Testprep数学精解
RECALL THAT A TRIANGLE IS A RIGHT TRIANGLE IF AND ONLY IF THE SQUARE OF THE
LONGEST SIDE IS EQUAL TO THE SUM OF THE SQUARES OF THE SHORTER SIDES (PYTHAG
OREAN THEOREM). HENCE, (1) IMPLIES THAT THE TRIANGLE IS A RIGHT TRIANGLE. SO
THE AREA OF THE TRIANGLE IS (6)(8)/2. NOTE, THERE IS NO NEED TO CALCULATE T
HE AREA--WE JUST NEED TO KNOW THAT THE AREA CAN BE CALCULATED. HENCE, THE AN
SWER IS EITHER A OR D.
TURNING TO (2), WE SEE IMMEDIATELY THAT WE HAVE A RIGHT TRIANGLE. HENCE, AGA
IN THE AREA CAN BE CALCULATED. THE ANSWER IS D.
EXAMPLE 3: IS P < Q ?
(1) P/3 < Q/3
(2) -P + X > -Q + X
MULTIPLYING BOTH SIDES OF P/3 < Q/3 BY 3 YIELDS P < Q.
HENCE, (1) IS SUFFICIENT. AS TO (2), SUBTRACT X FROM BOTH SIDES OF -P + X >
-Q + X, WHICH YIELDS -P > -Q.
MULTIPLYING BOTH SIDES OF THIS INEQUALITY BY -1, AND RECALLING THAT MULTIPLY
ING BOTH SIDES OF AN INEQUALITY BY A NEGATIVE NUMBER REVERSES THE INEQUALITY
, YIELDS P < Q.
HENCE, (2) IS ALSO SUFFICIENT. THE ANSWER IS D.
EXAMPLE 4: IF X IS BOTH THE CUBE OF AN INTEGER AND BETWEEN 2 AND 200, WHAT I
S THE VALUE OF X?
(1) X IS ODD.
(2) X IS THE SQUARE OF AN INTEGER.
SINCE X IS BOTH A CUBE AND BETWEEN 2 AND 200, WE ARE LOOKING AT THE INTEGERS
:
WHICH REDUCE TO
8, 27, 64, 125
SINCE THERE ARE TWO ODD INTEGERS IN THIS SET, (1) IS NOT SUFFICIENT TO UNIQU
ELY DETERMINE THE VALUE OF X. THIS ELIMINATES CHOICES A AND D.
NEXT, THERE IS ONLY ONE PERFECT SQUARE, 64, IN THE SET. HENCE, (2) IS SUFFIC
IENT TO DETERMINE THE VALUE OF X. THE ANSWER IS B.
EXAMPLE 5: IS CAB A CODE WORD IN LANGUAGE Q?
(1) ABC IS THE BASE WORD.
(2) IF C IMMEDIATELY FOLLOWS B, THEN C CAN BE MOVED TO THE FRONT OF THE CODE
WORD TO GENERATE ANOTHER WORD.
FROM (1), WE CANNOT DETERMINE WHETHER CAB IS A CODE WORD SINCE (1) GIVES NO
RULE FOR GENERATING ANOTHER WORD FROM THE BASE WORD. THIS ELIMINATES A AND D
..
TURNING TO (2), WE STILL CANNOT DETERMINE WHETHER CAB IS A CODE WORD SINCE N
OW WE HAVE NO WORD TO APPLY THIS RULE TO. THIS ELIMINATES B.
HOWEVER, IF WE CONSIDER (1) AND (2) TOGETHER, THEN WE CAN DETERMINE WHETHER
CAB IS A CODE WORD:
FROM (1), ABC IS A CODE WORD.
FROM (2), THE C IN THE CODE WORD ABC CAN BE MOVED TO THE FRONT OF THE WORD:
CAB.
HENCE, CAB IS A CODE WORD AND THE ANSWER IS C.
UNWARRANTED ASSUMPTIONS
BE EXTRA CAREFUL NOT TO READ ANY MORESINTOSA STATEMENT THAN WHAT IS GIVEN.
?THE MAIN PURPOSE OF SOME DIFFICULT PROBLEMS IS TO LURE YOUSINTOSMAKING AN U
NWARRANTED ASSUMPTION.
IF YOU AVOID THE TEMPTATION, THESE PROBLEMS CAN BECOME ROUTINE.
EXAMPLE 6: DID INCUMBENT I GET OVER 50% OF THE VOTE?
(1) CHALLENGER C GOT 49% OF THE VOTE.
(2) INCUMBENT I GOT 25,000 OF THE 100,000 VOTES CAST.
IF YOU DID NOT MAKE ANY UNWARRANTED ASSUMPTIONS, YOU PROBABLY DID NOT FIND T
HIS TO BE A HARD PROBLEM. WHAT MAKES A PROBLEM DIFFICULT IS NOT NECESSARILY
ITS UNDERLYING COMPLEXITY; RATHER A PROBLEM IS CLASSIFIED AS DIFFICULT IF MA
NY PEOPLE MISS IT. A PROBLEM MAY BE SIMPLE YET CONTAIN A PSYCHOLOGICAL TRAP
THAT CAUSES PEOPLE TO ANSWER IT INCORRECTLY.
THE ABOVE PROBLEM IS DIFFICULT BECAUSE MANY PEOPLE SUBCONSCIOUSLY ASSUME THA
T THERE ARE ONLY TWO CANDIDATES. THEY THEN FIGURE THAT SINCE THE CHALLENGER
RECEIVED 49% OF THE VOTE THE INCUMBENT RECEIVED 51% OF THE VOTE. THIS WOULD
BE A VALID DEDUCTION IF C WERE THE ONLY CHALLENGER (YOU MIGHT ASK, "WHAT IF
SOME PEOPLE VOTED FOR NONE-OF-THE-ABOVE?" BUT DON’T GET CARRIED AWAY WITH FI
NDING EXCEPTIONS. THE WRITERS OF THE GMAT WOULD NOT SET A TRAP THAT SUBTLE).
BUT WE CANNOT ASSUME THAT. THERE MAY BE TWO OR MORE CHALLENGERS. HENCE, (1)
IS INSUFFICIENT.
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