Finding Dimensions of a Rectangle

Video Explanation

Question

In a rectangle, if length is increased by 3 m and breadth is decreased by 4 m, the area is reduced by 67 sq m. If length is decreased by 1 m and breadth is increased by 4 m, the area is increased by 89 sq m. Find the dimensions of the rectangle.

Solution

Step 1: Let Variables

Let length = \(x\) m

Let breadth = \(y\) m

Original area = \(xy\)

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Step 2: Form Equations

First condition:

\[ (x+3)(y-4) = xy – 67 \]

Expand:

\[ xy – 4x + 3y – 12 = xy – 67 \]

Cancel \(xy\):

\[ -4x + 3y – 12 = -67 \]

\[ -4x + 3y = -55 \quad (1) \]

— Second condition:

\[ (x-1)(y+4) = xy + 89 \]

Expand:

\[ xy + 4x – y – 4 = xy + 89 \]

Cancel \(xy\):

\[ 4x – y – 4 = 89 \]

\[ 4x – y = 93 \quad (2) \]

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Step 3: Solve Linear Equations

Multiply (2) by 3:

\[ 12x – 3y = 279 \quad (3) \]

Add (1) and (3):

\[ (-4x + 3y) + (12x – 3y) = -55 + 279 \]

\[ 8x = 224 \]

\[ x = 28 \]

Substitute into (2):

\[ 4(28) – y = 93 \]

\[ 112 – y = 93 \]

\[ y = 19 \]

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Step 4: Final Answer

\[ \text{Length} = 28 \text{ m}, \quad \text{Breadth} = 19 \text{ m} \]

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Verification

Check 1: \((31)(15) = 465 = 532 – 67\) ✔

Check 2: \((27)(23) = 621 = 532 + 89\) ✔