The ratio of the circumference of two circles is `4:5.` Find the ratio of their radii.

Answer

To solve the problem, we need to find the ratio of the radii of two circles given the ratio of their circumferences. Here's the step-by-step solution:

### Step 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.

### Step 2: Set up the ratio of the circumferences
Let \( r_1 \) be the radius of the first circle and \( r_2 \) be the radius of the second circle. According to the problem, the ratio of the circumferences of the two circles is given as \( 4:5 \). Therefore, we can write:
\[ \frac{C_1}{C_2} = \frac{4}{5} \]
Substituting the formula for circumference, we have:
\[ \frac{2 \pi r_1}{2 \pi r_2} = \frac{4}{5} \]

### Step 3: Simplify the equation
The \( 2 \pi \) in the numerator and denominator cancels out:
\[ \frac{r_1}{r_2} = \frac{4}{5} \]

### Step 4: Write the ratio of the radii
From the simplified equation, we find that the ratio of the radii \( r_1 \) to \( r_2 \) is:
\[ r_1 : r_2 = 4 : 5 \]

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

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