Question

The area of a rectangle gets reduced by 9 sq. units if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and breadth by 2 units, then the area is increased by 67 sq. units. Find the length and breadth of the rectangle.

Answer 1

Given:

• Let the length of the rectangle be (x) units.
• Let the breadth of the rectangle be (y) units.

Step-wise calculation:

1. According to the first condition:

If length is reduced by 5 units and breadth is increased by 3 units, the area decreases by 9 sq. units.

So, the area equation is:

(x – 5)(y + 3) = xy – 9 

Expanding the left side:

xy + 3x – 5y – 15 = xy – 9 

Simplify:

3x – 5y – 15 = –9 

3x – 5y = 6   ...(1)

2. According to the second condition:

If length is increased by 3 units and breadth is increased by 2 units, the area increases by 67 sq. units.

So, the area equation is:

(x + 3)(y + 2) = xy + 67 

Expanding the left side:

xy + 2x + 3y + 6 = xy + 67 

Simplify:

2x + 3y + 6 = 67

2x + 3y = 61   ...(2)

3. Now solve equations (1) and (2):

3x – 5y = 6

2x + 3y = 61

Multiply (1) by 3 and (2) by 5 to eliminate (y):

9x – 15y = 18   ...(3) 

10x + 15y = 305   ...(4)

Add (3) and (4):

9x – 15y + 10x + 15y = 18 + 305 

19x = 323 

`x = 323/19`

x = 17

4. Substitute (x = 17) into equation (1): 

3(17) – 5y = 6 

51 – 5y = 6 

–5y = 6 – 51

–5y = –45 

`y = 45/5`

y = 9

Length of the rectangle: 17 units.

Breadth of the rectangle: 9 units.