$$1\frac{3}{4} - 1\frac{1}{5} + 1\frac{5}{8} = x$$

### Concept & Strategy This problem tests your speed and accuracy in adding and subtracting mixed fractions with unlike denominators. The most efficient approach is the Split Method: decouple the integers from the proper fractions. This keeps the arithmetic lightweight and reduces the chance of calculation errors compared to converting everything to improper fractions. ### Step-by-Step Solution * **Step 1:** Separate the integers and fractions. Integers: $(1 - 1 + 1)$ Fractions: $(\frac{3}{4} - \frac{1}{5} + \frac{5}{8})$ * **Step 2:** Calculate the integer sum. $1 - 1 + 1 = 1$ * **Step 3:** Calculate the fraction sum by finding the Least Common Multiple (LCM). The denominators are 4, 5, and 8. The lowest number divisible by all three is 40. Convert $\frac{3}{4}$ to $\frac{30}{40}$ (multiply by 10/10) Convert $\frac{1}{5}$ to $\frac{8}{40}$ (multiply by 8/8) Convert $\frac{5}{8}$ to $\frac{25}{40}$ (multiply by 5/5) Combine them: $\frac{30 - 8 + 25}{40} = \frac{47}{40}$ * **Step 4:** Simplify the improper fraction. $\frac{47}{40} = 1\frac{7}{40}$ * **Step 5:** Add the integer result from Step 2 to the fraction result from Step 4. Total $= 1 + 1\frac{7}{40} = 2\frac{7}{40}$ ### Exam Strategy & Shortcut If you convert to improper fractions first ($\frac{7}{4} - \frac{6}{5} + \frac{13}{8}$), you end up dealing with numerators like 70, 48, and 65 over 40. While manageable, it adds cognitive load. The split method keeps your intermediate numbers well under 30, allowing for faster mental arithmetic. ### Common Pitfall Forgetting to convert the improper fraction $\frac{47}{40}$ back into a mixed number before adding it to the integer total. A student might get $1$ from the integers and see $\frac{47}{40}$, get confused, and incorrectly guess an answer or misalign the final addition. ### Final Answer **Therefore, the correct answer is 2\frac{7}{40}.**