Question

New Clarendon Park is undergoing renovations to its gardens. One garden that was originally a square is being adjusted so that one side is doubled in length, while the other side is decreased by three meters.
The new rectangular garden will have an area that is 25% more that the original square garden. Write an equation that could be used to determine the length of a side of the original square garden.
Determine the area, in square meters, of the new rectangular garden.

Answer 1

The answer is 80 square meters.

The square area is expressed as:
A = a²,
where A is the area of the square, and a is the side of the square.

The rectangle area is expressed as:
A₁ = a₁ · b₁,
where A₁ is the area of the rectangle, and a₁ and b₁ are the sides of the rectangle.

After renovations, square garden becomes rectangular.

One side is doubled in length:
a₁ = 2a

The other side is decreased by three meters.
b₁ = a - 3

The new area is 25% than the original square garden:
A₁ = A + 25%A =
     = A + 25/100·A
     = A + 1/25·A
     = a² + 1/25·a²
     = a² + 0.25·a²
     = 1.25·a²

If the starting equation is:
A₁ = a₁ · b₁

Thus, the equation is:
1.25a² = 2a·(a - 3)
1.25a² = 2a · a - 2a · 3
1.25a² = 2a² - 6a

Therefore, the equation that could be used to determine the length of a side of the original square garden is:
2a² - 6a = 1.25a²


Now, we will solve the equation:
2a² - 6a = 1.25a²
2a² - 1.25a² - 6a = 0
0.75a² - 6a = 0
⇒ a(0.75a - 6) = 0

From here, one of the multiplier must be zero - either a or (0.75a - 6). Since a could not be zero, (0.75a - 6) is:
0.75a - 6 = 0
0.75a = 6
a = 6 ÷ 0.75
a = 8

If the side of the square is 8, then the area of the rectangle is
A₁ = 1.25 · a²
A₁ = 1.25 ·8²
A₁ = 1.25 · 64
A₁ = 80

Therefore, the area of the new rectangle garden is 80 square meters.