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.