Question

the area of a rectangle gets reduced by 8sq units,if its length is reduced by 5units and breadth is increased by 3units.if we increase the length by 3units and breadth by units,the area increases by 76 square units.find the dimension of the rectangle​

Answer 1

$\large{\underline{\bf{\purple{Correct\:Question:-}}}}$

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

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$\huge{\underline{\bf{\red{Solution:-}}}}$

• Let the length be x unit's
• Let the breadth be y unit's
• Area = xy sq. units

$\bf\:\underbrace{\pink{According\:To\:1st\: condition}}$

✦The area of rectangle gets reduced by 9 units if it's length reduced by 5 and breadth increased by 3 units.

So,

$\leadsto \rm\:\:xy-9=(x-5)(y+3)$

$\leadsto \rm\:\:xy - 9 = x(y + 3) - 5(y + 3) \\ \\\leadsto \rm\:\:{ \cancel{xy }}- 9 = { \cancel{xy }}+ 3x - 5y - 15 = 0 \\ \\\leadsto \rm\:\:3x - 5y - 15 + 9 = 0 \\ \\ \leadsto \bf\:\:3x - 5y - 6...............(i)$

$\bf\:\underbrace{\pink{According\:To\:2nd\: condition}}$

✦ If we increase Length by 3 unit's and breadth by 2 units the area increases by 67 units.

So,

$\leadsto \rm\:\:xy + 67 = (x + 3)(y + 2) \\ \\\leadsto \rm\:\:xy + 67 = x(y + 2) + 3(y + 2) \\ \\ \leadsto \rm\ { \cancel{xy}} + 67= \:{ \cancel{xy }}+ 2x + 3y + 6 \\ \\ \leadsto \rm\:\:2x + 3y + 6 - 67 = 0 \\ \\ \leadsto \bf\:\:2x + 3y - 61 = 0................(ii)$

Multiply equation (i) by 2 and equation (ii) by 3 we get,

$\leadsto \rm\:\:6x - 10y = 12..........(iv) \\\leadsto \rm\:\:6x + 9y = 183...........(v)$

$\bf\:\underbrace {\blue{By \: using\: elimination\: method\:}}$

solving (iv) and (v)

6x - 10y = 12

6x + 9y = 183

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⠀⠀⠀-19y = -171

⠀⠀⠀⠀⠀y = 171/19

⠀⠀⠀⠀⠀$\bf{\green{y=9}}$

Now,

putting value of y in eq.(i)

3x - 5y = 6

$\leadsto \rm\:\:3x-5(9)=6$

$\leadsto \rm\:\:3x-45=6$

$\leadsto \rm\:\:3x=6+45$

$\leadsto \rm\:\:3x=51$

$\leadsto \rm\:\:$x = 51/3

$\leadsto\bf{\green{x=17}}$

So,

The $\bf{\purple{length}}$ of the rectangle is $\bf{\purple{9\: units}}$ and $\bf{\purple{breadth}}$ of rectangle is $\bf{\purple{17\: unit's}}$

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Answer 2

$\large\underline\mathrm{Solution}$

• Let the Length be x
• Let the breadth by y
• Area = xy

A.T.Q 1st conditions.

• The area of a rectangle gets reduced by 8sq units,if its length is reduced by 5units and breadth is increased by 3units.

so,

$\implies$ xy - 9 = (x - 5)(y + 3)

$\implies$ xy - 9 = x(y + 3) - 5(y + 3)

$\implies$ -9 = xy + 3x - 5y - 15 = 0

$\implies$ 3x - 5y - 6. .....(1)

2nd conditions.

• if we increase the length by 3units and breadth by units,the area increases by 76 units.

so,

$\implies$ xy + 67 = (x + 3)(y + 2)

$\implies$ xy + 67 = x(y + 2) + 3(y + 2)

$\large\underline\mathrm{To \: find}$ 67 = 2x + 3y - 61 = 0. ....(2)

Multiply eq. (1) by 2 and eq. (2) by 3. we get,

$\implies$ 6x - 10y = 12. ....(3)

$\implies$ 6x + 9y = 183. ....(4)

$\implies$ 6x - 10y = 12 / 6x + 9y = 183

$\implies$ -19y = -171

$\implies$ y = 9

Now, putting value of y in eq (1)

$\implies$ 3x - 5y = 6

$\implies$ 3x - 5(9) = 6

$\implies$ 3x - 45 = 6

$\implies$ 3x = 51

$\implies$ x = 51/6

$\implies$ x = 17

$\implies$ x = 17, y = 9

So, The Length of the rectangle is 9 unit's and breadth of the rectangle is 17 unit's.