Now， consider （2） alone. Since Incumbent I received 25，000 of the 100，000 vo
tes cast， I necessarily received 25% of the vote. Hence， the answer to the q
uestion is “No， the incumbent did not receive over 50% of the vote.” Therefo
re， （2） is sufficient to answer the question. The answer is B.
Note， some people have trouble with （2） because they feel that the question
asks for a “yes” answer. But on Data Sufficiency questions， a “no” answer is
just as valid as a “yes” answer. What we're looking for is a definite answe
r.
CHECKING EXTREME CASES
？When drawing a geometric figure or checking a given one， be sure to include
drawings of extreme cases as well as ordinary ones.
Example 1： In the figure to the right， AC is a chord and B is a point on the
circle. What is the measure of angle x？
Although in the drawing AC looks to be a diameter， that cannot be assumed. A
ll we know is that AC is a chord. Hence， numerous cases are possible， three
of which are illustrated below：
In Case I， x is greater than 45 degrees； in Case II， x equals 45 degrees； in
Case III， x is less than 45 degrees. Hence， the given information is not su
fficient to answer the question.
Example 2： Three rays emanate from a common point and form three angles with
measures p， q， and r. What is the measure of q + r ？
It is natural to make the drawing symmetric as follows：
In this case， p = q = r = 120， so q + r = 240. However， there are other draw
ings possible. For example：
In this case， q + r = 180. Hence， the given information is not sufficient to
answer the question.
Problems：
1. Suppose 3p + 4q = 11. Then what is the value of q？
（1） p is prime.
（2） q = -2p
（1） is insufficient. For example， if p = 3 and q = 1/2， then 3p + 4q = 3（3）
+ 4（1/2） = 11. However， if p = 5 and q = -1， then 3p + 4q = 3（5） + 4（-1） = 1
1. Since the value of q is not unique， （1） is insufficient.
Turning to （2）， we now have a system of two equations in two unknowns. Hence
， the system can be solved to determine the value of q. Thus， （2） is suffici
ent， and the answer is B.
