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

### Concept & Perimeter and Radius Proportionality For any circle, the perimeter (circumference) is directly proportional to its radius ($C = 2\pi r$). Therefore, the ratio of the perimeters of two circles is equal to the ratio of their radii. $$ \frac{C_{outer}}{C_{inner}} = \frac{R}{r} $$ ### Step-by-Step Solution * **Establish the ratio:** Let the outer radius be $R$ and the inner radius be $r$. $\frac{2\pi R}{2\pi r} = \frac{23}{22} \Rightarrow \frac{R}{r} = \frac{23}{22}$ * **Introduce a common multiplier:** Let $R = 23x$ and $r = 22x$. * **Use the width information:** The width of the path is the difference between the outer and inner radii. $\text{Width} = R - r = 5$ m. $23x - 22x = 5 \Rightarrow x = 5$ * **Calculate inner radius and diameter:** Inner radius $r = 22x = 22 \times 5 = 110$ m. Inner diameter $d = 2r = 2 \times 110 = 220$ m. ### Exam Strategy & Shortcut Recognize that the difference in ratio parts ($23 - 22 = 1$ part) corresponds to the actual width of $5$ m. Therefore, $1 \text{ part} = 5$ m. The inner radius represents $22$ parts, so $r = 22 \times 5 = 110$. The diameter is double that, $220$ m. ### Common Pitfall A frequent error is stopping after calculating the inner radius ($110$ m) and selecting it as the answer, forgetting that the question specifically asks for the **diameter**. ### Final Answer Therefore, the correct answer is **220 m**.