A train travels at a speed of 30 km/hr for 12 minutes and at a speed of 45 km/hr for the next 8 minutes. The average speed of the train for this journey is (S.S.C., 2005)

### Concept & Average Speed Calculation To find the average speed when given speeds and times, first use $Distance = Speed \times Time$ to find the distance of each segment, then apply the master average speed formula. Time must be in hours. $$Average\ Speed = \frac{d_1 + d_2}{t_1 + t_2}$$ ### Step-by-Step Solution * **Step 1: Calculate distance for the first segment.** * Speed $= 30$ km/hr, Time $= 12$ mins $= \frac{12}{60}$ hours $= \frac{1}{5}$ hours. * Distance 1 $= 30 \times \frac{1}{5} = 6$ km. * **Step 2: Calculate distance for the second segment.** * Speed $= 45$ km/hr, Time $= 8$ mins $= \frac{8}{60}$ hours $= \frac{2}{15}$ hours. * Distance 2 $= 45 \times \frac{2}{15} = 3 \times 2 = 6$ km. * **Step 3: Calculate Total Distance and Total Time.** * Total Distance $= 6 + 6 = 12$ km. * Total Time $= 12$ mins $+ 8$ mins $= 20$ mins $= \frac{20}{60}$ hours $= \frac{1}{3}$ hours. * **Step 4: Calculate Average Speed.** * Average Speed $= \frac{12}{1/3} = 12 \times 3 = 36$ km/hr. ### Exam Strategy & Shortcut Notice that the distances for both segments are equal ($6$ km each). When distances are equal, the average speed is the harmonic mean of the speeds: $\frac{2xy}{x+y} = \frac{2 \times 30 \times 45}{30 + 45} = \frac{2700}{75} = 36$ km/hr. However, calculating the total distance and time is often just as fast. ### Common Pitfall Multiplying the speed by the time in minutes without converting to hours (e.g., $30 \times 12 = 360$), which fundamentally breaks the unit relationship. ### Final Answer Therefore, the correct answer is **36 km/hr**.