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
t_{1} = time travelled by the first train
t_{2} = time travelled by the second train
Q: How does t_{1} relate to t_{2}?
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
t_{1} = t_{2} + 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 
d_{1} 
= 
d_{2} 

r_{1}t_{1} 
= 
r_{2}t_{2} 
from the equation d = rt 
(30 mph) t_{1} 
= 
(60 mph) t_{2} 
rates are stated in the problem 
(30 mph) (t_{2} + 1hr) 
= 
(60 mph) t_{2} 
from step 2 
Step 4: Solve the equation
(30 mph) (t_{2} + 1 hr) 
= 
(60 mph) t_{2} 

(30 mph )(t_{2} + 1 hr) 
= 
(60 mph ) t_{2} 
units cancel 
t_{2} + 1 hr 
= 
2 t_{2} 
divide by 30 
t_{2} + 1 hr  t_{2} 
= 
2 t_{2}  t_{2} 

1 hr 
= 
t_{2} 

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
d_{total} = total distance travelled by both trains
d_{1} = total distance travelled by the first train
d_{2} = 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.
d_{total} 
= 
d_{1} + d_{2} 

d_{total} 
= 
r_{1}t_{1} + r_{2}t_{2} 
from the equation d = rt 
Step 4: Solve the equation
d_{total} 
= 
r_{1}t_{1} + r_{2}t_{2} 


= 
(80 mph)t_{1} + (105 mph)t_{2} 


= 
(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 .
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