calculator ba actuary

8/6/2019 Calculator BA Actuary

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REV IEW OF CALCULATOR FUNCTIONSFOR THE TEXAS INSTRUMENTS BA II PLUS@

S am uel B rov erm an , U niversity of T oro ntoThis n ot e p re se nts a re vi ew o f c alc ula to r fi na nc ia l fu nc tio ns fo r th e Tex asIn strumen ts BA II PLUS c alc ula to r. T his n ote, in clu din g a n umbe r o f th eex ample s u sed a s illu stra tio ns, is re prin ted w ith p erm is sio n from th e 3r dedition of the book Mathem atics of Investm ent and Credit, by S.B roverm an. A lso, several exam ples from SOA/CAS m ath of financeexams (old Course 2) will be presented illustrating the use of thecalculator.A d eta ile d g uid eb oo k fo r th e o peratio n o f a nd fu nc tio ns a va ila ble o n th eBA II P LUS can be fo und at the follow ing internet site:h ttp: //education .t i.com/us/g loballgu ides.html#f inance . I t wi ll be assumedth at y ou h av e a va ila ble a nd h av e rev iewe d th e a pp ro priate g uid e b oo k fo rth e c alc ula to r th at y ou a re u si ng .Financial functions w ill be review ed in the order that the related conceptsare covered in Chapters 1 to 8 of Mathem atics of Investm ent and C redit.Some numerical values will be rounded off to fewer decimals than areac tu ally d isp la yed in th e c alc ulato r d is pla y.It w ill be assum ed that unless indicated otherw ise, each new keystrokes eq uenc e s ta rts w ith c le ar r eg is te rs . Ca lc ula to r r eg is te rs a re c le are d w iththe key st roke sequences

j2ndllC LR WORKI ICE/C I an d

12ndl lCLR TVMl lcE /c l.It w ill also be assum ed that the calculator is operating in US date form atand US commas and decimals format, w ith the display showing 9d ecim als. T he se a re th e d efa ult s ettin gs fo r th e c alc ulato r, b ut th ey c an b echanged in the "FORM AT" work sheet, which is accessed with thek ey stro ke s eq uenc e 12ndII FORMATI. A lth ough t he number o f d ec ima lsto display is set to 9, in the exam ples below it w ill often be the case thatd olla r amounts a re w ritte n a s ro unded to t he n ea re st . 0 1 .


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CHAIN (CHN) AND ALGEBRAIC OPERATING (ADS) SYSTEM MODESWhen the calculator is operating in chain calculation m ode, the usuala lg eb ra ic o rd er o f o pe ra tio ns is n ot r espe cte d. For in st an ce , th e k ey str ok esequence 1+ 2 x 3 I;] results in an answ er of 9. This is true because thecalculation of 1+ 2 is performed first, resulting in 3, which is thenm ultiplied by 3, resulting in 9. When the calculator is in AOS mode, there su lt o f th e k ey st ro ke s eq uenc e above w ill b e 7 . This is tru e b ec au se in th eh ie ra rchy o f a lgeb ra ic ope ra tion s, mu lt ip li ca ti on i s done befor e add it ion, so2 x 3 is calculated first, resulting in 6, and then the addition operation isapplied resulting in I + 6, w hich is 7. T he order of operations m ode can besel ec ted in the "FORMAT" work shee t.

ACCUMULATED AND PRESENT VALUES OF A S INGLE PAYMENTUSING A COMPOUND INTEREST RATEAcc umulate d v alu es a nd p re se nt v alu es o f sin gle p aymen ts u sin g a nn ua l(or m ore general periodic) effective interest rates can be determ inedusing the cal cu la to r f unct ions a s descr ib ed below.

ACCUMULATED VALUE:We use E xam ple 1 .1 to illu strate th is fu nctio n.A deposit of 1000 m ade at tim e 0 grow s at effective annual interest rate 9% .The accumulated value at the end of 3 years is 1000(1.09)3 =1,295.03.T his can be found using the calculator in tw o w ays.1. W e use standard arithmetic operators in standard calculator mode

w ith th e fo llowin g k ey stro ke s.1.09 [Z] 3 I;] @ 1000 I;]The screen should d isp lay 1 ,295 .029.In this function , y=I.09 and x=3.

2. W e use tim e value of m oney functions (T VM ).12ndilPN I ill I (this sets 1 c om pounding period per year).12nd ilQUITI ( th is r eturns cal cu la to r to s tandard- ca lcul ator mode )


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UNKNOWN INTEREST RATE:A s an ex am ple of solving fo r th e in terest rate, w e con sid er E xam ple 1 .5 (c).A n in itial inv estm ent of 25 ,0 00 at effective ann ual rate of in terest i g rows to1,000,000 in 25 years. Then 25,000(1+i)25 = 1,000, 000, from which wege t i=(40)1/25 -1=.1590(15.90%). This can be found using thecalcu lato r power fu nctio n w ith the follow in g key stro kes:40 I.2J .04 [;]g 1 [;], th e s cree n s ho uld d is pla y 0 .1 58 99 72 34 .Using f in anci al f unct ion s, t he key st roke sequence solving for i is12ndilPN I ill 1 12ndilQ UITI2 50 00 !PV ll0 00 00 0 1 +/-IIFVI25 [EJ ICPTI IINIThe s cr ee n should d is pl ay 15.8 9972344 ( th is is th e % measure ).

UNKNOWN T IME P ER IOD:As an exampl e o f sol vin g f or a n unknown time p erio d, s up po se th at a n in itia lin ve stment o f 1 00 a t month ly compound r ate o f in te re st i grow s to 300 in nmonth s a t month ly in te re st r ate i=.75%. Then 100(1.0075t =300, fromw hich w e get n = I n l ~ ~ 7 5 = 147.03 m onths. This can be found using thecal cu la to r ILNI funct ion.Us ing f in anci al f unct ion s, t he key st roke sequence solving for n is\2ndIIPN \ ill 1 12ndilQUITI1 00 Ip vl 3 00 1+/-IIFVI.75 IIN llcPT\ [EJ.The sc re en sh ou ld d is pla y 1 47 .0 30 26 . S lig htly more th an 1 47 mon th s o fc ompo un din g w ill b e re qu ire d. T he c alc ula to r re tu rn s a v alu e o f n basedon com pounding including fractional periods, so that the value of147.03026 means tha t 100(1.0075)14703026 300.


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C onversion from nom inal annual to effective annual interest rate:We apply the follow ing sequence of keystrokes.

12 ndilICONVI (NOM= ap pears ),24 IENTER I (the nominalrate) ill ill (CIY=appears)121ENTER I (n um ber of com po un din g p erio ds ),ill ill (EFF= appears) ICPTI.

The screen should display 26.82. W e have converted the nom inal annualinterest rate of 24% (keyed in as 24) compounded monthly (keyed in as1 2) to the equ iv alen t effectiv e an nu al interes t rate of2 6.8 2% .C onversion from effective annual to nom inal annual interest rate:

12ndilICONVI il l (EFF= appears )26.82IENTER! il l (CIY= appears)12 IENTER ! ill (N OM= appears)ICPTI

The scre en sh ou ld d isp la y 2 3.9 96 6 (ro un d to 2 4). We h av e c on ve rte d th ee ffe ctiv e a nn ua l in te re st rate o f 2 6.8 2% (k ey in 2 6.8 2) to th e e qu iv ale ntnominal annua l i nte re st ra te compounded month ly (k ey in 12) o f24% .

N om in al D iscou nt R atesThe nom inal annual discount rate com pounded m times per year can befound from the effective annual rate of interest and vice-versa usingc alc ula to r fu nc tio ns as illu stra te d b elow.A nom inal annual discount rate of .09 (9% ) com pounded quarterly isequ iv alent to an e ffe cti ve annua l ra te o f in te re st o f i = . 0953 (9.53%). Therelationship i = (1 - d~4)r4 -1 can be used, or the equivalent rate can befound in th e f ollow ing way s.


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U se the keystroke sequence 12ndilPNI 4 IENTER I12ndIIQUIT I. T hissets the num ber of com pounding periods per year to 4 and returns tostandard calculation functions. It is im portant to do this step first,e nte rin g th e n umber o f c ompound in g p er io ds i n th e y ea r.I. We first fin d th e e qu iv alen t e ffec tiv e a nn ua l ra te o f in te re st from th eg iv en nom in al a nnua l ra te o f d is co un t.

4 1+ /-1 [E J 9 1+ /-llw ll [ f Y ] ICPTIIFVIT he d isp la y s ho uld re ad -1 .0 95 3; w e in te rp re t th is a s in dic atin g th at th ee ffe ctiv e annua l ra te o f in te re st is 9 .5 3% . We hav e c alc ula te d

( 1 )p ( -.09 )41+ -N = 1+ 4 = FV = -1 .0 953,where P = 4 w as entered w ith the PN function, N = -4 w as entered w ith41+/-1 [EJ,and 1=-.09 (or d=9%) wasenteredwith91+/-llwl.2. W e now find the equivalent nom inal annu al rate of discount from theg iven e ff ec tive annua l r at e o f int er es t.

4 1+ /- 1 [E J, Key in 1 .0 953ll+ /- IIFV I,K ey in llpvl, K ey inIC PT IIIN I.T he display should read -9.00; this is the negative of the equivalentn om in al a nn ua l ra te o f d is co un t c ompo un de d 4 tim es p er y ea r.Note that when we enter FV, we enter - (l+EFF).

LEVEL PAYMENT ANNUITY VALVAnONT he accum ulated value and present value of a level paym ent annuity-immediate can be found using calculator functions. C lear calculatorr eg is te rs b ef ore s ta rtin g th e k ey str ok e s eq uenc e. The c alc ula to r s ho uld b ein s tandard cal cu la to r mode .


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Accumu la ted Valu e o f Annuitv -Immedia te :A deposit of 1000 is made at the end of each year for 20 years. Thedeposits earn interest at an effective annual rate of interest of 4% . Theaccumulated value of the deposits at the tim e of (and including) the 20thdeposit is 1000s2O].o4 = 1000[ (1.~~:O-l] = 29,778. This can be foundusi ng the cal cu la to r.U se the keystroke sequence 12ndllPNIl IENTER I12nd[

IQUIT ! (th is

sets the num ber of com pounding periods per year to 1 and returns tos ta nd ard c alc ula tio n fu nc tio ns ). It is impor ta nt to d o th is s te p firs t, s o th atth e compound in g p erio d corre sp onds to th e p aymen t p erio d.2 0 IR J41IN I 0 !PVllOOO IPMT !lcP T!IFV!T he display should read -29,778.08, the negative of the accum ulatedannuity value .Pre sent Valu e o f Annuity -Immedia te :Payments of SO w ill be made at the end of each month for 10 years. Themonthly com pound interest rate is % % . The present value of the annuityone month before the first payment is made is

SO~.OO75 =SO[1 ;~~5 J=3 ,947 .08(1 0 y ears, 1 2mo nth s pe r y ea r).The calculator should still have /2nd II PN I set to 1 (if not, use thekeystroke sequence outlined above to set PN to 1). This m eans that therate entered as IN will be a rate per period. In this example the period isone m onth. W e use the follow ing sequence of keystrokes.120 lli] .7S!IN IIEN TER I 50 IPMTIIE NT ER I0 IFVllcPTllpv lT he display should read -3,947.08, the negative of the present value ofthe annuity.


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In th e g en era l eq ua tio n f or a n a nn uity -immed iate PV = PMT. am i if any 3of the 4 variables PV, PMT, N, i (in %) are given, then the calculatorfunctions can be used to solve for the 4th variable. The sam e is true for theequation PMT . SNji = FV (keep in mind that when PMT is entered aspositive, F V or PV are returned as negative, and vice-versa).F indin2 th e Payment Amount:A loan of 1000 is to be repaid with m onthly paym ents for 3 years at acom pound m onthly interest rate of .5% . The monthly paym ent is Kwhere 1000 = Ka361 005' so that K = alOOO = 30.42.. '361.005T his can be found using the follow ing sequence of keystrokes:36 lliJ .5 IINIIENTERI 1000 Ipvl 0 IFVllcPTllpM TI

The d is pl ay s hould r ead -30.42.F in din 2 the Interest R ate:Suppose that the loan payment is 35 for a 36 payment loan of amount1000 and the interest rate is to be found. Then 1000 = 35ll36Ji' There is noalgeb raic so lu tio n for i. T he follow ing keystrokes give us i.3 6lliJ 1 00 0 1 +/-1 Ip vl 3 5 IPMTI 0 IFVllcPTllw lThe display should read 1.31(% ). That is the effective rate of interest permonth.F ind in2 the Number o fPavments :We w ill u se Examp le 2 .1 3 to illu stra te th e ca lc ula to r fu nc tio n fo r fin din gthe unknown num ber of paym ents. In Exam ple 2.13, Sm ith wishes toaccum ulate 1000 by m eans of sem iannual deposits earning interest atnominal annual rate i(2) = .08, with interestcreditedsemiannually.In part (a) of Example 2.13, Smith makes deposits of 50 every sixmon ths. W e w ish to solve fo r n in the equation 1 000 = 5 0 . snJ.04


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10 l liJ (10 annua l paymen ts ),8 I IIY I (nominalannual in teres t rate of 8%),I 1+/-l lpMTllcPTllpvl.The d isplay should read 6.637 , the present value of th e an nuity . T ofind the accum ulated value of the annuity, we continue with thefollowingkeystrokes: 0 Ipvl!cPTllFvl

The d is play s hould r ead 14.655 , t he accumu la ted valu e o f th e annuit y.N ote that if w e fin d the equiv alent effective annual rate of interest first,i = .082432 , w e could have found the annuity value as follow s w ithoutsetting CN to 4. This is done in the following way (CN an d PN ar eboth set to I).

1 0 l liJ (10 annua l paymen ts ),8 .2 432 lIlY I (ef fective annual interes t rate) ,I 1+/-l lpMTllcPTllpvl

The display should read 6.637, the present value of the annuity.2. Annuity-immediate of I per m onth for 5 years at effective annualrate of interest 6% . The equivalent one-month interest rate is

j = ( 1 . 0 6 ) 1 1 1 2 - I = .0 0 4 8 6 8 and the present value of the annuityone m onth before the first paym ent is a60lj = 5 1.9 24 . The a nn uityvalue can be found using financial functions with the followingsequence of keys trokes./2ndl [ f Z Y ] 12 I ENTER I (th is se ts 12 paymen ts p er y ea r),il l 1 I ENTER I ( th is sets CN=I inter es t convers ion per iod per yea r) ,then 12ndIIQUlTI.5 j2ndllxPN IlliJ (5 x 12 = 60 m onthly paym ents),6 lIlY I (ef fective annual interes t rate) ,I 1+/-l lpMTllcPTllpvl

The display should read 51.924, the present value of the annuity.


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w e cannot use these calculator functions to find (Is)-,., but since theni lnume ra to r o f (Is)-,. is s-,. - n, w e can find s-,. f ir st , t hen sub tr ac t n,nil nil niland then divide by i.

Finding (Ds)nJi :35(1.04i5 -sm04Suppose that we wish to find (Ds)m04 = .04 . W e firstfind the numerator with the following keystrokes. The calculatorpaym ent m ode should be set to END .

35 lli] 41w liENTER I 35 1+ /-llpvl1 IPMTI ICPTIIFVI

The display should read 64.4609, w hich is 35(1.04)35 - sm04.In this sequence of keystrokes w e have created an initial paym ent receivedof 35 at time 0 (

IPV I ), and a series of 35 payments of 1 each paid out at the

end of each year. The net accum ulated value at the end of 35 years is35(1.04)35 -s351.04 = 64 .4609 ( lFVI).

Then,64.4609(Ds)35104= .04 = 1 ,611.52 .

Note that w e could have incorporated division by .04 into the keystrokesequence by keying in 35 g .04 B 1+/-llp vl in ste ado f351+/-llpvl, and by keying i n 1g .04 B IPMTI instead o f 1 ]PMTI.It is also possible to use the cashflow worksheet to find present andaccumulated values of a series of non-level payments. There is alim itation that allow s an initial paym ent, CFo, and up to 24 morepaymen t amount s.This w ould be a less efficient w ay of finding (Ia )20108' for ins tance.


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The value -2,011.56 is returned (we m ust key in IC PTllpM T! in orderto use the amort izat ion func tions) .T o find OB12 u se th e f ollowin g k ey str ok es .

1 2n d[IAMORTI (th is o pen s th e am ortizatio n w ork sh eet).The display reads PI =. Key in 12 IENTER! ill.The display reads P2 =. K ey in 12IENTER[.T hen u sin g il l again, the display should read B AL = 248, 292.01 .This is OB12'Using il l again gives the display PRN= -148.25;this is -PR12'Using il l again gives the display IN T= -1,863.30; this is -112'To find 113+114+...+123+124 and 1i3+1i4+",+P23+P24 use thefollow ing keystrokes w hile still in the am ortization w orksheet (w e leavea w orksheet using the 12nd!IQUlT I sequence of keystrokes).Use ill until PI= appears again and enter 13 (key in 13 I ENTER I ).Then use il l and P2= appears and we enter 24 (key in 24 [ENTER [).The next use of il l given us B AL = 246, 473.79 , the outstanding balanceat the end of the period to tim e 24 months).ill again gives us PRN = -1,868,21; this is -(1i3 + 1i4 + . .. + P23 + P24) ,th e n eg ativ e to ta l amoun t o f p rin cip al p aid in p ayments 1 3 t o 2 4 (th esecond year) .ill again gives us INT= -22,270.46; this is -(113 + 114+... + 123 + 124)'the negative total amount of interest paid in payments 13 to 24.Note that 1i3 + 1i4 +... + P23 + P2 4 = OB12 - OB24 cou ld b e fo un d fromOB12 an d OB24, an d

1 13 +1 14 + ...+ 12 3 + 12 4= 12K - (PR13+PRI4 + ... +PR23+PR24)= 12K -(OB12-0B24) '


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Bond P rice and Y ield B etw een C ouD on D ates:T he bond exam ples considered so far have had valuation take place on acoupon date. It is also possible to use the bond w orksheet functions tofind the price (given the yield) or the yield (given the price) of a bond ata ny tim e, o n o r be tw een co upo n d ates .We u se E xample 4 .2 to illu stra te th e v alu atio n o f a bo nd b etw een c ou po ndates. A bond has face am ount 100, w ith an annual coupon rate of 10%and c ou po ns pa yab le s em i-a nnu ally. T he b ond matu res o n Ju ne 1 8,2 01 0and is purchased on A ugust 1,2000 at a yield rate of 5% (nom inal annualyield com pounded sem i-annually). The quoted purchase price fromExamp le 4 .2 is 1 38 .6 0. T his c an b e fo un d u sin g th e fo llowin g k eys tro ke sin the bond work shee t.SDT=8-01-2000 (en ter 8 .0100),CPN=10 , RDT=6-18-2010 ( en te r 6 .1810) ,RV=100 , ACT, 2N, YLD=5 dENTER I m ust be used after each entry).A t PR1=, u se ICPT I to ca lcu la te th e p rice .T he d is pla y sh ould re ad 1 38.6 0. Note th at this is the q uote d pric e whic hexcludes the accrued coupon. T he accrued coupon am ount is found atAI=.If a price had been entered instead of a yield rate, we could havecomputed the y ie ld .

INTERNAL RATE OF RETURN AND NET PRESENT VALUEThe internal rate of return for a series of payments received andpaym ents m ade can be found in a couple of different w ays, dependingupon the nature of the series of paym ents. W hen w e consider a levelp aymen t a nn uity w ith o r w ith ou t a b allo on p aymen t at th e tim e o f the la sta nnuity p aymen t, we c an ente r v alu es in to th e v aria ble slliJ, Ipvl, IPMTI and IFvl, and then use !cPTI Iwl to find thei nt er es t r at e wh ich sat is fi es the r el at ionship PV = PM!'. a ;;/j+ FV . vi .The interna l rat e of return is ).


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EXAMPLES FROM SOAJCAS EXAM FM/2(FORMERLYCOURSE2 COMPOUNDNTEREST)

L::. 20 03, # 33 (A nnu ity V aluatio n) 8A t an effective annual interest rate of i, i > 0" both of the follow inga nn uitie s h av e a p re sen t v alu e o f X:(i) A 2 0-y ea r a nn uity -immed ia te w ith an nu al p ayme nts o f 5 5(ii) A 3 0-y ea r a nn uity -immed ia te w ith a nn ua l p aymen ts th at p ay s 3 0 p eryear for the first 10 years, 60 per year for the second 10 y ears, and 90p er y ea r fo r th e fin al 1 0 y ears.Calculate X(A ) 575 (B) 585 (C) 595 (D) 605 (E) 615I SOLUTION IThe s erie s o f c ashflows re pre se ntin g th e d iff ere nc e b etween ( i) a nd ( ii) i sa series of 10 paym ents of 55 - 30 = 25 each, follow ed by a series of 10payments of 55 - 60 = - 5 each, follow ed by a series of 10 paym ents of-90 each. The interest rate that m akes the present value of this seriese qu al to 0 is fo un d u sin g th e IRR fu nc tio n as fo llows.K ey in [g] 0 IENTER ! il l 25 IENTER ! il l 10 IENTER I ,il l 5 I+/-IIENTERI ill 10 IENTER\,ill 90 1+ /-1IENTER I il l 10 IENTERI ,IIRRllcP TI. T he display shou ld read IRR = 7.177.T his is the interest rate per year. Set PlY and CIY to 1. T he follow ingk ey stro ke s g iv e u s th e v alu e o f X1 2n dilQU IT I1 2n dllCLR TVMI20 ll iJ 7 .1 77 !w l 5 51+/-l lpMTllcPT llp vl .The d is pl ay shoul d r ea d 574 .7 4. An swe r: A


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'-=' 2003, #26 (Dec reas ing AnnUi ty~ (Exercise 2.3.13)1000 is deposited into F und X; which earns an effective annual rate of6% . A t the end of each year, the interest earned plus an additional 100 iswithdrawn from the fund. At the end of the tenth year, the fund isd ep le te d. The annua l w ith dr awa ls o f in te re st a nd p rin cip al a re d epos ite din to Fund Y, which earns an effective annual rate of 9% . D eterm ine thea ccumu la te d v alu e o f Fund Yat the end of year 10.(A) 1 519 (B) 1 819 (C) 2085 (D) 2273 (E ) 2 431I SOLUTION IThe 1 0 d ep os its to F un d Yare 160, 154, 148, .. ., 112, 106.These can be entered as 10 separate cash flow s in the IC FI w orksheet.T he N PV function at 9% w ill give a present value (one year before thedeposit of 160) of 880.59. Then the FV at the end of 10 years will be2,085. A nsw er: C

~ November 2001,#16(Increasinf AnnUi~~ (Exercise2.3.12)O lga buys a 5-year increasing annuity for X O lga w ill receive 2 at theend of the first month, 4 at the end of the second month, and for eachmon th th erea fte r th e pa yment in cre as es b y 2 . T he n om ina l in te re st ra te is9% convert ib le quar te rly. Calcu la te X(A) 2680 (B) 2730 (C) 2780 (D) 2830 (E) 2880I SOLUTION IWith monthly r at e} , X = 2(Ia)601J'We are given 3-m onth rate .0225, so that (1 +})3 = 1 .0225, and the re fo re ,) = .007444. T he num erator of (Ia)601i can be found by the follow ingkeystrokes.12ndllBGNI[M J ~ 12ndilQUITI12ndllPNI12 IENTER I [I] ICN I 4 IENTER I1 2n dIIQU IT I,60 [H] I 1+/-llpM TI 9jINI60 IFVllcPTI!pvlThi s r esults in th e d is pla y re ad in g PV = 10.1587.Then g .007444 ~ 2 re su lts in 2 ,7 29 o n th e d is play . Answer: B


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