A circular path is made from two concentric circular rings in such a way that the smaller ring when allowed to roll over the circumference of the bigger ring, it takes three full revolutions. If the area of the pathway is equal to n times the area of the smaller ring, then n is equal to

Radius of smaller ring = r units

Radius of bigger ring = R units

According to the question,

2SR = 3 × 2Sr

⇒ R = 3r

Area of shaded part (path

$=\pi \mathrm{R}^{2}-\pi r^{2}=\pi(3 r)^{2}-\pi r^{2}$

$=9 \pi r^{2}-\pi r^{2}=8 \pi r^{2}$

$\therefore 8 \pi r^{2}=n \cdot \pi r^{2}$

$\Rightarrow n=\frac{8 \pi r^{2}}{\pi r^{2}}=8$