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## Train Problems - Sample Math Practice Problems

The math problems below can be generated by MathScore.com, a math practice program for schools and individual families. References to complexity and mode refer to the overall difficulty of the problems as they appear in the main program. In the main program, all problems are automatically graded and the difficulty adapts dynamically based on performance. Answers to these sample questions appear at the bottom of the page. This page does not grade your responses.

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### Complexity=1, Mode=sameDirHrsPassed

Solve. If asked for time, a proper answer looks like this: 1:35am

 1 A train leaves Cairo at 3:00 am, averaging 30 mph.Another train headed in the same direction leaves Cairo at 6:00 am, averaging 60 mph.To the nearest tenth, how many hours after the second train leaves will it overtake the first train? 2 A train leaves Madrid at 8:00 pm, averaging 60 mph.Another train headed in the same direction leaves Madrid at 12:00 am, 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 7:00 pm, averaging 80 mph.Another train headed in the same direction leaves Venice at 10:00 pm, averaging 100 mph.To the nearest minute, at what time will the second train overtake the first train? 2 A train leaves Taipei at 3:00 am, averaging 50 mph.Another train headed in the same direction leaves Taipei at 4:00 am, averaging 70 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 Brussels at 11:00 am, averaging 60 mph.Another train headed in the opposite direction leaves Brussels at 1:00 pm, averaging 90 mph.To the nearest mile, how far are the two trains from each other at 3:00 pm? 2 A train leaves Denver at 12:00 am, averaging 60 mph.Another train headed in the opposite direction leaves Denver at 1:00 am, averaging 70 mph.To the nearest mile, how far are the two trains from each other at 3:00 am?

### Complexity=4, Mode=oppDirTime

Solve. If asked for time, a proper answer looks like this: 1:35am

 1 A train leaves Austin for a nearby city at 2:15 am, averaging 80 mph.Another train leaves the nearby city for Austin at 3:30 am, averaging 110 mph.If the nearby city is 2855 miles from Austin, to the nearest minute, at what time will the two trains pass each other? 2 A train leaves Venice for a nearby city at 9:00 am, averaging 55 mph.Another train leaves the nearby city for Venice at 10:15 am, averaging 65 mph.If the nearby city is 788.75 miles from Venice, 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 Florence at 9:45 pm, averaging 50 mph.Another train headed in the same direction leaves Florence at 11:30 pm, averaging 60 mph.To the nearest tenth, how many hours after the second train leaves will it overtake the first train? 2 A train leaves Florence at 6:00 am, averaging 80 mph.Another train headed in the same direction leaves Florence at 9:00 am, averaging 105 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 Los Angeles at 6:45 pm, averaging 85 mph.Another train headed in the same direction leaves Los Angeles at 9:30 pm, averaging 105 mph.To the nearest minute, at what time will the second train overtake the first train? 2 A train leaves Cairo at 2:45 pm, averaging 30 mph.Another train headed in the same direction leaves Cairo at 3:15 pm, averaging 45 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 Rome at 7:15 am, averaging 95 mph.Another train headed in the opposite direction leaves Rome at 8:30 am, averaging 115 mph.To the nearest mile, how far are the two trains from each other at 10:30 am? 2 A train leaves Barcelona at 9:45 am, averaging 85 mph.Another train headed in the opposite direction leaves Barcelona at 12:15 pm, averaging 95 mph.To the nearest mile, how far are the two trains from each other at 1:15 pm?

### Complexity=8, Mode=oppDirTime

Solve. If asked for time, a proper answer looks like this: 1:35am

 1 A train leaves Geneva for a nearby city at 3:04 pm, averaging 37 mph.Another train leaves the nearby city for Geneva at 7:30 pm, averaging 55 mph.If the nearby city is 624.03333333333 miles from Geneva, to the nearest minute, at what time will the two trains pass each other? 2 A train leaves Florence for a nearby city at 11:47 pm, averaging 44 mph.Another train leaves the nearby city for Florence at 12:30 am, averaging 68 mph.If the nearby city is 367.53333333333 miles from Florence, 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 Rome at 4:54 am, averaging 88 mph.Another train headed in the same direction leaves Rome at 5:45 am, averaging 108 mph.To the nearest minute, at what time will the second train overtake the first train? 2 A train leaves Buenos Aires at 9:04 am, averaging 98 mph.Another train headed in the same direction leaves Buenos Aires at 1:15 pm, averaging 113 mph.To the nearest tenth, how many hours after the second train leaves will it overtake the first train?

### Complexity=10, Mode=mix

Solve. If asked for time, a proper answer looks like this: 1:35am

 1 A train leaves Barcelona at 3:15 am, averaging 30 mph.Another train headed in the opposite direction leaves Barcelona at 4:45 am, averaging 41 mph.To the nearest mile, how far are the two trains from each other at 5:45 am? 2 A train leaves Brussels at 8:15 am, averaging 53 mph.Another train headed in the opposite direction leaves Brussels at 12:45 pm, averaging 71 mph.To the nearest mile, how far are the two trains from each other at 2:45 pm?

### Complexity=1, Mode=sameDirHrsPassed

Solve. If asked for time, a proper answer looks like this: 1:35am

1A train leaves Cairo at 3:00 am, averaging 30 mph.
Another train headed in the same direction leaves Cairo at 6:00 am, averaging 60 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 + t2

r1(3 + t2) = r2t2
30(3 + t2) = 60t2
90 + 30 t2 = 60t2
90 = 30t2
t2 = 3

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 3 hours × 30 mph = 90 miles.

The second train travels at a relative rate of 30mph faster than the first train and it starts 90 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 = 90 ÷ 30 = 3
2A train leaves Madrid at 8:00 pm, averaging 60 mph.
Another train headed in the same direction leaves Madrid at 12:00 am, 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

### Complexity=2, Mode=sameDirTime

Solve. If asked for time, a proper answer looks like this: 1:35am

1A train leaves Venice at 7:00 pm, averaging 80 mph.
Another train headed in the same direction leaves Venice at 10:00 pm, averaging 100 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 = 3 + t2

r1(3 + t2) = r2t2
80(3 + t2) = 100t2
240 + 80t2 = 100t2
240 = 20t2
t2 = 12

Now we must use that to determine the time the second train overtakes the first.
Adding the time passed to 10:00 pm we get 10:00 am

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 3 hours× 80 mph = 240 miles.

The second train travels at a relative rate of 20mph 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 ÷ 20 = 12

Now we must use that to determine the time the second train overtakes the first.
Adding the time passed to 10:00 pm we get 10:00 am

2A train leaves Taipei at 3:00 am, averaging 50 mph.
Another train headed in the same direction leaves Taipei at 4:00 am, averaging 70 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
50(1 + t2) = 70t2
50 + 50t2 = 70t2
50 = 20t2
t2 = 2.5

Now we must use that to determine the time the second train overtakes the first.
2.5 hrs can be converted to hours and minutes. It should be 2 hrs and 0.5 × 60 = 30 min.
Adding the time passed to 4:00 am we get 6:30 am

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 1 hour× 50 mph = 50 miles.

The second train travels at a relative rate of 20mph faster than the first train and it starts 50 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 = 50 ÷ 20 = 2.5

Now we must use that to determine the time the second train overtakes the first.
2.5 hrs can be converted to hours and minutes. It should be 2 hrs and 0.5 × 60 = 30 min.
Adding the time passed to 4:00 am we get 6:30 am

### Complexity=3, Mode=oppDirDist

Solve. If asked for time, a proper answer looks like this: 1:35am

1A train leaves Brussels at 11:00 am, averaging 60 mph.
Another train headed in the opposite direction leaves Brussels at 1:00 pm, averaging 90 mph.
To the nearest mile, how far are the two trains from each other at 3:00 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 11:00 am and 3:00 pm = 4
Let t2 = time between 1:00 pm and 3:00 pm = 2

dtotal = 60t1 + 90t2
dtotal = 60 × 4 + 90 × 2
dtotal = 240 + 180
dtotal = 420
2A train leaves Denver at 12:00 am, averaging 60 mph.
Another train headed in the opposite direction leaves Denver at 1:00 am, averaging 70 mph.
To the nearest mile, how far are the two trains from each other at 3:00 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 12:00 am and 3:00 am = 3
Let t2 = time between 1:00 am and 3:00 am = 2

dtotal = 60t1 + 70t2
dtotal = 60 × 3 + 70 × 2
dtotal = 180 + 140
dtotal = 320

### Complexity=4, Mode=oppDirTime

Solve. If asked for time, a proper answer looks like this: 1:35am

1A train leaves Austin for a nearby city at 2:15 am, averaging 80 mph.
Another train leaves the nearby city for Austin at 3:30 am, averaging 110 mph.
If the nearby city is 2855 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 = 1.25 + t2

d = r1(1.25 + t2) + r2t2
2855 = 80(1.25 + t2) + 110t2
2855 = 100 + 80t2 + 110t2
2755 = 190t2
t2 = 14.5

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.
14.5 hours can be converted into hours and minutes. It is 14 hours and 0.5× 60 = 30 min.
Adding this amount of time to 3:30 am yields 6:00 pm as the time.
2A train leaves Venice for a nearby city at 9:00 am, averaging 55 mph.
Another train leaves the nearby city for Venice at 10:15 am, averaging 65 mph.
If the nearby city is 788.75 miles from Venice, 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 = 1.25 + t2

d = r1(1.25 + t2) + r2t2
788.75 = 55(1.25 + t2) + 65t2
788.75 = 68.75 + 55t2 + 65t2
720 = 120t2
t2 = 6

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 10:15 am yields 4:15 pm as the time.

### Complexity=5, Mode=sameDirHrsPassed

Solve. If asked for time, a proper answer looks like this: 1:35am

1A train leaves Florence at 9:45 pm, averaging 50 mph.
Another train headed in the same direction leaves Florence at 11:30 pm, averaging 60 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 = 1.75 + t2

r1(1.75 + t2) = r2t2
50(1.75 + t2) = 60t2
87.5 + 50 t2 = 60t2
87.5 = 10t2
t2 = 8.8

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 1.75 hours × 50 mph = 87.5 miles.

The second train travels at a relative rate of 10mph faster than the first train and it starts 87.5 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 = 87.5 ÷ 10 = 8.8
2A train leaves Florence at 6:00 am, averaging 80 mph.
Another train headed in the same direction leaves Florence at 9:00 am, averaging 105 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 + t2

r1(3 + t2) = r2t2
80(3 + t2) = 105t2
240 + 80 t2 = 105t2
240 = 25t2
t2 = 9.6

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 3 hours × 80 mph = 240 miles.

The second train travels at a relative rate of 25mph 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 ÷ 25 = 9.6

### Complexity=6, Mode=sameDirTime

Solve. If asked for time, a proper answer looks like this: 1:35am

1A train leaves Los Angeles at 6:45 pm, averaging 85 mph.
Another train headed in the same direction leaves Los Angeles at 9:30 pm, averaging 105 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
85(2.75 + t2) = 105t2
233.75 + 85t2 = 105t2
233.75 = 20t2
t2 = 11.6875

Now we must use that to determine the time the second train overtakes the first.
11.6875 hrs can be converted to hours and minutes. It should be 11 hrs and 0.6875 × 60 = 41 min.
Adding the time passed to 9:30 pm we get 9:11 am

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 2.75 hours× 85 mph = 233.75 miles.

The second train travels at a relative rate of 20mph 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 ÷ 20 = 11.6875

Now we must use that to determine the time the second train overtakes the first.
11.6875 hrs can be converted to hours and minutes. It should be 11 hrs and 0.6875 × 60 = 41 min.
Adding the time passed to 9:30 pm we get 9:11 am

2A train leaves Cairo at 2:45 pm, averaging 30 mph.
Another train headed in the same direction leaves Cairo at 3:15 pm, averaging 45 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 = 0.5 + t2

r1(0.5 + t2) = r2t2
30(0.5 + t2) = 45t2
15 + 30t2 = 45t2
15 = 15t2
t2 = 1

Now we must use that to determine the time the second train overtakes the first.
Adding the time passed to 3:15 pm we get 4:15 pm

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 0.5 hours× 30 mph = 15 miles.

The second train travels at a relative rate of 15mph faster than the first train and it starts 15 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 = 15 ÷ 15 = 1

Now we must use that to determine the time the second train overtakes the first.
Adding the time passed to 3:15 pm we get 4:15 pm

### Complexity=7, Mode=oppDirDist

Solve. If asked for time, a proper answer looks like this: 1:35am

1A train leaves Rome at 7:15 am, averaging 95 mph.
Another train headed in the opposite direction leaves Rome at 8:30 am, averaging 115 mph.
To the nearest mile, how far are the two trains from each other at 10:30 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 7:15 am and 10:30 am = 3.25
Let t2 = time between 8:30 am and 10:30 am = 2

dtotal = 95t1 + 115t2
dtotal = 95 × 3.25 + 115 × 2
dtotal = 308.75 + 230
dtotal = 538.75
dtotal = 539 (rounded)
2A train leaves Barcelona at 9:45 am, averaging 85 mph.
Another train headed in the opposite direction leaves Barcelona at 12:15 pm, averaging 95 mph.
To the nearest mile, how far are the two trains from each other at 1:15 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 9:45 am and 1:15 pm = 3.5
Let t2 = time between 12:15 pm and 1:15 pm = 1

dtotal = 85t1 + 95t2
dtotal = 85 × 3.5 + 95 × 1
dtotal = 297.5 + 95
dtotal = 392.5
dtotal = 393 (rounded)

### Complexity=8, Mode=oppDirTime

Solve. If asked for time, a proper answer looks like this: 1:35am

1A train leaves Geneva for a nearby city at 3:04 pm, averaging 37 mph.
Another train leaves the nearby city for Geneva at 7:30 pm, averaging 55 mph.
If the nearby city is 624.03333333333 miles from Geneva, 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 = 4.4333333333333 + t2

d = r1(4.4333333333333 + t2) + r2t2
624.03333333333 = 37(4.4333333333333 + t2) + 55t2
624.03333333333 = 164.03333333333 + 37t2 + 55t2
460 = 92t2
t2 = 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.
Adding this amount of time to 7:30 pm yields 12:30 pm as the time.
2A train leaves Florence for a nearby city at 11:47 pm, averaging 44 mph.
Another train leaves the nearby city for Florence at 12:30 am, averaging 68 mph.
If the nearby city is 367.53333333333 miles from Florence, 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 = 0.71666666666667 + t2

d = r1(0.71666666666667 + t2) + r2t2
367.53333333333 = 44(0.71666666666667 + t2) + 68t2
367.53333333333 = 31.533333333333 + 44t2 + 68t2
336 = 112t2
t2 = 3

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 12:30 am yields 3:30 am as the time.

### Complexity=9, Mode=mix

Solve. If asked for time, a proper answer looks like this: 1:35am

1A train leaves Rome at 4:54 am, averaging 88 mph.
Another train headed in the same direction leaves Rome at 5:45 am, averaging 108 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 = 0.85 + t2

r1(0.85 + t2) = r2t2
88(0.85 + t2) = 108t2
74.8 + 88t2 = 108t2
74.8 = 20t2
t2 = 3.74

Now we must use that to determine the time the second train overtakes the first.
3.74 hrs can be converted to hours and minutes. It should be 3 hrs and 0.74 × 60 = 44 min.
Adding the time passed to 5:45 am we get 9:29 am

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 0.85 hours× 88 mph = 74.8 miles.

The second train travels at a relative rate of 20mph faster than the first train and it starts 74.8 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 = 74.8 ÷ 20 = 3.74

Now we must use that to determine the time the second train overtakes the first.
3.74 hrs can be converted to hours and minutes. It should be 3 hrs and 0.74 × 60 = 44 min.
Adding the time passed to 5:45 am we get 9:29 am

2A train leaves Buenos Aires at 9:04 am, averaging 98 mph.
Another train headed in the same direction leaves Buenos Aires at 1:15 pm, averaging 113 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 = 4.1833333333333 + t2

r1(4.1833333333333 + t2) = r2t2
98(4.1833333333333 + t2) = 113t2
409.96666666667 + 98 t2 = 113t2
409.96666666667 = 15t2
t2 = 27.3

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 4.1833333333333 hours × 98 mph = 409.96666666667 miles.

The second train travels at a relative rate of 15mph faster than the first train and it starts 409.96666666667 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 = 409.96666666667 ÷ 15 = 27.3

### Complexity=10, Mode=mix

Solve. If asked for time, a proper answer looks like this: 1:35am

1A train leaves Barcelona at 3:15 am, averaging 30 mph.
Another train headed in the opposite direction leaves Barcelona at 4:45 am, averaging 41 mph.
To the nearest mile, how far are the two trains from each other at 5:45 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 3:15 am and 5:45 am = 2.5
Let t2 = time between 4:45 am and 5:45 am = 1

dtotal = 30t1 + 41t2
dtotal = 30 × 2.5 + 41 × 1
dtotal = 75 + 41
dtotal = 116
2A train leaves Brussels at 8:15 am, averaging 53 mph.
Another train headed in the opposite direction leaves Brussels at 12:45 pm, averaging 71 mph.
To the nearest mile, how far are the two trains from each other at 2:45 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 8:15 am and 2:45 pm = 6.5
Let t2 = time between 12:45 pm and 2:45 pm = 2

dtotal = 53t1 + 71t2
dtotal = 53 × 6.5 + 71 × 2
dtotal = 344.5 + 142
dtotal = 486.5
dtotal = 487 (rounded)