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# A rectangular field with width x is marked with paint

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Intern
Joined: 04 Oct 2017
Posts: 40
A rectangular field with width x is marked with paint  [#permalink]

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04 May 2018, 01:54
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16
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Difficulty:

65% (hard)

Question Stats:

61% (02:21) correct 39% (02:13) wrong based on 107 sessions

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A rectangular field with width x is marked with paint into 4ths and 7ths that are perpendicular to the long side of the field. In terms of x, what is the largest difference between any two consecutive 4th and 7th marks on this side of the field? (Assume there are no painted marks at the beginning or end of the long side of the field).

A. $$\frac{1x}{28}$$

B. $$\frac{1x}{14}$$

C. $$\frac{3x}{28}$$

D. $$\frac{1x}{7}$$

E. $$\frac{5x}{28}$$
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Joined: 22 May 2016
Posts: 3055
A rectangular field with width x is marked with paint  [#permalink]

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04 May 2018, 07:48
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parthjain68 wrote:
A rectangular field with width x is marked with paint into 4ths and 7ths that are perpendicular to the long side of the field. In terms of x, what is the largest difference between any two consecutive 4th and 7th marks on this side of the field? (Assume there are no painted marks at the beginning or end of the long side of the field).

A. $$\frac{1x}{28}$$

B. $$\frac{1x}{14}$$

C. $$\frac{3x}{28}$$

D. $$\frac{1x}{7}$$

E. $$\frac{5x}{28}$$

Attachment:

field4ths7ths.png [ 50.32 KiB | Viewed 1448 times ]

Sketching is quick - less than a minute to solve.
(I drew only the number line, wrote the integer values,
and did not calculate distances to the right of 14, which divides the length
into two mirror images.)

Let $$x$$ = the field's LENGTH = $$28$$ (LCM of 4 and 7)

Divide the field's length into one-fourth increments
Values will be
7, 14, 21

Divide the field further into one-seventh increments
Values will be
4, 8, 12, 16, 20, 24

Simply check the distances between two consecutive 4th and 7th marks

Distances = 3, 2, and 1

Distance of 3 is the greatest -- but it is a distance of $$\frac{3}{28}$$
$$x = 28$$
Greatest distance: $$\frac{3}{28}x$$

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Joined: 04 Oct 2017
Posts: 40
Re: A rectangular field with width x is marked with paint  [#permalink]

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04 May 2018, 08:45
Dear generis,

I really liked your explanation. Thanks for the same, especially the part where you did not even calculate values to the right of 14 because it creates a mirror image.

I didn't understand 2 specific points of the question.

1. First, it says that the width is x and it is divided into 4ths and 7ths. You took the length but it is perfectly fine as it will make sense if we take width = x.

2. Second, it says that the width is divided into 4ths and 7ths THAT ARE PERPENDICULAR to the longer side. HOW IS THIS POSSIBLE? If the width is divided into 4ths and 7ths, won't the divisions be be PARALLEL to the longer side i.e. length instead of being perpendicular to the longer side?

Would really appreciate your thoughts on the same.

Parth Jain
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Joined: 02 Sep 2009
Posts: 56226
Re: A rectangular field with width x is marked with paint  [#permalink]

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04 May 2018, 09:56
3
1
parthjain68 wrote:
A rectangular field with width x is marked with paint into 4ths and 7ths that are perpendicular to the long side of the field. In terms of x, what is the largest difference between any two consecutive 4th and 7th marks on this side of the field? (Assume there are no painted marks at the beginning or end of the long side of the field).

A. $$\frac{1x}{28}$$

B. $$\frac{1x}{14}$$

C. $$\frac{3x}{28}$$

D. $$\frac{1x}{7}$$

E. $$\frac{5x}{28}$$

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A rectangular field with width x is marked with paint  [#permalink]

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04 May 2018, 09:59
1
1
parthjain68 wrote:
Dear generis,

I really liked your explanation. Thanks for the same, especially the part where you did not even calculate values to the right of 14 because it creates a mirror image.

I didn't understand 2 specific points of the question.

1. First, it says that the width is x and it is divided into 4ths and 7ths. You took the length but it is perfectly fine as it will make sense if we take width = x.

2. Second, it says that the width is divided into 4ths and 7ths THAT ARE PERPENDICULAR to the longer side. HOW IS THIS POSSIBLE? If the width is divided into 4ths and 7ths, won't the divisions be be PARALLEL to the longer side i.e. length instead of being perpendicular to the longer side?

Would really appreciate your thoughts on the same.

Parth Jain

parthjain68 , good questions. Spot on. They are related.
The logic that underlies your point about parallel lines propelled me to make x = longer side.

The bottom line: precisely because the lines must be perpendicular
to the longer side (IMO the most restrictive condition),
I concluded that "width" was either a typo or a generic term for "span between two points,"
especially because the question uses "long side" instead of "length."

That is, because the lines must be perpendicular to the longer side,
width = span = longer side = what we usually call "length"

"WIDTH": Probably a generic term for distance between two points

• The problem could have said breadth
I have seen the word "breadth" used to describe a span that
in other parts of the world is called "width"

If the author meant that "x MUST = the short side"
given that "shorter side" is usually referred to as width,
and that breadth would be clearer (this, coming from a North American!)
then why not write "x = the short side"?
Something fishy is going on.

• the question specifies "long side of the field." Unusual. Possibly a hint.
The question contains width, long side, and lines perpendicular to the long side.

The specificity of "the long side" logically cuts both ways.

The specification could mean that long side is NOT width.
Because "width" typically means the shorter side,
this conclusion seems to be the most natural.
But . . .GMAT? Forget about natural. Traps abound.

On the other hand, using the term "long side" is unusual.
Why not just call it "length"?

If a question writer were trying to confuse, perhaps s/he
would use a word such as "width" that connotes span . . .

. . . while simultaneously using the distinctive term "long side" to hint
that "width" does not necessarily mean "the short side."
Again, why not just call the longer side "length"?!
Answer: perhaps the longer side is not length

The clincher? "Perpendicular" lines.
The dividing lines are perpendicular to the long side.
That condition seems to be the most restrictive.

As far as I can tell, those lines must appear this way:

|____|____|____|____|____|
------------LONG SIDE------------

It makes no sense to try to divide what by implication is "the short side."
(I tried for about five seconds. At that point, I decided "width" was
a generic term or a mistake.)

I see no way to use lines perpendicular to the long side
unless the long side = x

And I see no way to justify "longer side = x" without
one of two conclusions:
1) "width" is a typo; or

2) the author of the question deliberately used odd terminology
and left hints to suggest we must think outside the box.
(Way outside the box, IMO. )

I hope that helps.
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Re: A rectangular field with width x is marked with paint  [#permalink]

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04 May 2018, 20:55
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Dear generis,

Thank you so much for your super-detailed and amazing reply. The forum says that I cannot give more than 1 Kudos. Haha!

1. The part where you switched to dividing the length within 5 seconds of deciding that it was a typo probably comes with experience. I couldn't think of it and got really confused between dividing the length or the width.

Thanks a lot anyway.

Parth Jain
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Joined: 09 Sep 2013
Posts: 11646
Re: A rectangular field with width x is marked with paint  [#permalink]

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14 Jun 2019, 04:49
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Re: A rectangular field with width x is marked with paint   [#permalink] 14 Jun 2019, 04:49
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