The ratio of the volumes of two spheres is 8 : 27. If r and R are the radii of sphere respectively, then find the (R - r) : r.

Answer

To solve the problem, we need to find the ratio \((R - r) : r\) given that the ratio of the volumes of two spheres is \(8 : 27\). Let's break it down step by step.

### Step 1: Understand the volume formula for spheres
The volume \(V\) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \(r\) is the radius of the sphere.

### Step 2: Set up the ratio of the volumes
Let the radius of the smaller sphere be \(r\) and the radius of the larger sphere be \(R\). According to the problem, the ratio of their volumes is:
\[ \frac{V_1}{V_2} = \frac{8}{27} \]
Substituting the volume formula, we have:
\[ \frac{\frac{4}{3} \pi r^3}{\frac{4}{3} \pi R^3} = \frac{8}{27} \]

### Step 3: Simplify the equation
The \(\frac{4}{3} \pi\) cancels out from both sides:
\[ \frac{r^3}{R^3} = \frac{8}{27} \]

### Step 4: Take the cube root of both sides
To solve for the ratio of the radii, we take the cube root:
\[ \frac{r}{R} = \frac{\sqrt[3]{8}}{\sqrt[3]{27}} = \frac{2}{3} \]

### Step 5: Express \(R\) in terms of \(r\)
From the ratio \(\frac{r}{R} = \frac{2}{3}\), we can express \(R\) as:
\[ R = \frac{3}{2} r \]

### Step 6: Find \(R - r\)
Now, we need to find \(R - r\):
\[ R - r = \frac{3}{2} r - r = \frac{3}{2} r - \frac{2}{2} r = \frac{1}{2} r \]

### Step 7: Find the ratio \((R - r) : r\)
Now we can express the ratio \((R - r) : r\):
\[ (R - r) : r = \frac{1}{2} r : r = \frac{1}{2} : 1 \]
This simplifies to:
\[ (R - r) : r = 1 : 2 \]

### Final Answer
Thus, the ratio \((R - r) : r\) is:
\[ \boxed{1 : 2} \]