The ratio of the radii of two circles is 5:3. Find the ratio of their circumferences, 

Answer

To find the ratio of the circumferences of two circles given the ratio of their radii, follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Ratio of Radii**:
We are given that the ratio of the radii of two circles is 5:3. This means:
\[ \text{Radius of Circle 1} : \text{Radius of Circle 2} = 5 : 3 \]

2. **Assign Variables to the Radii**:
Let the radius of the first circle be \( R \) and the radius of the second circle be \( r \). From the ratio, we can express this as:
\[ \frac{R}{r} = \frac{5}{3} \]

3. **Use the Formula for Circumference**:
The formula for the circumference \( C \) of a circle is given by:
\[ C = 2\pi \times \text{radius} \]
Therefore, the circumferences of the two circles can be expressed as:
- Circumference of Circle 1: \( C_1 = 2\pi R \)
- Circumference of Circle 2: \( C_2 = 2\pi r \)

4. **Set Up the Ratio of the Circumferences**:
Now, we need to find the ratio of the circumferences:
\[ \frac{C_1}{C_2} = \frac{2\pi R}{2\pi r} \]

5. **Simplify the Ratio**:
The \( 2\pi \) in the numerator and denominator cancels out:
\[ \frac{C_1}{C_2} = \frac{R}{r} \]

6. **Substitute the Ratio of Radii**:
From step 2, we know that \( \frac{R}{r} = \frac{5}{3} \). Therefore:
\[ \frac{C_1}{C_2} = \frac{5}{3} \]

7. **Conclusion**:
The ratio of the circumferences of the two circles is:
\[ \text{Circumference of Circle 1} : \text{Circumference of Circle 2} = 5 : 3 \]