The diameter of the front wheel of an engine is 2x cm and what of rear wheel is 2y cm. To cover the same distance, find the number of times the rear wheel will revolve when the front wheel revolves 'n' times.

Options

A. A)`(n)/(x y)` times
B. B)`(y n)/(x)` times
C. C)`(n x)/(y)` times
D. D)`(x y)/(n)` times

Answer

To solve the problem, we need to find the number of times the rear wheel revolves when the front wheel revolves 'n' times. We will use the relationship between the distance covered by the wheels and their diameters.

### Step-by-Step Solution:

1. **Understand the Relationship Between Revolutions and Distance:**
The distance covered by a wheel in one revolution is equal to its circumference. The circumference (C) of a circle is given by the formula:
\[ C = \pi \times \text{diameter} \]

2. **Calculate the Circumference of the Front Wheel:**
The diameter of the front wheel is \(2x\) cm. Therefore, the circumference of the front wheel (C_front) is:
\[ C_{\text{front}} = \pi \times 2x = 2\pi x \text{ cm} \]

3. **Calculate the Distance Covered by the Front Wheel in n Revolutions:**
If the front wheel revolves \(n\) times, the total distance (D) covered by the front wheel is:
\[ D = n \times C_{\text{front}} = n \times 2\pi x = 2n\pi x \text{ cm} \]

4. **Calculate the Circumference of the Rear Wheel:**
The diameter of the rear wheel is \(2y\) cm. Therefore, the circumference of the rear wheel (C_rear) is:
\[ C_{\text{rear}} = \pi \times 2y = 2\pi y \text{ cm} \]

5. **Set the Distance Covered by the Rear Wheel Equal to the Distance Covered by the Front Wheel:**
Let \(k\) be the number of revolutions of the rear wheel. The distance covered by the rear wheel when it revolves \(k\) times is:
\[ D_{\text{rear}} = k \times C_{\text{rear}} = k \times 2\pi y \]

6. **Equate the Distances:**
Since both wheels cover the same distance, we can set the distances equal to each other:
\[ 2n\pi x = k \times 2\pi y \]

7. **Simplify the Equation:**
We can cancel \(2\pi\) from both sides of the equation:
\[ n x = k y \]

8. **Solve for k:**
Rearranging the equation gives us:
\[ k = \frac{n x}{y} \]

### Final Answer:
The number of times the rear wheel will revolve when the front wheel revolves \(n\) times is:
\[ k = \frac{n x}{y} \]