Math Practice Problems - Train Problems

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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=2, Mode=sameDirTime

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

Complexity=3, Mode=oppDirDist

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

Complexity=4, Mode=oppDirTime

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

Complexity=5, Mode=sameDirHrsPassed

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

Complexity=6, Mode=sameDirTime

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

Complexity=7, Mode=oppDirDist

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

Complexity=8, Mode=oppDirTime

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

Complexity=9, Mode=mix

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

Complexity=10, Mode=mix

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

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 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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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 11:00 am and 3:00 pm = 4 Let t₂ = time between 1:00 pm and 3:00 pm = 2 d_total = 60t₁ + 90t₂ d_total = 60 × 4 + 90 × 2 d_total = 240 + 180 d_total = 420

#	Problem	Correct Answer	Your Answer
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?
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 12:00 am and 3:00 am = 3 Let t₂ = time between 1:00 am and 3:00 am = 2 d_total = 60t₁ + 70t₂ d_total = 60 × 3 + 70 × 2 d_total = 180 + 140 d_total = 320

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 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.

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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 7:15 am and 10:30 am = 3.25 Let t₂ = time between 8:30 am and 10:30 am = 2 d_total = 95t₁ + 115t₂ d_total = 95 × 3.25 + 115 × 2 d_total = 308.75 + 230 d_total = 538.75 d_total = 539 (rounded)

#	Problem	Correct Answer	Your Answer
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?
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 9:45 am and 1:15 pm = 3.5 Let t₂ = time between 12:15 pm and 1:15 pm = 1 d_total = 85t₁ + 95t₂ d_total = 85 × 3.5 + 95 × 1 d_total = 297.5 + 95 d_total = 392.5 d_total = 393 (rounded)

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 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.

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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

#	Problem	Correct Answer	Your Answer
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?
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 3:15 am and 5:45 am = 2.5 Let t₂ = time between 4:45 am and 5:45 am = 1 d_total = 30t₁ + 41t₂ d_total = 30 × 2.5 + 41 × 1 d_total = 75 + 41 d_total = 116

#	Problem	Correct Answer	Your Answer
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?
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 8:15 am and 2:45 pm = 6.5 Let t₂ = time between 12:45 pm and 2:45 pm = 2 d_total = 53t₁ + 71t₂ d_total = 53 × 6.5 + 71 × 2 d_total = 344.5 + 142 d_total = 486.5 d_total = 487 (rounded)

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