Question

If the length of a rectangle is decreased by 2 metres and the breadth increased by 2 metres the area would increase 8 square metres.If the length is decreased by 3 metres and the breadth decreased by 1 metre the area would decrease by 27 square metres .What are the length and breadth?
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Answer 1

Solution :

Let the length of the Rectangle be l m.

And the Breadth of the Rectangle be b m.

We know the formula for area of a rectangle I.e

$\boxed{\bf{A = L \times B}}$

Where :

• L = Length
• B = Breadth
• A = Area

Using the above formula and substituting the values in it, we get :

$:\implies \bf{A = L \times B}$

$:\implies \bf{A = l \times b}$

$:\implies \bf{A = lb}$

Hence the area of the Rectangle is lb.

Equation for the 1st case :

According to the Question , when the length is decreased by 2 m (i.e, l - 2) and the breadth is increased by 2 m (i.e, b + 2) , the area increases by 8 m².

$\underline{:\implies \bf{A + 8 = (l - 2) \times (b + 2)}}$

$:\implies \bf{lb + 8 = (l - 2) \times (b + 2)}$

$:\implies \bf{lb + 8 = lb + 2l - 2b - 4}$

$:\implies \bf{8 + 4 = lb + 2l - 2b - lb}$

$:\implies \bf{12 = 2l - 2b}$

By dividing the Equation by

$:\implies \bf{\dfrac{12}{2} = \dfrac{2l}{2} - \dfrac{2b}{2}}$

$:\implies \bf{6 = l - b}$

$:\implies \bf{l - b = 6}$⠀⠀⠀⠀⠀⠀Eq.(i)

Equation for the 2nd case :

According to the Question , when the length is decreased by 3 m (i.e, l - 3) and the breadth is decreased by 1 m (i.e, b - 2) , the area decreased by 27 m².

$\underline{:\implies \bf{A + 8 = (l - 2) \times (b + 2)}}$

$:\implies \bf{lb - 27 = (l - 3) \times (b - 1)}$

$:\implies \bf{lb - 27 = lb - l - 3b + 3}$

$:\implies \bf{-27 - 3 = lb - l - 3b - lb}$

$:\implies \bf{-30 = - l - 3b}$

$:\implies \bf{-30 = - (l + 3b)}$

$:\implies \bf{\not{-}30 = \not{-} (l + 3b)}$

$:\implies \bf{30 = l + 3b}$

$:\implies \bf{l + 3b = 30}$⠀⠀⠀⠀⠀⠀Eq.(ii)

Now by subtracting Eq.(ii) from Eq.(i) , we get :

$:\implies \bf{[(l - b) - (l + 3b) = 6 - 30}$

$:\implies \bf{(l - b - l - 3b) = 6 - 30}$

$:\implies \bf{\not{l} - b - \not{l} - 3b) = 6 - 30}$

$:\implies \bf{- b - 3b = 6 - 30}$

$:\implies \bf{-4b = -24}$

$:\implies \bf{\not{-}4b = \not{-}24}$

$:\implies \bf{4b = 24}$

$:\implies \bf{b = \dfrac{24}{4}}$

$:\implies \bf{b = 6}$

$\boxed{\therefore \bf{b = 6}}$

Hence the Breadth of the Rectangle is 6 m.

Now by putting the value of b in the eq.(i) , we get :

$:\implies \bf{l - b = 6}$

$:\implies \bf{l - 6 = 6}$

$:\implies \bf{l = 6 + 6}$

$:\implies \bf{l = 12}$

$\boxed{\therefore \bf{l = 12}}$

Hence the length of the Rectangle is 12 m.