If the circumfernece of two circles are in the ratio `4 : 5`, What is the ratio of their radii?

Options

A. `5:4`
B. `2:6`
C. `4:5`
D. `2:3`

Answer

To solve the problem, we need to find the ratio of the radii of two circles given that their circumferences are in the ratio of 4:5.

### Step-by-Step Solution:

1. **Understand the relationship between circumference and radius**:
The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle.

2. **Set up the ratio of the circumferences**:
Let the radius of the first circle be \( R_1 \) and the radius of the second circle be \( R_2 \). According to the problem, the circumferences of the two circles are in the ratio of 4:5. Therefore, we can write:
\[ \frac{C_1}{C_2} = \frac{4}{5} \]

3. **Substituting the circumference formula**:
Substitute the formula for circumference into the ratio:
\[ \frac{2\pi R_1}{2\pi R_2} = \frac{4}{5} \]

4. **Cancel out common terms**:
The \( 2\pi \) in the numerator and denominator cancels out:
\[ \frac{R_1}{R_2} = \frac{4}{5} \]

5. **Express the ratio of the radii**:
From the above equation, we can express the ratio of the radii as:
\[ R_1 : R_2 = 4 : 5 \]

### Final Answer:
The ratio of the radii of the two circles is \( 4 : 5 \).

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