Math Skill: Train Problems
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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
The problem asks for the total distance travelled by the two trains when it is 11:30am. Our equation solves for the total distance so the answer to the problem is .

### Complexity=1, Mode=sameDirHrsPassed

Solve. If asked for time, a proper answer looks like this: 1:35am
 1.   A train leaves San Francisco at 6:00 pm, averaging 60 mph.Another train headed in the same direction leaves San Francisco at 10:00 pm, averaging 90 mph.To the nearest tenth, how many hours after the second train leaves will it overtake the first train? 2.   A train leaves Minneapolis at 11:00 am, averaging 70 mph.Another train headed in the same direction leaves Minneapolis at 3:00 pm, averaging 90 mph.To the nearest tenth, how many hours after the second train leaves will it overtake the first train?

### Complexity=2, Mode=sameDirTime

Solve. If asked for time, a proper answer looks like this: 1:35am
 1.   A train leaves Venice at 10:00 pm, averaging 80 mph.Another train headed in the same direction leaves Venice at 11:00 pm, averaging 110 mph.To the nearest minute, at what time will the second train overtake the first train? 2.   A train leaves Paris at 11:00 am, averaging 50 mph.Another train headed in the same direction leaves Paris at 1:00 pm, averaging 80 mph.To the nearest minute, at what time will the second train overtake the first train?

### Complexity=3, Mode=oppDirDist

Solve. If asked for time, a proper answer looks like this: 1:35am
 1.   A train leaves Madrid at 5:00 pm, averaging 50 mph.Another train headed in the opposite direction leaves Madrid at 9:00 pm, averaging 80 mph.To the nearest mile, how far are the two trains from each other at 1:00 am? 2.   A train leaves Las Vegas at 1:00 pm, averaging 60 mph.Another train headed in the opposite direction leaves Las Vegas at 4:00 pm, averaging 80 mph.To the nearest mile, how far are the two trains from each other at 6:00 pm?

### Complexity=4, Mode=oppDirTime

Solve. If asked for time, a proper answer looks like this: 1:35am
 1.   A train leaves Taipei for a nearby city at 3:45 pm, averaging 80 mph.Another train leaves the nearby city for Taipei at 7:00 pm, averaging 100 mph.If the nearby city is 440 miles from Taipei, to the nearest minute, at what time will the two trains pass each other? 2.   A train leaves Kansas City for a nearby city at 8:15 pm, averaging 90 mph.Another train leaves the nearby city for Kansas City at 10:45 pm, averaging 120 mph.If the nearby city is 4215 miles from Kansas City, to the nearest minute, at what time will the two trains pass each other?

### Complexity=5, Mode=sameDirHrsPassed

Solve. If asked for time, a proper answer looks like this: 1:35am
 1.   A train leaves Brussels at 10:15 pm, averaging 35 mph.Another train headed in the same direction leaves Brussels at 1:30 am, averaging 45 mph.To the nearest tenth, how many hours after the second train leaves will it overtake the first train? 2.   A train leaves Cairo at 3:30 pm, averaging 55 mph.Another train headed in the same direction leaves Cairo at 7:45 pm, averaging 85 mph.To the nearest tenth, how many hours after the second train leaves will it overtake the first train?

### Complexity=6, Mode=sameDirTime

Solve. If asked for time, a proper answer looks like this: 1:35am
 1.   A train leaves Madrid at 10:45 am, averaging 50 mph.Another train headed in the same direction leaves Madrid at 1:30 pm, averaging 80 mph.To the nearest minute, at what time will the second train overtake the first train? 2.   A train leaves Denver at 9:30 am, averaging 60 mph.Another train headed in the same direction leaves Denver at 11:45 am, averaging 85 mph.To the nearest minute, at what time will the second train overtake the first train?

### Complexity=7, Mode=oppDirDist

Solve. If asked for time, a proper answer looks like this: 1:35am
 1.   A train leaves Florence at 2:45 pm, averaging 50 mph.Another train headed in the opposite direction leaves Florence at 3:00 pm, averaging 80 mph.To the nearest mile, how far are the two trains from each other at 4:00 pm? 2.   A train leaves Rome at 8:30 am, averaging 70 mph.Another train headed in the opposite direction leaves Rome at 11:30 am, averaging 85 mph.To the nearest mile, how far are the two trains from each other at 12:30 pm?

### Complexity=8, Mode=oppDirTime

Solve. If asked for time, a proper answer looks like this: 1:35am
 1.   A train leaves Prague for a nearby city at 7:32 am, averaging 96 mph.Another train leaves the nearby city for Prague at 11:45 am, averaging 113 mph.If the nearby city is 5420.8 miles from Prague, to the nearest minute, at what time will the two trains pass each other? 2.   A train leaves Austin for a nearby city at 2:55 pm, averaging 41 mph.Another train leaves the nearby city for Austin at 6:15 pm, averaging 63 mph.If the nearby city is 604.66666666667 miles from Austin, to the nearest minute, at what time will the two trains pass each other?

### Complexity=9, Mode=mix

Solve. If asked for time, a proper answer looks like this: 1:35am
 1.   A train leaves Florence at 9:25 am, averaging 59 mph.Another train headed in the same direction leaves Florence at 11:45 am, averaging 72 mph.To the nearest minute, at what time will the second train overtake the first train? 2.   A train leaves Austin for a nearby city at 6:13 am, averaging 86 mph.Another train leaves the nearby city for Austin at 10:30 am, averaging 104 mph.If the nearby city is 760 miles from Austin, to the nearest minute, at what time will the two trains pass each other?

### Complexity=10, Mode=mix

Solve. If asked for time, a proper answer looks like this: 1:35am
 1.   A train leaves Berlin for a nearby city at 7:10 pm, averaging 86 mph.Another train leaves the nearby city for Berlin at 8:00 pm, averaging 100 mph.If the nearby city is 680 miles from Berlin, to the nearest minute, at what time will the two trains pass each other? 2.   A train leaves Las Vegas for a nearby city at 10:29 am, averaging 70 mph.Another train leaves the nearby city for Las Vegas at 12:45 pm, averaging 82 mph.If the nearby city is 710 miles from Las Vegas, to the nearest minute, at what time will the two trains pass each other?

### Complexity=1, Mode=sameDirHrsPassed

Solve. If asked for time, a proper answer looks like this: 1:35am
1A train leaves San Francisco at 6:00 pm, averaging 60 mph.
Another train headed in the same direction leaves San Francisco at 10:00 pm, averaging 90 mph.
To the nearest tenth, how many hours after the second train leaves will it overtake the first train?
Solution
We begin by noting that the distance travelled by both trains must be the same when the second train overtakes the first.
Distance = Rate × Time
Therefore, r1t1 = r2t2

Let t2 = time that the second train is travelling.
Let t1 = time that the first train is travelling alone + time that it's travelling with the second train = 4 + t2

r1(4 + t2) = r2t2
60(4 + t2) = 90t2
240 + 60 t2 = 90t2
240 = 30t2
t2 = 8

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 4 hours × 60 mph = 240 miles.

The second train travels at a relative rate of 30mph faster than the first train and it starts 240 miles behind the first train.
Thus we can find the time it will take to overtake the first train by relating distance rate and time once again.

Time = Distance ÷ Rate
Time = 240 ÷ 30 = 8
2A train leaves Minneapolis at 11:00 am, averaging 70 mph.
Another train headed in the same direction leaves Minneapolis at 3:00 pm, averaging 90 mph.
To the nearest tenth, how many hours after the second train leaves will it overtake the first train?
Solution
We begin by noting that the distance travelled by both trains must be the same when the second train overtakes the first.
Distance = Rate × Time
Therefore, r1t1 = r2t2

Let t2 = time that the second train is travelling.
Let t1 = time that the first train is travelling alone + time that it's travelling with the second train = 4 + t2

r1(4 + t2) = r2t2
70(4 + t2) = 90t2
280 + 70 t2 = 90t2
280 = 20t2
t2 = 14

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 4 hours × 70 mph = 280 miles.

The second train travels at a relative rate of 20mph faster than the first train and it starts 280 miles behind the first train.
Thus we can find the time it will take to overtake the first train by relating distance rate and time once again.

Time = Distance ÷ Rate
Time = 280 ÷ 20 = 14

### Complexity=2, Mode=sameDirTime

Solve. If asked for time, a proper answer looks like this: 1:35am
1A train leaves Venice at 10:00 pm, averaging 80 mph.
Another train headed in the same direction leaves Venice at 11:00 pm, averaging 110 mph.
To the nearest minute, at what time will the second train overtake the first train?
Solution
We begin by noting that the distance travelled by both trains must be the same when the second train overtakes the first.
Distance = Rate × Time
Therefore, r1t1 = r2t2

Let t2 = time that the second train is travelling.
Let t1 = time that the first train is travelling alone + time that it's travelling with the second train = 1 + t2

r1(1 + t2) = r2t2
80(1 + t2) = 110t2
80 + 80t2 = 110t2
80 = 30t2
t2 = 2.6666666666667

Now we must use that to determine the time the second train overtakes the first.
2.6666666666667 hrs can be converted to hours and minutes. It should be 2 hrs and 0.66666666666667 × 60 = 40 min.
Adding the time passed to 11:00 pm we get 1:40 am

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 1 hour× 80 mph = 80 miles.

The second train travels at a relative rate of 30mph faster than the first train and it starts 80 miles behind the first train.
Thus we can find the time it will take to overtake the first train by relating distance rate and time once again.

Time = Distance ÷ Rate
Time = 80 ÷ 30 = 2.6666666666667

Now we must use that to determine the time the second train overtakes the first.
2.6666666666667 hrs can be converted to hours and minutes. It should be 2 hrs and 0.66666666666667 × 60 = 40 min.
Adding the time passed to 11:00 pm we get 1:40 am

2A train leaves Paris at 11:00 am, averaging 50 mph.
Another train headed in the same direction leaves Paris at 1:00 pm, averaging 80 mph.
To the nearest minute, at what time will the second train overtake the first train?
Solution
We begin by noting that the distance travelled by both trains must be the same when the second train overtakes the first.
Distance = Rate × Time
Therefore, r1t1 = r2t2

Let t2 = time that the second train is travelling.
Let t1 = time that the first train is travelling alone + time that it's travelling with the second train = 2 + t2

r1(2 + t2) = r2t2
50(2 + t2) = 80t2
100 + 50t2 = 80t2
100 = 30t2
t2 = 3.3333333333333

Now we must use that to determine the time the second train overtakes the first.
3.3333333333333 hrs can be converted to hours and minutes. It should be 3 hrs and 0.33333333333333 × 60 = 20 min.
Adding the time passed to 1:00 pm we get 4:20 pm

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 2 hours× 50 mph = 100 miles.

The second train travels at a relative rate of 30mph faster than the first train and it starts 100 miles behind the first train.
Thus we can find the time it will take to overtake the first train by relating distance rate and time once again.

Time = Distance ÷ Rate
Time = 100 ÷ 30 = 3.3333333333333

Now we must use that to determine the time the second train overtakes the first.
3.3333333333333 hrs can be converted to hours and minutes. It should be 3 hrs and 0.33333333333333 × 60 = 20 min.
Adding the time passed to 1:00 pm we get 4:20 pm

### Complexity=3, Mode=oppDirDist

Solve. If asked for time, a proper answer looks like this: 1:35am
1A train leaves Madrid at 5:00 pm, averaging 50 mph.
Another train headed in the opposite direction leaves Madrid at 9:00 pm, averaging 80 mph.
To the nearest mile, how far are the two trains from each other at 1:00 am?
Solution
Since both trains are travelling in opposite directions, their total distance apart is the sum of the distances they each travelled.
Distance = Rate × Time
Therefore, dtotal = r1t1 + r2t2

Let t1 = time between 5:00 pm and 1:00 am = 8
Let t2 = time between 9:00 pm and 1:00 am = 4

dtotal = 50t1 + 80t2
dtotal = 50 × 8 + 80 × 4
dtotal = 400 + 320
dtotal = 720
2A train leaves Las Vegas at 1:00 pm, averaging 60 mph.
Another train headed in the opposite direction leaves Las Vegas at 4:00 pm, averaging 80 mph.
To the nearest mile, how far are the two trains from each other at 6:00 pm?
Solution
Since both trains are travelling in opposite directions, their total distance apart is the sum of the distances they each travelled.
Distance = Rate × Time
Therefore, dtotal = r1t1 + r2t2

Let t1 = time between 1:00 pm and 6:00 pm = 5
Let t2 = time between 4:00 pm and 6:00 pm = 2

dtotal = 60t1 + 80t2
dtotal = 60 × 5 + 80 × 2
dtotal = 300 + 160
dtotal = 460

### Complexity=4, Mode=oppDirTime

Solve. If asked for time, a proper answer looks like this: 1:35am
1A train leaves Taipei for a nearby city at 3:45 pm, averaging 80 mph.
Another train leaves the nearby city for Taipei at 7:00 pm, averaging 100 mph.
If the nearby city is 440 miles from Taipei, to the nearest minute, at what time will the two trains pass each other?
Solution
Note that the total distance travelled between the two trains must equal the distance between the two cities.
Distance = Rate × Time
Therefore d = r1t1 + r2t2.

Let t2 = Time travelled by the second train.
Let t1 = Time travelled by the first train alone + Time travelled with second train = 3.25 + t2

d = r1(3.25 + t2) + r2t2
440 = 80(3.25 + t2) + 100t2
440 = 260 + 80t2 + 100t2
180 = 180t2
t2 = 1

We must use the amount of time that passes after the second train leaves to determine the time at which the trains pass by each other.
Adding this amount of time to 7:00 pm yields 8:00 pm as the time.
2A train leaves Kansas City for a nearby city at 8:15 pm, averaging 90 mph.
Another train leaves the nearby city for Kansas City at 10:45 pm, averaging 120 mph.
If the nearby city is 4215 miles from Kansas City, to the nearest minute, at what time will the two trains pass each other?
Solution
Note that the total distance travelled between the two trains must equal the distance between the two cities.
Distance = Rate × Time
Therefore d = r1t1 + r2t2.

Let t2 = Time travelled by the second train.
Let t1 = Time travelled by the first train alone + Time travelled with second train = 2.5 + t2

d = r1(2.5 + t2) + r2t2
4215 = 90(2.5 + t2) + 120t2
4215 = 225 + 90t2 + 120t2
3990 = 210t2
t2 = 19

We must use the amount of time that passes after the second train leaves to determine the time at which the trains pass by each other.
Adding this amount of time to 10:45 pm yields 5:45 pm as the time.

### Complexity=5, Mode=sameDirHrsPassed

Solve. If asked for time, a proper answer looks like this: 1:35am
1A train leaves Brussels at 10:15 pm, averaging 35 mph.
Another train headed in the same direction leaves Brussels at 1:30 am, averaging 45 mph.
To the nearest tenth, how many hours after the second train leaves will it overtake the first train?
Solution
We begin by noting that the distance travelled by both trains must be the same when the second train overtakes the first.
Distance = Rate × Time
Therefore, r1t1 = r2t2

Let t2 = time that the second train is travelling.
Let t1 = time that the first train is travelling alone + time that it's travelling with the second train = 3.25 + t2

r1(3.25 + t2) = r2t2
35(3.25 + t2) = 45t2
113.75 + 35 t2 = 45t2
113.75 = 10t2
t2 = 11.4

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 3.25 hours × 35 mph = 113.75 miles.

The second train travels at a relative rate of 10mph faster than the first train and it starts 113.75 miles behind the first train.
Thus we can find the time it will take to overtake the first train by relating distance rate and time once again.

Time = Distance ÷ Rate
Time = 113.75 ÷ 10 = 11.4
2A train leaves Cairo at 3:30 pm, averaging 55 mph.
Another train headed in the same direction leaves Cairo at 7:45 pm, averaging 85 mph.
To the nearest tenth, how many hours after the second train leaves will it overtake the first train?
Solution
We begin by noting that the distance travelled by both trains must be the same when the second train overtakes the first.
Distance = Rate × Time
Therefore, r1t1 = r2t2

Let t2 = time that the second train is travelling.
Let t1 = time that the first train is travelling alone + time that it's travelling with the second train = 4.25 + t2

r1(4.25 + t2) = r2t2
55(4.25 + t2) = 85t2
233.75 + 55 t2 = 85t2
233.75 = 30t2
t2 = 7.8

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 4.25 hours × 55 mph = 233.75 miles.

The second train travels at a relative rate of 30mph faster than the first train and it starts 233.75 miles behind the first train.
Thus we can find the time it will take to overtake the first train by relating distance rate and time once again.

Time = Distance ÷ Rate
Time = 233.75 ÷ 30 = 7.8

### Complexity=6, Mode=sameDirTime

Solve. If asked for time, a proper answer looks like this: 1:35am
1A train leaves Madrid at 10:45 am, averaging 50 mph.
Another train headed in the same direction leaves Madrid at 1:30 pm, averaging 80 mph.
To the nearest minute, at what time will the second train overtake the first train?
Solution
We begin by noting that the distance travelled by both trains must be the same when the second train overtakes the first.
Distance = Rate × Time
Therefore, r1t1 = r2t2

Let t2 = time that the second train is travelling.
Let t1 = time that the first train is travelling alone + time that it's travelling with the second train = 2.75 + t2

r1(2.75 + t2) = r2t2
50(2.75 + t2) = 80t2
137.5 + 50t2 = 80t2
137.5 = 30t2
t2 = 4.5833333333333

Now we must use that to determine the time the second train overtakes the first.
4.5833333333333 hrs can be converted to hours and minutes. It should be 4 hrs and 0.58333333333333 × 60 = 35 min.
Adding the time passed to 1:30 pm we get 6:05 pm

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 2.75 hours× 50 mph = 137.5 miles.

The second train travels at a relative rate of 30mph faster than the first train and it starts 137.5 miles behind the first train.
Thus we can find the time it will take to overtake the first train by relating distance rate and time once again.

Time = Distance ÷ Rate
Time = 137.5 ÷ 30 = 4.5833333333333

Now we must use that to determine the time the second train overtakes the first.
4.5833333333333 hrs can be converted to hours and minutes. It should be 4 hrs and 0.58333333333333 × 60 = 35 min.
Adding the time passed to 1:30 pm we get 6:05 pm

2A train leaves Denver at 9:30 am, averaging 60 mph.
Another train headed in the same direction leaves Denver at 11:45 am, averaging 85 mph.
To the nearest minute, at what time will the second train overtake the first train?
Solution
We begin by noting that the distance travelled by both trains must be the same when the second train overtakes the first.
Distance = Rate × Time
Therefore, r1t1 = r2t2

Let t2 = time that the second train is travelling.
Let t1 = time that the first train is travelling alone + time that it's travelling with the second train = 2.25 + t2

r1(2.25 + t2) = r2t2
60(2.25 + t2) = 85t2
135 + 60t2 = 85t2
135 = 25t2
t2 = 5.4

Now we must use that to determine the time the second train overtakes the first.
5.4 hrs can be converted to hours and minutes. It should be 5 hrs and 0.4 × 60 = 24 min.
Adding the time passed to 11:45 am we get 5:09 pm

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 2.25 hours× 60 mph = 135 miles.

The second train travels at a relative rate of 25mph faster than the first train and it starts 135 miles behind the first train.
Thus we can find the time it will take to overtake the first train by relating distance rate and time once again.

Time = Distance ÷ Rate
Time = 135 ÷ 25 = 5.4

Now we must use that to determine the time the second train overtakes the first.
5.4 hrs can be converted to hours and minutes. It should be 5 hrs and 0.4 × 60 = 24 min.
Adding the time passed to 11:45 am we get 5:09 pm

### Complexity=7, Mode=oppDirDist

Solve. If asked for time, a proper answer looks like this: 1:35am
1A train leaves Florence at 2:45 pm, averaging 50 mph.
Another train headed in the opposite direction leaves Florence at 3:00 pm, averaging 80 mph.
To the nearest mile, how far are the two trains from each other at 4:00 pm?
Solution
Since both trains are travelling in opposite directions, their total distance apart is the sum of the distances they each travelled.
Distance = Rate × Time
Therefore, dtotal = r1t1 + r2t2

Let t1 = time between 2:45 pm and 4:00 pm = 1.25
Let t2 = time between 3:00 pm and 4:00 pm = 1

dtotal = 50t1 + 80t2
dtotal = 50 × 1.25 + 80 × 1
dtotal = 62.5 + 80
dtotal = 142.5
dtotal = 143 (rounded)
2A train leaves Rome at 8:30 am, averaging 70 mph.
Another train headed in the opposite direction leaves Rome at 11:30 am, averaging 85 mph.
To the nearest mile, how far are the two trains from each other at 12:30 pm?
Solution
Since both trains are travelling in opposite directions, their total distance apart is the sum of the distances they each travelled.
Distance = Rate × Time
Therefore, dtotal = r1t1 + r2t2

Let t1 = time between 8:30 am and 12:30 pm = 4
Let t2 = time between 11:30 am and 12:30 pm = 1

dtotal = 70t1 + 85t2
dtotal = 70 × 4 + 85 × 1
dtotal = 280 + 85
dtotal = 365

### Complexity=8, Mode=oppDirTime

Solve. If asked for time, a proper answer looks like this: 1:35am
1A train leaves Prague for a nearby city at 7:32 am, averaging 96 mph.
Another train leaves the nearby city for Prague at 11:45 am, averaging 113 mph.
If the nearby city is 5420.8 miles from Prague, to the nearest minute, at what time will the two trains pass each other?
Solution
Note that the total distance travelled between the two trains must equal the distance between the two cities.
Distance = Rate × Time
Therefore d = r1t1 + r2t2.

Let t2 = Time travelled by the second train.
Let t1 = Time travelled by the first train alone + Time travelled with second train = 4.2166666666667 + t2

d = r1(4.2166666666667 + t2) + r2t2
5420.8 = 96(4.2166666666667 + t2) + 113t2
5420.8 = 404.8 + 96t2 + 113t2
5016 = 209t2
t2 = 24

We must use the amount of time that passes after the second train leaves to determine the time at which the trains pass by each other.
Adding this amount of time to 11:45 am yields 11:45 am as the time.
2A train leaves Austin for a nearby city at 2:55 pm, averaging 41 mph.
Another train leaves the nearby city for Austin at 6:15 pm, averaging 63 mph.
If the nearby city is 604.66666666667 miles from Austin, to the nearest minute, at what time will the two trains pass each other?
Solution
Note that the total distance travelled between the two trains must equal the distance between the two cities.
Distance = Rate × Time
Therefore d = r1t1 + r2t2.

Let t2 = Time travelled by the second train.
Let t1 = Time travelled by the first train alone + Time travelled with second train = 3.3333333333333 + t2

d = r1(3.3333333333333 + t2) + r2t2
604.66666666667 = 41(3.3333333333333 + t2) + 63t2
604.66666666667 = 136.66666666667 + 41t2 + 63t2
468 = 104t2
t2 = 4.5

We must use the amount of time that passes after the second train leaves to determine the time at which the trains pass by each other.
4.5 hours can be converted into hours and minutes. It is 4 hours and 0.5× 60 = 30 min.
Adding this amount of time to 6:15 pm yields 10:45 pm as the time.

### Complexity=9, Mode=mix

Solve. If asked for time, a proper answer looks like this: 1:35am
1A train leaves Florence at 9:25 am, averaging 59 mph.
Another train headed in the same direction leaves Florence at 11:45 am, averaging 72 mph.
To the nearest minute, at what time will the second train overtake the first train?
Solution
We begin by noting that the distance travelled by both trains must be the same when the second train overtakes the first.
Distance = Rate × Time
Therefore, r1t1 = r2t2

Let t2 = time that the second train is travelling.
Let t1 = time that the first train is travelling alone + time that it's travelling with the second train = 2.3333333333333 + t2

r1(2.3333333333333 + t2) = r2t2
59(2.3333333333333 + t2) = 72t2
137.66666666667 + 59t2 = 72t2
137.66666666667 = 13t2
t2 = 10.589743589744

Now we must use that to determine the time the second train overtakes the first.
10.589743589744 hrs can be converted to hours and minutes. It should be 10 hrs and 0.58974358974359 × 60 = 35 min.
Adding the time passed to 11:45 am we get 10:20 pm

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 2.3333333333333 hours× 59 mph = 137.66666666667 miles.

The second train travels at a relative rate of 13mph faster than the first train and it starts 137.66666666667 miles behind the first train.
Thus we can find the time it will take to overtake the first train by relating distance rate and time once again.

Time = Distance ÷ Rate
Time = 137.66666666667 ÷ 13 = 10.589743589744

Now we must use that to determine the time the second train overtakes the first.
10.589743589744 hrs can be converted to hours and minutes. It should be 10 hrs and 0.58974358974359 × 60 = 35 min.
Adding the time passed to 11:45 am we get 10:20 pm

2A train leaves Austin for a nearby city at 6:13 am, averaging 86 mph.
Another train leaves the nearby city for Austin at 10:30 am, averaging 104 mph.
If the nearby city is 760 miles from Austin, to the nearest minute, at what time will the two trains pass each other?
Solution
Note that the total distance travelled between the two trains must equal the distance between the two cities.
Distance = Rate × Time
Therefore d = r1t1 + r2t2.

Let t2 = Time travelled by the second train.
Let t1 = Time travelled by the first train alone + Time travelled with second train = 4.2833333333333 + t2

d = r1(4.2833333333333 + t2) + r2t2
760 = 86(4.2833333333333 + t2) + 104t2
760 = 368.36666666667 + 86t2 + 104t2
391.63333333333 = 190t2
t2 = 2.0612280701754

We must use the amount of time that passes after the second train leaves to determine the time at which the trains pass by each other.
2.0612280701754 hours can be converted into hours and minutes. It is 2 hours and 0.061228070175439× 60 = 4 min.
Adding this amount of time to 10:30 am yields 12:34 pm as the time.

### Complexity=10, Mode=mix

Solve. If asked for time, a proper answer looks like this: 1:35am
1A train leaves Berlin for a nearby city at 7:10 pm, averaging 86 mph.
Another train leaves the nearby city for Berlin at 8:00 pm, averaging 100 mph.
If the nearby city is 680 miles from Berlin, to the nearest minute, at what time will the two trains pass each other?
Solution
Note that the total distance travelled between the two trains must equal the distance between the two cities.
Distance = Rate × Time
Therefore d = r1t1 + r2t2.

Let t2 = Time travelled by the second train.
Let t1 = Time travelled by the first train alone + Time travelled with second train = 0.83333333333333 + t2

d = r1(0.83333333333333 + t2) + r2t2
680 = 86(0.83333333333333 + t2) + 100t2
680 = 71.666666666667 + 86t2 + 100t2
608.33333333333 = 186t2
t2 = 3.2706093189964

We must use the amount of time that passes after the second train leaves to determine the time at which the trains pass by each other.
3.2706093189964 hours can be converted into hours and minutes. It is 3 hours and 0.27060931899642× 60 = 16 min.
Adding this amount of time to 8:00 pm yields 11:16 pm as the time.
2A train leaves Las Vegas for a nearby city at 10:29 am, averaging 70 mph.
Another train leaves the nearby city for Las Vegas at 12:45 pm, averaging 82 mph.
If the nearby city is 710 miles from Las Vegas, to the nearest minute, at what time will the two trains pass each other?
Solution
Note that the total distance travelled between the two trains must equal the distance between the two cities.
Distance = Rate × Time
Therefore d = r1t1 + r2t2.

Let t2 = Time travelled by the second train.
Let t1 = Time travelled by the first train alone + Time travelled with second train = 2.2666666666667 + t2

d = r1(2.2666666666667 + t2) + r2t2
710 = 70(2.2666666666667 + t2) + 82t2
710 = 158.66666666667 + 70t2 + 82t2
551.33333333333 = 152t2
t2 = 3.6271929824561

We must use the amount of time that passes after the second train leaves to determine the time at which the trains pass by each other.
3.6271929824561 hours can be converted into hours and minutes. It is 3 hours and 0.62719298245614× 60 = 38 min.
Adding this amount of time to 12:45 pm yields 4:23 pm as the time.