Width - Shaded Portion of Square / Partial Solution

[Guest]

A smaller square is placed inside a larger square. The larger square is 6 inches on each side. A portion of the smaller square is shaded and is 4/9 of the area of each square. Determine the width of the shaded portion.

A partial solution is included with problem:

Let x = length of shaded portion in either figure.

Total shaded area in each is 4 * x * 6 - x / 2

How do I proceed from this point ?

[Guest]

Please reply to my question. Thanks.

[Guest]

Missing information...
What is shape of shaded portion
and it cannot be 4/9 of the small square and also 4/9 of the large square

[Guest]

"Missing information...
What is shape of shaded portion"

The shape of shaded portion is a rectangle.

Error in post:

The large square is 6 inches on each side. The shaded portion is 4/9 of the area of the square. There is an unshaded white square in each corner of the larger square.

I have an illustration but I am not sure how to post it.

[Guest]

You are only adding to the confusion
A white square taken from each corner of the large square leaves a cross shape inside the large square..?????

[Guest]

The width of the shaded rectangles are equal to the length of the unshaded squares. One unshaded square is in each corner. The shaded rectangles are on each side and top and bottom of the larger square and the unshaded squares are on each side of the rectangles. There are 4 shaded rectangles.

[Guest]

Is this a different question.?
you have forgotten about the bit in the middle
and the other bit that is 4/9 the area of something else

[Guest]

The square in the middle is unshaded. The total shaded area is 4/9 of the whole square.

[Guest]

Well then, The problem is totally different that described on first post.
So it has 4 unshaded squares at the corners and an unshaded square in the middle

so its now a simple problem

[Guest]

And the shaded portions is now 4 separate portions

I assume you want to now find the dimensions of each shaded portion

The answer is.... either each is 2 inch by 2 inch square or a 4 inch by 1 inch rectangle

which gives 4/9 of the large square shaded

[Guest]

Please post how you got the answer.

[Guest]

A rough diagram of how I understand your problem is....
It may not look very square due to proportional fonts etc, but should give the idea of a square of dimensions.....
(x + y + x) by (x + y + x) = 6 by 6 inches

<-x-+<------y----->+-x->

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Let the shaded rectangles have width = x and length = y

and whole diagram is a square 6 by 6 inches

Then you are telling me 2x + y = 6 .....the size of the large square
This gives us that y = 6 -2x

Also you say the area shaded is 4/9 of the large square >>> 4/9 x 36 = 16

So 4 times xy = 16 OR 4*(xy) = 16 giving xy = 4

Therefore x(6-2x) = 4

This gives.... -2x^2 + 6x = 4 OR -2x^2 + 6x - 4 = 0

Divide by factor of 2

gives -x^2 +3x -2 = 0 ...solve for x by factoring

(-x+2)(x+1) = 0

Either -x+2=0 OR x+1=0 gives

Either -x=-2 gives x=2 OR x=1

When x = 2 .....y= 6-2x = 2 so each shaded is a square of 2 by 2

When x = 1 .....y= 6-2x = 4 so each shaded is a rectangle of 4 by 1

and 4 times 2 by 2 is 16 square inches equals 4/9 of large square

and 4 times 4 by 1 is 16 square inches equals 4/9 of large square


........I thought you should have been able to make some attempt to solve it yourself........

[Guest]

To see the diagram more clearly as a square

Copy and paste the diagram into windows notepad and look at in "Courier" font or any other non proportional font

[Guest]

I have located the problem in a different location with this solution :

Let x = length of shaded portion.

Total shaded area = 4 * x * 6-x/2

4x * 6-x/2 = 4/9 * 36 = 16 *

x^2 - 6x + 8 = 0 **

(x-2) (x-4) = 0

x = 2 , x = 4

* I know this step is similar to your solution.

** I don't understand this step.

[Guest]

Its a different problem, the equations are different from the start. The solution in the post with the text diagram is for the diagram because I was unable to decipher your question as posted.
Post the QUESTION correctly so that it makes sense to read, then you may get the correct solution.

[Guest]

The large square is 6 inches on each side. The shaded portion is 4/9 of the area of the square. Determine width of the shaded portion.

The problem has two illustrations.

"A white square taken from each corner of the large square leaves a cross shape inside the large square..?????"

Your question is an accurate description of the first illustration.


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An accurate rough diagram of the second illustration.

[Guest]

This is a solution for the cross shaped sheded.

"<-x-+<------y----->+-x->

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The above diagram illustrates the case whre there are 4 unshaded squares in the corners and the rest is shaded.
The area shaded is shape of a cross. copy and paste it in windows notepad and use courier font to see it in a better shape.

The shaded area is still 4/9 of the large square area.

Now the starting equation is ..... y^2 + 4xy = 16

and substituting in y as from before gives....
(6 - 2x)^2 + 4x(6 - 2x) = 16

36 +4x^2 - 24x + 24x - 8x^2 = 16

36 - 4x^2 = 16

-4x^2 = -20
x^2 = 5

X = sqroot(5) = 2.236 inches

Y = (6 - (2 x 2.236)) = (6 - 4.472) = 1.528 inches

That means 4 unshaded squares at the corners each 2.236 x 2.236
That gives total area unshaded = 4 x 2.236 x 2.235 = 4 x 5 = 20 = 5/9 of the original square unshaded, so 4/9 is shaded.

[Guest]

x^2 - 6x + 8 = 0 **

I don't understand this step in the solution with problem.

[Guest]

Your question in the last post is about the solution you provided. But you seem to not be able to describe the question in a way that I can understand what the problem is asking or wanting worked out. That is why I had to make my own assumptions and do a solution based on that. And I provided 2 diagrams and 2 solutions for 2 different situations.

Below is from your last post.......
Let x = length of shaded portion.......what do you mean by the length of the shaded portion, You cannot describe it to me so how do I know what it is. It one post you say it is shape of a cross with 4 corners not shaded.....How can this have length.???

Total shaded area = 4 * x * 6-x/2 ......this means nothing to me except you can describe it in words. Why would you have x/2...?? minus half the length..????.....I assume there should be some brackets to make this more readable

4x * 6-x/2 = 4/9 * 36 = 16 *.........Also I assume some brackets missing here as well, but I don't know where the equation comes from.
I can assume it maybe should be..... 4x(6 -x)/2 = 16....But I dont know what it relates to in the problem question.
gives(24x - 4x^2)/2 = 16
gives 12x - 2x^2 = 16
gives 6x - x^2 = 8
Rearranging gives 6x - x^2 - 8 = 0
OR -x^2 + 6x - 8 = 0
divide across by -1 gives .... x^2 - 6x + 8 = 0
Which is what you have.....??????? The equations are probably wrong to start with because I do not know what the question is. If this is the answer you are looking for please post a diagram of the square and the shaded portion so that I can understand the question.

All I have done here is manipulate the algebra to see if I could arrive at the final equation you had, but I dont know what shape of shaded area they refer to...????. Wen typing out algebra it is very important to make proper use of brackets.

x^2 - 6x + 8 = 0 **

When doing any of these type of questions you need to first understand what the facts of the question are. Then logically in descriptive language terms describe what the question is asking you to find or work out. Then you need to put this together logically in your mind so that you can work through it one step at a time and end up with the answer.
Then and only then you should set out the question mathematically in the form of mathematical statements or equations to follow the logical thinking you had set out to do. Then solve the equations.

[Guest]

I have located a link to the problem :

https://archive.org/stream/intermediate ... 3/mode/2up

Maybe that will be a significant help.

Thanks again.