The ratio of circumference and diameter of a circle is 22 : 7. If the circumference be $1\frac{4}{7}$m. then the radius of the circle is

### Concept & Circle Proportions The ratio of the circumference ($C$) of a circle to its diameter ($D$) is a constant, mathematically defined as $\pi$ (approximated as $\frac{22}{7}$). We can use this ratio directly to find the diameter and consequently the radius. $$\frac{C}{D} = \pi \approx \frac{22}{7}$$ $$Radius = \frac{Diameter}{2}$$ ### Step-by-Step Solution * Given: The ratio $\frac{C}{D} = \frac{22}{7}$. * The circumference ($C$) is given as a mixed fraction: $1\frac{4}{7}$ m. * Convert the mixed fraction to an improper fraction: $C = \frac{1 \times 7 + 4}{7} = \frac{11}{7}$ m. * Set up the proportion: $\frac{\frac{11}{7}}{D} = \frac{22}{7}$. * Multiply both sides by $D$: $\frac{11}{7} = \frac{22}{7} \times D$. * Solve for $D$: $D = \frac{11}{7} \times \frac{7}{22} = \frac{11}{22} = \frac{1}{2}$ m. * The radius ($r$) is half of the diameter: $r = \frac{D}{2} = \frac{\frac{1}{2}}{2} = \frac{1}{4}$ m. ### Exam Strategy & Shortcut Recognize that $\frac{11}{7}$ is exactly half of $\frac{22}{7}$. Since the circumference is halved, the diameter must also be halved (from a conceptual 1 unit to $\frac{1}{2}$ unit). The radius is half of that, resulting in $\frac{1}{4}$ instantly. ### Common Pitfall A highly common mistake is stopping after calculating the diameter ($\frac{1}{2}$ m) and selecting it as the answer, forgetting that the question specifically asked for the radius. ### Final Answer Therefore, the correct answer is **$\frac{1}{4}$m**.