If Xiluva puts two oranges in each basket, four oranges are in excess. If she puts five oranges in each basket, one basket is in excess. How many oranges and baskets has Xiluva?
1. Let \( o \) represent the number of oranges and \( b \) represent the number of baskets.
2. According to the problem, if Xiluva puts two oranges in each basket, four oranges are in excess. This can be expressed as:
   \[
   2b + 4 = o
   \]
3. Additionally, if she puts five oranges in each basket, one basket is in excess. This can be expressed as:
   \[
   5(b - 1) = o
   \]
4. We now have a system of two equations:
   \[
   \begin{cases}
   2b + 4 = o \\
   5(b - 1) = o
   \end{cases}
   \]
5. To solve this system, we can set the two expressions for \( o \) equal to each other:
   \[
   2b + 4 = 5(b - 1)
   \]
6. Simplify and solve for \( b \):
   \[
   2b + 4 = 5b - 5
   \]
   \[
   4 + 5 = 5b - 2b
   \]
   \[
   9 = 3b
   \]
   \[
   b = 3
   \]
7. Substitute \( b = 3 \) back into one of the original equations to find \( o \). Using \( 2b + 4 = o \):
   \[
   2(3) + 4 = o
   \]
   \[
   6 + 4 = o
   \]
   \[
   o = 10
   \]

The final answer is \( \boxed{b = 3} \) and \( \boxed{o = 10} \).