1. **State the problem:** Evaluate the expression \[ \left(-\frac{1}{4}\right) \times \left(\frac{5}{8} + 1 \frac{7}{8}\right) \div 2 \frac{2}{5} + \frac{5}{6} \] using BIDMAS (Brackets, Indices, Division and Multiplication, Addition and Subtraction). 2. **Convert mixed numbers to improper fractions:** - \(1 \frac{7}{8} = \frac{15}{8}\) - \(2 \frac{2}{5} = \frac{12}{5}\) 3. **Simplify inside the brackets:** \[ \frac{5}{8} + \frac{15}{8} = \frac{20}{8} = \frac{5}{2} \] 4. **Multiply by \(-\frac{1}{4}\):** \[ -\frac{1}{4} \times \frac{5}{2} = -\frac{5}{8} \] 5. **Divide by \(\frac{12}{5}\):** \[ -\frac{5}{8} \div \frac{12}{5} = -\frac{5}{8} \times \frac{5}{12} = -\frac{25}{96} \] 6. **Add \(\frac{5}{6}\):** Find common denominator for \(\frac{25}{96}\) and \(\frac{5}{6}\): \[ \frac{5}{6} = \frac{80}{96} \] \[ -\frac{25}{96} + \frac{80}{96} = \frac{55}{96} \] 7. **Final answer:** \[ \boxed{\frac{55}{96}} \] This is the evaluated value of the expression using BIDMAS rules.