A boat can travel $36$ km upstream in $5$ hours. If the speed of the stream is $2.4$ kmph, how much time will the boat take to cover a distance of $78$ km downstream? (in hours) [United India Insurance Co. Ltd. (UIICL) Assistant (Online) Exam, 2015]

### Concept & Speed Conversions Find the boat's still water speed using the upstream speed and stream speed, then use that to calculate the downstream time. $$ \text{Upstream Speed} = \text{Boat Speed} - \text{Stream Speed} $$ $$ \text{Downstream Speed} = \text{Boat Speed} + \text{Stream Speed} $$ ### Step-by-Step Solution * Calculate upstream speed ($U$): Distance = $36$ km, Time = $5$ hours. $U = \frac{36}{5} = 7.2$ kmph. * Given stream speed ($V$) = $2.4$ kmph. * Find boat speed in still water ($B$): Since $U = B - V$, we have $7.2 = B - 2.4$. $B = 7.2 + 2.4 = 9.6$ kmph. * Calculate downstream speed ($D$): $D = B + V = 9.6 + 2.4 = 12$ kmph. * Calculate time for $78$ km downstream: $\text{Time} = \frac{\text{Distance}}{D} = \frac{78}{12} = 6.5$ hours. ### Exam Strategy & Shortcut Notice the relationship $D = U + 2V$. This allows you to skip finding $B$ entirely. $D = 7.2 + 2(2.4) = 7.2 + 4.8 = 12$ kmph. The time is immediately $78 / 12 = 6.5$ hours. ### Common Pitfall Adding the stream speed only once to the upstream speed, mistaking the result for the downstream speed rather than just finding the still water speed first. ### Final Answer Therefore, the correct answer is **(b) 6.5**.