Question

If the length of a rectangle is reduced by 50% and the breadth is increased by 25%, it takes the
shape of a square. Find the ratio of (1) the areas of the rectangle and the square, and
(ii) the perimeters of the rectangle and the square.​

Answer 1

Step-by-step explanation:

Given :-

If the length of a rectangle is reduced by 50% and the breadth is increased by 25%, it takes the

shape of a square.

To find :-

1) Find the ratio of the areas of the rectangle and the square?

2)Find the perimeters of the rectangle and the square ?

Solution :-

Let the length of a rectangle be l units

Let the breadth of the rectangle be b units

We know that

Area of a rectangle = lb sq.units --------(1)

If the length is reduced by 50% then the new length

=> l-(50%of l)

=> l -(50/100)×l

=> l-(1/2)×l

=>l-(l/2)

=> (2l-l)/2

=> l/2 units

And

If the breadth is increased by 25% then the new breadth

=> b+(25% of b)

=> b+(25/100)×b

=> b+(1/4)×b

=> b+(b/4)

=> (4b+b)/4

=> 5b/4 units

Area of the new rectangle

=> (l/2)×(5b/4)

=> (l×5b)/(2×4)

=>(5lb)/8 sq.units ---------(2)

Given that

If the length of a rectangle is reduced by 50% and the breadth is increased by 25%, it takes the

shape of a square.

=> Area of the new rectangle = Area of a square

=> Area of a square = (5lb)/8 sq.units

Ratio of the rectangle and a square

=> lb : (5/8)lb

=> lb/(5lb/8)

On cancelling 'lb' then

=> 1:5/8

=> 1/(5/8)

=> 1/×(8/5)

=> 8/5

=> 8:5

We know that

Perimeter of a rectangle = 2(l+b) units

Measurements of a square are l/2 units and 5b/4 units

but

All sides are equal in a square

=> l/2 = 5b/4

=> l = (5b/4)×2

=> l = 5b/2 units or

=> 2l = 5b

=> b = 2l/5 units

We know that

Perimeter of a square = 4a units

=> P = 4×l/2

=> P = 2l units

Perimeter of the square = 2l units

Now,

Ratio of the Perimeters of the rectangle and the square

=> 2(l+b) : 2l

=> (l+b) : l

=> (l+(2l/5)) : l

=>(5l+2l)/5 :l

=> (7l/5):l

=>(7l/5)/l

=> 7l/5l

On cancelling 'l' then

=> 7/5

=> 7:5

Answer:-

1) The ratio of the areas of the rectangle and the square is 8:5

2)The perimeters of the rectangle and the square is 7:5

Used formulae:-

Rectangle :-

• Area of a rectangle = lb sq.units

• Perimeter of a rectangle = 2(l+b) units

• Where,l = length of a rectangle

• b = breadth of the rectangle

Square:-

• All sides are equal in a square

• Perimeter of a square = 4a units

• Area of a square = a² sq.units

• Where,a = Side of a square

• a:b can be written as a/b