the length of a rectangle exceeds its breadth by 4 cm. if length and breadth are each increased by 3cm, the area of the new recangle will be 81cm square more than that of the given rectangle find the length and breadth of the given rectangle

Let the length and breadth of the original rectangle be \( l \) cm and \( b \) cm, respectively. We are given two key conditions:

1. The length exceeds the breadth by 4 cm:
   \[
   l = b + 4
   \]

2. When the length and breadth are each increased by 3 cm, the area of the new rectangle becomes 81 cm² more than the area of the original rectangle:
   \[
   (l + 3)(b + 3) = lb + 81
   \]

### Step 1: Write the equation for the new area
The area of the new rectangle is given by:
\[
A_{\text{new}} = (l + 3)(b + 3)
\]
The area of the original rectangle is:
\[
A_{\text{original}} = l \times b
\]
From the given, we have:
\[
A_{\text{new}} = A_{\text{original}} + 81
\]

### Step 2: Expand and set up the equation
Using the area relationship:
\[
(l + 3)(b + 3) = lb + 81
\]
Expanding both sides:
\[
lb + 3b + 3l + 9 = lb + 81
\]
Cancel \( lb \) from both sides:
\[
3b + 3l + 9 = 81
\]
Simplify:
\[
3b + 3l = 72
\]
\[
b + l = 24
\]

### Step 3: Use the length-breadth relationship
Substitute \( l = b + 4 \) into \( b + l = 24 \):
\[
b + (b + 4) = 24
\]
Simplify:
\[
2b + 4 = 24
\]
\[
2b = 20
\]
\[
b = 10
\]

### Step 4: Find the length
Now, substitute \( b = 10 \) into \( l = b + 4 \):
\[
l = 10 + 4 = 14
\]

### Final Answer:
The length of the rectangle is 14 cm, and the breadth is 10 cm.

Would you like further clarification or more details?

Here are 5 related questions:
1. How do we calculate the area of a rectangle?
2. What happens if both dimensions are increased by the same value in a rectangle?
3. How would the problem change if the area difference were 100 cm² instead of 81 cm²?
4. What is the relationship between perimeter and area of a rectangle?
5. Can we apply a similar approach to solve problems with squares?

**Tip:** In geometry, always start by defining variables for unknowns and translating word problems into equations.

The length of a rectangle exceeds its breadth by 4 cm. If the length and breadth are each increased by 3 cm, the area of the new rectangle will be 81 cm² more than that of the given rectangle. Find the length and breadth of the given rectangle.

This problem involves finding the length and breadth of a rectangle where the length exceeds the breadth by 4 cm, and an area increase is given. Learn how to solve this using linear equations, suitable for students in Grades 7-9.

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