Calculating Required Train Speed

The problem asks for the new speed required to complete a journey in a shorter time, given the initial speed and time. The distance of the journey remains constant.

Understanding Speed, Distance, and Time

The relationship between speed ($S$), distance ($D$), and time ($T$) is given by the formula:

$ D = S \times T $

Since the distance ($D$) for the journey is constant, the speed ($S$) is inversely proportional to the time ($T$). This means if the time decreases, the speed must increase proportionally to cover the same distance.

We can express this relationship as:

$ S_1 \times T_1 = S_2 \times T_2 $

Where:

• $S_1$ is the initial average speed.
• $T_1$ is the initial time taken.
• $S_2$ is the new required speed.
• $T_2$ is the new required time.

Applying the Formula

Given:

• Initial speed ($S_1$) = 42 km/h
• Initial time ($T_1$) = 45 min
• New time ($T_2$) = 35 min

We need to find the new speed ($S_2$).

Since both times are in minutes, we can use them directly in the formula $S_1 T_1 = S_2 T_2$, as the units of time will cancel out.

Calculating the New Speed

Substitute the known values into the formula:

$ 42 \text{ km/h} \times 45 \text{ min} = S_2 \times 35 \text{ min} $

Now, solve for $S_2$:

$ S_2 = \frac{42 \text{ km/h} \times 45 \text{ min}}{35 \text{ min}} $

Simplify the calculation:

$ S_2 = \frac{42 \times 45}{35} \text{ km/h} $

Divide 45 and 35 by their greatest common divisor, 5:

$ S_2 = \frac{42 \times 9}{7} \text{ km/h} $

Divide 42 by 7:

$ S_2 = 6 \times 9 \text{ km/h} $

$ S_2 = 54 \text{ km/h} $

Conclusion

To reduce the journey time from 45 minutes to 35 minutes, the train must run at an average speed of 54 km/h.