Date Data Suff

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hese guestlons, you am to classlfy each problem accordmg toe five flxed answeT chmces, rather than finfl a solqtlon to the problemach problepl conszsts of a questlon and two statements Xou are to decldeàether the mfprmatlon m each statement alone 1é sufficltnt to answer theuestlon.ot, lf nmther 1s, whether the lsformatlon m thq $wo statements

ether zs suftkmntTlfe followmg pages mclude test-takmg stratemes, sumple questlons,''- ''- 4

a detmled explanahon of every problem These zxplanatlons presentosglble prob lem-solvlng strategles for th: examples

est-tckiqg Strctegies for Dctc Suffzciêncy

Do not wasr: valuable tlme solvmg a probleml you areonly to determlne rhether sufficlent mformatlon ts gtvento solve tlw problem Fzrst conszder statement (1) aadstatcment (2) separately and determlne whether eachalone glves sufficxent lllformatzon to solve the problem

Be sure todisregard the fnfonnatlon gwvn m statement (1) ,when you evaluate the lnformatlon glven m statement-tz)

If etther, or /0th, of the statements gzve sufficlent lnfor-maùon to aolve the pmblem, çhck on the oval correspondjpgto the descsêtlon of wluch statementts) glve sufficlûntlnformatlon to solve the problem If not, conslder tlte

mformatlnh ,1p both statement (1) and statement (2)'n en chck osi the oval correspondlng to the descnptlonof wàethey the statemenis TOGEXHER pve sufficlentmfotrmatlgn to solve the problem

Remember th'at when you are doterm lnmg whether thrre

xs sufficie'nt mfprmatlonztb. nswer a quesllqn of the form t'Wvtlylhe ywlpzç-w' 9'' the lpfpvmayjqn gwen' . ..must be suffislqni-'io hnd oc aqd Mnly-p-p..q value for y

Belng able io detcrmlne mlnlnmm or maxlYum faluesor an answer of the form y = .z: + 2 ts <ot sufflczeht,because sqch answers constltute a range of values. ratherhall ''tlte value of y,'' '

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Wlwn geometnc figures are lnvolved, be very' cerelulnot to mak* unwprrantvpfassumptlgns based on thefigures Flgures are not necessarlly drawn to scalc,they are genera'lzed figures showlpg httle more thanznfexsectlng hne segments and thç betweenness of

pomts, angles, aqd reglons So, for example, lf a Tiguredescnbed as a rectangle looks llke a s'uare you inay 'hotconclude that lt 1s, m fact, a square Just by looklng atthe figute

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This data sufflclency,problem conslsts Of a question and tWO state- .. .

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ments, Iabeled (1) and (2), In whlch certaln data are glven.You have , , .to declde wheth4r the data glven ln the statements are sufficlent for . -

answerlng the questlon. Uslng the data given In the statements plus . .- .) your knowledge of mathematics and everyday fact Jsuch as the

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' number of days ln July or the m eanlng of countercl

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.statement (1) ALONE Is sufflcient, but statement (2) alone is not.. .

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sufficlent to answer the questlon asked;

C statement (2) ALONE Is sufficient, but statement (1) alone ls notsufflclent to answer the questlon asked,

. BOTH statements (1) and (2) TOGETHER are sufficlent to answer, the quebtlon asked; but NEITHER statèment ALONE sufflclent

. EACH statement ALONE Is sufflclent to answer the questlon asked,

.statements (1) @nd (2) TOGETHER are NOT sufficlent to answer the.

'

questlon asked, and additional data specific to the problem are

y needed

Numbers, AII num bers used are real numbers.

' 'Flgures. A figure accompanylng a data sufflciency problem w1ll confdrm .

to the Informatlon glven in the questlon, but FIII not necessarlly .

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-234- :

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Llnes shown as strplght can be assumed lo be stratlght and . . , o r

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z rojsm iss a,jYou may assume that the posltlons Of poln. ts, angtes, reglons, n; ,j

JS y ' yyyj w,s /@C ps 9RS - j etc , exlst In the Orde)r shoWn and that angle measures are. .. q V( ! 7 j ,.rpmter than Zero - ' 'ê -.k'? ''u' '

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Note, In data sufflclency prp, b lems that ask for the value of ra quantlt y , , ;$kp.'

y .krso.y-..r,,mtj-: y mk

the data glv. en In the statements are stlfsflclent only when It ls '''';s., , 'j-7C.'jts ):-'y ..j. jjyjjs j .posslble to dete,rmlne exactly one num erlcal valuq, for the quantlty ,; x'- fuc:--l-k'-ty,4,'

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Example: . ,

In L'PQR, what is the value of x ? p . .

(1) PQ = PR x ..a .>... ojsmjss

y* z2) y = 40 Q a pjoctjon,

Explanatlon: Accordlng to statemqnt (1), PQ = PR , therefore, LPQR Isisogoeles and y= z Since x + y + z= 180, It follows that x + 2y = 18Q.

Slnce statement (1) does not glve a value for y , you cannot answer thequestion using statement (1) alone. Accordlng to statement (2), y = 40,therefore, x+ z.= 140. Since statement (2) does not glve a value for z,

you cannot answer the question uslng statement (2) élone Uslng bothstatements together, since x + 2y = 180 and the value of y is given,

you can flnd tpe value of x .Therefore, BOTH statements (1) and (2)T OG ETHER are sufficient to answer the question, but NEITHER

stàtement ALONE is sufflclent

To review these dlrections for subsequent questlons of this type, cllck on.7 '

HELP

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xM e-lon j # Anaw4r .<

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#A A SUFFICIENCY SAMPkE QUESTIONS

!Sfatement (1) ALONE ls suffimqat, but gtatement (2) alohe Js not sufficzehtB Statemvnt (2) ALONE ls suffimenq but statemont (1) alon'e ls lzot stzftielentC . BOTH slatements TOGETHER are suftklent, but NEITHER statement ALONE >s suftimentD EACH statemeht ALONE ls sufficlentE Statements (1) and (2) TOGETHER ar: NOT sufficzont

1 At a certaln plcmc, qaçh of the guests was servedelther a smgle sçdop or a double scoop of zce cream

I

How many of the guests were served a double scoopof 1ce creAmg

1

(1) At the pwntc, 60 percent of the guesls wvreserved a double scoop of 1ce cream

(2) A total of 120 scoops of 1ce cream wereserved to all the guegts at the plcnlc

2 By what percent was the price of a certam candy barlncreased?

(1) The pflce of the cgndy bF was mcreased by5 cents

(2) The pnce of the candybar after the zncreaqewas 45 cepts ' ? ''

5 A thoroughly blended blscmt mlx mclades only flourand baklng powder What ls the ratlq of the numberof gramk of bakmg powder to the numbez of gramsof flour ln the mlx?

(1) Exactly 9 9 grams pf flour are contalned m10 grams of tlle m1x

(2) Exactly 0 3 gram of baklng powdzr ls contalnedm 30 grams' of tlle nux

t

'

6 If a real estate agent recelved a commlsslon of6 pereent of the àeillng pnce of a certaln lmuse,what was the selhng prlce of tlze àouse?

(1) The sellmg prlce mlnus !he real estate agent'scommlsslon was $84,'600

(2) n e selllng pnce was 250 percent of the onglnalpurchase pnce of $36,000'

7... . EJL.-h t the value of k4 .2

(1) x = - jxj(2) xl = 4

o

o zo

8 hat ls the value of z ln the tnangle aboveg

j x metvrs

3 A clrcplar tub llas a band palnted around 1ts clrcum-ference, as shown above W hat ls fhe surface area oftlus pamted bandg

(1) .); = 0 5

(2) The helgkt of the tub ls 1 metçrl ,.

4 Is lt true thût a > b ?

(1) la > lb(2) a + c > b + c

(1) .x + )' = 139(2) y + z = 108

l.'!i

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A Siatement (1) ALONE ls sufficlent. but statement (2) alone ls hot sufticlentB Statement (2) ALONE ts sufficlent, but statement (1) alone ls not stzffimentC BOTH statements TOGETHER are sufficlent, but NEITHER statement ALONE 1: sufficlentD EACH statement ALONE ls sufhmem

E Statements (1) and (2) TOGETHBR are NX gufficlent

9 A certaln bakery gellg rye bread ln l6-ounce loaves

nd 24-ounce loaves, and a11 loavos of the same jlzese fl the same pnce per loafzregardless of tllenumber-ti-f-fl'aeK'lyecllasedzn at Is the pnce of a24-ounce loaf of rye bread m tG bakery?

The total pnce of a lGounce loaf and a24-(mnce loaf of tllls bread 1: $2 40

(2) The total pnce of two 16-pu,ce loaves andone 24-ounce loaf öf thls bread 1s $3 40

10 If = n' what ls the value of .x ?)'

(1) yn = 10

1y = 40 and n = -4

11, If m and n are consecutlve posltlve lntegerxs, lsm greatet than n 9

e %.hbm - 1 and n + 1 are consecutwe posmve zz >( %

7 mz- b s'r N.ntegers kyr> p

w * 'W %. 1 3 22)m ls an even lnteqejy. ke * ,.

ro A *Yq ..r'- If .t ls equal to one of the numbers -y , j', or y ,ï e$ *,'7+'-X- v S'A'--' gtk : wllat ls the value of x ? - -- y çt7v.- xw A,q

NrJ.2 a and Sand re among those people who sold

ffl tlckets t ra1 #..'oney for Club X lf Paula 1 < x < 1aan andy sold a total of 100 of the hckets, how 4 2

many of the tickels dld Paula $e1l? 3(2) < x < j

32116

14 E1 and L egph represent smgle dlgtts ln the

declmal above, what dtpt does Q repres4nt?

(1) When the declmal ls rounded to the nearesttenth. 3 2 ts the result

(2) Wlzen the declmd ls rounded to the nearest'--x hundredth, 3 24 ls the result,

%

15 A certaln company currently has how manyemployeçsg

I

(1) If 3 addltlocal eëployees are hlred by the

conlpany and aI1 pf tho presem employeesremaln, tlwre w111 be at least 70 employees lnthe company

(2) If no addmonal employees are lured by thecompany and 3 of the present employeesreslgn, there w11l be ftwer than 15 employeesm tlw company

!' as many of the raffle tlckets as1) Sandy sold3

Paula d1d

(2) Saady sold 8 percent of a1l the raffle ttckets soldfor Club X '

13 Is the mteger n odd?

(1) n is dlvlslble by 3(2) a ls dlvlslble by 5

17 f a, !), and c are mtegerg, is a - b + c greatertha'n a + b - c ?

(1) b ts negatlve

(2) c ls posmve

18 If x + 2, + 1 = y - x, what ls the value of .x '?

(1) yl = 9(2) J = 32

wf

l e

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A Statement (1) ALONV ls sufficlent, but statement (2) alone ls not sufficl:ntB Statemqnt (2) ALONE ls suffictents but statement (1) alone Ts not suftklentC BOTH statements TOGETHER are sufficlent,ybut NEITHER statement ALONE ls sufhclentb EACH statement ALONE fs suffic:ent

E Statementg (1) and (2) TOGETHBR are NOT suflklent

19 lf n ls an lnteger, then n ls dlvlslble by how m any

posltwe lntegersg f

(1) a ls the product of two dlfferent prlmenumbûrs

(2) n and 23 am each dlvlslble by the samenumber of positwe lntçgersl

20 How many mlles long is the rolae from Hough onto Callaharf?

It w111 lake 1 hour less tlme to travel the entlreroute at an average tate of 55 mlles per hourthan at an average rate of 50 mlles per hour

It w1ll take 1 1 hoursf to travel 4he firsy half ofthe route at à.n avqrage rate of 25 mlles perhour

21 If x and y are posltjve, what ls thevalue'of.': 9

(l) .g = 3 927:

(2) y = 2 279

22 John and Davld eazh recelved a salary mcreaseW hlch one recmved the greater dollar lncrease?

(1) John's salaty lncreased 8 percent

(2) Davld'j salary lncreased 5 percent

23 Carlotta can drlve from her home to her officç byone of two posszble routes. If she must also returnby one of these routes, what ls the distatlce of jheshorter route? *

(1) Fhcn she dnves froz her home to her officeby the shorter roure and returns hy theIonger route, s'e drlves a total of 42kllometers ,

(2) When she drlves ''hoth ways, from her home toher offlce 1nd back by the longer route, shedrlves ?. total of 46 kilometers

24 If r ànd .% are posltwe lntegers, r ls what percent

of s ?

25 A shlrt and a palr of glbves çost a total of $41 70Hpw much does the palr of gloves cost?

(l) The shlrt costp twlce as ktuch as the gloves

(2) The shlrt costs $27 80

26 W hat ts the number of 360-degree rotatl'ons thata bzcycle wheel made whlle rolhng 100 meters m astratght ine wttlmut sllpplng? ,

(1) The dlameter of the blcyyle wheel, lncludlng ttlettre, was 0 5 meter

(2) The wheel made twenty 360-degree rotatlons.

. per mlpute

27 ' hat ls thç value of the sum of a hkt of n oddlntegersg

..F'

(1) n = 8

(2) The square of the number of mtegers on the hst '-1464 't

x f

18 lf a certyln ainmated cartoon conslsls of a total of17,280 frames on film, how mahy mlnutes w1ll lttake to run the rcartoong

(1) Th e oartoon runs wlthout mterruptlon at therate of 24 frames per second

(2) lt takes 6 tlmes as long to run the cartoon as lttakes t: rkwmd the film. and lt takes a totalof 1' mlnutes to do both

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A Statement (1) ALONE zs suflkjont, but statemont (2) alone ls not suffimentB statement (2) ALONE ls sufhclent, but statemeq: (1) alqne i: not suflimentC BOTH statemelgs TOGETHER are sufficlent, buy NEITHBR statement ALONE ls sufficlent

D EACH statement ALQNE ls sufhclent!Statements (1) and (2) TOGETHER a:e NOT stlfficlent

29 W hat was the average number of mries per gallonof g#sollne for a car dunng a certaln trlp?

(1) The total cost of the gasolme used by the carfor the 180-jmle trlp was $12 00

(2) The cost of the gasolme used by the car for tlletrlp was $1 20 per gallon

.&30 If x and y are posltlve, ls - greater than 1 ?

.Y

(1) .'Kk > 1

(2) x - y > 0

31 In LPQR, lf PQ = x, QR = x + 2, andPR = y, wàlch of tlw threamn f LP #has grsatçj-tsdegree measure?< < - v v -

( 1) y = x + 3

(2) .< = 2.

32 Is the prlme number p equal to 37 7

(1) # = n2 + 1. where n ls alt mteger(2) p2 Is greater than 200

33 The only contents of a parcel are 25 pllotograpb&and 30 negatlves W hat ls the total welght, inounces, of the parcel's contents?

(1 ) The welsht of each plmtograpk ls :3 times tlwwelght of each neggtlve

(2) The total welght of l of tlw photographs and 21 I

of the negatlves ls ounce'

h

38 A snm bf $200,000 from aecertaln t was dtvldedamöng a spouse and three chlldren ow much oftlle estatv dtd the younvest chlld reoiwe?

%

1 f the sum from the1 ) 'rhe spouse recelved e2 i

wofstate, and the oldest chlld received 4

tlw remmader

(2) EacN of the two qounger cluldten rec:we4$12,500 more than the oldest chlld qnd$62,500 les: tpan the spouse

. 2

W

34 If 2. and w represent the length and wtdth, respec-ttvely, of the reçtahgle above, what 1s the penmetzr?

(1) 22 + w = 40

(2) 2.+ w = 25

35 W hat is the ratlo of x to y ?

(1) x ls 4 more tllan twlce y(2) The ratlo of 0 5.. to 2y ls 3 to 5

36 If x, yb :nd z are tllree lntegers, are they consecu-tlvo mtegers?

(1) z - x = 2

(2) x < y < z

37 W hat 1s the value of .v ?

(1) - @ + X = .f - y

(2) x + )' = 2

-1/10 -






A Statement (1) AIa ONE is suffitient, but gtatemyxt (2) alone ls hot safsçlent..B Statement (2) ALO E is sufticlent, but statement (1) alone ls nöt jufflclent'C BOTH statemetlts TOGETHER are sufficlent', but NBITHER stytement ALONE ls sufficientD EACH statement ALONE ls.sufficlent - -

E Statements (1) and (2) TOOETHER are NOT suflklent

39 If the Lmcoln tgbrary's total expendlture for books,penodlcalj, an' newspapers laqt year was $35.,000,how much of the expendlture was for booksg

The expendzture för newspapers :as 40 percentgreater than the expend-ture foryenodlcals

t A

(2) The total of the expendlture for periodicals andnçwkpapers was 25 perc.nt less than thevxpendlture fo< books .

40 The symbol V represents olmzpf tlle followmg

opeqatlons' addftlon, subtractlon, multlphcatlon,or dlyision rW hat ls the vglue Qf 3 V 2 ?

(1) 0 V 1 = 1

(2) 1 V 0 = 1

41 The regular pnce for canned soup *a$ teduceddunng a sale How muc'h money could one havesaved by purchasmg a dozen 7-ounce cans of soupat the reduced prlce rather than at thè regular prlceg

(1) The regular pnce for the 7-ounce cans was 3 fora dollar

(2) The teduced prlce for the 7-ounce cans Fas 4f dullar ' 'r a

42 lf on a fishmz ttlp J1m and Tom eavh cau'jht some

fish, whlch one caught mor'e fish?:h -

(1) J1m caught -g. as many tish as Tom3 u .-

(2) After Tom stopped fishlng, J1m contmuedfishmg unttl he had caught 12 fish

43 If 5.1 + 3), = 17, what ls the value of .r 9

(1) ,x ls a posltlve integer(2) y = 4x

44 Yesterday Nan parked hzr car at a certam p'arklnggarage that tharges mbre for the flrst hour than for

each qdditlonal hour lf Nan's total parklng chargeat the garage yegterdày was $3 75 for how manyhours of parking was she charged'

Parklng chargçs at tie gârage .re $0 75 for thefirsy llout and $0 50 for each additlonal 'ouror fractlon of an hour

(1) If the charge for the first llour had been $1 00,Nan's total parhng chargc would have been$ 0 0

45 lf r and s Are lntrgeto, Is r + s dlvlslble.by 3 ?

(1) - s ls dlvlslble by 3(2) r ls dwjslble by 3

Q

# z

46 Wlmt ls the Vdlus of the circle above wlthcenter O ?

(1) The ratlo of OP to PQ ls 1 to 2(2) P ls the midpolnt of chord AB

47 A certmn 4-l1ter solutp: of vmegar ahd waterconslsts of x hters of vrmegar and y llters of waterHow many llters of vlnegâr does the solutloncontam g

x 3

4 8

51-

4 8

K

-241-








A statement (1) ALONE ls sufficlent, but stafement (2)?alonc ls not sufhclentB Statement (2) ALONE ls sufficlent. but statement (1) alone ls not suftklentC BOTH statements TOGBTHER are sufflclent, but NEITHER statement ALONE ls suffidentD EACH statement ALONE ls sufhclent

E Statements (1) and (2) TOGETHER are NOT suffiment

48 Is .& < 0 ?

(1) - 2x > 0(2) x3< 0

49 Of tlle 230 smgle-famlly homes buzlt m Clty X iastyec, llow many were ocvuplvd At the end of theyear?

(1) Of all slngle-famlly homes m Clty X,90 perctnt were occupted at the end of lastyear

(2) A total of 7,200 slngle-farmly homes tn Clty Xwere occupled at thewnd of last year .

50 Does the prodact Jkmn equal 1 ?

k-1- = 11)

1 1) = - and m = -

n

51 How many of the boys ln a group of 100 chtldrenhave brown halr?

(1) Of the chlldren ln the group, 60 percent havebrown halr

(2) Of the chlldren ln, the group, 40 are yoys52 Is tlze perlmeter of square S greater than the

perlmeter of equllateral trlangle T 9

(1) The ratlo of the length of a slde of S to thelength of a sttl. of T ls 4 5

(2) The sum of the lengths of a slde of S and aslde of T ls 18

53 If p and q are posmve lntegers and pq = 24,what ts the value of p ?

.% s an mteger1) t

6L ls an lnteger2)2

4

54 If x # 0, what ls the value of q ?I

(1) # = q

(2) .: = 3

5J. From May 1, 1960, to M:y 1, 1975, tlw closmg pnceof a share of stock X doubleed Wâat was 4heclosmg pnce of a share of stock X on M ay 1, 1960 9

(1) From May 1, 1975. to May 1, 1984, the clostngpnce Qf a sllare of stock X doubled

From M ay 1, 1975, to May 1, 1984, the closlngpnce of a share of stock X mcreased by

$4 50 *

56 If # ls a posltlve mteger, ls 'C- an lnteger?

(1) # 1, fhe square of an lnteger

VJ'- the square of an lnteger2) ls.

57 If Q ls an lnteger between 10 :nd 100, whay ls thevalue of Q 9

(1) One of Q's dtgits ls 3 more than the otlmr. andthe sum of lts dlglts ls 9

(2) Q < 50

58 If dzglt h ls the hundredths' dtglt ln the declmal# = 0 1h6, what ls the value of d, rounded to thenearest tenth?

11) d <4

(2) h < 5

59 what 1s thc value of xl - y2 7

x - y = y + 2

1

(2) A - 'Y = .Y + Y

-242-






A Statement (1) ALONE lstsufficienklbut statement (2) alone ls not sttfficlentB Statemdnt (2) ALONB ls sufficlent, but statement (1) âlrme ls no! gufficientC BOTH statem' ents TOGETHER are sufflclent; but NEITHER stgtement ALONE Js sufficlentD EACH statement ALONE ls sttfficlent

E Statements (1) atld (2) TOGETHER are NOT #ufficlent.

E

60 If o represents one of the opetatlons +, - .

and x, ls k e (2 + mj = (k e 2) + (k o m) for a1llmmbers k, 2., and m 9

(l) k o 1 ls not equal to 1 p k for somenumbers k

(2) o represepts gubtractTon

61 Commlttee m:mber W wants to sclïedule a one-hour meetlng on Thursd4y for htmself an4 threeother commlttee memherss X F, and Z lg 'therea one-hotlr penod on Thursdyy that ls open for al1four mumbvrs? '

On Thursday W' ,and X''have an, open penod

from 9 ()U a m to 12 ()0 noon(2) On Thursday. F has an' opeh perlod from

10 00 a m to 1 00 p.m. and Z has àn openpenod from 8 00 a m to 11 Q0 a m

) .A

62, lf JAçk's ahd Kate's anlmal salAtxs m 1985 wereeach 10 percens htgher t1)a11 thçy respectlve annualsalanes ln t984, what was Jack'j anquat salary m1984 9 z

?

(1)4 'hç gsm of Jack's anb Kate's annual salanes m1984 was $50.000 . ï '

(2) The stlm of Jack's apd Kate's anqual salanes ln1985 was $55,Q00

63 W hat zs the value of . 9

( 1 ) .!: + 1 = 2 - 3x1

. ' (2) = 2.1

64 How many newspapers werelsold at' certam news-stand today?

(1) A total of 100 newspapers were sold at thenewsstand yesterday, 10 fewer timn tmce tllenumber sold tod4y

(2) The number of newspApers sold at the news-stand yesterduy wqs 45 more tlzan tllenumber sold toda '.

65 How much dld a certam telephone call costg

(1) The call lasted 53 mlnutes

(2) The cost for the flrst 3 mmutes was 5 tlmes thecost for each addltlonal mtnute

h î

66 A certaln pxpressway h s exlts J. K, fa, and M ,ln thttt order' Whdt 17 the road dlstance fromx1t K to extt L ? '

' k

The road dlstânce from exlt J to exit L 1e21 kilometers

(2) The road dlstance-from exg K to exxt M ls2)6 kllometers '

67 Two cars, S an4 F, each eaveled a dlstdàce of50 mlles Dld car # use more gasohne than car F ?

(1) Cars S and T traveled the çntlrç dlstancv at,tlle rates 4f 55 mlles pe# hour and 50 mtlesper hqur, respectively.

(2) For the entlre dlgtance, câr S traveled 20 nulesper gallon Qf gasollne and car T traveled25 nules per galton of gasollne

68 If n ls a posltlve mteger, ls rl odd?

(1) 3a ls od4

(2) n + 3 ls even

69 Doel lm - .5 = 0 9l

(1) m # 0

(2) 6m = 9n

70 If xy <.3s is x < 1 7

(1) y > 3

(2) .* < 3

-1 3-








A Statement (1) ALONE ls Fufticlent, but staltment (2) alene ls npt sufficlentB Statement (2) ALONE lg sgfficlent, but Matement (1) alone ls pot sufficlentC BOTH statements TOGETHER are sumczent, but NEITHER statqment ALONE Is suffidzentD EACH statement ALON,B js sufflcientE Statements (1) and (2) TOGETHBR ar: NOT sufficlent

7 1 Eaçh of the eggs ln a bowl ls dyed red. Qr green! orblue If one egg js to be removed at random, what lsth: probablhty that the egg w11l be greeng

(1) There are 5 red eggs ln the bowl

(2) Thç probablllty that the egg wl1l be blue ls)

72 ls the average (anthmetic mean) of 1<:and zgreater than 20 9

The average (arlthmetlc mean) of 2x and 2),ls 48

(2) x = 3)?$

73 M arcla's bucket can hold a maxlmum of how manyltters of watvr?

(1) 'fhe bucket currently contalns 9 hters of water

(2) If 3 hters of water are added to the bllcket whenlt ls half full of water, the nmount of water ln

i

the bucket wlll mcrease by

Jk

11 1

,x1-

IH IV

*

76 If ab # 0, tn wllat qua4rant of tlie coordmatesystem above does polnt (4, b) 1149

(1) (b, J) lles ln quadrant IV.

(2) (a, -ô) lms ln quadrant 111.

77 From 1984 to 1987, the value of formgn goodsconsumed annually m the Ualyed Stat:s mcreased bywhat percentg

ln 1984 the value of forezga goods consumedconstltuted 19 8 percent of the total value ofgoods consumed 111 the Umted Stateg that

year

(2) In 1987 the value of formgn goods consumedconststuted 22 7 pement of the total value of tgoods consumed ln the Unlted States thatyear

A ?8 If x, y, and z are pogltlve. ls .< = zz

L1) 2 =AZ

(2) : =

79 lf .x and y are posltlve lntegers and xD = xb - 3,

what ls the value of xy 9

@

b14 In the tnangle above, does :2 + bl = c2 9

(1) .x + z = 90

(2) x = y

73 W hat ls the value of the posltwe mteger n ?

(l) a4 < 25

(2) n # a2

( 1) .v = 2(2) .:3 = 8

wir4zk.






A 8tatement (1) ALONB is suftkwnt, h?t .statemçnt (2) alono ts tmt Aùfficle'rjt

B Statement (2) ALONE ls Auftjclent, b4t statefnent (1) aloàe ls not sufficmntç BOTH statements TOGETHVR are suflklent, but NEITHER ytatement ALONB ls suffidzentD BACH statement ALONE ls suftklent xE Xstatements (1) and (2 TOGETHER are NOT sufflcyçnt

1

80 If k and n are mtegers, zs n dlvlslble by 7 ?

(1) n v 3 = lk(2) 2* - 4 ls dlvzslble by 7

8'1 If x and y are întegers âtld

y = t.v +31+/- x1 does y equa'l 7 ?

(1) .x < 4(2) .;r > - 3

82 lf 1 < # < 2, ls tllr teltths' dlptkof the de'lmalrepresentahon of d equal to 9 9

( l ) . # ,+ 0 0 1y 2(2) d + 0 05 > 2

q .''

83 Mw partlmpants 1h a racç conàlstod of ) teams wltlt3 runners on each tem A teâm wàs awarded 6 - npomts lf one .of 1ts nmpcrg t'iniàhed m nth place.where 1 f n 5 5 If a11 of the nmners finljhed thertce 4nd lf there were lm tjes, wag each teamawarded at leqst ono pmht?

(1) No yeam was awarded morè than a total of6 polnts

f

(2) No palr of teammates finlshed m censequtlve

plâces among the top five jlaces

84 lf x +j? + z > 0, ls z > 1 9

(1) z > .z'+ y + 1(2) x + y + 1 < 0

85 How many mtegvrs n are th re such tllay r < n < J 9

(1) s - r = 5:

(2) r and s at.4 not mtegers

86 A total of 9 women and 12 men resldç m the

21 apm ments that ate m a cerymn apartmentbmtdlng, one person to each apartment, If a polltaker ls to select one of the apyrtments at random,what ls tlm probablllty that tlle resldent of the apart-ment selected wlll be a woman who ts a studentg

(1) Of thè wdmpn, 4 a1:0 students.

(2) Of the wemen, 5 are not students *t

87 Is x greater thg.n 1 8 9

(1) x > 1 7

(2) x > 1 9

88 Hosqs X ân4 F slmultaneously fill an emptyswlmmlng pool that llas : capamty of 50,000 htersIf the flow m eqch hose ls mdepeqdent of the flow lnthe other hose, how màny hours wlll lt take to t111the pool?

(1) Hose X alone would take 28 hours to fill thepool

(2) Hose F alone would take 36 hours to fll1 thepoolm

>. k

Z

O

89. In tlze ftgute above. tf lmes (; atld m are parallel,what ls the valtw of x 9

(l) y = 120

(2) z = 60

90 lf wx and y are lntegers, what ls the value of y ?

(1) ly = 27

(2) x = :2

-245-








A Statement (1) ALONE ls sufflcleht, but statement (2) alone ls not sufficlentB Statement (2) ALONE ls sufficlen't, but statement (1) alone ls not suffimentC BOTH statements TOGBTHBR are sufficleht, but NEITHER statemenl ALONE ls sufficlent

D EACH statement ALONE ls sufflclent

E Statements (1) and (2) TOGETHER are NOT sufficlent

(c, +

P a(a, #)

96 If P and Q are each clrcular reglons. what ls th4radms of the larger of these reglons?

The area of P plus the area of Q ts equalto 90l

(2) The larger clmular reglon has a radlus that ls3 tlmes the radlus of the smaller clrcular

reglon

97 Is z less than 0 9

(1) xz > 0 and ),z < 0

(2) .e > 0

98 If the tojal prfce of n equally pmced shares of acertaln stock was $12,000, what was the prlçe pershare of the stock?

(1) If the pnce per sharc of the stock had been $1more. the total prlce of the n shares wouldhave been $300 more

(2) If the prwe per share ol the stock had been $2less, the total prme of the n sharts would

haye been 5 percent less

99 What ls the ratlo of .&

z = 1 and Jry = 32

2 -x = 2 a n d .!. = -1J' y 4

91 In the l'gure above, segments PR and QR areeach parallel (o one of the rectangular coordlnate

axes Is the ratlo of tbe length of QR to the lengt:of PR equal to 1 ?

(1) c = 3 and d = 4

(2) a = - 2 and b = - 1

92 In a school electlon. lf cach of the 900 voters votedfor elther Edlth or Josd (but not both). what percentof the female voters ln thls electlon voted for Josd?

(1) Elgkty percent of the female voters voted forEdlth

(2) Slxt# percent of the malg voters voted for Josë

93 Duflng weck F, how much dtd àt cqst, per mlle,for tlle gasohne used by car X ?

(1) Dunng week W, car X used gasollne thatcost $1 24 per gallolz:

(2) Durmg week F, car X was dnven 270 mlles

94 If r and s are lntegers, ls r dlvlslble by 7 9

(1) The product rs ls dlvlslble by 7

(2) s ls not dwlslble by 7 ..

m 595 If - = - , what ls the value of m + n ?

n 3

(1) m > 0

(2) lm + n = 26

100 W hat ls Rlcky's age now9

(1) Rlcky ls now twtce as old as he was exactly8 years ago

(2) Rlcky's slster Teresa ls now 3 tlmcs as old asRlcky was exactly 8 years ago

101 Is xy > 5 9

(1) 1 f .. f 3 and 2 K y K 4

(2) .* + .# = 5

-246-






A Statoment (1) ALONE zs eufficlent, but statemenl (2) alone ls hot sufpclentB Statement (2) ALONB ls sufticlent, but statement (1) alone ts not suffim4ntC BOTH statements TOOETHER are sufflment, but NEITHER statement ALONE ls suffimehtD EACH statement ALONE ls sufficmnt

E Stàtements (l) and (2) TOGETHER are NOT sufficlont

...) p ''.

1û2 ln year X. 8 7 percent of the men m tlw labor forcewere unemployed lp June compared wlth à 4 percgntm M ay If tlle number of men m thç labor force wasthe same for bqtll months. how many men wereunemployed ln Jutle of that 8:r9

(1) In May of year X, the number of unemployedx men in the labor force wAs 3 36 mllhon

(2) In year m 120,0Q.0 more mdnzln the laborforce wbre unemployed m Jut:e thrn 1rt M:y

1

103 If the average (anthmetlc mean) of 4 bumbers ls 50,how many of the numhers are greayer tltqn 50 9

(1) None of the four numbers ls dqual to 502) 'lWo of the numbers are equal tù 25 %

&

104 On Mondyy mormng a certaln machme ran ccmtmu-ously at a unzfprm rate to fill à protluchon orde: At .what tlme d1d 14 completely fijl the order thatmornmgg

n e machmv began filllng the order at 9 30 a m

1 of the order hy2) 'lrhe machme had tilled2

j '0,30 a m and of the order by 11 10 a m

j

105 If n + k = m, what ls the value. of k 9

108 ln the flgure aboves segments Rà and TIJ repre-sent two posltlons of tlle same ladder leanlng agalnstthe sldo JF of a wall n e Iength of 7V Ts howmucll greater than the length of RF 9

k

'

ti 19 metei1) The length of 1t' ls(2) The length of 47 ls 5 meters

107 Can the posltwe znteger tl be wnttm as the sum oftwo dlfferent posltwe pnme numberé?N

(1) n ls greater than :!(2) n l: odd

U

f

45@ 60@k A k

109 If both .x and z are ntmzero numbersa what ls the

lue of -% ?a.x

(1) ,e = 62) / = xb ,

110. lf x = 0 rstu. whère r, s, 1, and u eacll repre-sent a nonzero dtglt of x, whwat ls the value of x 9

(1) n = 10(2) m + 10 = n

106 Town T has 20,000 resldents, 60 percent of whom

afe female. Whét percent of tllq resldents were born1n' Town T 9

(1) The numbqr of female resldents who wereborn ln Town T ls tquce the number of maleresldents who were not born m Town T

(2) The number of female resldençs who were not

born ln Town F ls twlçe the number pffemale resldents who were boim ln Town F

(1) r = 3. - 21 = 64(2) The product of r and u ls equal to the

product of s and t

#

.a4y.








A Statement (1) ALONE is sufsclent but Matement (2) alone ts not juffimentB Statement (2) ALONE ls suuticjent, but statçment (1) alont ls not suf:mentC BOTH statements TOGBTHER are sufficlent, but NEITHER statement ALONE ts sufficlentD ZACH statpment ALONB js sufficlentE Statements (1) and (2) TOGETHER are N'T sufficlent-

111 What were the gross revenues from tlcket sales for

a certam 5lm dunng the second week ln whlch ltwas shown?

(1) Gross revenues dunng te second week wore$1 5 mllllon less than durlng the first week

Gross revenuçs durlng tile tlurd week were$2 9 mllhon Jejs than dunhg the first week

112 W hat number ls 15 percelt of x 9

(1) 18 is 6 percent of xl

(2) à- bf x 19 200

113 How many books dces m cardo have?

If m cardo had 15 fewer books, he would haveonly half as many as he actllally has

(2) - Rlcazdo has twme as many tktmn books asnonfictlon books

1 14 W hat was the amount of mûney donated to acertaln chantyg

(1) Of the amount donated, 40 percent came fromcorporate donatlons.

(2) Of the amount donated, $1 5 mllhon came fromnoncorporate donatlons

115 ls .x a negatwe numberg

(1) 9.x > 10x(2) x + 3 ls posltwe

116 John took a.test that had 60 qeestlons ngmberedfrom 1 to 60 How many of the questlons d!d heanswer correctly?

(1) The number of qtlestlons he answered correctly

ln the first half of the test was 7 more thanthe ntzmb:r he answered correctly ln tllesecond llalf of tlle test

1 of tlw bdd-numbered2) He answered:

1 17 The perimetzr of a rectangular garden ls 360 feet

What ls the length of the gardeng

(1) The length of the garden ls twlce the wldth(2) The dlfference between the length and wjdth of

the garden ls 60 feet

118 If 2*(5/) = t, what ls the value of t t

(1) x m rl + 3(2) 2, = 32

1 19 In AXFZ, what ls tlle leagtb of FZ 9

(1) The length of XY ls 3(2) The lvngth of X% ls 5

120 If Jl'l's average (arlthmetlc mean) score for tlzree* games of bowllng was 168, what was her lowest

score?

(1) Jlll's hlghest score was 204

(2) The sum of Tlll's two hlghest scores was 364

121 An empty rectangular swimmmg pool has unlform

de/h How long wlll tt tAe to fill th: pool wlthwater?

(l) Water w1l1 be pumped m at the rate of240 gallons per hour (1 cublc foot =7 5 gallons)

(2) The pool Is 60 feet long and 25 f:et wtde

122 ln Jefferson Sçhool, 300 students study French orSpanlsh or both If 100 of these studeats do n6tstudy French, how many of these syvdents studyb0th French and Spamsh?

(1) Of the 300 students, 60 do not study Spanlsh(2) A total of 240 of the students study Spanlsh

!. f tjwuesttons correctly andj o

even-numbered qqestlons conw tly

-248-






X$

A Statement (1) ALONE Is suffimbnt, but statement (3) alone ls not sum ment.B Statement (2) ALUNE ls sufficzent, but statement (1) alone ls not sufficlexit 'C BQTH slatemenls TOGETHBR are sufficjent, hut MEITHER statemqnt ALONB u suffimentD EACH staçement ALONE zs sufflclentE Statements (1) and (2) TOGETHER are NOT sufficlent

123 ls the value of n closer/to 50 than to. 75 9

(1) 75 - n > n - 50(2) n > 60

124 In the equatlon x2 + hx + 12 = 0, x ls Ik varl-able and b ls a constant W hat ls the value öf b ?

(1) .r - 3 ls a factor of x2 + bx ;+ 12(2) 4 ts a root of the equatloh xl + bx + 12 = 0

125 If Juan had a dodorAs appmntment on a certalnday, was the appomjmçnt on a Wednesday?

(1) Exactly 60 hours before the apppmtment, lt wasM onday

The appolntment was between 1 00 p m and9 00 p m

126 W hat ls the value of b + c 9A

(1) ab + cd + ac + bd = 6

(2) J + # = 4

127 If m lg an lnteger, zs m odd 9

(1) .!!. ls not an even lnteger2

(2) m - 3 zs an even imeger

129 At the begznnmg of last mopth, a statlonèry sçore

had m stock 250 wntmg pads. whkh had cost thestore $0 75 each Dorlng the same month, th'eustommade only one purchase of wntlng plds What wasthe total amount spent by the store on the wntlngpads tt had tn stock at the epd of last monthg

(1) Last hmnth tlle stgre pufehased 150 wntmgpads for $0 80 çach

#

(2) Last month the tntal revenue from t'he sale ofwntlng pads was $180

130 lf y ls an 'mteger, ls / dwlslble by 9 %

(1) y Ss dlvlslble by 4

(2) .Y ls dwlslble by 6

131 W hat was the ratlo of the number of cars to thenumber of trucks produced by Company X lastyear?

(1) Last year, lf th' humber of cars produced- s by Company X had been 8 percent greater)

the number (jf cars produced would havebeen 150 percent of theonumber of truçksproduced by Company X

Last year Company X produced 565,000 carsand 406,800 trucks

132 How lony dld lt take Betty to dnve nonstop on atnp from her home to Denver, Coloradog

lf Betty's average speed for the trlp had been

11 tlmes as fast, the trtp would havé tkken

2 hours

(2) Betty's average speed for the tnp was 50 mllesllour 'er

Xi

y .F

Z

128 What ls the mrclzmferelwerbf the clrcle above wlthcenter O ?

(1) The perlmeter of hOXZ ls 20 + 10 W(2) The Iength of arc #FZ ls 51

-249-








A Statement (1) ALONE $s guffiotent. but statement (2) alone ls not suffimentB Statement (2) ALONE ls sufficlent, but Matement (1) alone Ig not suft cientC BOTH statement: TOGBTHER are suffimeht, but NEITHER statement ALONB ls sufficlentD EACH statement ALONB ls sufficlent.E Statements (1) altd (2) TOGETHER are NOT sufhclent

P

X

O Q

A B C D

136 Jn the flgure above, ls CD > BC 1

(1) A9 = 20(2) A# = CD

137 How many more men than women art m tht mom?

(1) There ls a total cf 29 women and men m tlleroom

(2) The number of men m the room equals tllesquare of tlle number of women ln the room

100 - n38 Ifn ls an mteget ls an integer?

a

(1) n > 4(2) n2 = 25l

3139 Last Frlday a certaln shop sold . of the sweaters in

4 ;

1ts lnventory Each sweater sold for $20 What was

the total revenue last Fnday from the sale of lhese

sweaters?

133 In the rectangular coorthnate system above,

lf OP < PQ. ls the area of regton 0PQgreater than 48 9

(1) The coordmates of polnt P are (6v 8)(2) The coordmates of polnt Q are (13, 0)

2

a=1 x- + .--x 3x

134 In the expressjon above, lf xn # 0, what 15 thevalue of S ?

(1) .t= 2a

12) n =ï

135 If S 1à the lnfimte sequenceSL = 9, ss'z = 99, ,5f3 = 999, , Sk = 1'0k- 1, ,

ls every term m J dwlslble by t11* pnme number p ?

(1) p ls greater than 2

(2) At least one term m sequence S ls dlwslble by p(

'

(1) When fhe shop opened last Fndayg there were160 sweaters m 1ts mventory

(2) All but 40 sweaters m the shop's lnventory weresold last Frlday.

140 A Jar contalns 30 marbleg, of whlch 20 are red and10 are blue If 9 of the mqrbles are romoved, howmany of the mvbles left ln the Jar are redg

(1) Of the marbles removed. tlle ratlo of thenumber of red ones to the number of bltïeones ls 2 1

(2) Of the flrst 6 marbles removed, 4 are red

-250-






#

A Statement (1) ALONE ls slufficzcnt. but statement (2) alone ls not,stzfficleot,B Statement (2) ALONE ls suffimelt, but stdtement (1) alone ls not sufficjentC BOTH statements TOGETHBR are sufflcjent, but NEJTHER statemcnt ALONE ls su-fflclentD EACH statement ALONE ls sufflcient .

E Statements (1) Anll (2) TOGBTHER 4re NOT suffiment

4

Q @'.Y z

*

146 Whlle on a stratght road, car X and car F aretravellng at dlffemnt constant rates If car X ls now

1 mlle alleadpf car i', how many mmutçs fromnow wl1l car X be 2 mlles ahead of car F 9

Y

Car X ls travelmg at 50 mlle,s per hour anllcar F ls travellng at 40 mlles per hoùr

12) 3mlnutes ago car X wms - nule ahead of car F41 Is the trtanglo above equllateral?

(1) x = )?(2) z = 60

k

1.

142 If w + z' = 28, what ls the value of wz ?

(1) w and z are posltlve mtçgers42) w and z are conseoutlve odd lntqgers

l

147 In what year was Ellcn born?

T 11) Ellen's brother Pete, Who ls 1 - years older tjan

Ellen, was born m 1956In 1975 Elien turned 18 years old

: z'

148 Is 21 greater than 100 r1

W = 8

12) < 0 01F

ï t

149 What 1s tke number of female employees mCompany X9

(1) If Company X were to hlre 14 more peopleand a11 of these people were fcmales, the ratloof the number of male employees to thenumber of female employees would then be16 to 9

(2) Company X hàs 105 mbre male employeesthan femalc cmployees q

k% %

A <

<143 Wlll the first 10 volumbs of a zo-volume

encyclopedla f1t uprlght m the bookrack shownabove?

k 5

(1) .x = 50 centlmeters

(2) Twelve of the volûmes have an averxge

(anthmetlc mean) thlckness of 5 centtmeters

144 Is tz.x = 3 - bx 9

(1) x(a + b) = 3

(2) a = b = 1 5 and x = 1

145 W hat îs the value of the m teger .x 9

(1) x ls a pnme number

(2) 31 f x 4 37

N

150 Is the lnteger .;r dlvlslble by 36 9

(1) x ls dlvlslblc by 14

(2) x ls dtvlslble by 9

-2.,-








A

A Statement (1) ALONE ls sufficlent, but statement (2) alone ls not gufficlentB Statement (2) ALONE ls sufsclent, but statement (1) alone ls not sufficientC BOTH statements TOGETHER are sufficlent, but NEITHER st>tement ALONE ls sufficlentD EACH statement ALONE ls suffiment 'E Statemvnts (1) and (2) TOGETHER are NOT suftklent

What ls the average (ant:metjc mean) of ) and k 9

(1) Tke average (anthmetlc mean) of J + 2 andk + 4 lg 11

The average (anthmetlc mean) of J, k, and 14ls 10

154 In a certaln office, 50 pmrçent of the employees are

college graduates and 60 percent of the employeesare over forty years old If 30 percent of those overforty have master's degrees. how many of theemployees over forty have master's degrees?

(1) Exactly l00 of the employees are college gradu-ates

Of the employees forty yeAs old or less,25 perceny have master's degrees

155 Is xy < 6 ?

x < 3 and y < 2

1 2(2) y < x < y and y2 < 64

152 W hat 1: the value of 4 - b ?

(1) a = b + 4(2) (a - :)2 = 16

153 ls rst = 1 ?

( 1) rs = 1(2) st = 1

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A Statement (1) ALONE ls suff lent, but statement (2) alone ls not suflklentB Statemen4 (2) ALONE ls suflklent, but statem:nt (1).;a1oae ls lmt sufficlept.C BOTfI statements TOGETHER are sufficlent, but NEITHER statement ALONE ls sufficlentD EACH statement ALONE is suflklent ,

E Sutements (1) and (2) TOGETHER are NW sufhclent

The regular prlce per mgkt-ounce can of brand Xsoup ls $0 37, regardless of the number of canspurchaged What amount w11l be saved on tfm purchaseof 3 elght-otmce ians of brand X jotlp lf theregular pnce Ss reduced?

(1) At the reduced pnce, 3 elght-ounce cans ofbrand X soup w1l1 cost $0 99

(2) The gmount that F.1ll be suved on eack mght-ouncecan of brand X soup purchased at the reduced

pnce ls $0 04

57 Does Joe welgh more th% Tlm?

(1) Tlm's-welght ls 80 percent of poe's welght(2) Joe's wmght ts 125 percent of Tlm's welght'

i

p2 all odd lntegerg

(1) p ls an odd lnteger

(2) t- ls an odd mtegerl

, W hat ls the value o,f xy 9

(1) .x + y zz:i10.(2) .r - y = 6

Elena recelves a salazy plus a comlmsslon that ls equalto a fixed percentuge öf her sales revenue W lmt wastll total 4f Blena's salary and comm'lsslon last monthg

(1) Elena's montltly salary ls $1,000(2) Elena''s commljszon ls 5 percent of her sAles

revenue

162 What ls the avcrage (antllmetlc mean) of x, y. and z ?

(1) x+y = 5(2) )? + z = 7

l63 Chan and Mlekodwve separate cars along the entlrelength of a certam route If Chan m ade the ttap ln 15nunutes, how many mmutes d)d zt take Mleko to

make tlze sami trlp?i 31) Mleko's average speed for the trlp was - of Chan's

average speed

(2) Tlze route ls 14 nules long

165 W hat ls the value of the two-dlglt mteger x 9

(1) Tlw àum of tlze two dlglfs zs 3 '(2) x ls dwlslblc by 3 <

166 ls the number ,< between 0 2 aàd 0 7 9

(1) 5601 < 280(2) 700x > 280

167 Ig .x an lnteger?

x

(1) 'j- ls an mteger

(2) 7...7 ls an mteger

11 1

k

m IV

1 Pomt @, y) lles ln rhlch quadrallt of the rectacgularcoordlnate system shown aboveg

(1) x + y < 0(2) .'ç = 4 and )? = -7 *

$

168 A swlm club tlmt sold only mdm dual and falmlymembœshlps charged $390 for an mdwtdualmembeirslup If tlw dub's total revenuo from member-shlps was $480,000, what w:s the clzarge for a fgmllymembernhlp?

1(1) 'Ihe revenue from lndwidual membershlps was -k- ofthe total revenue from membershlps

)(2) 'l'he dub sold 1 5 tlmes as many famlly membershtps

as mdwldual membershps

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1httpu//www.neeviaecom






A Statement (1) ALONE lg suflklent, but statement (2) alone ls not suffimentB Statement (2) ALONE ls sumclent, but sutement (1) alone ls not sufclent.C BOTH statements TOGETHER are sufficlent, but NEITHER statement ALONE ls suëclentD EACH statcment ALONE ls suflklent

E Statements (1) and (2) TOGETHER are NW suflklentI

169 If x y, and z are posltwe nuvbers, ls a > y > z ?

(1) xz > yz

(2) l'j > l':

170 Can thc posltlve lntcger p be expressed as the productof twq mtegerg, each of whlch ls greater th0 1 ?

(1) 31 <p < 37(2) P ts 0dd

171 Currently there are 50 plcture books on each shelf m

the chlldren's sbctlon of a llbrary If tlwse books

were to be placed on smallor shelves wlt.h 30 plcturebooks on each shelf, how many of the smaller -shelves would be needed to hold all of tllese books?

(1) n e number of smaller shclves needed ls 6 morethan the cuaent number of shdves

(2) Currently there are 9 shelves m the chlldren'ssectlon

172 Is y = 6 9

(1) :2 = 36(2) yl - 7), + 6 = 0

S 45m T

11lI6nmll

l 5m - u

W

174 Quadnlateral RSTU showrt above ls a slte plaa for aparkmg lot m whlch slde RU ls pazallel to slde STand RU lg longer th% ST W hat ls th: area of theparkmg lot?

(l) RU = 80 meters

(2) VU = 20./9 meters

175 If xv = -6, whay s tllg value of .vt.x + y) ?

(1) . -y = 5

(2) .2 = 18176 If the 11st pncz of a new car was $12,300, what was the

cost of the car to the dealer?

(1) ne cost to the dealer was equal to 80 perceftt of thehst pnce

(2) 'fhe car was sold for $11,070, whlch was 12 5percent more than the cost to the dealer

177 If p, q, A; y, and z are dt/erent poslttve lntegers,whlch of the 5ve lntegers ls the meianz

(1) # + .t < q(2) y < z

E D C B A

G H 1

Lobby

M L j' J

N 0 P Q R

*

173 n e figure above represents the floor plan of anart gallery that has a lobby mld 18 rooms lf Llsagoes from the lobby lnto room A at the same tlme

tllat Paul goes from the lobby lnto room R, and eachgoes thrtmgh a1l of the rooms m successlons entenng byone door and exltlng by tlw other, whlch room wlll theybe m at the same t1me9

(1) Llsa spends 21 mlnutes ln each room and Paulspends 3.: mlnutes ln each room

(2) Llsa spends 10 mmutes less tlme m each room tàanPaul does

-254-






A Shtement (1) ALONE ls suzclent, but stptement f2) alone ls not suftklentB Sutement (2) AEONE js suffklent, but statement (1) alone ls not suMczent.C BW H statementg TOGETHMR are suflkfent, but NEITHER stat:ment ALONE ls su*.cient

b ' -....

àEACH statepent ALONE is sutlklent

E Statements (1) and (2) TOGETHER are NOT sumclent

A certam employee ls pald $6 per hottr for an1hovr workday Ifthe employee ls pa'td 12

tlmes thls rate for tllhe Aorked m excegs qf

8 hollrs durmg a .smgle day, how many hours

dtd the employee work today? '

(1) The çmployee was pald $18 mom f4r lzolzrsworked today tllan for hours worked

yesterday . '

(2) Yesterday the. employee worked 8 hours

lf n ls a. member of the set(33, 36, 38, 39, 41, 421,

what ls the vallm of n 9

(1) n ls even(2) )$ ls a multtple of 3

0 W hat ls tlle viue of .: ?

(1) N + 1 = 0(2) (;K + 1)2 = ,z2

A B C D

1 In tlle figure above, what ls the length of AD ?

(1) AC = 6(2) BD i= 6

82 A retatler purclfased a.felevplon get fUr p.&'percent less t11a11 1ts ltst pdce, and Qlec

sqld lt for y percent Iess than its 11st price'

What wa'j the hst pnce of tlle.televlslon setz

(1) .x = 15(2) .x - y = 5

Is ;t2 greater than x ?(1) XQ ls greatcr th= 1

(2) x ls greater than -1

r s1X4 W hat ls the value of y + - 9

r+s(1) = 5

(2) r + s t= 10

1. 1's a85 If x y and z at'e rmmbe s ls c =

(1) The, average (arltlunetlc mean) of a; )J and z ls 6(2) .K = -y '

B

C

A

186 The clrcular bâse of an above-ground swlmmmgpool hes m a levtl yard andpst touches twostratght sldes of a fence M polnts / and #, as

sllowh m the sglizre above Pomt C ls on theground,where the two sldvs of the fence meetHow far from tlle center of the pool''s base lspomt A ?

(1) The baée has area 250 squm'e feet(2) 'I'he center of the base ls 20 feet from pomt C

187 l.11 1979 M:. Jacbon bottght a total of n sharesef stock X atld. Mrs lackson bopght a total of

300 shares of stock X If the couple held a11of tllelr respechve sltams throvghom 1980, *4Mr lackson's 1980 (hvtdends on lus n shares

totaled $150, whal was the total amount of

Mrs Jgckson's 1980 dlvldepds on àer 300 sharesg(1) 1h 1980 the annual dwldend on each sharc of

stùck X was $2 15'

(2) Jn 1979 Mt Jackson bought a total of 200 shares ofstock X.

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A Statement (1) AI ,ONE ls suflklent. but statement (2) alone ls not suëment9 Statement (2) AI-ONE 1s succlent, bvt statement (1) alon.e ls not suëctent, 'C BGITI statements TOGETHER are suKclent, but NEITHER statementAlAm B ls suE clentD EACH statement AI,ONE ls suffklent

E Statements (1) and (2) TOGETHER are NJF suftklent

188 If Sara's age ls exactly twlce Blll's age, what lsSara's agez

(1) Four years ago, Sara's age was exactly 3tlmes Blll's age

(2) Elght years from now, Sara's age wlll beexadly 1 5 tlmes Blll's age

%.z

189 W hat Ss the value of - ?yz

2x. and z - -1) .A: = 2 5

x 5 % 1 1- = - VG - = .2)z 2 z 10

190 A11lnsmte sequence of posmve lntegers ls calledJn 'talpha sequence'' lf the number of even

lrtegers m the sequence ls finlte If S ls anlnfimte sequence of posltlve mtegers, Is S analpha sequeMceg

(1) n e flrst ten mtegers ln S are even(2) An mfimto number of mtegerg m S are odd

191 If ay > 0, does (x - 1)(y - 1) = 1 ?

(1) x + y = xy(2) x = z ,

192 After wlnmng 50 percent of the srst 20 games 1t

played. Team A won all of tlte remalmng gameszt played W hat wAs the total Immber of gamesthat T/am A won?

(1) Team A played 25 games altogetlter(2) Team A won 60 percent of a1l the gamçs lt played

CANCELATION FEES

Days Pnor Percent of

to m parture Package Pnce

46 or more 10%

45-31 35%

30-16 50%

15-5 65%

4 or fewer 100%

194 n e table above shows the cancelatlon feeschedule thgt a trayel agzncy uses to dbtemnlnethe fee charged to a tourlst who cancels a mppnor to departure If a tounst caaceled a trlp wltha package pncç of $1,700 and a depnrture date ofSeptember 4, on what day was the trlp canceled?

(1) n e cancelatlon fee was $595(2) lf tlm trlp ltad been*canceled tme day lator. the

cmltelatlon fee would have been $255 more

195 ls 51 less than 1,000 ?

(1) 5*+. > 3,000

(2) 51-1 = 51 - 500

196 If Hans purchased a pmr of skls and a skl Jacket, whatwas tle coM of the skls?

(A)The ratlo of the cost of the skls to the cost of theJacket was 5 to 1

(2) n e total cost of the skls and thelacket wa: $360

19t' Is x < ), ?

(1) z <y(2) z < x

E1

+

191 In tlte adàtlon problem aboves each of tlle symbols

Q, à, and 'k.represena a posltwe dlglt If

IZI < i, what ls the value of L ?

(1) W = 4

(2) EEI = 1-256-






A Statement (l) ALONE zs sufficlqpt, but stutemept (2) alone ls not suflktent.B Statement (2) ALONE ls suffictent, but statemûnt (1) alone 1s. aot stlfficlent.C B,orfll'statements TOGETHBR.are sufficlent, but NEITIIER statement ALONE ls snfficlentD EACH statement ALONE ls suflielent

E Statements (1) and (2) TOGETIIER are NOT sufficlent

$

19* lf a cerlntn clfy ls losmg 12 porcent of ltas dally watersupp.ly eàch day because of wâter-mmn breaks,whaf ls the dollar cost to.thé clty per day for thlsloss?

(1) The clty's .mly water supply ls 350 nulhongallons.

(2) 'lnhe cost tô the clyy for each 12,000 gallonsbofwater Iost ls $2, 203

199 Machlne X nms at a copstant rate gnd produces a lotconllytmg of 100 gans lp 2. hotlrs Hor mucll le,s hmewould lt t@e to produçq tlw lot of è4as lf botà

màchings X and F werû run smmkaneeusly?J

(1) Both machlne;s X and F prcduce the same number of,cans per hour ' /

(2) 1t takes machme X tmce as 1p2g to pmdcce the lot ofcans as lt yakes machines X and Frgnnlng

Flmttltaneously to pmduce the 1ot 204

Th e table abovo shows the dls> ce, m kilometers,by tlle most dsred rovte, between any two of th

fottr clûes. R, S, T; and U For example, thedlstqnce betwçtn Cltyx# and Clty U ls62 kilompters F# t ls the value of x 9

(1) By the mpst dlrect route, the dtstanc: between S andT is twlcc thç dlstanee between S and R

(2) By thg most dlrect route, the dlstance between FandU ls 1 5* tlmes the dlstance between R and T

R (î T UR ' 0 y x 62

S y 0 54 7) .x % 0 69

U 62 75 69' () -

ï '

200 lf x and y are posmve, what ls the valtlç j)f x 9

(1) 200 percent of .: equals 400 percentwof y(2) .x)) ls the square of a posmve lntegqr

Buckets X and Fcontmned oàly water and

1bucket F was -j' full If all of tlle water ln buckçt Xwas then poured mto bucket l', what fractlon .f

the capaclty of )' was then lilled *1th watei?g' A

(1) Before the water from X wa: poured. X was#

1- fujj

(2) X nnd Fhâve the same capamty

205 lf n ls aq lnteger, ls n + 2 a pnme numbçr?

(1) nas a#nme number(2) n + 1 ls not a pnme number

206 Is x rbetween 0 and 1 9,

(1) .z2 ls less than K(2) 73 ls posmve

207 Dtd Sally pay less th% x dollars, includlng salestax, for her blcycleg

(1) Tlw phce Sally pmd for the blcycle wms (0 9)2dollfs, excludlng the 1. percens sales taxu

(2) 'l'lw pnce Sally pmd for the blcycle was $170,ti 1ès tax ixclu 1ng sâ

Xzj. 'Y

B C

201 In tlw fkure above, what is thc measure of ZASC 9

(1) BX blsects ZXBY and SFllsects ZXBC(2) The measure of ZABX As 40%

202 If - 10.< k < 10, ls # > 0 9

1 >()1)k

(2) P > 9

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4

A Statement (1) ALONE ls sumcint, bet statement (2) alone lg not suëclent.B Statement (2) ALONE ls suëclent, but statement (1) alone ls not sumctentC BW H statements TOGETHK are suflkyent, buà NEI'I'HEF stat:ment ALONE ls sumclentD EACH statement ALONE ls stlëment

E Statements (l) mld (2) TOGETHER are NW sumclent

208 ls the posltwe square root of x anilnteger?

(1) .x = a4 and 1, Is an lnteger(2) x = 16

-..: : .. : . : .: : : : r0

209 If the successlve tzck marks shown on the numberhne above are equally spaced and lf x and y are

the numberà desjgnating the end polnts of lntervalsas shown, what ls fhe vAlve of y ?

l1).1 = Y

22) )rx = -

31

210 In a certam semor clttzens' club, arj more tlzan 'jj' of tlze

members over 75 years of age?

(1) Exactly 60 percent of the femple members are1

oveè 60 years of age, and, of these, - are over 75years of age

(2) Exactly 10 male members are over 75 years of age

211 If l # 0. ls r greater thnn aro?

(1) rl = 12(2) r + l = 7

212 If x ts an lnteger, ls y an mteger?

(1) n e average (anthmeûc mean) of 1, y, and y - 2ls x

(2) n e average (m thmetlc mean) of x and .y lsg. an mteger

+

@#*

< @*

j . .1 . .I

213 The lnslde of a rectangular carton ls 48 cenqmeters Iong,32 centlmeters wlde, and 15 centimeters btgh The

carton ls slled to capaclty wltll k ldegtyal cylmdncalcans of fwit that stand upnght ln rows and colunms, asmdlcated ln the Egure ahbve lf the cahs are 15centimeters lughi what ls tlle value of k ?

''

(l) Each of the cans has a ra*us of 4 cendmeters(2) 6 of the cans ht exactly along the lehgth of thç

carton

O d + 0 what ls the value of R ')14 If R = an y,,4

(1) x = 3

(2) x = 2y

215 Is the posmve lnteger n a multlple Jf 24 ?

(1) n ls a multlple of 4(2) n ls a multple of 6

f

d w

216 W hat 3s the area of the rectangular reglon above?

(1) .C + w = 6(2) #2 = 20

-258-






1 s

A Statement (t) ALONE ls suftklent, but statement (2) ualon: ls no( suëclentB Statement (2) ALONE ls guffklent, but stateritentrtl) alone ls not sufficlentC BO'I'H statements TOGZTHER are suflklent, but NEITHER statement ALONE ts suflktentD EACH statement ALONE ls sumclent .

E Statements (1) and (2) TOGETHER are NW suflklent

17 lf Aaronl Lee, and Totly have a total of $36, howmtlch money does Tony have?

1(1) Tony has twlce ag mudh money as Lee and y,as

much as Aaron z

(2) The sum of the gmounts of mopey thatrropyand.Lce have ls half the amount that Aaronhas

18 If n Is a posltlve mtcger, Is (l1e valuc of b - a at lehasttwlce the vàlue of 3* ,- 2/ ? '

(1) c = 2'' + 1 and b = 3?7 + l(2) n = 3

19 Tlze pnce per share of stock X mcreased by

10 percent Mver th: same tlme penod that the pnceper slzare of stock Fdecreased by 10 porc4nti'rhereduced pnce per share of stock i' was whqt

-

' gq r!ercent of tlle orlglnal pnce per Ghare of tock X

(1) The mcreaged pnce per share of stock # wascqual to the ongmal pnce per sharc of .stock F

(2) The mcrease m the pnce per share of stock X was10 -- tlze decrease m thc pnèe per share of stock F1

.6

,Any dpclmal that has only a limte number of non-zero dlglts ls a termmatmg declmal For example,

24, 0 82, and 5 096 are three tcrmmatmg decjmalsr

If r and s arc ppsmve lntegers and the ratlo - ls

L a termlnatmg declmul?xpressed as a declmal,ls

s(1) 90 < r < 100(2) s ::H: 4

f

whàt ts t:e value of mteger n 'i

223

224

225

226

Are at least 10 percerjt of the people m Country X who

are 65 years otd or olderemployed?

(1) Ip Country X, lt 3 percent of thepopulatlon ls 65 years oId or older

(2) Jn Country X, of tlle populatlon65 ycars old or older, 20 percent

of the men and 10 percent of thef 'd 'omen are emp o#e

% pIf x ls a posttwe number les's tha11 1.0, ls z greater

than tlze avemge (antltmepc ineall of .'r aryd 10 9J

( 1) On the numbetahnv, z j.s qloserto 10 than lt ls to .': u

(2) E = 5.:

lf y ts greater thm1 1 10 percel!y of ,,ls y jteater than 75 ''

h

(1) .r <, 75(2) )' -x = 10

#

% .

o ' 2. o ' D ot .V cD

In tnangle A#C above, whàt ls the lengthof sfde BC 9

(1) Llne segmentAD has 'length 6

(2) x = 36

(1) lz(&+1) zz 6(2) 22'z = 16

2 If .x ls a posltlve lnteger, ls 'Cx an lntçgerg( 1) 4. ls an mteger

(2) 3.1 ls not an integer

-259-








A Statement (1) ALONE ls sumclent but statemvnt (2) aloae ls not sumclentB Statement (2) ALONE Ts sumclent but statement (1) alone 1: not suëmentC BW'H statements TOGETX R are suffkient, but NEITHER statement ALONB lg stlmclentD BACH statement ALONE ls sumelent

E Statements (1) and (2) TOGETHBR are NW suëclent

227 If the two floors m a certmn bmldmg are 9 feet apart,how many steps pre there ln a set of sfnlm tllat extends

from tlle flrst floor to the second floor of the byldmg?

2 foot hlgh1) Each step 1s4

(2) Fmzth step js 1 foot wide

228 A bookgtore that sells used books sells each of 1ts paper-back books for a certam pnce and each of 1ts hardcover

bxks for a cerœn pnce lf Joe, Mana, and Paul boughtbooks m thl: stom. how much dld M ana pay for 1 paper-back book and 1 hardcover beok?

(1) Joe bought 2 paperback books and3 hardcover books for $12 50 '

(2) Paul bought 4 paperback books and6 hardcover books for $25 (X)

229 lf S ts a set of four nnmbers w, x, y, and z, 1stl)e range of the numbers ln S grea/r th% 2 9

(1) w- z,> 2(2) z xs thb least number m S

230 A box contams only rred chlps, whlte chlps, and blueclups If a chlp ls randomly selected from the box, whatls the probablhty that the clup wlll be elther whlte orblue?

(1) n e probablhty that the clup will be

1lue ls5

(2) The probablhty that the chlp wll1 be1

red ls j

?

231 Of the 4,8* votets who vote;d for or agamst ResoluhonKt 1,8* were Democratg and 3,0% were RepubhcansW hat was tl)e total number of female voters who votedfûr Resoluuon K ?

!

.:! 21) of the Democrats and g of the Repubhcans4

voted for Resolupon K

1(2) .j' of dze Democrqts who voted for Resolutzon K

1 f the Republlcans who voted fornd o

2Resoluhcn rwere femnleg

232 lf n ls g posltwo mteger, ls n, - n dwlslble by 4 9

(1) n = lk + 1, where k ls an lnteger(2) n2 + n ls dlwslble by 6

X*W33 lfvx # - y, ls < > 1 ?

x + y

(1) x > 0(2) y < 0

234 If .'t ls a positlve mteger and w ls a negatwe

mkger, wllat ls the value of .xw ?

11) xw =2

(2) w = -1

235 A certam group of car dealershlps agreed to donate .Kdollars to a Red Cross chapter for each çar sold dunng a30-day penod W hat was the total Amount that was

expected to be donatedg

(1) A total cf 5* cars were expeçted to be sold(2) 60 more carg were sold than expected, so tllat thetotal amount actually donRted was $28,4+

-260.






*

/

A Statçment (1) ALONE xs suflimenl, but gtatemeht (2) alone 1g not suftklentB Statement (2) ALONE ls suëtxenq but statement (1) âlone ls not sumclent 'C BOTH staements TOGETHER are suëclent, but NEITIIER statementc uc ls suflkfentD BACH statemem ALONE ls summent z

E statements (l) and (2) TOGETM R-re NX sttflklent

36 Islxl = y - z ?

(1) vx + y ce z(2) x < 0

37 If-'r ls an mtegeT, ls 9' + 9-œ = b'q

(1) 31 + 3-* = b +.2(2) x > 0

Z

K N

r241 lf r and s are posltwe mtegers, ls - an lnteger?

s., A

(l) Evury factor of s ls also afactor of r

k

(2) Every phme factor of .: lsalso a pnme factor of r

242 On Jane': credlt card account, the average dall/balance,.

N'

for a 30-day hllmg cycle ls the 'verage (anthmetlcmean) of the d lly balances at the end of each of the 30days At the beglnmng of a certsln 30-day bllllng èycle,

Jane's credl. cyrd accobnt had a balance of $600 Janemade a payment of $300 on the account dunng thebllllng cyde If no other amounts were added to orsubtracted f'rom the gccoqnt dunng the bllllng cycle,

what was the average dmly balance on Jane's accountfor the bllllng cycleg

.' t .j . .

(1) Jatle s payment was credlted on the 21st day of tllebllllng cycle

(2) 'rhe average çlmly balance through the 25th day ofthe bllhng cycle was $540

243 lf the lnteger n js greater than 1, ls n equal to 2 9

(1) n has exactly two pesmve factorj

(2) The d'fference of auy two dlstlntt posltwefactors of zl ls odd '

## v38- ln the fgurr above, what ly the ratloMN

(l) Thg penmeter of rectangle KLMN ls 30 meters(2) The t.:e,e small rectangles have the same

dlmensions

39 lf n zs arl lnteger between 2 and 100 and lf n Js also tlwsquare of an mteger, whay ls tlle value of n ?

(1) h ls qven(2) The cube root of n ts atl mteger

0 If .x and y are posmve mtegers such that .y = 8z + 12,what ls the greatest common dlvlsot of x atld y ?

(1) x = 12/, where u ls an lnteger

(2) y = 1Q, where z ls an lnteger

244 Is x > y 9D

(1) x = ), + 2

(2) 'j- = )' - 1

1 k.45 ls < b - a '?

a - b

(1) a x:F b

(2) 1 < I a-b 1

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A

A Statement (l) ALONE Is sufficlont, but sfatement (2) alone ls not sufmentB Statement (2) ALONE ls sumclent, but statement (1) alone ls rjot suKment,C BW 'H statemen? TOGETV R are stlfclent, but NEITHER statemlnt ALONE ls snëcieatD BACH statement M ONE lg suflklentE Statemçnts (1) and (2) TOGETHBR are NW suëczent

1 1246 If rs # 0, ls - + - = 4 ?

r s

(1) r + s = 4rz(2) r = & *

247 If m and a are posttwe mtegers, ls 5-1Al an mteger?

(1) n > m + 15(2) n = m(m + 1)

F

1> r ?48 Is# :2+2

(1) # = r(2) r > 0

249 W hat ls t14: value of 5a2 + 4.x - 1 ?

(l) x(x + 2) = 0(2) x = 0

250. W hat ls the value of 2 - :4 9

(1) c? - :2 = 16(2) a + b = S

251 W hat zs the average (anthmehc meap) of tq b, and c ?

(l) a + lb + 3c = 10(2) 3/ + 2b + c = 14

252. Are posmve lntegers p and q both greater th% n 2

(1) p - q ls greater than n(2) q >p

253 h the rectangular coordlnate system, are the polnts (6 J)and (a, v) eqmdlstant from the ongln?

(1) r+ s = 1' (2) u = 1 - r and v = 1 - J

254 ls L (?n2 + n2 + k2) = arlj + yn + zk 2

z x1) -= -k m

Z =Z2)m n

255 lf x ls to be selected at random from set 2*, what zs

1 - 5 u () ?he probablhty that.x

(l) T ls a set of 8 mtegers(2) F ls contamed ln the set ofmtegers from 1 to 25, mcluslvç

. m + x m256 If m > 0 and n > 0, ls > - ?

n + x n

(1) m < n(2) .4 > 0

257 Does the mteger k have a factor p guch that 1 < # < k ?

(1) k> 4t(2) 13I + 2 K ks 13' + 13

258 W hat ls tlze valu: of xy ?

(1) ,Y = x+ 1(2) )/ = .'1 + 1

259 If v a114 w are Afferent mtegers, doe: v = 0 ?

(1) vMz = O(2) w = 2

2 j-x?60 ls (x - 5) =

(1) -ahI > 0

(2) 5 - x > 0

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A Statement (1) AI ,ONE ls suftiment, but statement (2) alone hs not suoclentB Statement (2) AI ,0N.E ls sufliment, but stdtement (1) alone ts not spflklentC BOTH statements TOGETHE R are sufllclent, but NElTllElt'jtatemelit ALONE ls suflklentD EACH statement ALONE ls suëment

E Statementj (1) and (2) TOGETHER are NOT suffictent

261 What ls th number of members of Club X w%o are atleast 35 years of age9 - -

-1 of the members of Club X are1)Exactly4 .

under 35 yeurs of age

(2) The 64 women m Club X conshtute40 percent of the club's membershlp

262 If zn = 1, what ls the value of z ?

(1) n ls a nonzero mteget(2) z > 0

263 Is m :g' n 9

(1) m + n < 0(2) mn < 0

264 Is ?.x - 3), < xl 9

(1) 2.: - 3: = - 2(2) . > 2 and y > 0

x 3 .j5 If = -,ls .'r less than y ?

2 y '

(1) y â 3(2) )' S 4

266 Marta bought several penclls If each penml was elther

a 23-cent penml or a zl-cent penctl, how many 23-centpènclls dld Marta buy9

(i) Marta bought a total of 6 pencllv(2) 'rhe total vwlue of 'the periclls Marta

bought was 130 centsX

67 For a certmn set of n numbers, whete n > 1, ls theaverage (anthmetlc mean) equal to the medlatl?

(1) If the n numbers m the se' are llsted m mcreasmg

order, then the dlfference between any palr ofsuccçsslve numbers ln the set ls 2

(2) The range of the n numbers m the set ls l(n -1)l

W ;

268 Herry pBrchased 3 ltems durmg a sale He recelved a20 percent dlscount off the regular pnce of the mostexpenslve ltetn and a 10 percent dlscount off the regular

pnee of each of the other 2 ltems Was $e total amountof the 3 discounts grèater than 15 percent of the sum of

the regular pnces pf the 3 ltemsg

(1) The regular pnce of the most expenslve ltemwas $50, and the regular pnçe of the next mostexpenslve ltem was $2U

(2) The regular pnce ofthe least expenslve ltemwas $15

269 W henevçr Martm has a restatlrant b11l wlth an amountbetween $10 and $99, he calculates the dollar amount ofthe tlp as 2 hmes the tens d1pt of the amount of hp blll Ifthe amount of Martm's most recent reslurant b1ll was

between $10 and $99, was the hp calculated by Mar:n onthls blll greater *m/15 percent of the amotmt of the blll?

(1) The amount of the 6111 was between $15 and $50(2) The hp calculated by Martm was $8

@

, 2

# R+- ' '' '' - J'''--'--+1v#*1

270 lf arc PQR above ls a semlclrcle, what ls the Iength ofdlameter #C 9

(1) a = 4(2) b = 1

r71 lf r an.d s are nonzero mtegtrs. ls - an mtegey?

$

(1) r - 1 = @ + I)4â' - 1)(2) r - s = 2.0

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A Statement (1) ALONE ls suflklent, but statement (2) alone ls not sumclentB Statement (2) ALONE ls snmclent. but statement (1) altme ls not suëmentC BOTH statements TOGE'IXER œe suKclent, but NBITIIER statementM ox ls suKdqnt.D EACH statement ALONB ls suëclentE Statements (1) and (2) TOGETHER are NX sumclent

x72 In the fractlon.where .< and z are positlve

lntegers, what ls the value of y 9

(1) The least comm.on denommqtorx 1

of - and ls 6'y

(2) .t = 1

273 If x and ,y are nonzero mtegers, ls x? < f ?

(1) X = :2(2) .y > 2

-(f. an mteger?74 If n ls an lnteger,ls 7

3n( 1) ls an mteger

5a(2) ls an mteger

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