Question : Each side of a cube is decreased by 25%. Find the ratio of the volumes of the original cube and the resulting cube.
Option 1: 8 : 1
Option 2: 27 : 64
Option 3: 64 : 1
Option 4: 64 : 27
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Correct Answer: 64 : 27
Solution :
If each side of a cube is decreased by 25%, the new side length = $a' = a - 0.25a = 0.75a$
The volume of the new cube $=V' = (a')^3 = (0.75a)^3$
The ratio of the volumes of the original cube and the resulting cube,
⇒ $\frac{V}{V'} = \frac{a^3}{(0.75a)^3} = \frac{1}{0.75^3} = \frac{100×100×100}{75×75×75} = \frac{64}{27}$
Hence, the correct answer is 64 : 27.
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