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
=> y² = 1
=> y = ±1
=> x = ±4

Not sufficient.

The correct answer is E.

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 360°. Thus this doesn't provide any new information.

Not sufficient.

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

Now sum of all the interior angles of an regular polygon with n sides is given by (n - 2)*(180°). 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 = 1440°
Therefore, (n - 2)*(180°) = 1440°
=> (n - 2) = 8
=> n = 10

Thus the polygon is a regular decagon!

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.

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 360°. Refer to the image below,

Sum of the external angles = Sum of angles marked in red
= (180° - x) + (180° - y) + (180° - z)
= 540° - (x + y + z)
= 540° - 180°
= 360°