Bob and David are two typists. One afternoon, they were each given 40 pages for typing. They divided the work equally but David finished 20 minutes before Bob who took 2 hours for the same. The next afternoon, they were again given 77 pages to type. However, this time they decided to divide the work such that they finished typing simultaneously. How many pages did Bob have to type?

### Concept & Proportional Work When workers operate for the exact same duration (finishing simultaneously), the amount of work they each complete is directly proportional to their efficiencies (typing speeds). ### Step-by-Step Solution * **Analyze the First Task (Efficiency Ratio):** They divided the work equally (doing $20$ pages each, though the exact number doesn't matter, just that the work was identical). Bob took $2$ hours ($120$ minutes). David finished $20$ minutes before Bob, so David took $120 - 20 = 100$ minutes. Ratio of time taken (Bob : David) = $120 : 100 = 6 : 5$. Efficiency (speed) is inversely proportional to time. Ratio of their efficiencies (Bob : David) = $5 : 6$. * **Analyze the Second Task (Simultaneous Finish):** Total pages to type = $77$. Since they finish simultaneously, the time spent is equal. Therefore, the pages they type will be split exactly in the ratio of their efficiencies. Pages typed ratio (Bob : David) = $5 : 6$. * **Calculate Bob's Share:** Bob's share of pages = $\frac{5}{5 + 6} \times \text{Total Pages}$ Bob's share = $\frac{5}{11} \times 77$ Bob's share = $5 \times 7 = 35$ pages. ### Exam Strategy & Shortcut Recognize that "finishing simultaneously" means dividing the total pool of work by their speed ratio. The speeds are inversely proportional to their times for identical tasks. Times are $120$ vs $100$ ($6:5$), so speeds are $5:6$. Bob gets $5$ parts out of $11$. $77 \times \frac{5}{11} = 35$. ### Common Pitfall A frequent mistake is applying the time ratio directly to the work distribution instead of the efficiency ratio, which would erroneously give Bob more pages ($42$) even though he is the slower typist. ### Final Answer Therefore, the correct answer is **35**.