Finding the Length of a Rectangular Field using Ratio and Cost

This problem requires us to determine the length of a rectangular field given the ratio of its length to breadth and the total cost of cultivating the field at a specific rate per square meter. The cost of cultivation is directly related to the area of the field.

Understanding the Problem

• The length and breadth of the rectangular field are in the ratio 4:3.
• The cost of cultivating the field is ₹600.
• The rate of cultivation is ₹2 per square meter (\( \text{m}^2 \)).
• We need to find the length of the rectangular field.

Step-by-Step Solution to Find Field Length

Step 1: Calculate the Area of the Field

The total cost of cultivation is the area of the field multiplied by the cost per square meter. Therefore, we can find the area by dividing the total cost by the rate per square meter.

Area \( = \frac{\text{Total Cost}}{\text{Rate per } \text{m}^2} \)

Area \( = \frac{₹600}{₹2/\text{m}^2} \)

Area \( = 300 \text{ m}^2 \)

The area of the rectangular field is \(300 \text{ m}^2\).

Step 2: Express Length and Breadth using the Given Ratio

The ratio of length to breadth is given as 4:3. Let the common ratio be \(x\). Then,

• Length \( = 4x \) meters
• Breadth \( = 3x \) meters

Step 3: Set up an Equation using the Area Formula

The area of a rectangle is given by the formula: Area \( = \text{Length} \times \text{Breadth} \).

We know the area is \(300 \text{ m}^2\), and we have expressed length and breadth in terms of \(x\). Substitute these values into the area formula:

\(300 \text{ m}^2 = (4x) \times (3x)\)

\(300 = 12x^2\)

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

Now, we need to solve the equation \(300 = 12x^2\) for the variable \(x\).

Divide both sides by 12:

\(x^2 = \frac{300}{12}\)

\(x^2 = 25\)

Take the square root of both sides to find \(x\):

\(x = \sqrt{25}\)

\(x = 5\)

Since length and breadth must be positive values, we take the positive square root, so \(x = 5\).

Step 5: Calculate the Length of the Field

We defined the length of the field as \(4x\). Now that we have the value of \(x\), we can calculate the length.

Length \( = 4x \)

Length \( = 4 \times 5 \)

Length \( = 20 \text{ m}\)

Let's also calculate the breadth for completeness:

Breadth \( = 3x \)

Breadth \( = 3 \times 5 \)

Breadth \( = 15 \text{ m}\)

To verify, let's check the area with these dimensions: Area \( = 20 \text{ m} \times 15 \text{ m} = 300 \text{ m}^2 \), which matches the area calculated from the cultivation cost.

The length of the rectangular field is 20 meters.

Revision Table: Rectangular Field Calculations

Additional Information: Rectangles and Area

A rectangle is a quadrilateral with four right angles. Opposite sides are equal in length. The area of a rectangle is the amount of space it covers, measured in square units.

• The formula for the area of a rectangle is \( \text{Area} = \text{Length} \times \text{Breadth} \).
• The perimeter of a rectangle is the total length of its boundaries, given by \( \text{Perimeter} = 2 \times (\text{Length} + \text{Breadth}) \).
• Ratios provide a way to relate two or more quantities. When a ratio is given, using a common variable (like \(x\)) helps in setting up equations based on other information (like area or perimeter).
• Cost calculations involving area often appear in problems related to fields, floors, or walls. The total cost is usually the area multiplied by the cost per unit area.