A boat takes $19$ hours for travelling downstream from point A to point B and coming back to a point C midway between A and B. If the velocity of the stream is $4$ kmph and the speed of the boat in still water is $14$ kmph, what is the distance between A and B?

### Concept & Total Time Equation When a journey involves multiple legs with different speeds, the total time is the sum of the time taken for each leg. $$ \text{Total Time} = \frac{D_1}{S_1} + \frac{D_2}{S_2} $$ ### Step-by-Step Solution * **Given:** Downstream journey is from A to B (let distance be $D$). Return journey is from B to C (midway), so distance is $\frac{D}{2}$. Stream speed $S = 4$ kmph, Boat speed $B = 14$ kmph. * Calculate speeds: Downstream speed = $14 + 4 = 18$ kmph. Upstream speed = $14 - 4 = 10$ kmph. * Set up the total time equation: $\frac{D}{18} + \frac{D/2}{10} = 19$ * Simplify the equation: $\frac{D}{18} + \frac{D}{20} = 19$ * Find a common denominator ($180$): $\frac{10D + 9D}{180} = 19$ $\frac{19D}{180} = 19$ * Solve for $D$: $D = 180$ km. ### Exam Strategy & Shortcut Once you get $\frac{D}{18} + \frac{D}{20} = 19$, you can factor out $D$: $D(\frac{1}{18} + \frac{1}{20}) = 19 \implies D(\frac{38}{360}) = 19 \implies D(\frac{19}{180}) = 19 \implies D = 180$. Fast fraction math saves valuable seconds. ### Common Pitfall Assuming the return journey is the full distance from B to A, ignoring the "midway" condition in the problem text, which leads to completely incorrect equations. ### Final Answer Therefore, the correct answer is **180 km**.