Finding Dimensions of a Rectangle

Video Explanation

Question

The area of a rectangle remains the same if the length is increased by 7 m and the breadth is decreased by 3 m. If the length is decreased by 7 m and the breadth is increased by 4 m, the area is decreased by 21 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 (area same):

\[ (x+7)(y-3) = xy \]

Expand:

\[ xy – 3x + 7y – 21 = xy \]

Cancel \(xy\):

\[ -3x + 7y – 21 = 0 \]

\[ -3x + 7y = 21 \quad (1) \]

— Second condition (area decreases by 21):

\[ (x-7)(y+4) = xy – 21 \]

Expand:

\[ xy + 4x – 7y – 28 = xy – 21 \]

Cancel \(xy\):

\[ 4x – 7y – 28 = -21 \]

\[ 4x – 7y = 7 \quad (2) \]

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Step 3: Solve Linear Equations

Add (1) and (2):

\[ (-3x + 7y) + (4x – 7y) = 21 + 7 \]

\[ x = 28 \]

Substitute into (2):

\[ 4(28) – 7y = 7 \]

\[ 112 – 7y = 7 \]

\[ 7y = 105 \]

\[ y = 15 \]

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Step 4: Final Answer

\[ \text{Length} = 28 \text{ m}, \quad \text{Breadth} = 15 \text{ m} \]

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Verification

Check 1: \((35)(12) = 420 = 28 \times 15\) ✔

Check 2: \((21)(19) = 399 = 420 – 21\) ✔