The ratio of radii of two circles is 3:4. Find the ratio of their circumferences.

Options

A. 1 : 2
B. 4 : 3
C. 3 : 4
D. 2 : 1

Answer

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

### Step-by-Step Solution:

1. **Identify the Ratio of Radii**:
We are given that the ratio of the radii of the two circles is \( R_1 : R_2 = 3 : 4 \).

2. **Express the Radii**:
Let the radius of the first circle \( R_1 = 3x \) and the radius of the second circle \( R_2 = 4x \), where \( x \) is a common multiplier.

3. **Formula for Circumference**:
The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi R \]
Therefore, the circumferences of the two circles can be expressed as:
\[ C_1 = 2\pi R_1 = 2\pi (3x) = 6\pi x \]
\[ C_2 = 2\pi R_2 = 2\pi (4x) = 8\pi x \]

4. **Find the Ratio of Circumferences**:
Now, we can find the ratio of the circumferences \( C_1 : C_2 \):
\[ \frac{C_1}{C_2} = \frac{6\pi x}{8\pi x} \]
The \( \pi x \) terms cancel out:
\[ \frac{C_1}{C_2} = \frac{6}{8} = \frac{3}{4} \]

5. **Conclusion**:
Therefore, the ratio of the circumferences of the two circles is \( C_1 : C_2 = 3 : 4 \).