Solving the Length of a Rectangular Field Using Ratio and Perimeter

When dealing with the dimensions of geometric shapes, it is often necessary to use given ratios and perimeter calculations to determine the exact measurements. In this article, we will explore how to find the length of a rectangular field given its ratio of length to breadth and its perimeter.

Understanding the Problem

The ratio of the length and breadth (L:B) of a rectangular field is given as 4:3. The perimeter of the field is 56 meters.

Step-by-Step Solution

Let's denote the length as L and the breadth as B. According to the given ratio, we can express them as:

L 4x

B 3x

Where x is a common multiplier. The formula for the perimeter ( P ) of a rectangle is given by:

P 2L 2B

We know the perimeter is 56 meters, so we can set up the equation:

2L 2B 56

Substituting the expressions for L and B:

2(4x) 2(3x) 56

This simplifies to:

8x 6x 56

14x 56

Dividing both sides by 14:

x 4

Now we can find the length:

L 4x 4 × 4 16 meters

The breadth is:

B 3x 3 × 4 12 meters

Verification

Let's verify the perimeter using the actual values of length and breadth:

P 2L 2B

P 2(16) 2(12) 32 24 56 meters

This confirms our solution is correct.

Solving by Another Method

Another way to solve the problem is by using the ratio directly. Assume the length is 4k and the breadth is 3k.

The perimeter of the rectangular field is given by:

2(L B) 2(4k 3k) 14k

Given the perimeter is 56 meters:

14k 56

k 56/14 4

Therefore, the length of the rectangular field is:

L 4k 4 × 4 16 meters

And the breadth is:

B 3k 3 × 4 12 meters

Conclusion

The length of the rectangular field is 16 meters, and the breadth is 12 meters. This solution demonstrates the use of ratio and perimeter formulas in solving geometric problems. Understanding these principles can be beneficial for various mathematical applications.