The area of a rectangle gets reduced by 9 sq. meters if its length is reduced by 5 meters and breadth is increased by 3 meters. If its length is increased by 3 meters and breadth is increased by 2 meters, its area increases by 67 sq. meters. Find the length and breadth of the rectangle.

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Points to Remember:

  • The problem involves solving a system of simultaneous equations.
  • We need to represent the area of the rectangle using variables for length and breadth.
  • The changes in area due to changes in length and breadth will provide the equations.

Introduction:

This question is a mathematical word problem requiring an analytical approach. It involves finding the dimensions (length and breadth) of a rectangle given information about how changes in its dimensions affect its area. The core concept is the formula for the area of a rectangle: Area = Length × Breadth. We will use this formula to create a system of two simultaneous equations and solve for the unknown length and breadth.

Body:

1. Defining Variables and Setting up Equations:

Let’s denote the original length of the rectangle as ‘l’ meters and the original breadth as ‘b’ meters. The original area is therefore lb square meters.

  • Equation 1: If the length is reduced by 5 meters (l-5) and the breadth is increased by 3 meters (b+3), the area reduces by 9 sq. meters. This translates to the equation: (l-5)(b+3) = lb – 9

  • Equation 2: If the length is increased by 3 meters (l+3) and the breadth is increased by 2 meters (b+2), the area increases by 67 sq. meters. This gives us the equation: (l+3)(b+2) = lb + 67

2. Solving the Simultaneous Equations:

Expanding the equations, we get:

  • Equation 1: lb + 3l – 5b – 15 = lb – 9 => 3l – 5b = 6
  • Equation 2: lb + 2l + 3b + 6 = lb + 67 => 2l + 3b = 61

Now we have a system of two linear equations with two variables:

3l – 5b = 6
2l + 3b = 61

We can solve this system using either substitution or elimination. Let’s use elimination. Multiply the first equation by 3 and the second equation by 5 to eliminate ‘b’:

9l – 15b = 18
10l + 15b = 305

Adding the two equations:

19l = 323
l = 17

Substitute the value of l (17) into either of the original equations (let’s use 2l + 3b = 61):

2(17) + 3b = 61
34 + 3b = 61
3b = 27
b = 9

3. Verifying the Solution:

Original area = lb = 17 * 9 = 153 sq. meters

  • Checking Equation 1: (17-5)(9+3) = 12 * 12 = 144 sq. meters (153 – 144 = 9, which is correct)
  • Checking Equation 2: (17+3)(9+2) = 20 * 11 = 220 sq. meters (220 – 153 = 67, which is correct)

Conclusion:

The length of the rectangle is 17 meters and the breadth is 9 meters. We successfully solved the problem by formulating a system of simultaneous equations based on the given information and solving for the unknowns. The solution was verified by substituting the values back into the original conditions. This problem highlights the practical application of algebraic techniques in solving real-world problems involving geometric shapes and their properties. Further, such problems encourage analytical thinking and problem-solving skills, crucial for various fields of study and professional endeavors. The accuracy of the solution underscores the importance of precise mathematical reasoning.

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