$\frac{-\frac{1}{2} - \frac{2}{3} + \frac{4}{5} - \frac{1}{3} + \frac{1}{5} + \frac{3}{4}}{\frac{1}{2} + \frac{2}{3} - \frac{4}{3} + \frac{1}{3} - \frac{1}{5} - \frac{4}{5}}$ is simplified to

### Concept & Strategy When faced with a long chain of fractional additions and subtractions, do not immediately try to find a massive Least Common Multiple (LCM) for all terms. Instead, group fractions that share the same denominator to simplify the expression into small integers before doing any final cross-multiplication. ### Step-by-Step Solution * **Given Expression:** * Numerator: $-\frac{1}{2} - \frac{2}{3} + \frac{4}{5} - \frac{1}{3} + \frac{1}{5} + \frac{3}{4}$ * Denominator: $\frac{1}{2} + \frac{2}{3} - \frac{4}{3} + \frac{1}{3} - \frac{1}{5} - \frac{4}{5}$ * **Step 1: Group and simplify the Numerator terms** * Group by denominator $2$ & $4$: $\left(-\frac{1}{2} + \frac{3}{4}\right) = \left(-\frac{2}{4} + \frac{3}{4}\right) = \frac{1}{4}$ * Group by denominator $3$: $\left(-\frac{2}{3} - \frac{1}{3}\right) = -\frac{3}{3} = -1$ * Group by denominator $5$: $\left(\frac{4}{5} + \frac{1}{5}\right) = \frac{5}{5} = 1$ * Total Numerator = $\frac{1}{4} - 1 + 1 = \frac{1}{4}$ * **Step 2: Group and simplify the Denominator terms** * Keep denominator $2$: $\frac{1}{2}$ * Group by denominator $3$: $\left(\frac{2}{3} - \frac{4}{3} + \frac{1}{3}\right) = \frac{2 - 4 + 1}{3} = -\frac{1}{3}$ * Group by denominator $5$: $\left(-\frac{1}{5} - \frac{4}{5}\right) = -\frac{5}{5} = -1$ * Total Denominator = $\frac{1}{2} - \frac{1}{3} - 1 = \frac{3 - 2}{6} - 1 = \frac{1}{6} - \frac{6}{6} = -\frac{5}{6}$ * **Step 3: Divide the simplified Numerator by the Denominator** * $\frac{\frac{1}{4}}{-\frac{5}{6}} = \frac{1}{4} \div -\frac{5}{6}$ * $\frac{1}{4} \times -\frac{6}{5} = -\frac{6}{20}$ * Simplify by dividing by $2$: $-\frac{3}{10}$ ### Exam Strategy & Shortcut Scan the equation quickly. Notice how $+\frac{4}{5}$ and $+\frac{1}{5}$ cleanly make a $1$, and $-\frac{2}{3}$ and $-\frac{1}{3}$ make a $-1$. These cancel each other out instantly, leaving only $-\frac{1}{2} + \frac{3}{4}$ in the numerator. Visual grouping like this turns a 2-minute calculation into a 20-second visual puzzle. ### Common Pitfall The worst mistake you can make here is finding the LCM for $2, 3, 4, 5$ (which is $60$) and multiplying every single fraction's numerator. E.g., $(-\frac{30}{60} - \frac{40}{60} + \frac{48}{60} \dots)$. This creates unnecessarily large numbers and almost guarantees a basic arithmetic or negative sign error under time pressure. Always look for natural groupings first. ### Final Answer **Therefore, the correct answer is $-\frac{3}{10}$.**