This is a problem that illustrates how solving equations using algebra can get a solution that may defy a false mathematical intuition.

Since the bat costs $1 more than the ball, and the two of them together cost $1.10, we would like to know the cost of the ball. We can set up an algebraic problem to solve this. Usually we let a variable represent what we are looking for.

Let x be the cost of the ball.

Then setup an equation that combines all the information given:

`x+(x+1)=1.10` since the ball plus the bat cost $1.10. Now simplify the left side.

`x+x+1=1.1` multiply by 10 to get rid of decimals

`10x+10x+10=11` collect like terms

`20x=11-10` simplify the right side

`20x=1` divide

`x=1/20` express as money

`x=0.05`

**The ball costs 5 cents.**

Let x be the cost of the ball.

Since the bat costs $1 more than the ball, in math form it is:

cost of the bat = x + 1

So the total cost of the bat and the ball is:

cost of the bat + cost of ball = 1 .10

`x + 1 + x = 1.10`

Combining like terms yields:

`2x+ 1 = 1.10`

Then, solve for x. To do so, subtract both sides by 1.

`2x+1-1=1.10-1`

`2x=0.10`

And divide both sides by 2.

`(2x)/2=0.10/2`

`x= 0.05`

**Hence, the ball costs $0.05 .**