George takes 8 hours to copy a 50-page manuscript while Sonia can copy the same manuscript in 6 hours. How many hours would it take them to copy a 100-page manuscript, if they work together?

### Concept & Scaled Work First, calculate the combined time it takes for both individuals to complete the base unit of work (the 50-page manuscript). Then, scale that time proportionally to the new target amount of work (the 100-page manuscript). ### Step-by-Step Solution * **Given:** George's time for 50 pages = 8 hours. Sonia's time for 50 pages = 6 hours. Target work = 100 pages. * **Calculation (For 50 pages):** George's 1-hour work = $\frac{1}{8}$ of the 50-page manuscript. Sonia's 1-hour work = $\frac{1}{6}$ of the 50-page manuscript. Combined 1-hour work = $\frac{1}{8} + \frac{1}{6}$ LCM of 8 and 6 is 24. Combined 1-hour work = $\frac{3}{24} + \frac{4}{24} = \frac{7}{24}$ * Time taken together to finish one 50-page manuscript = $\frac{24}{7}$ hours. * **Calculation (For 100 pages):** A 100-page manuscript is exactly double the work of a 50-page manuscript. Total time = $2 \times \left(\frac{24}{7}\right) = \frac{48}{7}$ hours. * Converting to a mixed fraction: $48 \div 7 = 6$ with a remainder of 6. Total time = $6\frac{6}{7}$ hours. ### Exam Strategy & Shortcut Use the $xy / (x+y)$ shortcut for combined time on a single unit of work. Time for 50 pages = $\frac{8 \times 6}{8 + 6} = \frac{48}{14} = \frac{24}{7}$ hours. Since 100 pages is double the work, simply multiply by 2: Total time = $\frac{48}{7} = 6\frac{6}{7}$ hours. ### Common Pitfall A frequent error is calculating the time for the 50-page manuscript ($\frac{24}{7}$ or $3\frac{3}{7}$ hours) and failing to double it for the 100-page target, leading to a completely incorrect final value. ### Final Answer Therefore, the correct answer is **$6\frac{6}{7}$**.