Equation - Periphery, Length / Solution - Explanation

[Guest]

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.

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.

[Guest]

Please reply to my question. Thanks.

[Guest]

You may not understand the solution
I don't understand the question
a rectangle does not have a diameter

Ask you sensible questions

[Guest]

"You may not understand the solution
I don't understand the question
a rectangle does not have a diameter"

Can you explain it by omitting the statement about diameter ?

[Guest]

Ask a sensible question

State the facts logically

[Guest]

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.


Error in statement : The pond's area is 1/9 part of the whole garden's area.

Here is my understanding of the problem:

In one corner of a rectangular garden is a pond.

The area of pond is 1/9 the area of the total garden's area.

The periphery of the garden is larger than the pond by 200 yards.

The length of the garden is increased by 3 yards and the width is increased by 5 yards.

By increasing the length and width the garden's area is increased by 645 square yards.


Determine periphery and length and width of garden.

[Guest]

And what about the rectangular pond that is is in one corner of the garden and approx. the diameter or width ???? of the graden. A rectangle can be any shape as long as it is rectangular in shape. And how approximate is approx.

[Guest]

The pond is also rectangular. The problem stated the pond had approx. the same diameter as the garden. Because it was stated that a rectangle has no diameter, I omitted that statement.

[Guest]

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


I am not clear on these steps of the solution.

[Guest]

You would need to try and explain what you are trying to work out.

Is the pond a circular (complete circle) pond within the garden placed near one corner, OR is it rectangular pond in a rectangular garden..??

OR is it a quadrant pond sitting in one of the corners of a rectangular garden....???

OR is it a rectangular pond within a circular garden....???

[Guest]

I am assuming the pond is also rectangular inside the rectangular garden.

[Guest]

Your last post.... "I am assuming the pond is also rectangular inside the rectangular garden."
This is useless information......a rectangle can be any shape or size. Don't post questions except you know the facts about the question.

From the first post..... "The pond is also rectangular with approx. the same diameter as the garden."

I am assuming this to mean...."The pond is also rectangular with the same shape as the garden."

Then the question simply makes more sense............

The facts now are........
It does not matter where the pond is in the garden, or even if it is in it.
The pond area is 1/9 of area of garden.
The perimeter of the garden is 200 yards more than the perimeter of the pond.
When the garden length is increased by 3 yards and the width is increased by 5 yards the new area is 645 square yards more than the original.

Let L = length of original garden and W = width of original garden.
Garden area is L x W = LW square yards.....Pond area is 1/9 times LW = 1/9LW
and... Pond length will be 1/3L and width will be 1/3W

Difference in perimeters will be
2(L + W) - 2(1/3L + 1/3W) = 200
4/3L + 4/3W = 200
L + W = 150
W = 150 - L .......eqn1
......................

Increase in garden area......
3(W + 5) + 5L = 645
3W + 5L = 630

Substitute from eqn1

450 - 3L + 5L = 630
2L = 180
L = 90 yards

This means that W = 60 yards

Check.........
Garden Area is 90 x 60 = 5400 sq yards....Pond area is 600 sq yards

New garden area is 93 x 65 = 6045
Difference is 645 sq yards

Pond length is 30 yards and width is 20 ....perimeter is 100 yards
Garden perimeter is 2 x (90 + 60) = 300 yards ....difference is 200 yards.

[Guest]

I understand the solution now.

Two questions :

I rechecked the problem statement about the diameter - the pond is a rectangle about the same diameter with the garden. Would that change the outcome ?


I had reversed x and y - x being the shorter side and y being the longer side. Would that change the outcome ?

[Guest]

Two questions : ...Your questions....


I rechecked the problem statement about the diameter - the pond is a rectangle about the same diameter with the garden. Would that change the outcome ?
Reply.... I assumed a typing error and the closest I assumed was..."same shape"...meaning similar in shape....as in similar triangles etc.

Diameter....normally refers to distance across a circle passing through the centre, or at least the distance across any object passing through its centre.
This doesnt have any meaning I can see for the question posted since the garden is rectangular and rectangular pond.....and does not help in any way.
What we need to know for the question is the relationship between the linear dimensions of the sides of the pond and the area of the pond.
We are told the area of the pond is 1/9 of the garden.....and this was not clear in the question either so I assumed it to be 1/9 of the original area.
We were also told the perimeter of the pond was 200 yards less than the garden but we did not know anything about the lengths of the pond sides or how they related to the garden. We could have been told the pond was circular then we could have worked out the perimeter form the area. Similarily if we had been told it was square the length of the sides would have been the sq root of the area. But no information was forthcoming so I assumed the pond to have a similar shape to the garden. As in similar triangles corresponding sides of the pond will then have same ratio of length to corresponding sides of the garden, and similarily the ratio of corresponding areas will be the "sides ratio squared". So 1/3L x 1/3W = 1/9LW. So if pond area is 1/9 then sides length is 1/3.



I had reversed x and y - x being the shorter side and y being the longer side. Would that change the outcome ?
Reply.... No, reversing x and y would not change the outcome of the solution provided you followed your work through correctly. x and y are just indicators for the unknowns and could be A and B OR L and W that I used.