Finding Area of a Rectangle

Video Explanation

Question

In a rectangle, if length is increased by 2 units and breadth is decreased by 2 units, the area decreases by 28 sq units. If length is decreased by 1 unit and breadth is increased by 2 units, the area increases by 33 sq units. Find the area of the rectangle.

Solution

Step 1: Let Variables

Let length = \(x\) units

Let breadth = \(y\) units

Original area = \(xy\)

—

Step 2: Form Equations

First condition:

\[ (x+2)(y-2) = xy – 28 \]

Expand:

\[ xy – 2x + 2y – 4 = xy – 28 \]

Cancel \(xy\):

\[ -2x + 2y – 4 = -28 \]

\[ -2x + 2y = -24 \]

\[ -x + y = -12 \quad (1) \]

— Second condition:

\[ (x-1)(y+2) = xy + 33 \]

Expand:

\[ xy + 2x – y – 2 = xy + 33 \]

Cancel \(xy\):

\[ 2x – y – 2 = 33 \]

\[ 2x – y = 35 \quad (2) \]

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Step 3: Solve Linear Equations

From (1):

\[ y = x – 12 \]

Substitute into (2):

\[ 2x – (x – 12) = 35 \]

\[ x + 12 = 35 \]

\[ x = 23 \]

Then:

\[ y = 23 – 12 = 11 \]

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Step 4: Find Area

\[ \text{Area} = xy = 23 \times 11 = 253 \]

—

Conclusion

\[ \text{Area of rectangle} = 253 \text{ square units} \]

Verification

Check 1: \((25)(9) = 225 = 253 – 28\) ✔

Check 2: \((22)(13) = 286 = 253 + 33\) ✔