Three plots having an area of 132, 204 and 228 square metres respectively are to be sub-divided into equal vegetable beds. If the breadth of a bed is 3 metres, find the maximum length that a bed can have

Options

A. 14 metres
B. 4 metres
C. 24 metres
D. 6 metres

Answer

To solve the problem step by step, we need to find the maximum length of a vegetable bed that can be created from the three plots of land with areas 132, 204, and 228 square meters, given that the breadth of each bed is 3 meters.

### Step 1: Identify the areas of the plots
The areas of the three plots are:
- Plot 1: 132 square meters
- Plot 2: 204 square meters
- Plot 3: 228 square meters

### Step 2: Find the HCF (Highest Common Factor) of the areas
To find the maximum area of a vegetable bed that can be created from these plots, we need to calculate the HCF of the three areas.

1. **Factors of 132**:
- 132 = 2 × 2 × 3 × 11 = 2² × 3 × 11
2. **Factors of 204**:
- 204 = 2 × 2 × 3 × 17 = 2² × 3 × 17
3. **Factors of 228**:
- 228 = 2 × 2 × 3 × 19 = 2² × 3 × 19

Now, we find the common factors:
- The common factors are 2² and 3.

Thus, the HCF is:
\[ HCF = 2² × 3 = 4 × 3 = 12 \]

### Step 3: Calculate the area of one vegetable bed
Since the HCF represents the maximum area of one vegetable bed, we have:
- Area of one bed = 12 square meters.

### Step 4: Relate area to dimensions of the bed
We know that the area of a rectangle (bed) is given by:
\[ \text{Area} = \text{Length} \times \text{Breadth} \]

Given that the breadth (B) of the bed is 3 meters, we can express the length (L) in terms of the area:
\[ 12 = L \times 3 \]

### Step 5: Solve for the length
To find the length, we rearrange the equation:
\[ L = \frac{12}{3} = 4 \text{ meters} \]

### Conclusion
The maximum length that a bed can have is **4 meters**.

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