SOLUTION: A passenger train can travel 325 mi in the same time a freight train takes to travel 200 mi. If the speed of the passenger train is 25 mi/h faster than the speed of the freight tr

Question 106209: A passenger train can travel 325 mi in the same time a freight train takes to travel 200 mi. If the speed of the passenger train is 25 mi/h faster than the speed of the freight train, find the speed of each.
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
To solve this problem, use the distance formula:
.
D = R*T
.
where D represents the distance traveled, R represents the rate or speed, and T represents
the amount of Time that the travel takes place.
.
Before you begin recognize that if the speed of the freight train is F mph, then the speed of
the passenger train is F + 25 mph.
.
Now we can write two equations. The distance that the passenger train travels is 325 miles.
The speed at which it travels is F + 25. And the time it travels is T. So its equation is:
.
325 = (F+25)*T
.
And the distance that the freight train travels is 200 miles. Its speed is F. And the time
it travels is also T. So its distance equation is:
.
200 = F*T
.
Suppose we solved both these equations for T. In the first equation we would divide both
sides by (F+25) and we would get that:
.

.
In the second equation we would solve for T by dividing both sides by F to get:
.

.
But the problem tells us that the two times ... the time the passenger train travels
is the same amount of time as the freight train travels. Therefore the two left sides (which
are both T) are equal. Therefore, we know that the two right sides must be equal also. So
we can set the two right sides equal and this equation becomes:
.

.
Let's multiply both sides by
.

.
Cancel the terms in the numerators with the same term in the denominator:
.

.
and the equation reduces to:
.

.
Multiply out the right side by multiplying 200 times each of the terms in the parentheses
to get:
.

.
Get rid of the 200F on the right side by subtracting 200F from both sides to reduce the
equation to:
.

.
Solve for F by dividing both sides by 125 and you have:
.

.
This tells you that the speed of the freight train is 40 mph. Since the speed of the passenger
train is 25 mph faster, its speed is 40 + 25 = 65 mph.
.
You can check these answers by noting that at 40 mph it will take the freight train 5
hours to go 200 miles. And at 65 mph in 5 hours the passenger train goes 5 * 65 = 325 miles.
Everything checks.
.
Hope this helps you to understand this problem and shows you a way of working through it
to get the answer.
.
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