$$6\frac{9}{13} - 4\frac{4}{11} + 2\frac{2}{5} = x$$

### Concept & Strategy This problem tests your stamina with fractions involving prime denominators. Because 13, 11, and 5 are all co-prime (they share no common factors), their Least Common Multiple (LCM) is simply their product. As always, separate the whole numbers from the fractions to prevent the numerators from ballooning to unmanageable sizes. ### Step-by-Step Solution * **Step 1:** Separate wholes and fractions. Whole Numbers: $6 - 4 + 2$ Fractions: $\frac{9}{13} - \frac{4}{11} + \frac{2}{5}$ * **Step 2:** Calculate whole numbers. $6 - 4 + 2 = 4$ * **Step 3:** Find the LCM for the fractions. LCM $= 13 \times 11 \times 5 = 143 \times 5 = 715$. *(Note: All options have a denominator of 715, confirming this step).* * **Step 4:** Convert each fraction to the common denominator. For $\frac{9}{13}$: Multiply by $\frac{11 \times 5}{11 \times 5} = \frac{55}{55}$. $\rightarrow \frac{9 \times 55}{715} = \frac{495}{715}$ For $\frac{4}{11}$: Multiply by $\frac{13 \times 5}{13 \times 5} = \frac{65}{65}$. $\rightarrow \frac{4 \times 65}{715} = \frac{260}{715}$ For $\frac{2}{5}$: Multiply by $\frac{13 \times 11}{13 \times 11} = \frac{143}{143}$. $\rightarrow \frac{2 \times 143}{715} = \frac{286}{715}$ * **Step 5:** Combine the fraction numerators. $\frac{495 - 260 + 286}{715} = \frac{235 + 286}{715} = \frac{521}{715}$ * **Step 6:** Combine wholes and fractions. Result $= 4 + \frac{521}{715} = 4\frac{521}{715}$ ### Exam Strategy & Shortcut Look at the whole numbers first: $6 - 4 + 2 = 4$. By simply doing this 2-second calculation, you can immediately eliminate Options C and D (which start with 9). Now you are choosing between $4\frac{596}{715}$ and $4\frac{521}{715}$. You only need to verify the unit digit of the fraction numerator. Numerator unit digit logic: $(9 \times 5) - (4 \times 5) + (2 \times 3) \rightarrow 5 - 0 + 6 = 11$, which ends in 1. Option B ends in 1 (521). Option A ends in 6 (596). You can confidently choose Option B without doing the full heavy multiplication. ### Common Pitfall Calculating the numerators ($9 \times 55$, $4 \times 65$, etc.) often leads to simple multiplication errors under time pressure. Attempting to solve this problem without using unit-digit elimination on the options takes substantially more time and carries a high risk of arithmetic mistakes. ### Final Answer **Therefore, the correct answer is 4\frac{521}{715}.**