Data Sufficiency

If A, B, and C are distinct points on the number line. Is the length of the line segment AB less than 10?


(1) The sum of the lengths of line segments AC and BC is 10

(2) The sum of the lengths of line segments AB and AC is more than 10


IMO B

Experts, please comment.

eitijan wrote:If A, B, and C are distinct points on the number line. Is the length of the line segment AB less than 10?


(1) The sum of the lengths of line segments AC and BC is 10

(2) The sum of the lengths of line segments AB and AC is more than 10

The average test-taker will assume that the points must be in alphabetic order:
A...B...C
Be sure to try NON-ALPHABETIC orderings.
The following cases satisfy both statements.

Case 1:
A=0..............B=8....C=9
Here:
AC+BC = 10, satisfying Statement 1.
AB+AC > 10, satisfying Statement 2.
In this case, AB < 10.

Case 2:
A=0...............C=8.............B=10
Here:
AC+BC = 10, satisfying Statement 1.
AB+AC > 10, satisfying Statement 2.
In this case, AB = 10.

Since the answer is YES in Case 1 but NO in Case 2, the two statements combined are INSUFFICIENT.

The correct answer is E.

eitijan wrote:If A, B, and C are distinct points on the number line. Is the length of the line segment AB less than 10?


(1) The sum of the lengths of line segments AC and BC is 10

(2) The sum of the lengths of line segments AB and AC is more than 10


IMO B

Experts, please comment.

Given: A, B, C are distinct points on the number line
Required: Is AB < 10

Statement 1: The sum of the lengths of line segments AC and BC is 10
We can place the points on the line in the following ways
Case 1:
----A-----B----C-----
In this case, since AC + BC = 10, hence AB < 10
Case 2:
----A-----C----B----
In this case, AC + BC = 10
Hence AB = 10
Two different answers,
Insufficient

Statement 2: The sum of the lengths of line segments AB and AC is more than 10
Again let us take the two cases

Case 1:
----A----B----C----
We are given that AB + AC > 10
Nothing can be said about AB

Case 2:
----A----C----B----
We are given that AB + AC > 10
Again, nothing can be said about AB.
Insufficient

Correct Option: E

Another conceptual way of thinking about it:

Length of AC = x
Length of BC = y
Length of AB = z

S1: x + y = 10
S2: x + z > 10

We can conclude that z > y, but we can't conclude that z < 10. (In fact, since z = x + y is one immediate possibility -- if C is in between A and B on the number line, we'd have to have z = x + y -- z = 10 is certainly possible.)