Charlie plans to sell bananas for forty cents and apples for fifty cents at his fruit stand, but Dave accidentally reverses the prices. After selling all their fruit they earn a dollar more than they would have with the original prices. How many more bananas than apples did they sell?

$\mathrm{(A) \ } 2 \qquad \mathrm{(B) \ } 4 \qquad \mathrm {(C) \ } 5 \qquad \mathrm{(D) \ } 10 \qquad \mathrm{(E) \ } 20$

1. Let \( x \) be the number of apples and \( y \) be the number of bananas they sold.
2. If they had sold the fruits at the original prices, they would have earned \( 0.4y + 0.5x \) dollars.
3. With the reversed prices, they earned \( 0.4x + 0.5y \) dollars.
4. According to the problem, the earnings with the reversed prices are one dollar more than the earnings with the original prices. Therefore, we have the equation:
   \[
   0.4x + 0.5y = 0.5x + 0.4y + 1
   \]
5. Simplify the equation by subtracting \( 0.4x \) and \( 0.4y \) from both sides:
   \[
   0.1y = 0.1x + 1
   \]
6. Divide both sides by 0.1 to isolate \( y \):
   \[
   y = x + 10
   \]
7. This equation tells us that the number of bananas \( y \) is 10 more than the number of apples \( x \). Therefore, the difference between the number of bananas and apples is:
   \[
   y - x = 10
   \]

The final answer is \( \boxed{10} \)