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Math Expert V
Joined: 02 Sep 2009
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Difficulty:   95% (hard)

Question Stats: 39% (02:31) correct 61% (02:49) wrong based on 2073 sessions

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The Official Guide For GMAT® Quantitative Review, 2ND Edition

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

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hrish88 wrote:
so doest it mean any width <= 39/4 is possible.

this is the 2nd last problem in OG.so i thought it would be difficult. No I mean ANY width is possible.

Let the the side of small square be $$x$$ and the big square $$y$$.

Given: $$\frac{x^2}{y^2-x^2}=\frac{25}{39}$$ --> $$\frac{x^2}{y^2}=\frac{25}{64}$$ --> $$\frac{x}{y}=\frac{5}{8}$$.

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.
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Math Expert V
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Re: hard problem OG Quant 2nd edition  [#permalink]

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1
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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

Why would ANY width of the strip be impossible?

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Posts: 165
Re: hard problem OG Quant 2nd edition  [#permalink]

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7
2
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
##### General Discussion
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Posts: 24
Re: hard problem OG Quant 2nd edition  [#permalink]

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so doest it mean any width <= 39/4 is possible.

this is the 2nd last problem in OG.so i thought it would be difficult. Manager  Joined: 25 Aug 2009
Posts: 92
Location: Streamwood IL
Schools: Kellogg(Evening),Booth (Evening)
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Re: hard problem OG Quant 2nd edition  [#permalink]

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1
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.
Math Expert V
Joined: 02 Sep 2009
Posts: 65807
A square wooden plaque has a square brass inlay in the  [#permalink]

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atish wrote:
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.

Are you saying that in real life everything has 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.
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Re: hard problem OG Quant 2nd edition  [#permalink]

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2
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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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.

Nice explanation.You rock man as always.
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Re: hard problem OG Quant 2nd edition  [#permalink]

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1
Bunuel wrote:
hrish88 wrote:
so doest it mean any width <= 39/4 is possible.

this is the 2nd last problem in OG.so i thought it would be difficult. No I mean ANY width is possible.

Let the the side of small square be $$x$$ and the big square $$y$$.

Given: $$\frac{x^2}{y^2-x^2}=\frac{25}{39}$$ --> $$\frac{x^2}{y^2}=\frac{25}{64}$$ --> $$\frac{x}{y}=\frac{5}{8}$$.

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?
Math Expert V
Joined: 02 Sep 2009
Posts: 65807
Re: hard problem OG Quant 2nd edition  [#permalink]

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2
1
mainhoon wrote:
Bunuel wrote:
hrish88 wrote:
so doest it mean any width <= 39/4 is possible.

this is the 2nd last problem in OG.so i thought it would be difficult. No I mean ANY width is possible.

Let the the side of small square be $$x$$ and the big square $$y$$.

Given: $$\frac{x^2}{y^2-x^2}=\frac{25}{39}$$ --> $$\frac{x^2}{y^2}=\frac{25}{64}$$ --> $$\frac{x}{y}=\frac{5}{8}$$.

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

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Hi Bunuel,

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=?

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

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appy001 wrote:
Hi Bunuel,

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=?

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 $$y$$.

Hope it's clear.
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Re: hard problem OG Quant 2nd edition  [#permalink]

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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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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?

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

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I didn't understand the statement "We are asked which value of (y-x)/2 is possible" . Can someone explain?
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Posts: 65807
Re: A square wooden plaque has a square brass inlay in the  [#permalink]

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1
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.

Hope it's clear.
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Posts: 65807
Re: A square wooden plaque has a square brass inlay in the  [#permalink]

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6
3
Why is it not y-x? Why do we calculate (y-x)/2?

Consider the diagram below:
Attachment: Wooden strip.png [ 2.75 KiB | Viewed 75038 times ]
As you can see the width of the wooden strip (the width of grey strip) is $$\frac{y-x}{2}$$.
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Re: hard problem OG Quant 2nd edition  [#permalink]

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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?

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

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Bunuel 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 ? Re: A square wooden plaque has a square brass inlay in the   [#permalink] 18 Sep 2012, 10:50

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