The rate of the express train is 15 mph faster than the freight train.
Let's grasp onto this piece of information.┬а
The rate at which the freight train goes will be labeled "x" (keep in mind rate is just the speed)
If the freight train has a speed of "x" then the express train which is going 15mph faster than the freight train can be represented as "x + 15mph."
Our result is in HOURS, but our speed is in Miles per Hour (Miles/Hour). Meaning we need to multiply our speeds by some time (t).
Let's set up two equations:┬а
Freight Train: 18 hours * x mph = distance
Express Train: 15 hours * (x mph + 15mph) = distance
They are both the same distance so we can set the two equations equal to each other like this:┬а
18 hours * x mph = 15 hours * (x mph + 15mph)
That right side is a bit of a thorn in our side, so let's simplify it by multiplying all the terms in the parenthesis by 15 hours (keep in mind the hours in the units cancel out).┬а
18 hours * x mph = 15x miles + 15 * 15 miles
18 hours * x mph = 15x miles + 225 miles
Let's simplify the left side ([tex] \frac{miles}{hour}*hour = miles [/tex])
18x miles = 15x miles + 225 miles
Get all the "x" terms on one side by subtracting 15x on both sides.┬а
18x - 15x = 225 miles
Simplify
3x = 225 miles
Divide by 3 on both sides to solve for x.┬а
225/3┬а
225 = 25*9
(25*9)/3 = 25*3 = 75mph
So "x" which we chose was the rate of the freight train is 75mph.┬а
Adding 15mph to the freight train rate gives you the express trains rate which is 90mph.┬а
Hope that helped.┬а
