A motor boat can travel at $10$ km/hr in still water. It travelled $91$ km downstream in a river and then returned taking altogether $20$ hours. Find the rate of flow of the river.

### Concept & Round Trip Time Formula For a round trip on a river where the distance is known, the total time is the sum of the downstream and upstream transit times. $$ \text{Total Time} = \frac{\text{Distance}}{B + S} + \frac{\text{Distance}}{B - S} $$ ### Step-by-Step Solution * **Given:** Boat speed ($B$) = $10$ km/hr, Distance ($d$) = $91$ km, Total time = $20$ hours. Let Stream speed be $S$. * Set up the total time equation: $\frac{91}{10 + S} + \frac{91}{10 - S} = 20$ * Factor out $91$ and find a common denominator: $91 \left[ \frac{(10 - S) + (10 + S)}{(10 + S)(10 - S)} \right] = 20$ * Simplify the numerator and the denominator using difference of squares: $91 \left[ \frac{20}{100 - S^2} \right] = 20$ * Cancel $20$ from both sides to simplify: $\frac{91}{100 - S^2} = 1 \implies 91 = 100 - S^2$ * Isolate $S^2$ and solve for $S$: $S^2 = 100 - 91 = 9$ $S = 3$ km/hr (since flow speed must be positive). ### Exam Strategy & Shortcut For this specific structure, simply plug the options into the equation $\frac{91}{10+S} + \frac{91}{10-S} = 20$. Try option (a) where $S=3$: $\frac{91}{13} + \frac{91}{7} = 7 + 13 = 20$. It matches perfectly! Plugging in options is significantly faster than solving the quadratic expansion. ### Common Pitfall Expanding the expression into a full quadratic equation and making a trivial arithmetic sign error, whereas simply canceling the $20$ from both sides immediately collapses it into a simple linear-style solve. ### Final Answer Therefore, the correct answer is **3 km/hr**.