The ratio between the diameters of two cirlces is 3:5. Find the ratio between their : (i) radii (ii) circumferences (iii) areas.

Answer

To solve the problem step by step, we will find the ratio between the radii, circumferences, and areas of two circles given the ratio of their diameters.

### Given:
The ratio of the diameters of two circles is \(3:5\).

### Step 1: Find the ratio of the radii
1. **Understanding the relationship between diameter and radius**:
- The diameter \(d\) of a circle is twice the radius \(r\). Thus, \(d = 2r\).

2. **Express the diameters in terms of radii**:
- Let the diameters of the two circles be \(d_1\) and \(d_2\).
- Given \(d_1:d_2 = 3:5\), we can express this as:
\[ \frac{d_1}{d_2} = \frac{3}{5} \]
- Substituting the relationship \(d = 2r\):
\[ \frac{2r_1}{2r_2} = \frac{3}{5} \]
- The \(2\) cancels out:
\[ \frac{r_1}{r_2} = \frac{3}{5} \]

**Ratio of radii**: \(3:5\)

### Step 2: Find the ratio of the circumferences
1. **Understanding the formula for circumference**:
- The circumference \(C\) of a circle is given by \(C = 2\pi r\).

2. **Express the circumferences in terms of radii**:
- Let the circumferences of the two circles be \(C_1\) and \(C_2\).
- Then:
\[ C_1 = 2\pi r_1 \quad \text{and} \quad C_2 = 2\pi r_2 \]
- The ratio of the circumferences is:
\[ \frac{C_1}{C_2} = \frac{2\pi r_1}{2\pi r_2} \]
- The \(2\pi\) cancels out:
\[ \frac{C_1}{C_2} = \frac{r_1}{r_2} \]
- From the previous step, we know:
\[ \frac{r_1}{r_2} = \frac{3}{5} \]

**Ratio of circumferences**: \(3:5\)

### Step 3: Find the ratio of the areas
1. **Understanding the formula for area**:
- The area \(A\) of a circle is given by \(A = \pi r^2\).

2. **Express the areas in terms of radii**:
- Let the areas of the two circles be \(A_1\) and \(A_2\).
- Then:
\[ A_1 = \pi r_1^2 \quad \text{and} \quad A_2 = \pi r_2^2 \]
- The ratio of the areas is:
\[ \frac{A_1}{A_2} = \frac{\pi r_1^2}{\pi r_2^2} \]
- The \(\pi\) cancels out:
\[ \frac{A_1}{A_2} = \frac{r_1^2}{r_2^2} \]
- From the previous step, we know:
\[ \frac{r_1}{r_2} = \frac{3}{5} \]
- Squaring both sides gives:
\[ \frac{r_1^2}{r_2^2} = \left(\frac{3}{5}\right)^2 = \frac{9}{25} \]

**Ratio of areas**: \(9:25\)

### Final Answers:
- Ratio of radii: \(3:5\)
- Ratio of circumferences: \(3:5\)
- Ratio of areas: \(9:25\)