Question

If the length of a rectangle is reduced by 5 units and its breadth is increased by 3 units,

then the area of the rectangle is reduced by 8 square units. If length is reduced by 3 units

and breadth is increased by 2 units, then the area of rectangle will increase by 67 square

units. Then find the length and breadth of the rectangle.​

Answer 1

GIVEN

If the length of a rectangle is reduced by 5 units and its breadth is increased by 3 units, then the area of the rectangle is reduced by 8 square units. If length is reduced by 3 units and breadth is increased by 2 units, then the area of rectangle will increase by 67 square

units.

TO FIND

Find the length and breadth of the rectangle.

SOLUTION

★Let the length be x and breadth be y

According to the given condition

✫ In first case

If the length of a rectangle is reduced by 5 units and its breadth is increased by 3 units, then the area of the rectangle is reduced by 8 square units

✞ Area of rectangle = length × breadth

➨ xy - 8 = (x - 5)(y + 3)

➨ x(y + 3)-5(y + 3) = xy - 8

➨ xy + 3x - 5y - 15 = xy - 8

➨ 3x - 5y = 15 - 8

➨ 3x - 5y = 7 ---(i)

_____________________

✫In second case

If length is reduced by 3 units and breadth is increased by 2 units, then the area of rectangle will increase by 67 square

✞ Area of rectangle = length × breadth

➨ xy + 67 = (x - 3)(y + 2)

➨ x(y + 2)-3(y + 2) = xy + 67

➨ xy + 2x - 3y - 6 = xy + 67

➨ 2x - 3y = 67 + 6

➨ 2x - 3y = 73 ---(ii)

_____________________

✬ Multiply (i) by 2 and (ii) by 3

• 6x - 10y = 14
• 6x - 9y = 219

✬ Subtract both the equations

➨ (6x - 10y) - (6x - 9y) = 14 - 219

➨ 6x - 10y - 6x + 9y = -205

➨ - y = -205

➨ y = 205

✬ Putting the value of y in eqⁿ (i)

➨ 3x - 5y = 7

➨ 3x - 5 × 205 = 7

➨ 3x - 1025 = 7

➨ 3x = 1025 + 7

➨ 3x = 1032

➨ x = 1032/3 = 344

Hence,

Required length = x = 344 unit

Required breadth = y = 205 unit

_____________________

Answer 2

Gɪᴠᴇɴ :-

If the length of a rectangle is reduced by 5 units and its breadth is increased by 3 units, then the area of the rectangle is reduced by 8 square units. If length is reduced by 3 units and breadth is increased by 2 units, then the area of rectangle will increase by 67 square units.

ᴛᴏ ғɪɴᴅ :-

• Length
• Breadth

sᴏʟᴜᴛɪᴏɴ :-

➦ Let length(l) be x units & breadth(b) be y units

Then,

➦ Area of Rectangle = length(l) × breadth(b)

➦ Area = xy unit²

Cᴏɴᴅɪᴛɪᴏɴ -1 :-

• Length = (x - 5) units

• Breadth = (y + 3) units

• Area = (xy - 8) unit²

We get,

→ (x - 5)(y + 3) = xy - 8

→ xy + 3x - 5y - 15 = xy - 8

→ xy - xy + 3x - 5y = 15 - 8

→ 3x - 5y = 7.

→ x = (7 + 5y) / 3. ---(1)

Cᴏɴᴅɪᴛɪᴏɴ -2 :-

• Length = (x - 3) units

• Breadth = ( y + 2) units

• Area = (xy + 67) unit²

We get,

→ (x - 3)(y + 2) = (xy + 67)

→ xy + 2x - 3y - 6 = xy + 67

→ xy - xy + 2x - 3y = 67 + 6

→ 2x - 3y = 73. ---(2)

Put value of (1) in (2) , we get,

$\implies \: 2x - 3y = 73 \\ \\ \\ \implies \: 2 \times( \frac{7 + 5y}{3} ) - 3y = 73 \\ \\ \\ \implies \: \frac{14 + 10y}{3} - 3y = 73 \\ \\ \\ \implies \: \frac{14 + 10y}{3} - \frac{3y \times 3}{1 \times 3} = 73 \\ \\ \\ \implies \: 14 + 10y - 9y = 73 \times 3 \\ \\ \\ \implies \: y = 219 - 14 = 205$

We get,

→ y = 205 units

Put y = 205 units in (2) , we get

→ 2x - 3y = 73

→ 2x - 3×205 = 73

→ 2x = 73 + 615

→ 2x = 688

→ x = 688/2

→ x = 344 units

Hence,

• Length = x = 344 units

• Breadth = y = 205 units