The diameter of the front and the rear wheels of a tractor are 63 cm and 1.54 m respectively. The rear wheel is rotating at `24(6)/(11)` revolutions per minute. Find :
the revolutions per minute made by the front wheel.

Answer

To solve the problem, we need to find the revolutions per minute (RPM) made by the front wheel of the tractor given the diameter of both wheels and the RPM of the rear wheel.

### Step 1: Understand the relationship between the wheels
Both the front and rear wheels of the tractor travel the same distance in the same time because they are connected to the same axle. Therefore, the linear distance traveled by both wheels in one minute will be equal.

### Step 2: Calculate the circumference of both wheels
The circumference \( C \) of a circle (wheel) can be calculated using the formula:
\[ C = \pi \times D \]
where \( D \) is the diameter of the wheel.

- **Front Wheel Diameter**: 63 cm = 0.63 m
- **Rear Wheel Diameter**: 1.54 m

Now, calculate the circumferences:
- **Circumference of Front Wheel**:
\[ C_{FW} = \pi \times 0.63 \]
- **Circumference of Rear Wheel**:
\[ C_{RW} = \pi \times 1.54 \]

### Step 3: Calculate the distance traveled by each wheel in one minute
Let \( x \) be the revolutions per minute of the front wheel. The distance traveled by the front wheel in one minute is:
\[ \text{Distance}_{FW} = x \times C_{FW} = x \times (\pi \times 0.63) \]

The rear wheel is rotating at \( 24 \frac{6}{11} \) RPM. First, convert this to an improper fraction:
\[ \text{RPM}_{RW} = 24 + \frac{6}{11} = \frac{264 + 6}{11} = \frac{270}{11} \]

The distance traveled by the rear wheel in one minute is:
\[ \text{Distance}_{RW} = \frac{270}{11} \times C_{RW} = \frac{270}{11} \times (\pi \times 1.54) \]

### Step 4: Set the distances equal to each other
Since both distances are equal:
\[ x \times (\pi \times 0.63) = \frac{270}{11} \times (\pi \times 1.54) \]

### Step 5: Cancel \( \pi \) from both sides
\[ x \times 0.63 = \frac{270}{11} \times 1.54 \]

### Step 6: Solve for \( x \)
To isolate \( x \), divide both sides by 0.63:
\[ x = \frac{\frac{270}{11} \times 1.54}{0.63} \]

### Step 7: Calculate the values
First, calculate \( \frac{270 \times 1.54}{11} \):
\[ 270 \times 1.54 = 415.8 \]
Now divide by 11:
\[ \frac{415.8}{11} = 37.8 \]
Now divide by 0.63:
\[ x = \frac{37.8}{0.63} \approx 60 \]

### Final Answer
The revolutions per minute made by the front wheel is approximately **60 RPM**.

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