A train can travel 50% faster than a car. Both start from point $A$ at the same time and reach point $B$ 75 kms away from $A$ at the same time. On the way, however, the train lost about 12.5 minutes while stopping at the stations. The speed of the car is (M.A.T., 2003)

### Concept & Speed Ratio When two vehicles cover the same distance, the ratio of the time taken is inversely proportional to the ratio of their speeds. $$Time \propto \frac{1}{Speed}$$ ### Step-by-Step Solution * **Given:** Distance = 75 km. Train is 50% faster than car. Train stoppage time = 12.5 mins = $\frac{12.5}{60}$ hours. * **Speed Ratio:** Let speed of car be $v$. Speed of train = $v + 0.5v = 1.5v$. Ratio of speeds (Car : Train) = $v : 1.5v = 2 : 3$. * **Time Ratio:** Ratio of time taken (without stoppages) = $3 : 2$. * **Formulate Equation:** The difference in actual travel time is exactly the stoppage time (since they start and arrive at the same time). * Time taken by Car - Travel time of Train = Stoppage time * $\frac{75}{v} - \frac{75}{1.5v} = \frac{12.5}{60}$ * $\frac{75}{v} - \frac{50}{v} = \frac{12.5}{60}$ * $\frac{25}{v} = \frac{12.5}{60}$ * $v = \frac{25 \times 60}{12.5} = 2 \times 60 = 120$ kmph. ### Exam Strategy & Shortcut Using ratios: Time ratio (Car:Train) = 3:2. Difference is 1 unit. This 1 unit is the time the train stopped to let the car catch up: 1 unit = 12.5 mins. Time taken by car = 3 units = $3 \times 12.5 = 37.5$ mins = $\frac{37.5}{60}$ hours. Speed of car = $\frac{\text{Distance}}{\text{Time}} = \frac{75}{\frac{37.5}{60}} = \frac{75 \times 60}{37.5} = 120$ kmph. ### Common Pitfall A common error is equating the train's total journey time to the car's, forgetting to account for the stoppage time algebraically. Always remember: Train moving time = Car time - Stoppage time. ### Final Answer Therefore, the correct answer is **120 kmph**.