X , Polygon & 12 meters

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goyalsau: Legendary Member; Posts: 866; Joined: Mon Aug 02, 2010 6:46 pm; Location: Gwalior, India; Thanked: 31 times

X , Polygon & 12 meters

by goyalsau » Sat Dec 25, 2010 5:08 am

What is the value of x?
(1) x/y = 4
(2) xy = 4

How many sides does a regular polygon have?
(1) The sum of its exterior angles = 360degrees
(2) The total sum of all the interior and exterior angles = 1,800degrees

If a rope is cut into three pieces of unequal length, what is the length of the shortest of these pieces of rope?
(1) The combined length of the longer two pieces of rope is 12 meters
(2) The combined length of the shorter two pieces of rope is 11 meters.

Saurabh Goyal
[email protected]
-------------------------

EveryBody Wants to Win But Nobody wants to prepare for Win.

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Source: Beat The GMAT — Data Sufficiency | Sat Dec 25, 2010 5:08 am

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pm; Location: Milpitas, CA; Thanked: 1854 times; Followed by:523 members; GMAT Score:770

by Anurag@Gurome » Sat Dec 25, 2010 7:46 am

goyalsau wrote:What is the value of x?
(1) x/y = 4
(2) xy = 4

Statement 1: x/y = 4 => x = 4y
For every real y there is a real x.
Thus infinite possible values of x.

Not sufficient.

Statement 2: xy = 4 => x = 4/y
For every real y there is a real x except y = 0, where x is undefined.
Thus infinite possible values of x.

Not sufficient.

1 & 2 Together: Replace x with 4y
=> (4y)*y = 4
=> yÂ² = 1
=> y = Â±1
=> x = Â±4

Not sufficient.

The correct answer is E.

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pm; Location: Milpitas, CA; Thanked: 1854 times; Followed by:523 members; GMAT Score:770

by Anurag@Gurome » Sat Dec 25, 2010 8:02 am

goyalsau wrote:How many sides does a regular polygon have?
(1) The sum of its exterior angles = 360 degrees
(2)

Statement 1: The sum of its exterior angles = 360 degrees
Sum of all the exterior angles of any regular polygon is 360Â°. Thus this doesn't provide any new information.

Not sufficient.

Statement 2: The total = 1800Â°
=> Sum of all the interior angles = 1800Â° - Sum of all the exterior angles = 1800Â° - 360Â° = 1440Â°

Now sum of all the interior angles of an regular polygon with n sides is given by (n - 2)*(180Â°). Thus we can find the number of sides of the polygon.

Sufficient.

The correct answer is B.

Note: The properties highlighted in blue are standard and proven properties of a regular polygon. For detail: https://www.mathsisfun.com/geometry/inte ... ygons.html and https://www.mathsisfun.com/geometry/inte ... ygons.html. The second link has an excellent animated proof for the sum of exterior angles!

For interested students: Let's calculate the number of sides.
Sum of all the internal angles = 1440Â°
Therefore, (n - 2)*(180Â°) = 1440Â°
=> (n - 2) = 8
=> n = 10

Thus the polygon is a regular decagon!

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GMAT/MBA Expert

Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pm; Location: Milpitas, CA; Thanked: 1854 times; Followed by:523 members; GMAT Score:770

by Anurag@Gurome » Sat Dec 25, 2010 8:37 am

goyalsau wrote:If a rope is cut into three pieces of unequal length, what is the length of the shortest of these pieces of rope?
(1) The combined length of the longer two pieces of rope is 12 meters
(2) The combined length of the shorter two pieces of rope is 11 meters.

Say the lengths of the three pieces are x meters, y meters and z meters, where 0 < x < y < z. We have to find x.

Now the quick and simple solution :

# Statement 1 gives (y + z) = 12 => x can be anything
# Statement 2 gives (x + y) = 11 => x and y can be anything
# Together 1 & 2 gives (x + 2y + z) = 23 => x, y and z can be anything

Thus correct answer is E.

Now the following solution is a detailed one which gives an insight of the "anything"s mentioned above. I agree that this much of detail is not necessary for this problem but may be you'll need it in some other cases!

Statement 1: The combined length of the longer two pieces of rope is 12 meters.

Thus, (y + z) = 12

Now for two unequal quantities, the larger one of them is always greater than the average (arithmetic mean) of them and the smaller one of them is always smaller than the average of them. Also their distance from the average is same. Thus value of y must be something less than 6 and value of z must something greater than 6 (and these two "something must be equal). Thus 0 < y < 6 < z.

As x < y, x must be less than 6 too. Therefore, 0 < x < 6

Not sufficient

Statement 2: The combined length of the shorter two pieces of rope is 11 meters.

Thus, (x + y) = 11

Applying the same logic as above, value of x must be something less than 5.5 and value of y must something greater than 5.5 (and these two "something must be equal). Thus 0 < x < 5.5 < y.

Not sufficient

1 & 2 Together: 0 < x < 5.5 < y < 6 < z

Not sufficient

The correct answer is E.

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goyalsau: Legendary Member; Posts: 866; Joined: Mon Aug 02, 2010 6:46 pm; Location: Gwalior, India; Thanked: 31 times

by goyalsau » Sat Dec 25, 2010 9:19 am

Anurag@Gurome wrote:
goyalsau wrote:How many sides does a regular polygon have?
(1) The sum of its exterior angles = 360 degrees
(2)
Statement 1: The sum of its exterior angles = 360 degrees
Sum of all the exterior angles of any regular polygon is 360Â°. Thus this doesn't provide any new information.

Not sufficient.

Statement 2: The total = 1800Â°
=> Sum of all the interior angles = 1800Â° - Sum of all the exterior angles = 1800Â° - 360Â° = 1440Â°

Now sum of all the interior angles of an regular polygon with n sides is given by (n - 2)*(180Â°). Thus we can find the number of sides of the polygon.

Sufficient.

The correct answer is B.

Note: The properties highlighted in blue are standard and proven properties of a regular polygon. For detail: https://www.mathsisfun.com/geometry/inte ... ygons.html and https://www.mathsisfun.com/geometry/inte ... ygons.html. The second link has an excellent animated proof for the sum of exterior angles!

For interested students: Let's calculate the number of sides.
Sum of all the internal angles = 1440Â°
Therefore, (n - 2)*(180Â°) = 1440Â°
=> (n - 2) = 8
=> n = 10

Thus the polygon is a regular decagon!

I did not know that sum of exterior angles of a regular polygon is add's up to 360. I already know about the sum of interior angle of a regular polygon is ( n - 2 ) * 180. but this exterior angle rule is new for me.

Thanks.

I was thinking about triangles so just wan't to make sure one thing.

As i know triangle is also a polygon but i think for triangle it should be 180, Even for a normal triangle but for a equilateral triangle it must be 360.

There is no such rule for a triangles, Its only only about regular polygons, Can we consider equilateral triangle or not????

Saurabh Goyal
[email protected]
-------------------------

EveryBody Wants to Win But Nobody wants to prepare for Win.

Quote

GMAT/MBA Expert

Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pm; Location: Milpitas, CA; Thanked: 1854 times; Followed by:523 members; GMAT Score:770

by Anurag@Gurome » Sat Dec 25, 2010 10:01 am

goyalsau wrote:I was thinking about triangles so just wan't to make sure one thing.

As i know triangle is also a polygon but i think for triangle it should be 180, Even for a normal triangle but for a equilateral triangle it must be 360.

There is no such rule for a triangles, Its only only about regular polygons, Can we consider equilateral triangle or not????

These rules are applicable for not only any regular polygon but for any convex polygon. Thus for any general triangle, the sum of all the external angles is 360Â°. Refer to the image below,