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Question
A rectangular garden has in one corner a pond. The pond's area is 1/9 part of the total 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.

2(x + y) = periphery of garden.

1/3[2(x + Y)] = periphery of pond.

2(x + y) - 2/3(x + y) = 200

2/3(x + y) = 100

x + y = 150

(y + 3) (x + 5) = xy + 645

3x + 5y = 630

3x + 3y = 450

2y = 180
y = 90

x = 150 - y = 60

periphery = 300 yards.

sides = 60 and 90 yards.

I don't understand the solution. Thanks.

Answer
The solution is unclear because the question was poorly phrased and contains an error.  For example, it states that
 "the pond's area is 1/9 part of the total garden"
and
 ⅓x and ⅓y = lengths of longer and shorter sides of pond
The second statement does not follow from the first!—unless the pond and garden are similar rectangles, which was not stipulated.

Later it states that "the pond is also rectangular with approx. the same diameter as the garden," which is impossible.

"The periphery of the garden exceeds the pond by 200 yards."
Is this an old textbook, or a text that is translated from another language? It is unusual to refer to "the periphery" instead of "the perimeter," especially since periphery often refers to a boundary, not the length of that boundary.

perimeter of garden = 2(x+y)
perimeter of pond = 2(⅓x+⅓y) = ⅔(x+y)
2(x+y) - ⅔(x+y) = 200
(2-⅔)(x+y) = 200
x+y = 200/(2-⅔) = 150

"if the longer side of the garden is increased by 3 yd and the shorter side by 5 yd, the area of the garden increases 645 yd˛"
Note that their equation
 (y+3)(x+5) = xy+645
is incorrect. Since the longer side is x, the left side of the equation should be (x+3)(y+5).

original area = xy
new area = (x+3)(y+5) = xy+5x+3y+15

xy+5x+3y+15 = xy + 645
5x+3y = 630
(2x+3x)+3y = 630
2x+3(x+y) = 630
2x+3(150) = 630
2x = 180
x = 90
y = 150-x = 60

The dimensions of the original garden were 90 yd by 60 yd. Its perimeter was 300 yd.

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