A solid 5x5x5 cube is composed of unit cubes. Each face of the large, solid cube is partially painted with gray paint, as shown.
What fraction of the entire solid cube's unit cubes have no paint on them? Express your answer as a common fraction.
I think the easiest way would be to count the cubes that do have paint on them and then subtract.
Each of the 8 vertices has paint on three sides of one cube* = 8 cubes
Each of the 6 faces has paint on one side of eight "center" cubes = 48 cubes
Total cubes with paint are 56 subtracted from 125 cubes in the block = 69 cubes with no paint
The fraction of cubes with no paint is 69 / 125
* Although each corner cube has paint on three sides, it's still just one cube.
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I think the easiest way would be to count the cubes that do have paint on them and then subtract.
Each of the 8 vertices has paint on three sides of one cube* = 8 cubes
Each of the 6 faces has paint on one side of eight "center" cubes = 48 cubes
Total cubes with paint are 56 subtracted from 125 cubes in the block = 69 cubes with no paint
The fraction of cubes with no paint is 69 / 125
* Although each corner cube has paint on three sides, it's still just one cube.
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