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

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Bunuel wrote:
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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L = length of wooden plaque
X - length of brass inlay

We are told that x^2/(L^2-x^2) = 25/39

39x^2 = 25L^2 - 25x^2
64x^2 = 25L^2
8x = 5L
x/L = 5/8

Now we know that the width of the frame is (L - x)/2

8x/5 = L

(8x/5 - x)/2 = (16x - 5x)/10*2

11x/20

or

x = 5L/8
(L - 5/8L)/2

= 3L / 16

Any values for x or L are possible because neither needs to be an integer.
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Re: A square wooden plaque has a square brass inlay in the  [#permalink]

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If the ratio of areas is given then the areas can be anything,
If the areas can be anything​ then the sides can be anything,
If the sides can be anything then the width can be anything.

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

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Bunuel wrote:
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

Let's say we want to create a plaque with a square brass inlay in the center, and we want the brass to wood ratio to be 25:39

Let's begin a square wooden board with ANY dimensions. Now place a square brass inlay in the middle of the wooden board, and keep adjusting the size of the brass inlay until we have a brass to wood ratio that is 25:39 At this point, if we shrink or expand the plaque . . . . . . the brass to wood ratio will remain at 25:39

So, as you can see, this plaque can be ANY size, which means the width of the wooden strip can have ANY measurement.

Cheers,
Brent
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Originally posted by GMATPrepNow on 15 Sep 2017, 12:19.
Last edited by GMATPrepNow on 16 Apr 2018, 12:34, edited 1 time in total.
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GMAT 1: 620 Q36 V39 GMAT 2: 650 Q47 V33 A square wooden plaque has a square brass inlay in the  [#permalink]

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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.

I have difficulties this this part $$\frac{x^2}{y^2-x^2}=\frac{25}{39}$$ --> $$\frac{x^2}{y^2}=\frac{25}{64}$$

How do you get it?
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Re: A square wooden plaque has a square brass inlay in the  [#permalink]

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Erjan_S 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.

I have difficulties this this part $$\frac{x^2}{y^2-x^2}=\frac{25}{39}$$ --> $$\frac{x^2}{y^2}=\frac{25}{64}$$

How do you get it?

$$\frac{x^2}{y^2-x^2}=\frac{25}{39}$$;

Cross multiply: $$39x^2=25y^2-25x^2$$;

Re-arrange: $$64x^2=25y^2$$;

$$\frac{x^2}{y^2}=\frac{25}{64}$$.

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

hahaha Bunuel this is your most shortest explanation great sense of humor! VP  D
Joined: 09 Mar 2016
Posts: 1230
Re: A square wooden plaque has a square brass inlay in the  [#permalink]

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Erjan_S 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.

I have difficulties this this part $$\frac{x^2}{y^2-x^2}=\frac{25}{39}$$ --> $$\frac{x^2}{y^2}=\frac{25}{64}$$

How do you get it?

hello generis i am honored to tag you and ask you three questions Q # 1: i reviewed whole thread and all explanations but cant understand why do we divide by 2 here $$\frac{y-x}{2}$$ ?

Q # 2: why do we subtract $$x^2$$from $$y^2$$ here in the denominator $$[m]\frac{x^2}{y^2-x^2}$$ ? we know that ratio is $$\frac{25}{39}$$, why to subtract ? Q # 3: my approach to to tho problem was simple i rounded 39 to 40 so i simply got ratio $$\frac{25}{40} = \frac{5}{8}$$ do you think its correct approach ?  Many thanks and have an awesome weekend ! P.S. by the way in your signature isnt there a SC issue ? it says "In the depths of winter, I finally learned
that within me there lay an invincible summer
."

but shouldnt we say so "In the depths of winter, I finally learned
that within me there is a thick layer of an invincible hot summer mood
" Senior SC Moderator V
Joined: 22 May 2016
Posts: 3558
A square wooden plaque has a square brass inlay in the  [#permalink]

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1
dave13 wrote:
Erjan_S wrote:
Bunuel wrote:
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.
I have difficulties this part $$\frac{x^2}{y^2-x^2}=\frac{25}{39}$$ --> $$\frac{x^2}{y^2}=\frac{25}{64}$$
How do you get it?
hello generis i am honored to tag you and ask you three questions Q # 1: SEE BELOW Q # 2: SEE BELOW Q # 3: SEE BELOW

Many thanks and have an awesome weekend ! P.S. by the way in your signature isnt there a SC issue ? it says "In the depths of winter, I finally learned
that within me there lay an invincible summer
."

but shouldnt we say so "In the depths of winter, I finally learned
that within me there is a thick layer of an invincible hot summer mood
"    OMG you are irrepressible.

Hi dave13 -
Your second question is hard to answer.
I cannot tell whether you are following the algebra.
Others have had a hard time. Maybe this write-up will help.

I am assigning capital letter variables in order to write as abbreviated an
Let P = the whole PLAQUE (which is a "big square"). Area of $$P = y^2$$
Let B = the square BRASS inlay (which is a "little square). Area of $$B = x^2$$
Let W = the WOODEN part (which is a uniform strip, like a "frame"). Area of $$W = y^2-x^2$$
Quote:
Q # 1: i reviewed whole thread and all explanations but cant understand why do we divide by 2 here $$\frac{y-x}{2}$$?

Short answer: because there are TWO border widths between $$y$$ and $$x$$. TAKE A LOOK:
Attachment: woodenborder.png [ 8.24 KiB | Viewed 1101 times ]

We are solving for the width of the border.
Does (side y) - (side x) = width of border?
No. $$(y - x)$$ = TWO times the width of the border

Why? On EACH side of $$x$$, there is one border width, $$w$$:
$$y = w + x + w$$
$$y - x = 2w$$
$$w = \frac{y-x}{2}$$

Try drawing the diagram. Let's say $$x = 2, y = 4$$, and width of wooden frame, W = 1 (write values in)
Does $$(y - x)$$ = width of wooden border? $$(y-x) = 2. W = 1$$. Not correct.
$$(y - x) = 2$$ * width of the wooden border.
Width of border therefore equals $$\frac{y-x}{2}=\frac{4-2}{2}=1$$
Quote:
Q # 2: why do we subtract $$x^2$$from $$y^2$$ here in the denominator $$\frac{x^2}{y^2-x^2}$$ ? we know that ratio is $$\frac{25}{39}$$, why to subtract ? Short answer: We subtract because
1) $$y^2 - x^2$$ equals the area of the wooden part, which
2) is the bottom part of the ratio given by the prompt
(I think you may be slightly confused about area of "the wooden part." It's the frame, the thin strip.)

We need this ratio: $$\frac{Area_{B}}{Area_{W}}$$, in which $$Area_{W} = y^2-x^2$$,
to get another ratio: side length of B to side length of P.

$$\frac{InnerSquareArea}{WoodenFrameArea} = \frac{25}{39}$$

1) inner square area of P = $$x^2$$
2) wooden frame area = $$(y^2 - x^2)$$
Think of the area of the wooden frame as a "shaded region."
You would find the shaded region's area by subtracting B's area $$(x^2)$$ from P's area $$(y^2)$$.

We go from $$\frac{area of B}{area of W}$$ to $$\frac{Side of B(=x)}{Side of P(=y)}$$ to border width= some y expression

Border width depends on length of $$y$$. That demonstration has steps

1) Use the given ratio of $$\frac{Barea}{Warea}$$ to find ratio between the areas of the squares
The given ratio is NOT between the areas of the two squares (that's what we need to find)

GIVEN ratio is $$\frac{B}{W} = \frac{25}{39}$$

Area of W = (Area of P - Area of B) (think of W as a shaded region)
Area of P = $$y^2$$
Area of B = $$x^2$$
Area of W = $$y^2 - x^2$$
Substitute that RHS for W in the original ratio.

$$\frac{B}{W}=\frac{x^2}{y^2-x^2}=\frac{25}{39}$$
To find the ratio of areas of squares (P and B), we need $$x^2$$ over $$y^2$$
Eliminate $$x^2$$ in the denominator. Cross multiply:

$$\frac{x^2}{y^2-x^2}=\frac{25}{39}$$
$$39x^2 = 25y^2 - 25x^2$$
$$64x^2=25y^2$$

DIVIDE both sides by $$y^2$$ and by $$64$$
$$\frac{x^2}{y^2}=\frac{25}{64}$$

(2) Use areas of squares to find ratio of side lengths of squares
Find ratio of side x to side y by taking the square root of $$\frac{x^2}{y^2}=\frac{25}{64}$$:
$$\frac{x}{y}=\frac{5}{8}$$

(3) Express width of the border in terms of $$y$$ only
At #2 we still have a ratio. Still no counting number.
Width of the border depends on $$y.$$
But $$y$$ can be any length. NO limit on values for y = no limit on border width
Bunuel 's next steps simply detail THAT there are no limits on the longer side.
He shows that every answer depends on $$y$$

Defining border width in terms of $$y$$
Recall that border width = $$\frac{y-x}{2}$$
Define everything in terms of $$y$$. Use the side ratio to do so
$$\frac{x}{y}=\frac{5}{8}$$
Multiply both sides by $$y$$
$$x = \frac{5}{8}y$$
Substitute that RHS for $$x$$

Border width calculation: $$\frac{y - x}{2} =\frac{y -\frac{5}{8}y}{2}$$

$$\frac{y-\frac{5}{8}y}{2}=(\frac{\frac{8}{8}y-\frac{5}{8}y}{2})=(\frac{\frac{3}{8}y}{2})$$

$$\frac{\frac{3}{8}y}{2}=(\frac{3}{8}y*\frac{1}{2})=\frac{3}{16}y$$

Border width equals $$\frac{3}{16}$$ of $$y$$

That's nice, but how long can $$y$$ be? $$y$$'s length can be ANY real number.

The width of the border, dependent on $$y$$'s length,
can be ANY value that is $$\frac{3}{16}$$ of an infinite set of possibilities.
Quote:
Q # 3: my approach to to tho problem was simple i rounded 39 to 40 so i simply got ratio $$\frac{25}{40} = \frac{5}{8}$$ do you think its correct approach ?  Excellent idea to try to find shortcuts. But . . . wrong ratio. You got lucky.
To find side length, given area of squares, you must take a square root. AND you must be solving for the right parts.
$$\frac{25}{39}$$ is the ratio of $$\frac{AreaB}{AreaW}$$
$$\frac{5}{8}$$ is ratio of $$\frac{SideB}{SideP}$$

Your progression seems to be: $$\frac{25}{39}= \frac{AreaB}{AreaP}=\frac{SideB}{SideW}$$
The ratio you derived is not correct: $$P\neq{W}$$
Tip: be clear about the parts of the ratios you set up. WHICH side to WHICH side, e.g.

I hope that helps. (whew!)
You get kudos for making me laugh about the SC issues in my signature. I am a reading fool (fanatic). Elegant prose uses metaphor. SC questions are hard but . . . most do not display elegant prose. BTW, the author was an astonishing human being. But did you REALLY go after my pick for translation of “Au milieu de l'hiver, j'apprenais enfin qu'il y avait en moi un été invincible”? Thou art brave. _________________
SC Butler has resumed! Get two SC questions to practice, whose links you can find by date, here.

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

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Anyone with an alternate explanation to this problem?

IanStewart
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My explanation would be just like Brent's - we're only given a ratio, so we can make the width anything we like just by making the diagram bigger or smaller.

If that's not convincing, imagine that it's possible the width is 1 inch. Then it must also be possible that the width is 2.54 units, because 1 inch is 2.54 centimeters. But if that's true, then the width can be anything at all, just by changing units. But I prefer the first way of thinking about the problem.
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Re: A square wooden plaque has a square brass inlay in the  [#permalink]

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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.

How do we know that we need y-x/2

Posted from my mobile device
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GMAT 1: 700 Q49 V37 Re: A square wooden plaque has a square brass inlay in the  [#permalink]

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hi nishthagupta

Check out the diagram. I hope it helps.

LS

nishthagupta 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.

How do we know that we need y-x/2

Posted from my mobile device

Attachments brass and wooden plate.png [ 15.19 KiB | Viewed 199 times ]

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