Train Problem
This topic applies Distance = Rate × Time. Review the basics of Distance, Rate, and Time here.
The general steps to follow for solving these problems are
Example 1: Trains travelling in the same direction
Solve. If asked for time, a proper answer looks like this: 1:35am A train leaves Prague at 6:45 pm, averaging 30 mph. Another train headed in the same direction leaves Prague at 7:45 pm, averaging 60 mph. To the nearest minute, at what time will the second train overtake the first train? Step 1: Determine what the problem is asking Q: What are you looking for? What is the problem asking? A: When the second train will overtake the first train, to the nearest minute Step 2: Assign variable(s) Since we are looking for an answer in terms of time, let us assign t1 = time travelled by the first train t2 = time travelled by the second train Q: How does t1 relate to t2? A: The first train started 1 hour before the second train, so the time travelled by the first train when it is taken over by the second train is t1 = t2 + 1 hr Step 3: Construct the equation We know that when the second train overtakes the first train, both trains have travelled the same distance. Since we know that distance = rate × time (or d = rt), we can construct the right equation. Distance travelled by the first train = Distance travelled by the second train d1 = d2 r1t1 = r2t2 from the equation d = rt (30 mph) t1 = (60 mph) t2 rates are stated in the problem (30 mph) (t2 + 1hr) = (60 mph) t2 from step 2 Step 4: Solve the equation (30 mph) (t2 + 1 hr) = (60 mph) t2 (30 mph )(t2 + 1 hr) = (60 mph ) t2 units cancel t2 + 1 hr = 2 t2 divide by 30 t2 + 1 hr - t2 = 2 t2 - t2 1 hr = t2 Step 5: Answer the problem The problem asks for the specific time when the second train overtakes the first train. From step 4, we have calculated that the second train travelled 1 hour before overtaking the first train. And from the problem, we know that the second train left the station at 7:45pm. After an hour after travelling, the time would be 8:45pm. Therefore, the second train overtakes the first train at 8:45pm. The answer to the problem is Example 2: Trains travelling in the opposite direction Solve. If asked for time, a proper answer looks like this: 1:35am A train leaves Las Vegas at 5:30 am, averaging 80 mph. Another train headed in the opposite direction leaves Las Vegas at 7:30 am, averaging 105 mph. To the nearest mile, how far are the two trains from each other at 11:30 am? Step 1: Determine what the problem is asking Q: What are you looking for? What is the problem asking? A: The total distance travelled by the two trains by 11:30am, to the nearest mile Step 2: Assign variable(s) Since we are looking for an answer in terms of distance, let us assign dtotal = total distance travelled by both trains d1 = total distance travelled by the first train d2 = total distance travelled by the second train Step 3: Construct the equation Since the two trains are travelling in opposite directions, their total distance apart is the sum of the distances they each travelled. dtotal = d1 + d2 dtotal = r1t1 + r2t2 from the equation d = rt Step 4: Solve the equation dtotal = r1t1 + r2t2 = (80 mph)t1 + (105 mph)t2 = (80 mph)(11:30am - 5:30am) + (105 mph)(11:30am - 7:30am) = (80 mph)(6 hr) + (105 mph)(4 hr) = 480 mi + 420 mi units cancel: (miles/hr)(hr) = mi = 900 mi Step 5: Answer the problem The problem asks for the total distance travelled by the two trains when it is 11:30am. Our equation solves for the total distance so the answer to the problem is .
Step 1: Determine what the problem is asking
Q: What are you looking for? What is the problem asking? A: When the second train will overtake the first train, to the nearest minute
Since we are looking for an answer in terms of time, let us assign t1 = time travelled by the first train t2 = time travelled by the second train Q: How does t1 relate to t2? A: The first train started 1 hour before the second train, so the time travelled by the first train when it is taken over by the second train is t1 = t2 + 1 hr
Q: How does t1 relate to t2? A: The first train started 1 hour before the second train, so the time travelled by the first train when it is taken over by the second train is t1 = t2 + 1 hr
We know that when the second train overtakes the first train, both trains have travelled the same distance. Since we know that distance = rate × time (or d = rt), we can construct the right equation. Distance travelled by the first train = Distance travelled by the second train d1 = d2 r1t1 = r2t2 from the equation d = rt (30 mph) t1 = (60 mph) t2 rates are stated in the problem (30 mph) (t2 + 1hr) = (60 mph) t2 from step 2
(30 mph) (t2 + 1 hr) = (60 mph) t2 (30 mph )(t2 + 1 hr) = (60 mph ) t2 units cancel t2 + 1 hr = 2 t2 divide by 30 t2 + 1 hr - t2 = 2 t2 - t2 1 hr = t2
The problem asks for the specific time when the second train overtakes the first train. From step 4, we have calculated that the second train travelled 1 hour before overtaking the first train. And from the problem, we know that the second train left the station at 7:45pm. After an hour after travelling, the time would be 8:45pm. Therefore, the second train overtakes the first train at 8:45pm. The answer to the problem is
From step 4, we have calculated that the second train travelled 1 hour before overtaking the first train. And from the problem, we know that the second train left the station at 7:45pm. After an hour after travelling, the time would be 8:45pm.
Therefore, the second train overtakes the first train at 8:45pm. The answer to the problem is
Example 2: Trains travelling in the opposite direction
Solve. If asked for time, a proper answer looks like this: 1:35am A train leaves Las Vegas at 5:30 am, averaging 80 mph. Another train headed in the opposite direction leaves Las Vegas at 7:30 am, averaging 105 mph. To the nearest mile, how far are the two trains from each other at 11:30 am? Step 1: Determine what the problem is asking Q: What are you looking for? What is the problem asking? A: The total distance travelled by the two trains by 11:30am, to the nearest mile Step 2: Assign variable(s) Since we are looking for an answer in terms of distance, let us assign dtotal = total distance travelled by both trains d1 = total distance travelled by the first train d2 = total distance travelled by the second train Step 3: Construct the equation Since the two trains are travelling in opposite directions, their total distance apart is the sum of the distances they each travelled. dtotal = d1 + d2 dtotal = r1t1 + r2t2 from the equation d = rt Step 4: Solve the equation dtotal = r1t1 + r2t2 = (80 mph)t1 + (105 mph)t2 = (80 mph)(11:30am - 5:30am) + (105 mph)(11:30am - 7:30am) = (80 mph)(6 hr) + (105 mph)(4 hr) = 480 mi + 420 mi units cancel: (miles/hr)(hr) = mi = 900 mi Step 5: Answer the problem The problem asks for the total distance travelled by the two trains when it is 11:30am. Our equation solves for the total distance so the answer to the problem is .
Q: What are you looking for? What is the problem asking? A: The total distance travelled by the two trains by 11:30am, to the nearest mile
Since we are looking for an answer in terms of distance, let us assign dtotal = total distance travelled by both trains d1 = total distance travelled by the first train d2 = total distance travelled by the second train
Since the two trains are travelling in opposite directions, their total distance apart is the sum of the distances they each travelled. dtotal = d1 + d2 dtotal = r1t1 + r2t2 from the equation d = rt
dtotal = r1t1 + r2t2 = (80 mph)t1 + (105 mph)t2 = (80 mph)(11:30am - 5:30am) + (105 mph)(11:30am - 7:30am) = (80 mph)(6 hr) + (105 mph)(4 hr) = 480 mi + 420 mi units cancel: (miles/hr)(hr) = mi = 900 mi
The problem asks for the total distance travelled by the two trains when it is 11:30am. Our equation solves for the total distance so the answer to the problem is .
