Testprep数学精解

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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 IS A POINT ON THE

CIRCLE. WHAT IS THE MEASURE OF ANGLE X?

ALTHOUGH IN THE DR

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