What fraction has the length of the rectangle been reduced?

[Natasha1]

Homework Statement

The width of a rectangle is increased by one tenth, but the area remains the same. By what fraction has the length of the rectangle been reduced?

The Attempt at a Solution

Length = x
Width = x + 1/10

Set equation:
L * W
x *(10x + x)
10x^2 + x^2

I am stuck... Please help...

 

[mjc123]

Why do you assume the length and width are initially equal? What is the width after increasing (It is not x + 1/10).
Try writing Before: length = L, width = W. After: length = Lx, width = Wy where x and y are constants. What is y? Therefore what is x?

 

[Natasha1]

Length = L*x
Width = W*y
where x and y are constants

W*(1/10) and L*x

 

[scottdave]

When i read this:

The width of a rectangle is increased by one tenth, but the area remains the same. By what fraction has the length of the rectangle been reduced?

I interpret as "the width is increased by 1/10 (of the original width)" or New_Width = Old_Width + Old_Width*(1/10)
I believe this is how the problem intends. Keep the area constant (set the new area equal to original).

 

[Natasha1]

original dimensions ---- w by l
original area = wl

new width = 9w/10
new length = L
L(9w/10) = lw
L = lw/(9w/10) = (10/9)l = 1.111..
so the length would have to increase which does not make sense as it says reduce in question.
The increase is 10/9 of the original or appr 11.1%

 

[scottdave]

original dimensions ---- w by l
original area = wl

new width = 9w/10

It looks like you are on the right track with the formulas, but you have the new width less than the original, while the problem states that it was increased.

 

[Natasha1]

new width = w + w/10
new length = L
L(11w/10) = lw
L = lw/(11w/10) = (10/11)l = 0.90909...
so the length would have to increase by 10/11 of the original or appr 9.09%

Is this correct please?

 

[scottdave]

Your numerical answer is correct, but the conclusion is not.
Since area is constant, if one dimension increases what happens to the other one? Think in extremes. If one dimension doubles, what happens to the other one?

 

[Natasha1]

so the length would have to decrease by 10/11 of the original or appr 9.09%