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The problem describes a cake with dimensions of 10 inches by 10 inches (since it's a square), frosted on the top and the four sides, and there are five people to share it. The goal is to cut the cake so that each person gets both the same volume of cake and an equal amount of frosting.

### Solution Breakdown:

1. **Volume Distribution**: 
   Since the cake is a square with a 10-inch side and the same height for every part of the cake, dividing the volume equally is simpler. The total cake volume can be divided into 5 equal parts, as each person should receive an equal amount.

2. **Frosting Distribution**:
   The cake is frosted on the top (10x10 = 100 square inches) and the four sides. If the height of the cake is \( h \), each side of the cake will have a frosted area of \( 10 \times h \) square inches, making the total area of the frosting on the sides \( 4 \times 10 \times h \).

The challenge is to divide both the top and the sides equally among the five people.

### How to Cut:
To ensure that both the volume and frosting are equally divided:
- **Cylindrical/Vertical Slices**: Cut the cake into vertical slices like a pizza, dividing it into 5 equal wedge-shaped pieces. These slices would extend from the center of the cake out to the edges.
- Each of these slices would have equal areas of frosting from both the top and the sides because the cake's top and sides will be proportionally distributed across all the slices. This ensures that each person gets 1/5th of the total surface area (frosting) and 1/5th of the cake's volume.

This solution guarantees that everyone gets an equal volume and an equal amount of frosting.

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Do you have any questions or need more details?

### Related Questions:
1. What is the total surface area of frosting on the cake?
2. How would the cutting method change if there were 6 people instead of 5?
3. How does the height of the cake affect the frosting distribution on the sides?
4. What if the cake was a rectangle instead of a square, how would you distribute the frosting equally?
5. Is there a way to divide the cake into more complex geometric shapes that still ensure equal volume and frosting?

### Tip:
When cutting a cake to share evenly, radial or wedge cuts from the center are an excellent method to ensure equal distribution of both volume and frosting.

How would you cut this cake so that each and every one of the five people at the party would get a piece that had not only the same volume, but the same amount of frosting?

Discover the mathematical approach to cutting a square cake so that 5 people each get the same volume and an equal amount of frosting. The solution involves geometry, focusing on volume distribution and surface area calculations. Ideal for students in Grades 5-7.

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