A man can row $\frac{3}{4}$ of a km against the stream in $11\frac{1}{4}$ minutes and returns in $7\frac{1}{2}$ minutes. Find the speed of the man in still water.

### Concept & Formula The speed of a man in still water is the average of his downstream and upstream speeds. $$ \text{Speed in Still Water} = \frac{\text{Downstream Speed} + \text{Upstream Speed}}{2} $$ Speed must be calculated in km/hr by converting fractional minute times into hours. ### Step-by-Step Solution - **Distance:** $\frac{3}{4} \text{ km}$. - **Upstream Time:** $11\frac{1}{4} \text{ mins} = \frac{45}{4} \text{ mins}$. Convert to hours by dividing by 60: $\frac{45}{4 \times 60} = \frac{45}{240} = \frac{3}{16} \text{ hr}$. - **Upstream Speed ($v$):** $v = \frac{\frac{3}{4}}{\frac{3}{16}} = \frac{3}{4} \times \frac{16}{3} = 4 \text{ km/hr}$. - **Downstream Time (returns):** $7\frac{1}{2} \text{ mins} = \frac{15}{2} \text{ mins}$. Convert to hours: $\frac{15}{2 \times 60} = \frac{15}{120} = \frac{1}{8} \text{ hr}$. - **Downstream Speed ($u$):** $u = \frac{\frac{3}{4}}{\frac{1}{8}} = \frac{3}{4} \times \frac{8}{1} = 6 \text{ km/hr}$. - **Speed in Still Water:** $\text{Speed} = \frac{6 + 4}{2} = 5 \text{ km/hr}$. ### Exam Strategy & Shortcut Instead of converting time to hours initially, you can find the speed in km/min and multiply the final result by 60. Upstream: $(\frac{3}{4}) / (\frac{45}{4}) = \frac{1}{15} \text{ km/min} \times 60 = 4 \text{ km/hr}$. Downstream: $(\frac{3}{4}) / (\frac{15}{2}) = \frac{1}{10} \text{ km/min} \times 60 = 6 \text{ km/hr}$. Average = $5 \text{ km/hr}$. ### Common Pitfall Mishandling the mixed fractions during division, especially when converting minutes to hours, causing cascading calculation errors. ### Final Answer Therefore, the correct answer is **5 km/hr**.