The value of $998\frac{2}{17} + 998\frac{3}{17} + 998\frac{5}{17} + 998\frac{8}{17} + 998\frac{16}{17}$ is

### Concept & Strategy When adding a series of mixed fractions, you can split each mixed number into its integer and fractional components. Adding the integers together separately from the fractions drastically simplifies the arithmetic. $$A\frac{B}{C} = A + \frac{B}{C}$$ ### Step-by-Step Solution Separate the integer parts from the fractional parts in the given expression: $$(998 + 998 + 998 + 998 + 998) + \left(\frac{2}{17} + \frac{3}{17} + \frac{5}{17} + \frac{8}{17} + \frac{16}{17}\right)$$ Since $998$ appears $5$ times, multiply it: $$\text{Sum of Integers} = 998 \times 5 = (1000 - 2) \times 5 = 5000 - 10 = 4990$$ Now, add the fractional parts. Since they all share the common denominator $17$, simply sum the numerators: $$\text{Sum of Fractions} = \frac{2 + 3 + 5 + 8 + 16}{17} = \frac{34}{17} = 2$$ Add the integer total and the fraction total together: $$4990 + 2 = 4992$$ ### Exam Strategy & Shortcut Count the number of terms ($5$). Multiply $998 \times 5 = 4990$. Looking at the options, the answer must be slightly more than $4990$. Option (a) is exactly $4990$, which ignores the fractions, so it is incorrect. The next logical step up is $4992$, making (b) the only plausible answer without even adding the fractions fully. ### Common Pitfall Students often attempt to convert each term into an improper fraction (e.g., $\frac{998 \times 17 + 2}{17}$), which leads to massive numbers, consumes too much time, and drastically increases the chance of calculation errors. ### Final Answer **Therefore, the correct answer is 4992.**