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These sample problems below for Train Problems were generated by the MathScore.com engine.

## Sample Problems For Train Problems

### Complexity=1, Mode=sameDirHrsPassed

Solve. If asked for time, a proper answer looks like this: 1:35am
 1.   A train leaves Austin at 3:00 pm, averaging 30 mph.Another train headed in the same direction leaves Austin at 4:00 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 Prague at 9:00 pm, averaging 30 mph.Another train headed in the same direction leaves Prague at 11:00 pm, averaging 50 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 Las Vegas at 12:00 pm, averaging 70 mph.Another train headed in the same direction leaves Las Vegas at 3:00 pm, averaging 80 mph.To the nearest minute, at what time will the second train overtake the first train? 2.   A train leaves Brussels at 12:00 am, averaging 60 mph.Another train headed in the same direction leaves Brussels 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 Bangkok at 4:00 am, averaging 40 mph.Another train headed in the opposite direction leaves Bangkok at 6:00 am, averaging 60 mph.To the nearest mile, how far are the two trains from each other at 8:00 am? 2.   A train leaves New York at 9:00 pm, averaging 70 mph.Another train headed in the opposite direction leaves New York at 1:00 am, averaging 80 mph.To the nearest mile, how far are the two trains from each other at 4:00 am?

### Complexity=4, Mode=oppDirTime

Solve. If asked for time, a proper answer looks like this: 1:35am
 1.   A train leaves Buenos Aires for a nearby city at 12:00 pm, averaging 35 mph.Another train leaves the nearby city for Buenos Aires at 3:00 pm, averaging 60 mph.If the nearby city is 2195 miles from Buenos Aires, to the nearest minute, at what time will the two trains pass each other? 2.   A train leaves Florence for a nearby city at 10:30 pm, averaging 50 mph.Another train leaves the nearby city for Florence at 11:00 pm, averaging 65 mph.If the nearby city is 1232.5 miles from Florence, 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 Tokyo at 9:45 pm, averaging 75 mph.Another train headed in the same direction leaves Tokyo at 11:15 pm, averaging 85 mph.To the nearest tenth, how many hours after the second train leaves will it overtake the first train? 2.   A train leaves Las Vegas at 12:30 pm, averaging 100 mph.Another train headed in the same direction leaves Las Vegas at 1:00 pm, averaging 115 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 Amsterdam at 1:45 am, averaging 65 mph.Another train headed in the same direction leaves Amsterdam at 2:30 am, averaging 95 mph.To the nearest minute, at what time will the second train overtake the first train? 2.   A train leaves Austin at 5:30 pm, averaging 80 mph.Another train headed in the same direction leaves Austin at 7:45 pm, averaging 105 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 Venice at 10:00 am, averaging 50 mph.Another train headed in the opposite direction leaves Venice at 12:30 pm, averaging 80 mph.To the nearest mile, how far are the two trains from each other at 1:30 pm? 2.   A train leaves Madrid at 9:15 pm, averaging 60 mph.Another train headed in the opposite direction leaves Madrid at 11:30 pm, averaging 90 mph.To the nearest mile, how far are the two trains from each other at 12:30 am?

### Complexity=8, Mode=oppDirTime

Solve. If asked for time, a proper answer looks like this: 1:35am
 1.   A train leaves Venice for a nearby city at 1:04 am, averaging 45 mph.Another train leaves the nearby city for Venice at 3:30 am, averaging 73 mph.If the nearby city is 817.5 miles from Venice, to the nearest minute, at what time will the two trains pass each other? 2.   A train leaves Prague for a nearby city at 12:51 am, averaging 94 mph.Another train leaves the nearby city for Prague at 4:15 am, averaging 106 mph.If the nearby city is 819.6 miles from Prague, 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 Minneapolis at 6:20 pm, averaging 42 mph.Another train headed in the same direction leaves Minneapolis at 9:30 pm, averaging 52 mph.To the nearest tenth, how many hours after the second train leaves will it overtake the first train? 2.   A train leaves Denver for a nearby city at 2:16 am, averaging 89 mph.Another train leaves the nearby city for Denver at 4:30 am, averaging 108 mph.If the nearby city is 620 miles from Denver, 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 Venice at 1:41 pm, averaging 35 mph.Another train headed in the same direction leaves Venice at 2:00 pm, averaging 65 mph.To the nearest tenth, how many hours after the second train leaves will it overtake the first train? 2.   A train leaves Denver at 1:28 pm, averaging 31 mph.Another train headed in the same direction leaves Denver at 4:00 pm, averaging 53 mph.To the nearest minute, at what time will the second train overtake the first train?

### Complexity=1, Mode=sameDirHrsPassed

Solve. If asked for time, a proper answer looks like this: 1:35am
1A train leaves Austin at 3:00 pm, averaging 30 mph.
Another train headed in the same direction leaves Austin at 4:00 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 + t2

r1(1 + t2) = r2t2
30(1 + t2) = 60t2
30 + 30 t2 = 60t2
30 = 30t2
t2 = 1

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 1 hour × 30 mph = 30 miles.

The second train travels at a relative rate of 30mph faster than the first train and it starts 30 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 = 30 ÷ 30 = 1
2A train leaves Prague at 9:00 pm, averaging 30 mph.
Another train headed in the same direction leaves Prague at 11:00 pm, averaging 50 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 = 2 + t2

r1(2 + t2) = r2t2
30(2 + t2) = 50t2
60 + 30 t2 = 50t2
60 = 20t2
t2 = 3

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 2 hours × 30 mph = 60 miles.

The second train travels at a relative rate of 20mph faster than the first train and it starts 60 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 = 60 ÷ 20 = 3

### Complexity=2, Mode=sameDirTime

Solve. If asked for time, a proper answer looks like this: 1:35am
1A train leaves Las Vegas at 12:00 pm, averaging 70 mph.
Another train headed in the same direction leaves Las Vegas at 3:00 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 = 3 + t2

r1(3 + t2) = r2t2
70(3 + t2) = 80t2
210 + 70t2 = 80t2
210 = 10t2
t2 = 21

Now we must use that to determine the time the second train overtakes the first.
Adding the time passed to 3:00 pm we get 12:00 pm

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 3 hours× 70 mph = 210 miles.

The second train travels at a relative rate of 10mph faster than the first train and it starts 210 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 = 210 ÷ 10 = 21

Now we must use that to determine the time the second train overtakes the first.
Adding the time passed to 3:00 pm we get 12:00 pm

2A train leaves Brussels at 12:00 am, averaging 60 mph.
Another train headed in the same direction leaves Brussels 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 = 4 + t2

r1(4 + t2) = r2t2
60(4 + t2) = 70t2
240 + 60t2 = 70t2
240 = 10t2
t2 = 24

Now we must use that to determine the time the second train overtakes the first.
Adding the time passed to 4:00 am we get 4:00 am

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 10mph 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 ÷ 10 = 24

Now we must use that to determine the time the second train overtakes the first.
Adding the time passed to 4:00 am we get 4:00 am

### Complexity=3, Mode=oppDirDist

Solve. If asked for time, a proper answer looks like this: 1:35am
1A train leaves Bangkok at 4:00 am, averaging 40 mph.
Another train headed in the opposite direction leaves Bangkok at 6:00 am, averaging 60 mph.
To the nearest mile, how far are the two trains from each other at 8:00 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 4:00 am and 8:00 am = 4
Let t2 = time between 6:00 am and 8:00 am = 2

dtotal = 40t1 + 60t2
dtotal = 40 × 4 + 60 × 2
dtotal = 160 + 120
dtotal = 280
2A train leaves New York at 9:00 pm, averaging 70 mph.
Another train headed in the opposite direction leaves New York at 1:00 am, averaging 80 mph.
To the nearest mile, how far are the two trains from each other at 4:00 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 9:00 pm and 4:00 am = 7
Let t2 = time between 1:00 am and 4:00 am = 3

dtotal = 70t1 + 80t2
dtotal = 70 × 7 + 80 × 3
dtotal = 490 + 240
dtotal = 730

### Complexity=4, Mode=oppDirTime

Solve. If asked for time, a proper answer looks like this: 1:35am
1A train leaves Buenos Aires for a nearby city at 12:00 pm, averaging 35 mph.
Another train leaves the nearby city for Buenos Aires at 3:00 pm, averaging 60 mph.
If the nearby city is 2195 miles from Buenos Aires, 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 + t2

d = r1(3 + t2) + r2t2
2195 = 35(3 + t2) + 60t2
2195 = 105 + 35t2 + 60t2
2090 = 95t2
t2 = 22

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 3:00 pm yields 1:00 pm as the time.
2A train leaves Florence for a nearby city at 10:30 pm, averaging 50 mph.
Another train leaves the nearby city for Florence at 11:00 pm, averaging 65 mph.
If the nearby city is 1232.5 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 = 0.5 + t2

d = r1(0.5 + t2) + r2t2
1232.5 = 50(0.5 + t2) + 65t2
1232.5 = 25 + 50t2 + 65t2
1207.5 = 115t2
t2 = 10.5

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.
10.5 hours can be converted into hours and minutes. It is 10 hours and 0.5× 60 = 30 min.
Adding this amount of time to 11:00 pm yields 9:30 am as the time.

### Complexity=5, Mode=sameDirHrsPassed

Solve. If asked for time, a proper answer looks like this: 1:35am
1A train leaves Tokyo at 9:45 pm, averaging 75 mph.
Another train headed in the same direction leaves Tokyo at 11:15 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 = 1.5 + t2

r1(1.5 + t2) = r2t2
75(1.5 + t2) = 85t2
112.5 + 75 t2 = 85t2
112.5 = 10t2
t2 = 11.3

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 1.5 hours × 75 mph = 112.5 miles.

The second train travels at a relative rate of 10mph faster than the first train and it starts 112.5 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 = 112.5 ÷ 10 = 11.3
2A train leaves Las Vegas at 12:30 pm, averaging 100 mph.
Another train headed in the same direction leaves Las Vegas at 1:00 pm, averaging 115 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 = 0.5 + t2

r1(0.5 + t2) = r2t2
100(0.5 + t2) = 115t2
50 + 100 t2 = 115t2
50 = 15t2
t2 = 3.3

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 0.5 hours × 100 mph = 50 miles.

The second train travels at a relative rate of 15mph 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 ÷ 15 = 3.3

### Complexity=6, Mode=sameDirTime

Solve. If asked for time, a proper answer looks like this: 1:35am
1A train leaves Amsterdam at 1:45 am, averaging 65 mph.
Another train headed in the same direction leaves Amsterdam at 2:30 am, averaging 95 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 = 0.75 + t2

r1(0.75 + t2) = r2t2
65(0.75 + t2) = 95t2
48.75 + 65t2 = 95t2
48.75 = 30t2
t2 = 1.625

Now we must use that to determine the time the second train overtakes the first.
1.625 hrs can be converted to hours and minutes. It should be 1 hrs and 0.625 × 60 = 38 min.
Adding the time passed to 2:30 am we get 4:08 am

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 0.75 hours× 65 mph = 48.75 miles.

The second train travels at a relative rate of 30mph faster than the first train and it starts 48.75 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 = 48.75 ÷ 30 = 1.625

Now we must use that to determine the time the second train overtakes the first.
1.625 hrs can be converted to hours and minutes. It should be 1 hrs and 0.625 × 60 = 38 min.
Adding the time passed to 2:30 am we get 4:08 am

2A train leaves Austin at 5:30 pm, averaging 80 mph.
Another train headed in the same direction leaves Austin at 7:45 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 = 2.25 + t2

r1(2.25 + t2) = r2t2
80(2.25 + t2) = 105t2
180 + 80t2 = 105t2
180 = 25t2
t2 = 7.2

Now we must use that to determine the time the second train overtakes the first.
7.2 hrs can be converted to hours and minutes. It should be 7 hrs and 0.2 × 60 = 12 min.
Adding the time passed to 7:45 pm we get 2:57 am

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 2.25 hours× 80 mph = 180 miles.

The second train travels at a relative rate of 25mph faster than the first train and it starts 180 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 = 180 ÷ 25 = 7.2

Now we must use that to determine the time the second train overtakes the first.
7.2 hrs can be converted to hours and minutes. It should be 7 hrs and 0.2 × 60 = 12 min.
Adding the time passed to 7:45 pm we get 2:57 am

### Complexity=7, Mode=oppDirDist

Solve. If asked for time, a proper answer looks like this: 1:35am
1A train leaves Venice at 10:00 am, averaging 50 mph.
Another train headed in the opposite direction leaves Venice at 12:30 pm, averaging 80 mph.
To the nearest mile, how far are the two trains from each other at 1:30 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 10:00 am and 1:30 pm = 3.5
Let t2 = time between 12:30 pm and 1:30 pm = 1

dtotal = 50t1 + 80t2
dtotal = 50 × 3.5 + 80 × 1
dtotal = 175 + 80
dtotal = 255
2A train leaves Madrid at 9:15 pm, averaging 60 mph.
Another train headed in the opposite direction leaves Madrid at 11:30 pm, averaging 90 mph.
To the nearest mile, how far are the two trains from each other at 12:30 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 9:15 pm and 12:30 am = 3.25
Let t2 = time between 11:30 pm and 12:30 am = 1

dtotal = 60t1 + 90t2
dtotal = 60 × 3.25 + 90 × 1
dtotal = 195 + 90
dtotal = 285

### Complexity=8, Mode=oppDirTime

Solve. If asked for time, a proper answer looks like this: 1:35am
1A train leaves Venice for a nearby city at 1:04 am, averaging 45 mph.
Another train leaves the nearby city for Venice at 3:30 am, averaging 73 mph.
If the nearby city is 817.5 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 = 2.43333333333 + t2

d = r1(2.43333333333 + t2) + r2t2
817.5 = 45(2.43333333333 + t2) + 73t2
817.5 = 109.5 + 45t2 + 73t2
708 = 118t2
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 3:30 am yields 9:30 am as the time.
2A train leaves Prague for a nearby city at 12:51 am, averaging 94 mph.
Another train leaves the nearby city for Prague at 4:15 am, averaging 106 mph.
If the nearby city is 819.6 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 = 3.4 + t2

d = r1(3.4 + t2) + r2t2
819.6 = 94(3.4 + t2) + 106t2
819.6 = 319.6 + 94t2 + 106t2
500 = 200t2
t2 = 2.5

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.
2.5 hours can be converted into hours and minutes. It is 2 hours and 0.5× 60 = 30 min.
Adding this amount of time to 4:15 am yields 6:45 am as the time.

### Complexity=9, Mode=mix

Solve. If asked for time, a proper answer looks like this: 1:35am
1A train leaves Minneapolis at 6:20 pm, averaging 42 mph.
Another train headed in the same direction leaves Minneapolis at 9:30 pm, averaging 52 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 = 3.16666666667 + t2

r1(3.16666666667 + t2) = r2t2
42(3.16666666667 + t2) = 52t2
133 + 42 t2 = 52t2
133 = 10t2
t2 = 13.3

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 3.16666666667 hours × 42 mph = 133 miles.

The second train travels at a relative rate of 10mph faster than the first train and it starts 133 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 = 133 ÷ 10 = 13.3
2A train leaves Denver for a nearby city at 2:16 am, averaging 89 mph.
Another train leaves the nearby city for Denver at 4:30 am, averaging 108 mph.
If the nearby city is 620 miles from Denver, 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 = 2.23333333333 + t2

d = r1(2.23333333333 + t2) + r2t2
620 = 89(2.23333333333 + t2) + 108t2
620 = 198.766666667 + 89t2 + 108t2
421.233333333 = 197t2
t2 = 2.13824027073

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.
2.13824027073 hours can be converted into hours and minutes. It is 2 hours and 0.138240270728× 60 = 8 min.
Adding this amount of time to 4:30 am yields 6:38 am as the time.

### Complexity=10, Mode=mix

Solve. If asked for time, a proper answer looks like this: 1:35am
1A train leaves Venice at 1:41 pm, averaging 35 mph.
Another train headed in the same direction leaves Venice at 2:00 pm, averaging 65 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 = 0.316666666667 + t2

r1(0.316666666667 + t2) = r2t2
35(0.316666666667 + t2) = 65t2
11.0833333333 + 35 t2 = 65t2
11.0833333333 = 30t2
t2 = 0.4

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 0.316666666667 hours × 35 mph = 11.0833333333 miles.

The second train travels at a relative rate of 30mph faster than the first train and it starts 11.0833333333 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 = 11.0833333333 ÷ 30 = 0.4
2A train leaves Denver at 1:28 pm, averaging 31 mph.
Another train headed in the same direction leaves Denver at 4:00 pm, averaging 53 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 = 2.53333333333 + t2

r1(2.53333333333 + t2) = r2t2
31(2.53333333333 + t2) = 53t2
78.5333333333 + 31t2 = 53t2
78.5333333333 = 22t2
t2 = 3.5696969697

Now we must use that to determine the time the second train overtakes the first.
3.5696969697 hrs can be converted to hours and minutes. It should be 3 hrs and 0.569696969697 × 60 = 34 min.
Adding the time passed to 4:00 pm we get 7:34 pm

Alternate Solution:
Distance = Rate × Time
Therefore, the total head start that the first train has is 2.53333333333 hours× 31 mph = 78.5333333333 miles.

The second train travels at a relative rate of 22mph faster than the first train and it starts 78.5333333333 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 = 78.5333333333 ÷ 22 = 3.5696969697

Now we must use that to determine the time the second train overtakes the first.
3.5696969697 hrs can be converted to hours and minutes. It should be 3 hrs and 0.569696969697 × 60 = 34 min.
Adding the time passed to 4:00 pm we get 7:34 pm

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