There are three farms, whose area is 288, 408 and 552 square metres, respectively. In which rows of equal length has to be made, if the width of each row is 4 meteres, then what will be the maximum length?

Options

A. 3 metre
B. 5 metre
C. 4 metre
D. 6 metre

Answer

To solve the problem, we need to find the maximum length of rows of equal length that can be made for three farms with areas of 288, 408, and 552 square meters, given that the width of each row is 4 meters.

### Step 1: Find the dimensions of each farm
To determine the maximum length of the rows, we first need to find the total area that can be allocated for the rows. Since the width of each row is 4 meters, we can express the area of each farm in terms of length and width.

For each farm:
- Area = Length × Width
- Given Width = 4 meters

Thus, we can express the length for each farm as:
- Length = Area / Width

Calculating the lengths:
1. For the first farm (Area = 288 m²):
\[ \text{Length}_1 = \frac{288}{4} = 72 \text{ meters} \]

2. For the second farm (Area = 408 m²):
\[ \text{Length}_2 = \frac{408}{4} = 102 \text{ meters} \]

3. For the third farm (Area = 552 m²):
\[ \text{Length}_3 = \frac{552}{4} = 138 \text{ meters} \]

### Step 2: Find the HCF of the lengths
Now, we need to find the highest common factor (HCF) of the lengths obtained from the three farms: 72, 102, and 138.

To find the HCF, we can use the prime factorization method:
1. **Prime factorization of 72**:
\[ 72 = 2^3 \times 3^2 \]

2. **Prime factorization of 102**:
\[ 102 = 2^1 \times 3^1 \times 17^1 \]

3. **Prime factorization of 138**:
\[ 138 = 2^1 \times 3^1 \times 23^1 \]

### Step 3: Identify common factors
Now, we identify the common factors from the prime factorizations:
- The common prime factors are \(2^1\) and \(3^1\).

### Step 4: Calculate HCF
The HCF can be calculated by taking the lowest power of the common prime factors:
\[ \text{HCF} = 2^1 \times 3^1 = 2 \times 3 = 6 \]

### Step 5: Conclusion
The maximum length of the rows of equal length that can be made for the three farms is **6 meters**.

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