Question

A larger wheel of diameter $50\mathrm{cm}$ is attached to a smaller wheel of diameter $30\mathrm{cm}$. Find the number of revolutions made by the smaller wheel, when the larger one makes $15$ revolutions.

Answer 1

Step 1: Determine the distance travelled by the smaller wheel in one revolution.

The circumference of a circle $\left(C\right)$ is calculated by $C=2\mathrm{\pi r}$ where $r$ represents the radius of the circle.

The radius of a circle $r$ is calculated by $r=\frac{d}{2}$ where $d$ represents the diameter of a circle.

Distance covered by smaller wheel in one revolution is equal to the circumference of a smaller wheel $\left(C\right)$.

$\begin{array}{rcl}& \Rightarrow & C=2\times \mathrm{\pi }\times \frac{30}{2}\\ & =& 30\mathrm{\pi }\mathrm{cm}\end{array}$

Step 2: Determine the distance travelled by the larger wheel in one revolution.

Distance covered by smaller wheel in one revolution is equal to the circumference of a larger wheel $\left(C\text{'}\right)$.

$\begin{array}{rcl}& \Rightarrow & C\text{'}=2\times \mathrm{\pi }\times \frac{50}{2}\\ & =& 50\mathrm{\pi }\mathrm{cm}\end{array}$

Step 3: Determine the number of revolutions made by the smaller wheel.

Let $x$ represents the number of revolutions made by the smaller wheel.

The distance travelled by a smaller wheel is equal to the distance travelled by the larger wheel.

Relation between no of revolution, circumference of a circle, and distance: $\mathrm{No}\mathrm{of}\mathrm{revolution}=\frac{\mathrm{Distance}}{\mathrm{Circumference}}$.

$$\begin{gathered} \Rightarrow x\times C=15\times C\text{'} \\ \Rightarrow x\times 30\mathrm{\pi }=15\times 50\mathrm{\pi } \\ \Rightarrow \mathrm{x}=\frac{15\times 50\mathrm{\pi }}{30\mathrm{\pi }}=25 \end{gathered}$$

Hence, the number of revolutions made by the smaller wheel is $25$.