Testprep数学精解
DIRECTIONS: EACH OF THE FOLLOWING DATA SUFFICIENCY PROBLEMS CONTAINS A QUEST
ION FOLLOWED BY TWO STATEMENTS, NUMBERED (1) AND (2). YOU NEED NOT SOLVE THE
PROBLEM; RATHER YOU MUST DECIDE WHETHER THE INFORMATION GIVEN IS SUFFICIENT
TO SOLVE THE PROBLEM.
THE CORRECT ANSWER TO A QUESTION IS
A IF STATEMENT (1) ALONE IS SUFFICIENT TO ANSWER THE QUESTION BUT STATEMENT
(2) ALONE IS NOT SUFFICIENT;
B IF STATEMENT (2) ALONE IS SUFFICIENT TO ANSWER THE QUESTION BUT STATEMENT
(1) ALONE IS NOT SUFFICIENT;
C IF THE TWO STATEMENTS TAKEN TOGETHER ARE SUFFICIENT TO ANSWER THE QUESTION
, BUT NEITHER STATEMENT ALONE IS SUFFICIENT;
D IF EACH STATEMENT ALONE IS SUFFICIENT TO ANSWER THE QUESTION;
E IF THE TWO STATEMENTS TAKEN TOGETHER ARE STILL NOT SUFFICIENT TO ANSWER TH
E QUESTION.
NUMBERS: ONLY REAL NUMBERS ARE USED. THAT IS, THERE ARE NO COMPLEX NUMBERS.
DRAWINGS: THE DRAWINGS ARE DRAWN TO SCALE ACCORDING TO THE INFORMATION GIVEN
IN THE QUESTION, BUT MAY CONFLICT WITH THE INFORMATION GIVEN IN STATEMENTS
(1) AND (2).
YOU CAN ASSUME THAT A LINE THAT APPEARS STRAIGHT IS STRAIGHT AND THAT ANGLE
MEASURES CANNOT BE ZERO.
YOU CAN ASSUME THAT THE RELATIVE POSITIONS OF POINTS, ANGLES, AND OBJECTS AR
E AS SHOWN.
ALL DRAWINGS LIE IN A PLANE UNLESS STATED OTHERWISE.
EXAMPLE:
IN TRIANGLE ABC TO THE RIGHT, WHAT IS THE VALUE OF Y?
(1) AB = AC
(2) X = 30
EXPLANATION: BY STATEMENT (1), TRIANGLE ABC IS ISOSCELES. HENCE, ITS BASE AN
GLES ARE EQUAL: Y = Z. SINCE THE ANGLE SUM OF A TRIANGLE IS 180 DEGREES, WE
GET X + Y + Z = 180. REPLACING Z WITH Y IN THIS EQUATION AND THEN SIMPLIFYIN
G YIELDS X + 2Y = 180. SINCE STATEMENT (1) DOES NOT GIVE A VALUE FOR X, WE C
ANNOT DETERMINE THE VALUE OF Y FROM STATEMENT (1) ALONE. BY STATEMENT (2), X
= 30. HENCE, X + Y + Z = 180 BECOMES 30 + Y + Z = 180, OR Y + Z = 150. SINC
E STATEMENT (2) DOES NOT GIVE A VALUE FOR Z, WE CANNOT DETERMINE THE VALUE O
F Y FROM STATEMENT (2) ALONE. HOWEVER, USING BOTH STATEMENTS IN COMBINATION,
WE CAN FIND BOTH X AND Z AND THEREFORE Y. HENCE, THE ANSWER IS C.
NOTICE IN THE ABOVE EXAMPLE THAT THE TRIANGLE APPEARS TO BE A RIGHT TRIANGLE
.. HOWEVER, THAT CANNOT BE ASSUMED: ANGLE A MAY BE 89 DEGREES OR 91 DEGREES,
WE CAN’T TELL FROM THE DRAWING. YOU MUST BE VERY CAREFUL NOT TO ASSUME ANY M
ORE THAN WHAT IS EXPLICITLY GIVEN IN A DATA SUFFICIENCY PROBLEM.
ELIMINATION
DATA SUFFICIENCY QUESTIONS PROVIDE FERTILE GROUND FOR ELIMINATION. IN FACT,
IT IS RARE THAT YOU WON’T BE ABLE TO ELIMINATE SOME ANSWER-CHOICES. REMEMBER
, IF YOU CAN ELIMINATE AT LEAST ONE ANSWER CHOICE, THE ODDS OF GAINING POINT
S BY GUESSING ARE IN YOUR FAVOR.
THE FOLLOWING TABLE SUMMARIZES HOW ELIMINATION FUNCTIONS WITH DATA SUFFICIEN
CY PROBLEMS.
STATEMENT CHOICES ELIMINATED
(1) IS SUFFICIENT B, C, E
(1) IS NOT SUFFICIENT A, D
(2) IS SUFFICIENT A, C, E
(2) IS NOT SUFFICIENT B, D
(1) IS NOT SUFFICIENT AND (2) IS NOT SUFFICIENT A, B, D
EXAMPLE 1: WHAT IS THE 1ST TERM IN SEQUENCE S?
(1) THE 3RD TERM OF S IS 4.
(2) THE 2ND TERM OF S IS THREE TIMES THE 1ST, AND THE 3RD TERM IS FOUR TIMES
THE 2ND.
(1) IS NO HELP IN FINDING THE FIRST TERM OF S. FOR EXAMPLE, THE FOLLOWING SE
QUENCES EACH HAVE 4 AS THEIR THIRD TERM, YET THEY HAVE DIFFERENT FIRST TERMS
:
0, 2, 4
-4, 0, 4
THIS ELIMINATES CHOICES A AND D. NOW, EVEN IF WE ARE UNABLE TO SOLVE THIS PR
OBLEM, WE HAVE SIGNIFICANTLY INCREASED OUR CHANCES OF GUESSING CORRECTLY--FR
OM 1 IN 5 TO 1 IN 3.
TURNING TO (2), WE COMPLETELY IGNORE THE INFORMATION IN (1). ALTHOUGH (2) CO
NTAINS A LOT OF INFORMATION, IT ALSO IS NOT SUFFICIENT. FOR EXAMPLE, THE FOL
LOWING SEQUENCES EACH SATISFY (2), YET THEY HAVE DIFFERENT FIRST TERMS:
1, 3, 12
3, 9, 36
THIS ELIMINATES B, AND OUR CHANCES OF GUESSING CORRECTLY HAVE INCREASED TO 1
IN 2.
NEXT, WE CONSIDER (1) AND (2) TOGETHER. FROM (1), WE KNOW "THE 3RD TERM OF S
IS 4." FROM (2), WE KNOW "THE 3RD TERM IS FOUR TIMES THE 2ND." THIS IS EQUI
VALENT TO SAYING THE 2ND TERM IS 1/4 THE 3RD TERM: (1/4)4 = 1. FURTHER, FROM
(2), WE KNOW "THE 2ND TERM IS THREE TIMES THE 1ST." THIS IS EQUIVALENT TO S
AYING THE 1ST TERM IS 1/3 THE 2ND TERM: (1/3)1 = 1/3. HENCE, THE FIRST TERM
OF THE SEQUENCE IS FULLY DETERMINED: 1/3, 1, 4. THE ANSWER IS C.
EXAMPLE 2: IN THE FIGURE TO THE RIGHT, WHAT IS THE AREA OF THE TRIANGLE?
(1)