Let the average speed of train B be $x$ miles per hour. Since train A is 20 mph faster, its average speed is $x + 20$ miles per hour.

Both trains cover their respective distances in the same time, so we can use the formula:

$\text{Time} = \frac{\text{Distance}}{\text{Speed}}$

For train A, the time taken is:

$\frac{240}{x + 20}$

For train B, the time taken is:

$\frac{180}{x}$

Since the times are equal, we can set these two expressions equal to each other:

$\frac{240}{x + 20} = \frac{180}{x}$

Now, solve for $x$. Cross-multiply to get:

$240x = 180(x + 20)$

Simplify:

$240x = 180x + 3600$

Subtract $180x$ from both sides:

$60x = 3600$

Solve for $x$:

$x = \frac{3600}{60} = 60$

So, the average speed of train B is $\boxed{60}$ miles per hour.

Would you like more details or have any further questions?

Here are 5 related questions:

1. What would be the speed of train A if train B's speed is 60 mph?
2. How do you calculate the time taken by train B to cover 180 miles at 60 mph?
3. If train A's speed were 30 mph more than train B's, how would that affect the answer?
4. How can you set up similar problems with different speeds and distances?
5. How can this type of problem be solved using graphical methods?

Tip: Cross-multiplying is a useful method when dealing with equal ratios or fractions in problems like this.