Finding the Length of a Rectangular Land Piece

This problem involves calculating the dimensions of a rectangular piece of land using information about its ratio of length to breadth and the cost of fencing its perimeter. We need to use the total fencing cost and the rate per meter to find the total length of the fence, which is the perimeter of the rectangle. Then, using the given ratio of length and breadth, we can find the actual dimensions.

Step 1: Calculate the Perimeter of the Land

The total cost of fencing the land is given as ₹2,880, and the rate of fencing is ₹15 per meter. The total length of the fence is the perimeter of the rectangular piece of land. We can find the perimeter by dividing the total cost by the rate per meter.

Perimeter = ₹2,880 \div ₹15/\text{m}

Let's calculate this:

\text{Perimeter} = \frac{2880}{15} \text{ m}

\text{Perimeter} = 192 \text{ m}

So, the perimeter of the rectangular land is 192 meters.

Step 2: Represent Length and Breadth using the Given Ratio

The question states that the length and breadth of the rectangular land are in the ratio 7 : 5. This means we can represent the length and breadth using a common multiplier, say \(x\).

• Let the length of the land be \(7x\) meters.
• Let the breadth of the land be \(5x\) meters.

Step 3: Set up an Equation using the Perimeter Formula

The formula for the perimeter of a rectangle is \(2 \times (\text{length} + \text{breadth})\). We know the perimeter is 192 meters, and we have represented the length as \(7x\) and the breadth as \(5x\). We can set up an equation:

\(2 \times (7x + 5x) = 192\)

Step 4: Solve the Equation for \(x\)

Now, let's solve the equation to find the value of \(x\).

\(2 \times (12x) = 192\)

\(24x = 192\)

To find \(x\), divide both sides by 24:

\(x = \frac{192}{24}\)

\(x = 8\)

The value of \(x\) is 8.

Step 5: Calculate the Length of the Land

We defined the length of the land as \(7x\). Now that we know \(x = 8\), we can find the length.

Length = \(7x\)

Length = \(7 \times 8\) meters

Length = \(56\) meters

Conclusion: Length of the Land

The length of the rectangular piece of land is 56 meters.

Revision Table: Key Concepts

Let's quickly review the key concepts used in solving this problem about rectangular land and fencing cost.

• Perimeter of a Rectangle: The total distance around the boundary of a rectangle. Formula is \(2 \times (\text{length} + \text{breadth})\).
• Ratio: A comparison of two quantities. If quantities are in ratio \(a : b\), they can be represented as \(ax\) and \(bx\), where \(x\) is a common factor.
• Cost Calculation: Total cost = Rate per unit \(\times\) Number of units. In this case, Total Fencing Cost = Rate per meter \(\times\) Perimeter.

Additional Information: Applications of Perimeter and Ratio

Understanding perimeter and ratio is crucial in various real-world applications, especially in geometry and measurement problems like finding the length of land.

• Fencing and Boundaries: Calculating perimeter is essential for determining the amount of material (like fencing wire, bricks for a wall) needed to enclose an area.
• Construction and Design: Architects and engineers use perimeter calculations for planning structures, walkways, and landscapes.
• Scaling and Proportions: Ratios are fundamental in scaling maps, blueprints, and models. They help maintain the correct proportions between different parts of an object or area.
• Dividing Quantities: Ratios are used to divide a quantity into parts according to specific proportions. For example, dividing an inheritance or mixing ingredients according to a recipe.

This problem combined the concepts of perimeter calculation and using ratios to find unknown dimensions, which is a common type of question in geometry and basic algebra.