Question

In a rectangle, if the length is increased by 3 metres and breadth is decreased by 4 metres, the area of the rectangle is reduced by 67 square metres. If length is reduced by 1 metre and breadth is increased by 4 metres, the area is increased by 89 sq. metres. Find the dimensions of the rectangle.

Answer 1

Given:

Let the original length = l metres and breadth = b metres.

When length increased by 3 and breadth decreased by 4, area is reduced by 67: (l + 3)(b – 4) = lb – 67.

When length reduced by 1 and breadth increased by 4, area is increased by 89: (l – 1)(b + 4) = lb + 89.

Step-wise calculation:

1. Expand both equations:

(l + 3)(b – 4) = lb + 3b – 4l – 12

= lb – 67 

⇒ 3b – 4l – 12 = –67 

⇒ 3b – 4l = –55   ...(1)

(l – 1)(b + 4) = lb + 4l – b – 4

= lb + 89 

⇒ 4l – b – 4 = 89

⇒ 4l – b = 93   ...(2)

2. Solve the linear system (1) and (2).

From (2): b = 4l − 93.

Substitute into (1): 3(4l – 93) – 4l = –55 

⇒ 12l – 279 – 4l = –55 

⇒ 8l – 279 = –55 

⇒ 8l = 224

⇒ l = 28 metres

3. Find b: b = 4l – 93

= 4(28) – 93 

= 112 – 93 

= 19 metres

4. Check: Original area = 28 × 19 = 532 m². 

(l + 3)(b – 4) = 31 × 15 = 465 = 532 – 67 

(l – 1)(b + 4) = 27 × 23 = 621 = 532 + 89

Length = 28 metres, Breadth = 19 metres.