Equation - Periphery, Length / Solution - Explanation

A rectangular garden has in one corner a pond. The pond's area is 1/9 part of the garden. The periphery of the garden exceeds the pond by 200 yards. The longer side of the garden is increased by 3 yards and the shorter side by 5 yards. The increase enlarges the garden by 645 square yards. The pond is also rectangular with approx. the same diameter as the garden. Determine periphery and length of each side of garden.

Let x = longer side
Let y = shorter side

1/3x and 1/3y = lengths of longer and shorter sides of pond. Since the pond is proportional to the garden, the solution assumes both sides are shorter by the same factor and because the pond's area is 1/9 the total, that factor must be the squareroot of 1/9 = 1/3

2(x + y) = periphery of garden. (Standard formula for perimeter of a rectangle)

1/3[2(x + Y)] = periphery of pond. (Same, although the solver combined a few terms. Clearer would be 2(1/3x + 1/3y))

2(x + y) - 2/3(x + y) = 200 (Perimeter of garden is 200 more than the perimeter of the pond)

2/3(x + y) = 100 (Simplification of the previous equation)

x + y = 150 (Simplification of the previous equation. We are going to use this below.)


(y + 3) (x + 5) = xy + 645 (This is the area part of the problem. If you increase the length by 3 (y+3) and the width by 5 (x+5), the total area will be 645 more than the initial area.)

3x + 5y = 630 (Lots of simplification of the previous equation)
(y+3)(x+5) = xy + 645
xy + 3x + 5y + 15 = xy + 645
3x + 5y = 630

3x + 3y = 450 (From the first equation, multiplying by 3.)

2y = 180 (Subtracting 3x+3y = 450 from 3x+5y = 630)
y = 90

x = 150 - y = 60

periphery = 300 yards.

sides = 60 and 90 yards