The sum of the radii of two different circles is 18.5 cm and the difference of their circumference is 22 cm. Find their radii

Answer

To solve the problem, we need to find the radii of two circles given the sum of their radii and the difference of their circumferences.

### Step-by-Step Solution:

1. **Define the Variables**:
Let the radius of the first circle be \( R \) and the radius of the second circle be \( r \).

2. **Set Up the Equations**:
From the problem statement, we have two equations:
- The sum of the radii:
\[ R + r = 18.5 \quad \text{(Equation 1)} \]
- The difference of their circumferences:
The circumference of a circle is given by \( C = 2\pi R \). Therefore, the difference of the circumferences can be expressed as:
\[ 2\pi R - 2\pi r = 22 \quad \text{(Equation 2)} \]

3. **Simplify Equation 2**:
Factor out \( 2\pi \) from Equation 2:
\[ 2\pi (R - r) = 22 \]
Now, divide both sides by \( 2\pi \):
\[ R - r = \frac{22}{2\pi} = \frac{11}{\pi} \quad \text{(Equation 3)} \]

4. **Substitute the Value of \( \pi \)**:
Using \( \pi \approx \frac{22}{7} \):
\[ R - r = \frac{11}{\frac{22}{7}} = \frac{11 \times 7}{22} = \frac{77}{22} = \frac{7}{2} \quad \text{(Equation 4)} \]

5. **Solve the System of Equations**:
Now we have two equations:
- Equation 1: \( R + r = 18.5 \)
- Equation 4: \( R - r = \frac{7}{2} \)

We can add these two equations:
\[ (R + r) + (R - r) = 18.5 + \frac{7}{2} \]
This simplifies to:
\[ 2R = 18.5 + 3.5 = 22 \]
Thus, dividing both sides by 2:
\[ R = 11 \text{ cm} \]

6. **Find the Value of \( r \)**:
Substitute \( R = 11 \) back into Equation 1:
\[ 11 + r = 18.5 \]
Solving for \( r \):
\[ r = 18.5 - 11 = 7.5 \text{ cm} \]

### Final Answer:
The radii of the two circles are:
- \( R = 11 \) cm
- \( r = 7.5 \) cm