1. ## Speed Problem

Train Speed.
A freight train requires 2.5 hours longer to make a 300-mile journey than an express train does. If the express averages 20 mph faster than the freight train, how long does it take the express train to make the trip?

This problem is different than any of the other speed/distance problems in my textbook and I need to know how to set the equation up. I've been scratching my head a while.

2. $\displaystyle Speed = \frac {distance}{time}$

Let's call the speed y and the time x.

The time for the freight train is x + 2.5
The speed for the express train is y + 20

$\displaystyle y = \frac {300}{x + 2.5}$

$\displaystyle y + 20 = \frac {300}{x}$

Let's multiply both of these equations by the denominators on the right hand side.

$\displaystyle y(x + 2.5) = 300$

$\displaystyle x(y + 20) = 300$

So, as both the equations are equal, both sides are equal.

$\displaystyle xy + 2.5y = xy + 20x$

We can subtract xy from both sides.

$\displaystyle 2.5y = 20x$

$\displaystyle y = 8x$

We can use 8x to express y from now on. Going back to the equation for the express train:

$\displaystyle 8x + 20 = \frac {300}{x}$

$\displaystyle x(8x + 20) = 300$

$\displaystyle 8x^2 + 20x = 300$

$\displaystyle 8x^2 + 20x - 300 = 0$

We now have a quadratic equation. I'll divide both sides by 4 to make it easier:

$\displaystyle 2x^2 + 5x - 75 = 0$

Time to factorise it:

$\displaystyle (2x + 15)(x - 5) = 0$

Now in real life we can't have a negative time, so x = 5

The original units were hours, and we have not converted these so it took the express train 5 hours to travel 300 miles.

3. Thank you so much Bruce!