When $\left( \frac{1}{2} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} \right)$ is divided by $\left( \frac{2}{5} - \frac{5}{9} + \frac{3}{5} - \frac{7}{18} \right)$, the result is

### Concept & Strategy To divide two complex bracketed expressions, solve each bracket independently first. When finding the Least Common Multiple (LCM) for multiple fractions, check if any fractions share a denominator or have easily combinable values to save calculation time. ### Step-by-Step Solution * **Evaluate the first bracket (Numerator):** * Expression: $\frac{1}{2} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6}$ * Find the LCM of 2, 4, 5, and 6, which is 60. * Convert fractions: $\frac{30}{60} - \frac{15}{60} + \frac{12}{60} - \frac{10}{60}$ * Combine: $\frac{30 - 15 + 12 - 10}{60} = \frac{17}{60}$ * **Evaluate the second bracket (Denominator):** * Expression: $\frac{2}{5} - \frac{5}{9} + \frac{3}{5} - \frac{7}{18}$ * Group fractions with the same or related denominators to simplify. * Group the 5s: $\frac{2}{5} + \frac{3}{5} = \frac{5}{5} = 1$ * Group the 9 and 18: $-\frac{5}{9} - \frac{7}{18}$ * Find common denominator (18): $-\frac{10}{18} - \frac{7}{18} = -\frac{17}{18}$ * Combine the groups: $1 - \frac{17}{18} = \frac{1}{18}$ * **Perform the final division:** * $\frac{17}{60} \div \frac{1}{18}$ * Convert division to multiplication by inverting the second fraction: $\frac{17}{60} \times 18$ * Simplify by dividing 18 and 60 by 6: $\frac{17 \times 3}{10} = \frac{51}{10}$ * Convert to mixed fraction: $5 \frac{1}{10}$ ### Exam Strategy & Shortcut Instantly spot that $\frac{2}{5} + \frac{3}{5} = 1$ in the divisor. This turns the messy second bracket into a trivial $1 - \frac{17}{18} = \frac{1}{18}$. Doing this visually removes half the workload of the problem, leaving only the first bracket to calculate properly. ### Common Pitfall Attempting to find a single massive LCM (like 90) for the entire second bracket without grouping first. Calculating $\frac{36}{90} - \frac{50}{90} + \frac{54}{90} - \frac{35}{90}$ is much more prone to arithmetic errors under time pressure than simply pairing off the obvious matches. ### Final Answer **Therefore, the correct answer is $5 \frac{1}{10}$.**