5 of \( 2 \frac{2}{5}\left[\frac{3}{4}-2 \frac{2}{3}\left\{\frac{3}{8}-\frac{1}{2}\left(\frac{1}{3}-\frac{1}{4}-\frac{1}{6}\right)\right\}\right] \)

To solve the expression \( 5 \times 2 \frac{2}{5}\left[\frac{3}{4}-2 \frac{2}{3}\left\{\frac{3}{8}-\frac{1}{2}\left(\frac{1}{3}-\frac{1}{4}-\frac{1}{6}\right)\right\}\right] \), we will break it down step by step. 1. **Convert the mixed number to an improper fraction**: \[ 2 \frac{2}{5} = 2 + \frac{2}{5} = \frac{10}{5} + \frac{2}{5} = \frac{12}{5} \] 2. **Calculate the innermost expression**: \[ \frac{1}{3} - \frac{1}{4} - \frac{1}{6} \] To combine these fractions, we need a common denominator. The least common multiple of 3, 4, and 6 is 12. \[ \frac{1}{3} = \frac{4}{12}, \quad \frac{1}{4} = \frac{3}{12}, \quad \frac{1}{6} = \frac{2}{12} \] Now, substituting these values: \[ \frac{4}{12} - \frac{3}{12} - \frac{2}{12} = \frac{4 - 3 - 2}{12} = \frac{-1}{12} \] 3. **Calculate the next part**: \[ \frac{1}{2}\left(\frac{1}{3} - \frac{1}{4} - \frac{1}{6}\right) = \frac{1}{2} \left(-\frac{1}{12}\right) = -\frac{1}{24} \] 4. **Now substitute back into the expression**: \[ \frac{3}{8} - \left(-\frac{1}{24}\right) = \frac{3}{8} + \frac{1}{24} \] To add these fractions, we need a common denominator. The least common multiple of 8 and 24 is 24. \[ \frac{3}{8} = \frac{9}{24} \] Now, substituting: \[ \frac{9}{24} + \frac{1}{24} = \frac{10}{24} = \frac{5}{12} \] 5. **Now calculate**: \[ 2 \frac{2}{3} \left\{\frac{3}{8} - \frac{1}{2}\left(\frac{1}{3} - \frac{1}{4} - \frac{1}{6}\right)\right\} = 2 \frac{2}{3} \cdot \frac{5}{12} \] Convert \( 2 \frac{2}{3} \) to an improper fraction: \[ 2 \frac{2}{3} = \frac{8}{3} \] Now multiply: \[ \frac{8}{3} \cdot \frac{5}{12} = \frac{8 \cdot 5}{3 \cdot 12} = \frac{40}{36} = \frac{10}{9} \] 6. **Substituting back into the main expression**: \[ \frac{3}{4} - 2 \frac{2}{3} \left\{\frac{3}{8} - \frac{1}{2}\left(\frac{1}{3} - \frac{1}{4} - \frac{1}{6}\right)\right\} = \frac{3}{4} - \frac{10}{9} \] To subtract these fractions, we need a common denominator. The least common multiple of 4 and 9 is 36. \[ \frac{3}{4} = \frac{27}{36}, \quad \frac{10}{9} = \frac{40}{36} \] Now substituting: \[ \frac{27}{36} - \frac{40}{36} = \frac{27 - 40}{36} = \frac{-13}{36} \] 7. **Final calculation**: \[ 5 \times 2 \frac{2}{5} \left[\frac{