The surface areas of two spheres are in the ratio `1 : 4`. Find the ratio of their volumes.

Explanation

To solve the problem of finding the ratio of the volumes of two spheres given that their surface areas are in the ratio of 1:4, we can follow these steps:

### Step 1: Understand the relationship between surface area and radius
The surface area \( S \) of a sphere is given by the formula:
\[ S = 4\pi r^2 \]
where \( r \) is the radius of the sphere.

### Step 2: Set up the ratio of the surface areas
Let the surface areas of the two spheres be \( S_1 \) and \( S_2 \). According to the problem, we have:
\[ \frac{S_1}{S_2} = \frac{1}{4} \]

### Step 3: Express the surface areas in terms of their radii
Let the radii of the two spheres be \( r_1 \) and \( r_2 \). Then we can express the surface areas as:
\[ S_1 = 4\pi r_1^2 \quad \text{and} \quad S_2 = 4\pi r_2^2 \]

### Step 4: Substitute the surface area expressions into the ratio
Substituting these into the ratio gives:
\[ \frac{4\pi r_1^2}{4\pi r_2^2} = \frac{1}{4} \]
The \( 4\pi \) cancels out, simplifying to:
\[ \frac{r_1^2}{r_2^2} = \frac{1}{4} \]

### Step 5: Take the square root to find the ratio of the radii
Taking the square root of both sides, we find:
\[ \frac{r_1}{r_2} = \frac{1}{2} \]

### Step 6: Use the ratio of the radii to find the ratio of the volumes
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3}\pi r^3 \]
Thus, the volumes of the two spheres can be expressed as:
\[ V_1 = \frac{4}{3}\pi r_1^3 \quad \text{and} \quad V_2 = \frac{4}{3}\pi r_2^3 \]

### Step 7: Set up the ratio of the volumes
Now we can set up the ratio of the volumes:
\[ \frac{V_1}{V_2} = \frac{\frac{4}{3}\pi r_1^3}{\frac{4}{3}\pi r_2^3} = \frac{r_1^3}{r_2^3} \]

### Step 8: Substitute the ratio of the radii into the volume ratio
Using the ratio of the radii we found earlier:
\[ \frac{r_1}{r_2} = \frac{1}{2} \]
We can cube this ratio:
\[ \frac{r_1^3}{r_2^3} = \left(\frac{1}{2}\right)^3 = \frac{1}{8} \]

### Step 9: Conclude the ratio of the volumes
Thus, the ratio of the volumes of the two spheres is:
\[ \frac{V_1}{V_2} = \frac{1}{8} \]

### Final Answer:
The ratio of their volumes is \( 1:8 \).