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805+ (Hard)|   Geometry|   Must or Could be True|                              
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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
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E
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A square wooden plaque has a square brass inlay in the center, leaving 15 Jan 2010, 06:14
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hrish88
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. :-D
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No, I mean ANY width is possible.

Let the the side of the small square be \(x\) and that of the big square be \(y\).



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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A square wooden plaque has a square brass inlay in the center, leaving 15 Jan 2010, 04:26
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hrish88
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
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Why would ANY width of the strip be impossible?

Answer: E.
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A square wooden plaque has a square brass inlay in the center, leaving 24 Mar 2010, 06:10
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hrish88
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
Show more

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.
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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. :-D
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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.
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A square wooden plaque has a square brass inlay in the center, leaving 15 Jan 2010, 10:44
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atish
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.
Show more

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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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.
Show more

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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Bunuel


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

Nice explanation.You rock man as always.
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Bunuel
hrish88
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. :-D

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.
Show more

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?
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Bunuel
hrish88
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. :-D

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?
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Yes, width can have any positive value: the larger the width is the larger the whole square would be.
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A square wooden plaque has a square brass inlay in the center, leaving 19 Sep 2010, 11:29
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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=?

Please enlighten me.
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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=?

Please enlighten me.
Show more

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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A square wooden plaque has a square brass inlay in the center, leaving 01 Apr 2012, 07:48
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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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mymbadreamz
I didn't understand the statement "We are asked which value of (y-x)/2 is possible" . Can someone explain?
Show more

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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mymbadreamz
Why is it not y-x? Why do we calculate (y-x)/2?
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Consider the diagram below:
Attachment:
Wooden strip.png
Wooden strip.png [ 2.75 KiB | Viewed 148833 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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This tells us that there is a particular ratio for the width of the wooden frame to the width of the brass inlay.

This means there is a f(x) = \((\frac{A}{B})(x)\). Any x is possible and will yield a corresponding f(x).
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Bunuel
hrish88
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. :-D

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.
Show more


Again,
This squestion is ambiguous.
Why cant anyone not interpret it as:
b^2/w^2 = 25/39
where b- is the side length for brass inlay,
and W - length of wooden box
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Bunuel
hrish88
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. :-D

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.


Again,
This squestion is ambiguous.
Why cant anyone not interpret it as:
b^2/w^2 = 25/39
where b- is the side length for brass inlay,
and W - length of wooden box
Show more

Not ambiguous at all (notice that the question is from OG).

The question says that "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 ...", so it should be b^2/(w^2-b^2)=25/39.

Consider the diagram below:

Only grey area is wooden, the remaining (white area) is brass.

Hope it's clear.
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A square wooden plaque has a square brass inlay in the center, leaving 16 Dec 2016, 12:08
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Hi, I did it another way, but in my solution none of the given value satisfies.

Take wooden (bigger area) =\(a^2\)
let x be the width.
so brass area = \((a-2x)^2\)
so,

\((25/39)=( (a-2x)^2 )/ (a^2)\)


1. x=1
\((a-2) = 5\)
so, a=7
and \(a^2 = 49\)

giving ratio : \(25/49\) which is not as given one.

trying all these, give different ratios.

what did i do wrong?
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