Is there an easier way to do this? "A freight train leaves a station at 4:00 p.m. traveling at 30 kilometers per hour. A passenger train leaves 1 hour later, traveling at 50 kilometers per hour. At what time will the passenger train overtake the freight train?

An easier way than what?

In these kind of problems you have to find which of the three variables, either time, distance or rate, are equal.

In this case, at the moment the 2nd train catches up to the first, they will have gone the same distance.

let the time taken for the 2nd train to catch the first be t hours
so distance covered by the slower train = 30(t+1)
distance covered by the faster train = 50t

50t = 30(t+1)
20t = 30
t = 3/2 hours or 1.5 hours

the faster train left at 5:00 and took 1.5 hours, so it will catch the slower at 6:30 pm

6:30p.m

To solve this problem, we can use a common formula to find the time it takes for one object to catch up to another. The formula is:

\[ \text{Time} = \frac{\text{Distance}}{\text{Relative Speed}} \]

In this case, the relative speed is the difference in speeds between the two trains. The passenger train is faster than the freight train, so its speed relative to the freight train is \(50 \, \text{km/h} - 30 \, \text{km/h} = 20 \, \text{km/h}\).

Now, let's assume that the time it takes for the passenger train to overtake the freight train is \(t\) hours. The freight train has already traveled for 1 hour, so its time is \(t + 1\) hours.

Next, we can set up the following equation:

\[ \frac{\text{Distance passenger train travels}}{\text{Relative Speed}} = \text{Time}\]

\[ \frac{50 \, \text{km/h} \cdot t}{20 \, \text{km/h}} = t + 1 \, \text{hour}\]

Now we can solve for \(t\):

\[ \frac{50t}{20} = t + 1\]

\[ 50t = 20t + 20\]

\[ 50t - 20t = 20\]

\[ 30t = 20\]

\[ t = \frac{20}{30} = \frac{2}{3} \, \text{hours}\]

To find the overtaking time in minutes, we can convert \(\frac{2}{3}\) hour to minutes:

\[ \frac{2}{3} \, \text{hour} \times 60 \, \text{minutes/hour} = \frac{40}{3} \, \text{minutes} \approx 13.33 \, \text{minutes}\]

Therefore, the passenger train will overtake the freight train in approximately 13.33 minutes. To find the exact time, we need to add the overtaking time to the scheduled departure time of the passenger train, which is 4:00 PM plus one hour.

So, the passenger train will overtake the freight train at 5:13 PM (approximately).