When the breadth of a rectangle is reduced by 4 cm (keeping the length constant),its area reduces by 72 `cm^2` and when the length is reduced by 4 cm (keeping the breadth constant) its area reduces by 64 `cm^2`. What is the perimeter of the rectangle? (in cm)

Options

A. 68
B. 72
C. 66
D. 76

Answer

To solve the problem step by step, we will define the variables, set up equations based on the information given, and then solve for the perimeter of the rectangle.

### Step 1: Define the Variables
Let:
- \( L \) = Length of the rectangle (in cm)
- \( B \) = Breadth of the rectangle (in cm)

### Step 2: Set Up the First Condition
When the breadth is reduced by 4 cm, the new breadth becomes \( B - 4 \). The area of the rectangle is given by:
\[ \text{Original Area} = L \times B \]
After the reduction in breadth, the area becomes:
\[ \text{New Area} = L \times (B - 4) \]
According to the problem, the area reduces by 72 cm²:
\[ L \times B - L \times (B - 4) = 72 \]
This simplifies to:
\[ L \times B - (L \times B - 4L) = 72 \]
\[ 4L = 72 \]
From this, we can solve for \( L \):
\[ L = \frac{72}{4} = 18 \, \text{cm} \]

### Step 3: Set Up the Second Condition
When the length is reduced by 4 cm, the new length becomes \( L - 4 \). The area now becomes:
\[ \text{New Area} = (L - 4) \times B \]
According to the problem, the area reduces by 64 cm²:
\[ L \times B - (L - 4) \times B = 64 \]
This simplifies to:
\[ L \times B - (L \times B - 4B) = 64 \]
\[ 4B = 64 \]
From this, we can solve for \( B \):
\[ B = \frac{64}{4} = 16 \, \text{cm} \]

### Step 4: Calculate the Perimeter
The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2(L + B) \]
Substituting the values of \( L \) and \( B \):
\[ P = 2(18 + 16) = 2 \times 34 = 68 \, \text{cm} \]

### Final Answer
The perimeter of the rectangle is \( 68 \, \text{cm} \).

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