The difference between the length and breadth of a rectangle is 23 m. If its perimeter is 206 m, then its area is (Section Officers', 2003)

### Concept & Formula This problem requires solving a system of linear equations derived from the perimeter and the difference between dimensions, followed by calculating the area. $$ \text{Perimeter} = 2(L + B) $$ $$ \text{Area} = L \times B $$ ### Step-by-Step Solution * **Given:** Difference between length and breadth: $L - B = 23$. Perimeter: $2(L + B) = 206$. * **Step 1:** Simplify the perimeter equation to find the sum of length and breadth. $L + B = \frac{206}{2}$ $L + B = 103$ * **Step 2:** Solve the system of equations. Equation 1: $L - B = 23$ Equation 2: $L + B = 103$ * Add both equations: $(L - B) + (L + B) = 23 + 103$ $2L = 126$ $L = 63\text{ m}$ * **Step 3:** Substitute $L$ back into Equation 2 to find $B$. $63 + B = 103$ $B = 103 - 63 = 40\text{ m}$ * **Step 4:** Calculate the area. $\text{Area} = 63 \times 40 = 2520\text{ m}^2$ ### Exam Strategy & Shortcut When you have the sum ($S = 103$) and the difference ($D = 23$) of two numbers, the larger number (Length) is simply $\frac{S + D}{2}$ and the smaller number (Breadth) is $\frac{S - D}{2}$. $L = \frac{103 + 23}{2} = 63$ $B = \frac{103 - 23}{2} = 40$ Then quickly multiply $63 \times 40 = 2520$. ### Common Pitfall A frequent error is forgetting to divide the perimeter by 2 before setting up the sum equation, erroneously using $L + B = 206$. Always remember the perimeter formula includes $2 \times$ the sum of dimensions. ### Final Answer Therefore, the correct answer is **$2520\text{ m}^2$**.