Each side of a cube is decreased by 25%. Find the ratio of the volumes of the original cube and the resulting cube.

Options

A. `8:1`
B. `27:64`
C. `64:1`
D. `64:27`

Answer

To find the ratio of the volumes of the original cube and the resulting cube after each side is decreased by 25%, we can follow these steps:

### Step 1: Understand the decrease in side length
The side length of the cube is decreased by 25%. This means that the new side length will be 75% of the original side length.

### Step 2: Express the decrease in fraction
25% can be expressed as a fraction:
\[ 25\% = \frac{25}{100} = \frac{1}{4} \]
Thus, if the original side length is \( s \), the new side length after a 25% decrease is:
\[ s' = s - \frac{1}{4}s = \frac{3}{4}s \]

### Step 3: Calculate the volume of the original cube
The volume \( V \) of a cube is given by the formula:
\[ V = s^3 \]
So, the volume of the original cube is:
\[ V_{\text{original}} = s^3 \]

### Step 4: Calculate the volume of the resulting cube
Using the new side length \( s' = \frac{3}{4}s \), the volume of the resulting cube is:
\[ V_{\text{resulting}} = (s')^3 = \left(\frac{3}{4}s\right)^3 = \frac{27}{64}s^3 \]

### Step 5: Find the ratio of the volumes
Now, we can find the ratio of the volumes of the original cube to the resulting cube:
\[ \text{Ratio} = \frac{V_{\text{original}}}{V_{\text{resulting}}} = \frac{s^3}{\frac{27}{64}s^3} = \frac{64}{27} \]

### Final Answer
Thus, the ratio of the volumes of the original cube to the resulting cube is:
\[ \frac{64}{27} \]

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