Train Problems Sample Problems

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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?

Answers

Complexity=1, Mode=sameDirHrsPassed

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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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, d_total = 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 d_total = 40t₁ + 60t₂ d_total = 40 × 4 + 60 × 2 d_total = 160 + 120 d_total = 280

#	Problem	Correct Answer	Your Answer
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?
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, d_total = 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 d_total = 70t₁ + 80t₂ d_total = 70 × 7 + 80 × 3 d_total = 490 + 240 d_total = 730

Complexity=4, Mode=oppDirTime

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

#	Problem	Correct Answer	Your Answer
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?
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.

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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, d_total = 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 d_total = 50t₁ + 80t₂ d_total = 50 × 3.5 + 80 × 1 d_total = 175 + 80 d_total = 255

#	Problem	Correct Answer	Your Answer
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?
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, d_total = 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 d_total = 60t₁ + 90t₂ d_total = 60 × 3.25 + 90 × 1 d_total = 195 + 90 d_total = 285

Complexity=8, Mode=oppDirTime

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

#	Problem	Correct Answer	Your Answer
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?
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.

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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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