The ratio of the outer and the inner perimeter of a circular path is 23 : 22. If the path is 5 metres wide, the diameter of the inner circle is :

Options

A. 110 m
B. 55m
C. 220 m
D. 230 m

Answer

To solve the problem, we need to find the diameter of the inner circle given the ratio of the outer and inner perimeter of a circular path and the width of the path.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We are given that the ratio of the outer perimeter (circumference) to the inner perimeter of a circular path is 23:22.
- The width of the path is 5 meters.

2. **Define Variables**:
- Let the radius of the inner circle be \( r \) meters.
- Therefore, the radius of the outer circle will be \( r + 5 \) meters (since the path is 5 meters wide).

3. **Formulate the Perimeters**:
- The perimeter (circumference) of the inner circle is given by \( C_{inner} = 2\pi r \).
- The perimeter (circumference) of the outer circle is given by \( C_{outer} = 2\pi (r + 5) \).

4. **Set Up the Ratio**:
- According to the problem, the ratio of the outer perimeter to the inner perimeter is:
\[ \frac{C_{outer}}{C_{inner}} = \frac{23}{22} \]
- Substituting the expressions for the circumferences:
\[ \frac{2\pi (r + 5)}{2\pi r} = \frac{23}{22} \]
- The \( 2\pi \) cancels out:
\[ \frac{r + 5}{r} = \frac{23}{22} \]

5. **Cross-Multiply to Solve for \( r \)**:
- Cross-multiplying gives:
\[ 22(r + 5) = 23r \]
- Expanding this:
\[ 22r + 110 = 23r \]
- Rearranging the equation:
\[ 110 = 23r - 22r \]
\[ 110 = r \]

6. **Finding the Diameter**:
- The diameter \( D \) of the inner circle is given by:
\[ D = 2r = 2 \times 110 = 220 \text{ meters} \]

### Final Answer:
The diameter of the inner circle is **220 meters**.