Testprep数学精解
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.
2. WHAT IS THE PERIMETER OF TRIANGLE ABC ABOVE?
(1) THE RATIO OF DE TO BF IS 1: 3.
(2) D AND E ARE MIDPOINTS OF SIDES AB AND CB, RESPECTIVELY.
SINCE WE DO NOT EVEN KNOW WHETHER BF IS AN ALTITUDE, NOTHING CAN BE DETERMIN
ED FROM (1). MORE IMPORTANTLY, THERE IS NO INFORMATION TELLING US THE ABSOLU
TE SIZE OF THE TRIANGLE.
AS TO (2), ALTHOUGH FROM GEOMETRY WE KNOW THAT DE = AC/2, THIS RELATIONSHIP
HOLDS FOR ANY SIZE TRIANGLE. HENCE, (2) IS ALSO INSUFFICIENT.
TOGETHER, (1) AND (2) ARE ALSO INSUFFICIENT SINCE WE STILL DON’T HAVE INFORM
ATION ABOUT THE SIZE OF THE TRIANGLE, SO WE CAN’T DETERMINE THE PERIMETER. T
HE ANSWER IS E.
3. A DRESS WAS INITIALLY LISTED AT A PRICE THAT WOULD HAVE GIVEN THE STORE A
PROFIT OF 20 PERCENT OF THE WHOLESALE COST. WHAT WAS THE WHOLESALE COST OF
THE DRESS?
(1) AFTER REDUCING THE ASKING PRICE BY 10 PERCENT, THE DRESS SOLD FOR A NET
PROFIT OF 10 DOLLARS.
(2) THE DRESS SOLD FOR 50 DOLLARS.
CONSIDER JUST THE QUESTION SETUP. SINCE THE STORE WOULD HAVE MADE A PROFIT O
F 20 PERCENT ON THE WHOLESALE COST, THE ORIGINAL PRICE P OF THE DRESS WAS 12
0 PERCENT OF THE COST: P = 1.2C. NOW, TRANSLATING (1)SINTOSAN EQUATION YIELD
S:
P - .1P = C + 10
SIMPLIFYING GIVES
..9P = C + 10
SOLVING FOR P YIELDS
P = (C + 10)/.9
PLUGGING THIS EXPRESSION FOR PSINTOSP = 1.2C GIVES
(C + 10)/.9 = 1.2C
SINCE WE NOW HAVE ONLY ONE EQUATION INVOLVING THE COST, WE CAN DETERMINE THE
COST BY SOLVING FOR C. HENCE, THE ANSWER IS A OR D.
(2) IS INSUFFICIENT SINCE IT DOES NOT RELATE THE SELLING PRICE TO ANY OTHER
INFORMATION. NOTE, THE PHRASE "INITIALLY LISTED" IMPLIES THAT THERE WAS MORE
THAN ONE ASKING PRICE. IF IT WASN’T FOR THAT PHRASE, (2) WOULD BE SUFFICIEN
T. THE ANSWER IS A.
4. WHAT IS THE VALUE OF THE TWO-DIGIT NUMBER X?
(1) THE SUM OF ITS DIGITS IS 4.
(2) THE DIFFERENCE OF ITS DIGITS IS 4.
CONSIDERING (1) ONLY, X MUST BE 13, 22, 31, OR 40. HENCE, (1) IS NOT SUFFICI
ENT TO DETERMINE THE VALUE OF X.
CONSIDERING (2) ONLY, X MUST BE 40, 51, 15, 62, 26, 73, 37, 84, 48, 95, OR 5
9. HENCE, (2) IS NOT SUFFICIENT TO DETERMINE THE VALUE OF X.
CONSIDERING (1) AND (2) TOGETHER, WE SEE THAT 40 AND ONLY 40 IS COMMON TO TH
E TWO SETS OF CHOICES FOR X. HENCE, X MUST BE 40. THUS, TOGETHER (1) AND (2)
ARE SUFFICIENT TO UNIQUELY DETERMINE THE VALUE OF X. THE ANSWER IS C.
5. IF X AND Y DO NOT EQUAL 0, IS X/Y AN INTEGER?
(1) X IS PRIME.
(2) Y IS EVEN.
(1) IS NOT SUFFICIENT SINCE WE DON’T KNOW THE VALUE OF Y. SIMILARLY, (2) IS
NOT SUFFICIENT. FURTHERMORE, (1) AND (2) TOGETHER ARE STILL INSUFFICIENT SIN
CE THERE IS AN EVEN PRIME NUMBER--2. FOR EXAMPLE, LET X BE THE PRIME NUMBER
2, AND LET Y BE THE EVEN NUMBER 2 (DON’T FORGET THAT DIFFERENT VARIABLES CAN
STAND FOR THE SAME NUMBER). THEN X/Y = 2/2 = 1, WHICH IS AN INTEGER. FOR AL
L OTHER VALUES OF X AND Y, X/Y IS NOT AN INTEGER. (PLUG IN A FEW VALUES TO V
ERIFY THIS.) THE ANSWER IS E.
6. IS 500 THE AVERAGE (ARITHMETIC MEAN) SCORE ON THE GMAT?
(1) HALF OF THE PEOPLE WHO TAKE THE GMAT SCORE ABOVE 500 AND HA