¡¡¡¡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£º 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£º
¡¡¡¡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£º 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£º
¡¡¡¡In this case£¬ p = q = r = 120£¬ so q + r = 240. However£¬ there are other draw
¡¡¡¡ings possible. For example£º
¡¡¡¡In this case£¬ q + r = 180. Hence£¬ the given information is not sufficient to
¡¡¡¡answer the question.
¡¡¡¡Problems£º
¡¡¡¡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.
