If the length of an edge of cube A is one-third the length of an edge of cube B, what is the ratio of the volume of cube A to the volume of cube B?

Answer

To find the ratio of the volume of cube A to the volume of cube B, we can follow these steps:

### Step 1: Define the edge lengths of the cubes
Let the length of an edge of cube B be \( X \). Since the length of an edge of cube A is one-third the length of an edge of cube B, we can express the edge length of cube A as:
\[ \text{Edge length of cube A} = \frac{1}{3}X \]

### Step 2: Calculate the volume of cube A
The volume \( V_A \) of a cube is given by the formula:
\[ V = \text{(edge length)}^3 \]
Thus, the volume of cube A can be calculated as:
\[ V_A = \left(\frac{1}{3}X\right)^3 = \frac{1}{27}X^3 \]

### Step 3: Calculate the volume of cube B
Using the same formula for the volume of cube B, we have:
\[ V_B = X^3 \]

### Step 4: Find the ratio of the volumes
Now, we can find the ratio of the volume of cube A to the volume of cube B:
\[ \text{Ratio} = \frac{V_A}{V_B} = \frac{\frac{1}{27}X^3}{X^3} \]
When we simplify this expression, we get:
\[ \text{Ratio} = \frac{1}{27} \]

### Step 5: Express the ratio in simplest form
The ratio of the volume of cube A to the volume of cube B can be expressed as:
\[ 1 : 27 \]

### Final Answer:
The ratio of the volume of cube A to the volume of cube B is \( 1 : 27 \).

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