Here's how to solve this classic riddle:

**1. Set up the equations:**

* Let 'b' be the cost of the bat.
* Let 'x' be the cost of the ball.

We know two things:
*  b + x = 1.10 (The total cost)
*  b = x + 1.00 (The bat costs $1.00 more than the ball)

**2. Substitute and solve for x:**

Since we know what 'b' equals from the second equation, we can substitute it into the first equation:

(x + 1.00) + x = 1.10

Now, combine the 'x' terms:

2x + 1.00 = 1.10

Subtract 1.00 from both sides:

2x = 0.10

Divide both sides by 2:

x = 0.05

**Answer:**

The ball costs **0.05** (or 5 cents).
