Math Practice Problems - Train Problems

Train Problems - Sample Math Practice Problems

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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 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?

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 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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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 5:00 pm and 1:00 am = 8 Let t₂ = time between 9:00 pm and 1:00 am = 4 d_total = 50t₁ + 80t₂ d_total = 50 × 8 + 80 × 4 d_total = 400 + 320 d_total = 720

#	Problem	Correct Answer	Your Answer
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?
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 1:00 pm and 6:00 pm = 5 Let t₂ = time between 4:00 pm and 6:00 pm = 2 d_total = 60t₁ + 80t₂ d_total = 60 × 5 + 80 × 2 d_total = 300 + 160 d_total = 460

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 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.

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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 2:45 pm and 4:00 pm = 1.25 Let t₂ = time between 3:00 pm and 4:00 pm = 1 d_total = 50t₁ + 80t₂ d_total = 50 × 1.25 + 80 × 1 d_total = 62.5 + 80 d_total = 142.5 d_total = 143 (rounded)

#	Problem	Correct Answer	Your Answer
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?
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 8:30 am and 12:30 pm = 4 Let t₂ = time between 11:30 am and 12:30 pm = 1 d_total = 70t₁ + 85t₂ d_total = 70 × 4 + 85 × 1 d_total = 280 + 85 d_total = 365

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 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.

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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.

#	Problem	Correct Answer	Your Answer
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?
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.

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