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 .
