Rectangle Reduction Scenario

We are given a rectangle. When its length is decreased by 10 m and its breadth by 5 m, it transforms into a square. This transformation also results in a loss of 650 m² from the original area.

Equations from Conditions

Let the original length of the rectangle be $L$ meters and the original breadth be $B$ meters.

The original area of the rectangle is $A_{orig} = L \times B$. After the reductions, the new length is $L-10$ and the new breadth is $B-5$. Since the new shape is a square, its sides must be equal:

$ L - 10 = B - 5 \quad (1) $

The new area is $A_{new} = (L-10)(B-5)$. The problem states that the area loss is 650 m²:

$ A_{orig} - A_{new} = 650 $ $ (L \times B) - (L-10)(B-5) = 650 \quad (2) $

Dimensions Solved

We solve the system of equations to find the original dimensions:

1. From equation (1), we express $L$ in terms of $B$: $ L = B - 5 + 10 $ $ L = B + 5 $
2. Substitute this expression for $L$ into equation (2): $ ((B+5) \times B) - ((B+5-10) \times (B-5)) = 650 $ $ (B^2 + 5B) - ((B-5) \times (B-5)) = 650 $
3. Expand and simplify the equation: $ (B^2 + 5B) - (B^2 - 10B + 25) = 650 $ $ B^2 + 5B - B^2 + 10B - 25 = 650 $ $ 15B - 25 = 650 $
4. Solve for $B$: $ 15B = 650 + 25 $ $ 15B = 675 $ $ B = \frac{675}{15} $ $ B = 45 \text{ m} $
5. Calculate $L$ using $L = B + 5$: $ L = 45 + 5 $ $ L = 50 \text{ m} $

Original Area Calculation

Now, calculate the area of the original rectangle using the found dimensions $L=50$ m and $B=45$ m:

$ \text{Original Area} = L \times B $ $ \text{Original Area} = 50 \text{ m} \times 45 \text{ m} $ $ \text{Original Area} = 2250 \text{ m}^2 $