12 Days of Christmas GMAT Competition - Day 1: A 10m stick is broken i

Given: A 10m stick is broken into four pieces of lengths a, b, c, and d by making three cuts on the stick.

Asked: Can the four pieces form a quadrilateral?

(1) d > c > b > a
Case 1: A quadrilateral is possible with sides d=4>c=3>b=2>a=1
Case 2: A quadrilateral can not be formed when d=7>c=1.2>b=1>a=.8
NOT SUFFICIENT

(2) d > a + b + c
A quadrilateral can not be formed since length of one side is greater than sum of other 3 sides.
If we lay other 3 sides in a straight line for maximum length, d can not join the 2 extremes.
SUFFICIENT

IMO B

A 10m stick is broken into four pieces of lengths a, b, c, and d by making three cuts on the stick. Can the four pieces form a quadrilateral?

This is very straight to the point question and can be solved by assuming values.

(1) d > c > b > a

Here there are 2 cases:

Case 1:
d = 9.4
c = 0.3
b = 0.2
a = 0.1
Clearly here d > a + b + c and thus a quadrilateral can not be formed.

Case 2:
d = 4
c = 3
b = 2
a = 1
With these values a quadrilateral can be formed.

Statement 1 is clearly INSUFFICIENT

(2) d > a + b + c

This statement is similar to the case 1 that we considered for statement A and is Clearly sufficient. With d > a + b + c, we can conclude with certainity that a quadrilateral can never be formed.

Hence, Statement 2 is clearly SUFFICIENT

Answer: Option B

Concept: In any n-sided polygon, the sum of any (n-1) sides must always be greater than the remaining side. This can be visualized as the remaining side being the straight line (hence shortest) distance between the two vertices.

Application: Statement 1 tells us that d>c>b>a.

Now let's take case 1: d=4, c=3, b=2, and a=1. In this case, each side is less than the sum of the remaining 3 sides. Possible Quadrilateral.

Let's take case 2: d=10, c=3, b=2, and a=1. In this case, "side d" is greater than the sum of the remaining 3 sides. Quadrilateral not possible.

Hence Statement 1 is insufficient in itself. But Statement 2 explicitly tells us that d>a+b+c which is a clear violation of a Quadrilateral's basic property as discussed above. Hence, it is sufficient to comment that such a Quadrilateral would not be possible.

Hence, option B.

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A 10m stick is broken into four pieces of lengths a, b, c, and d by making three cuts on the stick. Can the four pieces form a quadrilateral?


My approach is as follows:
The sum of 3 sides of quadrilateral should be greater than the fourth. This means a+b+c>d

(1) d > c > b > a
given a=1 b=2 c=3 d=4 it is possible to form quadrilateral since the sum of any 3 sides is greater then the fourth one;
given a=1.2 b=1.5 c=1.7 and d=5.6 it is impossible to form quadrilateral, as 1.2+1.5+1.7<5.6; Hence, st (1) is Not Sufficient

(2) d > a + b + c
This statement is the oppositte of a+b+c>d. For that reason, it is impossible to form quadrilateral. Sufficient

Answer: B

Question is can the sides form a quadrilateral?

The first statement can result in two possible answers:
CASE 1 Eg: 4,3,2,1 Here Quadrilateral possible

CASE 2 Eg: 6,2,0.55,0.45 Here, length of one side is greater than the sum of other three sides, so Quadrilateral not possible.

The second statement is self sufficient to answer the question of whether a Quadrilateral can be formed because there is only one possible case of "no Quadrilateral can be formed" when one side is greater than the sum of other three sides.

Hence, Correct Ans B (only II Sufficient)

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