Math Practice Online > free > lessons > Florida > 8th grade > Train Problems

Here are some tips for Train Problems, which aligns with Florida state standards:

Train Problem

This topic applies Distance = Rate × Time.
Review the basics of Distance, Rate, and Time here.

The general steps to follow for solving these problems are

1. Determine what the problem is asking
2. Assign variable(s)
3. Construct the equation such that there is only 1 unknown
4. Solve the equation for the unknown
5. Answer the question in the problem

Example 1: Trains travelling in the same direction

Solve. If asked for time, a proper answer looks like this: 1:35am
A train leaves Prague at 6:45 pm, averaging 30 mph.
Another train headed in the same direction leaves Prague at 7:45 pm, averaging 60 mph.
To the nearest minute, at what time will the second train overtake the first train?

Step 1:  Determine what the problem is asking

Q: What are you looking for? What is the problem asking?
A: When the second train will overtake the first train, to the nearest minute
Step 2:  Assign variable(s)
Since we are looking for an answer in terms of time, let us assign
t1 = time travelled by the first train
t2 = time travelled by the second train

Q:  How does t1 relate to t2?
A:  The first train started 1 hour before the second train, so the time travelled by the first train when it is taken over by the second train is
t1 = t2 + 1 hr

Step 3:  Construct the equation
We know that when the second train overtakes the first train, both trains have travelled the same distance. Since we know that distance = rate × time (or d = rt), we can construct the right equation.

 Distance travelled by the first train = Distance travelled by the second train d1 = d2 r1t1 = r2t2 from the equation d = rt (30 mph) t1 = (60 mph) t2 rates are stated in the problem (30 mph) (t2 + 1hr) = (60 mph) t2 from step 2

Step 4:  Solve the equation
 (30 mph) (t2 + 1 hr) = (60 mph) t2 (30 mph )(t2 + 1 hr) = (60 mph ) t2 units cancel t2 + 1 hr = 2 t2 divide by 30 t2 + 1 hr - t2 = 2 t2 - t2 1 hr = t2
The problem asks for the specific time when the second train overtakes the first train.

From step 4, we have calculated that the second train travelled 1 hour before overtaking the first train. And from the problem, we know that the second train left the station at 7:45pm. After an hour after travelling, the time would be 8:45pm.

Therefore, the second train overtakes the first train at 8:45pm.
The answer to the problem is

Example 2: Trains travelling in the opposite direction

Solve. If asked for time, a proper answer looks like this: 1:35am
A train leaves Las Vegas at 5:30 am, averaging 80 mph. Another train headed in the opposite direction leaves Las Vegas at 7:30 am, averaging 105 mph. To the nearest mile, how far are the two trains from each other at 11:30 am?

Step 1:  Determine what the problem is asking

Q: What are you looking for? What is the problem asking?
A: The total distance travelled by the two trains by 11:30am, to the nearest mile
Step 2:  Assign variable(s)
Since we are looking for an answer in terms of distance, let us assign
dtotal = total distance travelled by both trains
d1 = total distance travelled by the first train
d2 = total distance travelled by the second train
Step 3:  Construct the equation
Since the two trains are travelling in opposite directions, their total distance apart is the sum of the distances they each travelled.

 dtotal = d1 + d2 dtotal = r1t1 + r2t2 from the equation d = rt

Step 4:  Solve the equation
 dtotal = r1t1 + r2t2 = (80 mph)t1 + (105 mph)t2 = (80 mph)(11:30am - 5:30am) + (105 mph)(11:30am - 7:30am) = (80 mph)(6 hr) + (105 mph)(4 hr) = 480 mi + 420 mi units cancel: (miles/hr)(hr) = mi = 900 mi