A train started from station $A$ and proceeded towards station $B$ at a speed of $48$ km/hr. Forty-five minutes later another train started from station $B$ and proceeded towards station a at $50$ km/hr. If the distance between the two stations is $232$ km, at what distance from station A will the trains meet?

### Concept & Relative Speed with Head Start When two trains start at different times, calculate the distance covered by the first train during its "head start". Subtract this from the total distance to find the remaining gap. Then, use relative speed to find the time it takes to close the remaining gap. ### Step-by-Step Solution * **Given:** Total distance = $232$ km. Speed of Train from A = $48$ km/hr. Speed of Train from B = $50$ km/hr. Train A starts $45$ minutes ($3/4$ hour) early. * **Calculation:** Distance traveled by the train from $A$ in $45$ minutes = $48 \times \frac{3}{4} = 36$ km. * Remaining distance between the two trains when the train from $B$ starts = $232 - 36 = 196$ km. * Relative speed of the two trains (moving towards each other) = $48 + 50 = 98$ km/hr. * Time taken for them to meet after the train from $B$ starts = $\frac{196}{98} = 2$ hours. * The question asks for the meeting distance from station $A$. The train from $A$ traveled for $45$ minutes initially, plus $2$ hours more. * Distance covered by train from $A$ in the $2$ hours = $48 \times 2 = 96$ km. * Total distance from station $A$ = Head start distance + Distance traveled until meeting = $36 + 96 = 132$ km. ### Exam Strategy & Shortcut Find the distance remaining after the head start: $232 - (48 \times 0.75) = 196$. Find meeting time: $196 / (48 + 50) = 2$ hours. Calculate distance from A: Initial $36$ km + ($2 \times 48$) = $132$ km. ### Common Pitfall A common error is to calculate the distance traveled by train $A$ only during the simultaneous travel time ($96$ km), forgetting to add the initial head start distance ($36$ km). ### Final Answer Therefore, the correct answer is **$132$ km**.