation errors. Always handle the integers first ($2+4-1=5$), then quickly balance the fraction denominators mentally. ### Common Pitfall Students often convert mixed fractions to improper fractions (like $20/9$), which makes finding common denominators and the final division much more tedious and error-prone. Another mistake is improperly handling negative fractional remainders. ### Final Answer **Therefore, the correct answer is $4\frac{7}{9}$.**