Question

If the length of a rectangle is decreased by 5 units and its breadth increased by 2 units then the area of the rectangle decreases by 80 square units. If its length is increased by 10 units and its breadth decreased by 5 units then the area of the rectangle increases by 50 square units. Find the original length and breadth of the rectangle.​

Answer 1

Answer:

The original length and breadth of the rectangle are 40 units and 30 units respectively.

Step-by-step-explanation:

Let the original length of the rectangle be l units.

And the original breadth of the rectangle be b units.

We know that,

Area of rectangle ( A ) = Length * Breadth

∴ A = lb sq. units

From the first condition,

• Length = ( l - 5 ) units
• Breadth = ( b + 2 ) units
• Area = ( lb - 80 ) sq. units

∴ ( l - 5 ) ( b + 2 ) = ( lb - 80 )

⇒ lb + 2l - 5b - 10 = lb - 80

⇒ lb + 2l - 5b - lb = - 80 + 10

⇒ 2l - 5b = - 70

⇒ l - 2.5b = - 35

⇒ l = - 35 + 2.5b

⇒ l = 2.5b - 35 - - - ( 1 )

From the second condition,

• Length = ( l + 10 ) units
• Breadth = ( b - 5 ) units
• Area = ( lb + 50 ) sq. units

∴ ( l + 10 ) ( b - 5 ) = ( lb + 50 )

⇒ lb - 5l + 10b - 50 = lb + 50

⇒ lb - 5l + 10b - lb = 50 + 50

⇒ - 5l + 10b = 100

⇒ - l + 2b = 20

⇒ - ( 2.5b - 35 ) + 2b = 20 - - - [ From ( 1 ) ]

⇒ - 2.5b + 35 + 2b = 20

⇒ - 2.5b + 2b = 20 - 35

⇒ - 0.5b = - 15

⇒ 0.5b = 15

⇒ b = 15 ÷ 0.5

⇒ b = 30 units

By substituting b = 9 in equation ( 1 ), we get,

l = 2.5b - 35 - - - ( 1 )

⇒ l = 2.5 * 30 - 35

⇒ l = 75 - 35

⇒ l = 40 units

∴ The original length and breadth of the rectangle are 40 units and 30 units respectively.

Answer 2

Question:

• If the length of a rectangle is decreased by 5 units and its breadth increased by 2 units then the area of the rectangle decreases by 80 square units. If its length is increased by 10 units and its breadth decreased by 5 units then the area of the rectangle increases by 50 square units. Find the original length and breadth of the rectangle.

Answer:

• Original length and breadth of rectangle are 30 units and 80 units.

Explanation:

Given that:

• If the length of a rectangle is decreased by 5 units and its breadth increased by 2 units then the area of the rectangle decreases by 80 square units.

• If its length is increased by 10 units and its breadth decreased by 5 units then the area of the rectangle increases by 50 square units.

To Find:

• Original length and breadth of the rectangle?

Solution:

★ Formula used ::

$\large{\boxed{\sf{\pink{Area_{(rectangle)} = Length\:\times\:Breadth}}}}$

• Let original length of rectangle be L units

• And, original breadth of rectangle be B units

• So, area of rectangle is LB

According to 1st condition,

• If the length of a rectangle is decreased by 5 units and its breadth increased by 2 units then the area of the rectangle decreases by 80 square units.

Therefore,

• Length of rectangle = (L - 5) units

• Breadth of rectangle = (B + 2) units

• Area of rectangle = (LB - 80) square units

By plugging all values in formula we get,

↣ $\sf (L - 5)\:\times\:(B + 2) = LB - 80$

↣ $\sf L(B + 2) - 5(B + 2) = LB - 80$

↣ $\sf LB + 2L - 5B - 10 = LB - 80$

↣ $\sf LB + 2L - 5B - LB = - 80 + 10$

↣ $\sf \cancel{LB} - \cancel{LB} + 2L - 5B = -70$

↣ $\bf 2L - 5B = -70\quad - - - (1)$

According to 2nd condition,

• If its length is increased by 10 units and its breadth decreased by 5 units then the area of the rectangle increases by 50 square units.

Therefore,

• Length of rectangle = (L + 10) units

• Breadth of rectangle = (B - 5) units

• Area of rectangle = (LB + 50) square units

By plugging all values in formula we get,

↣ $\sf (L + 10)\:\times\:(B - 5) = LB + 50$

↣ $\sf L(B - 5) + 10(B - 5)) = LB + 50$

↣ $\sf LB - 5L + 10B - 50 = LB + 50$

↣ $\sf LB - 5L + 10B - LB = 50 + 50$

↣ $\sf \cancel{LB} - \cancel{LB} - 5L + 10B = 100$

↣ $\sf -5L + 10B = 100$

↣ $\sf -5(L - 2B) = 100$

↣ $\sf L - 2B = -{\cancel{\dfrac{100}{5}}}$

↣ $\sf L - 2B = -20$

↣ $\bf L = -20 + 2B\quad - - - (2)$

From (2) put in (1) we get,

↣ $\sf 2(-20 + 2B) - 5B = -70$

↣ $\sf -40 + 4B - 5B = -70$

↣ $\sf -40 + (-B) = -70$

↣ $\sf -40 - B = -70$

↣ $\sf -B = -70 + 40$

↣ $\sf \cancel{-}B = \cancel{-}30$

➼ $\bf\red{B = 30}$

$\large{\boxed{\sf{\therefore\:Breadth\:of\:rectangle = {\textsf{\textbf{30\:units}}}}}}$

Put B = 30 in (2) we get,

↣ $\sf L = -20 + (2\:\times\:30)$

↣ $\sf L = -20 + 60$

➼ $\bf\purple{L = 40}$

$\large{\boxed{\sf{\therefore\:Length\:of\:rectangle = {\textsf{\textbf{40\:units}}}}}}$

∴ Original length and breadth of rectangle are 30 units and 80 units.

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