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A square wooden plaque has a square brass inlay in the center, leaving a wooden strip of uniform width around the brass square. If the ratio of the brass area to the wooden area is 25 to 39, which of the following could be the width, in inches, of the wooden strip?

I. 1 II. 3 III. 4

(A) I only (B) II only (C) I and II only (D) I and III only (E) I, II , and III

Problem Solving Question: 175 Category:Geometry Area Page: 85 Difficulty: 600

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A square wooden plaque has a square brass inlay in the center ,leaving a wooden strip of uniform width around the brass square.if the ratio of the brass area to the wooden area is 25 to 39,which of the following could be the width ,in inches ,of the wooden strip.

I. 1 II. 3 III. 4

A.I only B.II only C.III only D.I and III only e.I,II and III

We are asked which value of \(\frac{y-x}{2}\) is possible. \(\frac{y-\frac{5}{8}y}{2}=\frac{3}{16}y=?\).

Well, expression \(\frac{3}{16}y\) can take ANY value depending on \(y\): 1, 3, 4, 444, 67556, 0,9, ... ANY. Basically we are given the ratios of the sides (5/8), half of their difference can be any value we choose, there won't be any "impossible" values at all.

Re: hard problem OG Quant 2nd edition [#permalink]

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15 Jan 2010, 09:49

let the width of the wooden part = w let the width of the brass part = b given brass area/wooden area = 25/39 area of brass part = b^2 area of wooden part = (b+2w)^2 - b^2 Simplifying 64b^2=25(b+2w)^2 8b=5b+10w b=10w/3

b has to be an integer or a terminating decimal. We can't have a width of 10/3 in real life (note the question doesn't ask for approximate width.) hence w has to be a multiple of 3. Answer is B.
_________________

let the width of the wooden part = w let the width of the brass part = b given brass area/wooden area = 25/39 area of brass part = b^2 area of wooden part = (b+2w)^2 - b^2 Simplifying 64b^2=25(b+2w)^2 8b=5b+10w b=10w/3

b has to be an integer or a terminating decimal. We can't have a width of 10/3 in real life (note the question doesn't ask for approximate width.) hence w has to be a multiple of 3. Answer is B.

Are you saying that in real life everything has the integer or terminating decimal length? Why cannot we have repeated decimal or even irrational number as width of something?

Take the square with side 1, diagonal would be \(\sqrt{2}\), it's not an integer or terminating decimal.

Also it's possible to divide the line segment into three equal parts, Google it and you find that it's quite easy.
_________________

Re: hard problem OG Quant 2nd edition [#permalink]

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15 Jan 2010, 11:25

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Quote:

Are you saying that in real life everything has the integer or terminating decimal length? Why can not we have repeated decimal or even irrational number as width of something?

Take the square with side 1, diagonal would be \(\sqrt{2}\), it's not an integer or terminating decimal.

Also it's possible to divide the line segment into three equal parts, google it and you find that it's quite easy.

That takes the question to a whole new dimension, I do understand what you are saying though. If the width of something is 10/3 that means you can never (accurately) measure it. The width of the brass square can never be measured practically, it can be only measured mathematically. If such a square was to be made, the creator would have to take a square of 10/10 dimension, divide it into 9 exactly equal parts and use one of them, he/she could never make just the square as it would be impossible to measure 10/3 inches. I do get the concept, but don't like the fact that a question can be based on it.
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Re: hard problem OG Quant 2nd edition [#permalink]

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15 Jan 2010, 13:09

Bunuel wrote:

Basically we are given the ratios of the sides (5/8), half of their difference can be any value we choose, there won't be any "impossible" values at all.

Hope it's clear.

Wow such a simple concept i must have left my brain someplace else.

Re: hard problem OG Quant 2nd edition [#permalink]

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24 Mar 2010, 06:10

5

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hrish88 wrote:

A square wooden plaque has a square brass inlay in the center ,leaving a wooden strip of uniform width around the brass square.if the ratio of the brass area to the wooden area is 25 to 39,which of the following could be the width ,in inches ,of the wooden strip.

I. 1 II. 3 III. 4

A.I only B.II only C.III only D.I and III only e.I,II and III

Area of brass square/ area of wooden strip = 25 /39 lets say length of the wooden plaque= l and length of the square brass = x then x^2 / (l^2 - x^2) = 25/39 =>39x^2 = 25l^2 - 25x^2 =>64x^2 = 25l^2 =>8x = 5l Width of wooden strip should be l-x =>x = 5l/8 so l -x = l = 5l/8 = 3l/8

Now 3l/8 could be any value depending on the value of l so answer is E.
_________________

We are asked which value of \(\frac{y-x}{2}\) is possible. \(\frac{y-\frac{5}{8}y}{2}=\frac{3}{16}y=?\).

Well, expression \(\frac{3}{16}y\) can take ANY value depending on \(y\): 1, 3, 4, 444, 67556, 0,9, ... ANY. Basically we are given the ratios of the sides (5/8), half of their difference can be any value we choose, there won't be any "impossible" values at all.

Hope it's clear.

To generalize then, since the answer does not seem to depend on the fact that the ration is 25/39, can it be said that regardless of what the ratio is, the width of strip can be ANYTHING?
_________________

We are asked which value of \(\frac{y-x}{2}\) is possible. \(\frac{y-\frac{5}{8}y}{2}=\frac{3}{16}y=?\).

Well, expression \(\frac{3}{16}y\) can take ANY value depending on \(y\): 1, 3, 4, 444, 67556, 0,9, ... ANY. Basically we are given the ratios of the sides (5/8), half of their difference can be any value we choose, there won't be any "impossible" values at all.

Hope it's clear.

To generalize then, since the answer does not seem to depend on the fact that the ration is 25/39, can it be said that regardless of what the ratio is, the width of strip can be ANYTHING?

Yes, width can have any positive value: the larger the width is the larger the whole square would be.
_________________

Thanx for the explanation. I didn't understand one part:

We are asked which value of \frac{y-x}{2} is possible. \frac{y-\frac{5}{8}y}{2}=\frac{3}{16}y=?

Please enlighten me.

We have that \(\frac{x}{y}=\frac{5}{8}\) --> \(x=\frac{5}{8}y\)

The width of the wooden strip would be \(\frac{y-x}{2}\), substitute \(x\): \(\frac{y-\frac{5}{8}y}{2}=\frac{3}{16}y\).

So the question is: which of the following could be the value of \(\frac{3}{16}y\)? Answer: expression \(\frac{3}{16}y\) can take ANY value depending on \(y\).

Re: hard problem OG Quant 2nd edition [#permalink]

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19 Sep 2010, 15:41

I actually created equation and substituted width value to conclude E. I should have simply thought straight like Bunuel and marked E in 15 sec.
_________________

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Re: hard problem OG Quant 2nd edition [#permalink]

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17 Jan 2011, 12:50

Bunuel wrote:

hrish88 wrote:

A square wooden plaque has a square brass inlay in the center ,leaving a wooden strip of uniform width around the brass square.if the ratio of the brass area to the wooden area is 25 to 39,which of the following could be the width ,in inches ,of the wooden strip.

I. 1 II. 3 III. 4

A.I only B.II only C.III only D.I and III only e.I,II and III

Why would ANY width of the strip be impossible?

Answer: E.

That was my rationale exactly. I got the answer correct, but the explanation in the book made me feel like I did not grasp the underlying math. Thanks Bunuel.

I didn't understand the statement "We are asked which value of (y-x)/2 is possible" . Can someone explain?

Question asks about the possible width, in inches, of the wooden strip.

Let the the side of small square be \(x\) and the big square \(y\), then the width of the wooden strip would be \(\frac{y-x}{2}\), which means that we are asked to determine the possible values of this exact expression.

Re: hard problem OG Quant 2nd edition [#permalink]

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01 Sep 2012, 07:48

Bunuel wrote:

hrish88 wrote:

A square wooden plaque has a square brass inlay in the center ,leaving a wooden strip of uniform width around the brass square.if the ratio of the brass area to the wooden area is 25 to 39,which of the following could be the width ,in inches ,of the wooden strip.

I. 1 II. 3 III. 4

A.I only B.II only C.III only D.I and III only e.I,II and III

Why would ANY width of the strip be impossible?

Answer: E.

Hi Bunuel, this was the most appropriate reason for the answer; however, can there be a case where such a condition("any possible width of the strip") might fail, provided there is no restriction on dimensions to be integral or non integral?
_________________

Whatever one does in life is a repetition of what one has done several times in one's life! If my post was worth it, then i deserve kudos

Re: A square wooden plaque has a square brass inlay in the [#permalink]

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18 Sep 2012, 11:50

Bunuel wrote:

mymbadreamz wrote:

Why is it not y-x? Why do we calculate (y-x)/2?

Consider the diagram below:

Attachment:

Wooden strip.png

As you can see the width of the wooden strip (the width of grey strip) is \(\frac{y-x}{2}\).

total length is 8x and the length of the countertop is 5x.so the one side length of the untiled area is w = 8x-5x/2 =3x/2 .Since x could be any value so the answer is E..Am I right Bunuel ?

gmatclubot

Re: A square wooden plaque has a square brass inlay in the
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18 Sep 2012, 11:50

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