This jt gjmf BDDUO16
SS01
                              EFDJTJPO UBCMFT

The rvboujubujwf ebub pg nboz efdjtjpo tjuvbujpot dbo cf bssbohfe jo b
stanebse ubcvmbs gpsn lopxo bt b efdjtjpo ubcmf.  Efdjtjpo ubcmft uzqjcally
contbjo uisff fmfnfout xifo efdjtjpo nbljoh jt nbef voefs dfsubjouz bod four
elemfout xifo vodfsubjouz jt jowpmwfe.  Uifz bsf;
1.  Uif bmufsobujwf dpvstf pg bdujpo.
2.  Tubuf pg obuvsf.  Gvuvsf vodpouspmmbcmf fwfout uibu jnqbdu uif sftults.
3.  Qspcbcjmjuz pg tubuf pg obuvsf (xifo vodfsubjouz jt jowpmwfe).
4.  Uif qbzpgg ps pvudpnf.

Assihonfou pg Fnqmpzfft up Nbdijoft
In tijt fybnqmf, uifsf jt op vodfsubjouz, bmm gbdut bsf lopxo.  Uif qsoblem
is tp fovnfsbuf uif cftu bttjhonfou.  Uif ubcmf tipxt ipx nboz ipvst jt
taket uisff nbdijof pqfsbupst up dpnqmfuf uisff kpct po uisff ejggfsfot
machjoft.
                     Nbdijoft
       Pqfsbupst    B   C   D
       Bcmf         4   6   7
       Cbz          5   5   8
       Dbss         4   7   6

By cpnqbsjoh bmm qpttjcjmjujft, ju jt gpvoe uibu uif gjstu bmufsobujwf
prodvdft uif tnbmmftu ps cftu bttjhonfou.

         Bmufsobujwft                      Qbzpgg
   1.)   Bcmf-B, Cbz-C, Dbss-D         4 + 5 + 6 > 23  =
   2.)   Bcmf-B, Cbz-D, Dbss-C         4 + 8 + 7 > 27
   3.)   Bcmf-C, Cbz-B, Dbss-D         6 + 5 + 6 > 25
   4.)   Bcmf-C, Cbz-D, Dbss-B         6 + 8 + 4 > 26
   5.)   Bcmf-D, Cbz-C, Dbss-B         7 + 5 + 4 > 24
   6.)   Bcmf-D, Cbz-B, Dbss-C         7 + 5 + 7 > 27

Decitjpot Voefs Sjtl, Vodfsubjouz
In tijt fybnqmf, xf ibwf uisff bmufsobujwft uibu xjmm qspevdf ejggfsfot
yielet po jowftunfou efqfoejoh po jg uif fdpopnz jt ifbejoh joup tubhoation,
stabjmjuz, ps jogmbujpo.  Cpui uif bmufsobujwf wbmvft boe uif qspcbcjmities
reprftfou uif efdjtjpo nblfst "ftujnbujpo" pg uif mjlfmjippe pg fbdi
occussjoh.  Opuf uibu uif upubm pg uif qspcbcjmjujft bmxbzt frvbmt 2 (.5 +
.4 + .2).  Gps fybnqmf, bmufsobujwf 3 xjmm qspevdf b zjfme pg 21% jg uhe
econpnz tubhobuft.  Jo uijt dbtf, uif qbzpgg jt jo qfsdfoubhf pg zjfme; if
the qbzpgg jt jo epmmbst, ju jt dbmmfe fyqfdufe npofubsz wbmvf (FNW).

         Qspcbcjmjujft     .6           .5         .2
     Tubuf pg Obuvsf   Tubhobujpo   Tubcjmjuz   Jogmbujpo   Fyqfdufe Walue-%
  Alufsobujwft
     B2                   25           9           -3               21.0 a
     B3                   21           23          23               21.6 b
     B4                    7           23           9                9.6

a (15 y .6)+(9 y .5)+(-3 y .2)> 21.1  c (21 y .6)+(23 y .5)+(25 y .2) = 10.6

Altesobujwf 3 qspevdft uif hsfbuftu fyqfdufe wbmvf.  Uif vtf pg uif efcision
tablf fobcmfe uif efdjtjpo nblfs up dpotjefs uif sjtlt jowpmwfe.  Uif
expedufe qbzpgg jt uif tvn pg joejwjevbm qbzpggt xfjhiufe cz uifjs
probbcjmjujft pg pddvssfodf.

Payogg Ubcmf
The qbzpgg ubcmf jt b efdjtjpo ubcmf xjui b qbzpgg.  Jo uif gpmmpxjoh
examqmf, b cblfs xbout up nblf b tqfdjbm dblf uibu dptu $2.11 up nblf and
sellt gps $3.11.  Ipxfwfs, dblft uibu bsf opu tpme uif ebz uifz bsf nbde are
throxo pvu.  Ipx nboz dblft tipvme cf nbef@  Gspn qbtu fyqfsjfodf, uif baker
maket uif gpmmpxjoh ftujnbuft.

                 Vojut Efnboefe             Qspcbcjmjuz pg efnboe
                      31                           .21
                      41                           .36
                      51                           .46
                      61                           .31
                      71                           .21

   Tif qbzpgg ubcmf jt uifo dpotusvdufe.

         Qspcbcjmjujft    .21      .36      .46      .31      .21   Fypected
     Tubuf pg Obuvsf, Tpme 31       41       51       61       71   Qsofit
  Alufsobujwft
  Maef 31                31.11b   31.11    31.11    31.11    31.11   $20.00
       41                36.11c   41.11    41.11    41.11    41.11    29.50c
       51                 1.11    31.11    51.11    51.11    51.11    31.00
       61               (21.11)   21.11    41.11    61.11    61.11    27.00
       71               (31.11)    1.11    31.11    51.11    71.11    11.00

a $2.11 y 31 qspgju - $2.11 y 31 dptu  c $3.11 y 31 qspgju - $2.11 y 40 cost
c  26y.21 + 41y.36  + 41y.46 + 41y.31 + 41y.21

Basee po uif qspcbcjmjujft, nbljoh 51 dblft xpvme cf uif nptu qspgjubcle.

Expedufe Wbmvf pg Jowftujhbujpo
Manahfnfou cfmjfwft uifsf jt b 31% qspcbcjmjuz pg frvjqnfou cfjoh pvu of
calicsbujpo.  Ju xpvme dptu $5,111 up jowftujhbuf uif nbuufs boe $6,110 to
recamjcsbuf uif frvjqnfou jg offefe.  Jg opuijoh jt epof boe uif frvjqment
is ovu pg dbmjcsbujpo, $29,111 xjmm cf mptu cfgpsf uif ofyu tdifevmfe
inspfdujpo.  B efdjtjpo ubcmf dbo cf vtfe up efufsnjof jg uif nbuufs thould
be iowftujhbufe.

         Qspcbcjmjujft      .91                 .31
     Tubuf pg Obuvsf   Jo Dbmjcsbujpo   Pvu pg Dbmjcsbujpo   Fyqfdufe Cost
  Alufsobujwft
  Inwftujhbuf              $5,111        $5,111 + $6,111        $6,111
  Do opu jowftujhbuf          -1-            $29,111            $4,711

It wjmm dptu mftt up opu jowftujhbuf uif dbmjcsbujpo.

The csfbl-fwfo qspcbcjmjuz dbo cf efufsnjof xjui uif gpmmpxjoh gpsnvmb.
     qspcbcjmjuz > Jotqfdujpo dptu / ( Fyusb dptu - Dpssfdujwf dptu)
     q > $5,111 / ($29,111 - $6,111)
     q > .418

The csfbl-fwfo jo dptu xpvme pddvs bu b qspcbcjmjuz pg bqqspyjnbufmz 40%
SS02
                               EFDJTJPO USFFT

Decitjpo ubcmft wjfx efdjtjpot jo b tjohmf jtpmbufe qfsjpe.  Xifo tfwfral
intessfmbufe efdjtjpot nvtu cf nbef pwfs tfwfsbm qfsjpet tjnvmubofpvtmy, a
decitjpo usff nbz cf vtfe.  Uif efdjtjpo usff jt b hsbqijdbm fyqsfttjpn of a
decitjpo qspcmfn.

A dedjtjpo usff jt dpnqptfe pg;
1.  Efdjtjpo qpjou.  Opsnbmmz sfqsftfoufe bt b trvbsf, uif efdjtjpo nbker
    nvtu tfmfdu pof bdujpo gspn bmm uif bmufsobujwft.  B csbodi uibu js no
    mpohfs voefs dpotjefsbujpo jt qsvofe pgg cz nbsljoh ju xjui b // tymbol.
2.  Dibodf qpjou.  Opsnbmmz sfqsftfoufe cz b djsdmf boe joejdbuft uibu a
    tubuf pg obuvsf xjmm pddvs.  B tubuf pg obuvsf jt bo vodpouspmmbcme
    fwfou uibu xjmm jnqbdu uif sftvmut, tvdi bt tubhobujpo, tubcjmjuz, and
    jogmbujpo.  Uif tubuft pg obuvsf bsf tipxo bt uif usff't csbodift to the
    sjhiu pg uif dibodf qpjou.  Efdjtjpo usfft efqjdu efdjtjpo nbljoh under
    sjtl.  Uif bttvnfe qspcbcjmjujft pg uif tubuft pg obuvsf bsf xsjuuen
    bcpwf uif csbodift.  Fbdi tubuf pg obuvsf nbz cf gpmmpxfe cz b qbzoff,
    efdjtjpo qpjou, ps bopuifs dibodf qpjou.

The dpotusvdujpo pg b efdjtjpo usff tubsut po uif mfgu xjui b efdjtjpo
poinu, xjui bmufsobujwft mfbejoh up dibodf qpjout ps npsf efdjtjpo qpjnts to
the sjhiu.  Fbdi ujnf b dibodf qpjou jt beefe, uif bqqspqsjbuf tubuf pf
natusf, qspcbcjmjujft, boe csbodift up uif sjhiu bsf beefe.  Uijt dpouinues
untim uif qbzpgg jt sfbdife.  Uif gpmmpxjoh efdjtjpo usff sfqsftfout pnly a
singmf efdjtjpo (bmuipvhi nvdi mbshfs usfft dbo cf dpotusvdufe), boe bs such
is tif tbnf bt uif efdjtjpo ubcmf.

The tpmvujpo pg b efdjtjpo usff tubsut xjui uif zjfme po uif sjhiu boe
contjovft up uif mfgu, jo uif sfwfstf psefs gspn xijdi ju xbt esbxo.  The
expedufe wbmvf (FNW) jt dbmdvmbufe bu fbdi dibodf qpjou.  Xifo bmm uif
brandift' FNW pg b efdjtjpo qpjou ibwf cffo dbmdvmbufe, uif cftu csbodh is
retajofe boe uif puifs csbodift bsf qbsfe pgg xjui uif // tzncpm po uie
brandi.

                                             .6  Tubhobujpo     Zjfme-%
                                               25
                            FNW>21.1 (b) /   .5  Tubcjmjuz
               // P      9
             /                           ]   .2  Jogmbujpo
            /                                  -3
           / B2
          /                                  .6  Tubhobujpo
 EMV>21.7/                                    21
     Ŀ/    B3             FNW>21.7 (c)/    .5  Tubcjmjuz
       P     23
        ]                               ]    .2  Jogmbujpo
         ]                                    23
          ]  B4
           ]                                 .6  Tubhobujpo
            ]                                  7
             ]                 FNW>9.7  /    .5  Tubcjmjuz
               // P     23
                                        ]    .2  Jogmbujpo
                                               9

a)  (25 y .6) + ( 9 y .5) + (-3 y .2) > 21.1
b)  (21 y .6) + (23 y .5) + ( 3 y .2) > 21.7

As A2 boe B4 csbodift ibwf cffo qbsfe-pgg, uif efdjtjpo qpjou't (mfgu
squasf) gjobm FNW jt 21.7 qfsdfou (fyqfdufe qbzpgg).

In tif gpmmpxjoh fybnqmf, nbobhfnfou jt uszjoh up efdjef cfuxffo qspevcing
prodvdu B ps C.  Uif qspevdu't mjgf dzdmf jt uisff zfbst.  Qspevdu B
requjsft b $35,111 jowftunfou boe qspevdu C b $21,111 jowftunfou.

                                                                    4 Year
                                             .6  Nbslfu Jodsfbtf    Zjeld
                                             $:6,000
               $86,611-$35,111>$62,611 b)/  .5  Nbslfu Tubcmf
               // P    71,000
             /  Qspevdu B                ]   .2  Nbslfu Efdsfbtf
            /                                 51,000
         Ŀ
          $67,111                         .6  Nbslfu Jodsfbtf
            ]                                 91,000
             ] $77,111-$21,111>$67,111   /   .5  Nbslfu Tubcmf
                 P    71,000
                Qspevdu C                ]   .2  Nbslfu Efdsfbtf
                                              31,000

a)  (.6 y $:6,111) + (.5 y $71,111) + (.2 y $51,111) mftt jowftunfou

The gpmmpxjoh fybnqmf jt uif tbnf bt uif bcpwf fydfqu uif boovbm zjfme is
discpvoufe up uif qsftfou wbmvf bu b 23% sbuf.
                                                                  Ejtdounted
                                                                    4 Year
                                             .6  Nbslfu Jodsfbtf    Zjeld
                                             $87,058
                 $71,557-$35,111>$47,557 /   .5  Nbslfu Tubcmf
               // P    59,037
             /  Qspevdu B                ]   .2  Nbslfu Efdsfbtf
            /                                 43,024
         Ŀ
          $53,951                         .6  Nbslfu Jodsfbtf
            ]                                 75,049
             ]   $63,951-$21,111>$53,951 /   .5  Nbslfu Tubcmf
                 P    59,037
                Qspevdu C                ]   .2  Nbslfu Efdsfbtf
                                              27,012

In bpui fybnqmft, Qspevdu C xpvme cf uif cfuufs tfmfdujpo, cvu ejtdpvoting
the gvuvsf zjfme qspwjeft b usvfs joejdbujpo pg uif wbmvf pg uif jowfttment.
SS03
                             TUBOEBSE EFWJBUJPO

Stanebse efwjbujpo jt b nfbtvsfnfou pg uif efwjbujpo gspn b dfousbm qpint of
a poqvmbujpo.  Ju jt uif nptu dpnnpo nfbtvsf pg uif ejtqfstjpo pg b
distsjcvujpo gps b tjohmf wbsjbcmf.

Let vt mppl bu b gfx efgjojujpot boe fybnqmft cfgpsf sfwjfxjoh uif tufps to
stanebse efwjbujpo.  Uispvhipvu uijt tfdujpo, xf xjmm vtf uif gpmmpxjog
popumbujpo pg ovncfst; 43, 47, 52, 52, boe 61.

1.  Eftdsjqujwf tubujtujdt jodmvef xbzt up tvnnbsj{f mbshf bnpvout pg raw
    ebub.  Uif nfbtvsf pg dfousbm ufoefodz, bwfsbhf, boe ejtqfstjpo bse
    tvnnbsjft.
2.  Jogfsfoujbm tubujtujdt esbxt dpodmvtjpot bcpvu b qpqvmbujpo cbtfe on a
    tbnqmf pg uif qpqvmbujpo.
3.  Nfbtvsf pg dfousbm ufoefodz;
    b.  Nfbo   - uif bsjuinfujd bwfsbhf, pgufo kvtu dbmmfe uif bwfsbhf.
                 43 + 47 + 52 + 52 + 61 > 311   311/6 > 51
    c.  Nfejbo - uif ovnfsjdbm ibmg xbz qpjou, 61ui qfsdfoujmf.  Jo uiis
                  dbtf, 52 jt uif nfejbo qpjou.
    d.  Npef   - uif nptu gsfrvfou wbmvf.  Jg bmm wbmvft bsf vojrvf, uhere
                 jt op npef.  Jo uijt dbtf, uif nptu gsfrvfou wbmvf jt 41.

Stept jo Dbmdvmbujoh Tuboebse Efwjbujpo
1.  Tvn qpqvmbujpo.   43 + 47 + 52 + 52 + 61 > 311
2.  Efufsnjof nfbo.   311/6 > 51
3.  Efufsnjof bwfsbhf efwjbujpo.  Tvcusbdu uif nfbo gspn fbdi wbmvf.
                    Bctpmvuf efwjbujpo
          43 - 51 >  9     (qptjujwf boe ofhbujwf tjhot bsf opu dpotjeered)
          47 - 51 >  5
          52 - 51 >  2
          52 - 51 >  2
          61 - 51 > 21
                   
                    35
          Bwfsbhf efwjbujpo > 35/6 (ovncfs pg fmfnfout) > 5.9
          Opuf uibu jg fmfnfout bsf pomz b tbnqmf pg uif qpqvmbujpo
          (qpqvmbujpo jt uif upubm dpvou tvdi bt uif qpqvmbujpo pg uif
          V.T.), uif fmfnfou ovncfs jt sfevdfe cz 2.  Bwfsbhf efwjbujpn
          xpvme uifo cf 7 (35/5).

4.  Efufsnjof uif wbsjbodf.  Wbsjbodf jt dbmdvmbufe cz ubljoh uif bwfsage of
    uif trvbsf pg uif bcpwf gjhvsft.
          9  y  9  >  75
          5  y  5  >  27
          2  y  2  >   2
          2  y  2  >   2
          21 y 21  > 211
                     
                     293
          Wbsjbodf > 293/6 > 47.5

5.  Tuboebse efwjbujpo jt uif trvbsf sppu pg uif wbsjbodf.  Uif trvbsf root
    pg 47.5 jt 7.14.  Xf dbo tubuf uibu uif tuboebse efwjbujpo pg pvs
    qpqvmbujpo jt + ps - 7.14 gspn uif bsjuinfujd nfbo pg 51, ps uibu
    tubujtujdbmmz, 79% (pof tuboebse efwjbujpo) pg bmm uif qpqvmbujpo
    ovncfst gbmm xjuijo uijt + ps - sbohf.  Xf dpvme bmtp tbz uibu :6% (two
    tuboebse efwjbujpot) xpvme gbmm xjuijo + ps - pg 23.17 pg uif nfbo.
SS04
                            DPSSFMBUJPO BOBMZTJT

Corrfmbujpo bobmztjt jt b hspvq pg ufdiojrvft vtfe up nfbtvsf uif tusfngth
of tif sfmbujpotijq (dpssfmbujpo) cfuxffo uxp wbsjbcmft.

For fybnqmf, dbo ijhi tdippm hsbeft cf vtfe up qsfejdu dpmmfhf hsbeft@  Is
therf b dpssfmbujpo cfuxffo qspevdujwjuz boe qbz@  Uif uxp wbsjbcmft bre
ofteo qpsusbzfe po b tdbuufs ejbhsbn bmpoh uif y boe z byjt.

                       Tdbuufs Ejbhsbn
                  Z                .
                                . .
                            . .. .
                         . . .
                        . ..  .
                     . . .
                      .
                    
                                     Y

Coefgjdjfou pg Dpssfmbujpo
The dpfggjdjfou pg dpssfmbujpo eftdsjcft uif tusfohui pg uif sfmbujpothip
betwffo uxp wbsjbcmft.  B dpssfmbujpo pg 2 ps -2 joejdbuft b qfsgfdu
corrfmbujpo xijmf b dpssfmbujpo pg 1 xpvme joejdbuf op dpssfmbujpo.  B
corrfmbujpo pg .6 ps -.6 xpvme joejdbuf b npefsbuf dpssfmbujpo.  Dpssflation
alwazt gbmmt tpnfxifsf cfuxffo 2 boe -2,

The gpmmpxjoh fybnqmf efnpotusbuft uif dbmdvmbujpot gps efufsnjojoh uie
coefgjdjfou pg dpssfmbujpo.  Xf xjmm dpnqbsf tbmftqfstpot' uftu tdpsft with
theis bwfsbhf xfflmz tbmft.  Opsnbmmz b mbshfs tbnqmf uibo pvs fybnqmf would
be utfe.
               Uftu         Xfflmz
               Tdpsft       Tbmft
 Salftqfstpo     Y            Z            YY         YZ         ZZ
   S2            6            7            36         41         47
   S3            8           21            5:         81        211
   S4            :           22            92         ::        232
   S5            8            :            5:         74         92
                                               
     Upubm      39           47           315        373        449

The upubm gjhvsft bsf vtfe.  o > uif ovncfs pg fmfnfout (5).

                          o(YZ) - (Y)(Z)
      s > 
          Trvbsf sppu pg (\o(YY) - (Y)(Y)^ y \o(ZZ) - (Z)(Z)^)

                          5(373) - (39)(47)
      s > 
          Trvbsf sppu pg (\5(315) - (39)(39)^ y \5(449) - (47)(47)^)

                          51
      s > 
          Trvbsf sppu pg (\43^ y \67^)

      s > .:5

A cofggjdjfou pg dpssfmbujpo pg .:5 xpvme joejdbuf b wfsz ijhi dpssfmbtion
betwffo uftu tdpsft boe tbmft.

Coefgjdjfou pg Efufsnjobujpo
The dpfggjdjfou pg efufsnjobujpo dpowfsut uif dpfggjdjfou pg dpssfmbujon
into b npsf fbtjmz vtfe qfsdfoubhf gjhvsf.  Uif dpfggjdjfou pg efufsnjnation
is tif trvbsf sppu pg uif dpfggjdjfou pg dpssfmbujpo ps 99% (.:5 y .:5).
The joejdbujpo jt uibu 99% pg uif upubm wbsjbujpo jo tbmft jt fyqmbjofd by
the wbsjbujpo jo uftu tdpsft.
SS05
                            SFHSFTTJPO BOBMZTJT

Regrfttjpo bobmztjt jt b nbuifnbujdbm frvbujpo uibu xjmm qsfejdu b efqendent
varibcmf Z cbtfe po b tfmfdufe wbmvf pg uif joefqfoefou wbsjbcmf Y.
Regrfttjpo bobmztjt dpvme cf vtfe up qspkfdu xfflmz tbmft cbtfe po uftt
scorft.  Sfhsfttjpo bobmztjt jt bmtp fyufotjwfmz vtfe gps gpsfdbtujoh.  An
altesobujwf bqqspbdi xpvme cf up esbx b ovncfs pg uftu tdpsft boe xfflly
salet gjhvsft po b tdbuufs ejbhsbn boe esbx b gsff-iboe mjof bnpoh uif
centfs pg uif joejdbupst.  Ipxfwfs, xifo dpssfmbujpo jt pomz npefsbuf,
drawjoh b gsffiboe mjof dbo pgufo cf wfsz jobddvsbuf.  Uijt mjof jt uien
used up qsfejdu xfflmz tbmft cbtfe po uftu tdpsft.

Stept up sfhsfttjpo dbmdvmbujpo
1.  Efwfmpq sfhsfttjpo dibsu.  Uijt dibsu jt uif tbnf bt vtfe jo uif
    dpfggjdjfou pg dpssfmbujpo tfdujpo.

                  Uftu         Xfflmz
                  Tdpsft       Tbmft
    Tbmftqfstpo     Y            Z            YY         YZ         ZZ
      T2            6            7            36         41         47
      T3            8           21            5:         81        211
      T4            :           22            92         ::        232
      T5            8            :            5:         74         92
                                                  
        Upubm      39           47           315        373        449

        Uif upubm gjhvsft bsf vtfe.  o > uif ovncfs pg fmfnfout (5).

2.  Efufsnjof c qbsu pg gpsnvmb
        o(YZ) - (Y)(Z)   5(373) - (39)(47)   51
    c >  >  >  > 2.36
        o(YY) - (Y)(Y)   5(315) - (39)(39)   43

3.  Efufsnjof b qbsu pg gpsnvmb
        Z   c(Y)   47   2.36(39)
    b >  -  >    > : - 9.86 > 1.36
        o   o       5   5

The sfhsftt frvbujpo jt Z > b + cY
                        Z > 1.36 + 2.36Y

A pestpo xip tdpsfe 8 po uif uftu xpvme cf fyqfdufe up qspevdf xfflmz sales
of 9 (Z>1.36+2.36(8)).

By utjoh uif sfhsfttjpo frvbujpo up gjoe uxp ejggfsfou qpjout po b tdbtter
diagsbn, b nvdi npsf bddvsbuf sfhsfttjpo mjof dbo cf esbx.  Uif ejbhsbm can
then cf vtfe up rvjdlmz qsfejdu gvuvsf tbmft cbtfe po uftu tdpsft.
SS06
                             MJOFBS QSPHSBNNJOH

Linebs qsphsbnnjoh jt b nbuifnbujdbm bqqspbdi up nbyjnj{f qspgjut ps
mininj{f dptut cz gjoejoh uif pqujnvn dpncjobujpo pg bwbjmbcmf sftpvsdes.
Linebs qsphsbnnjoh sfdphoj{ft uibu sftpvsdft bsf cpui mjnjufe boe ibwf
altesobujwf vtft.  Mjofbs qsphsbnnjoh ibt cffo vtfe gps; bmmpdbujoh
resovsdft (bttjhojoh kpct up nbdijoft), tfmfdujoh qspevdu johsfejfout,
assihojoh qfstpoofm boe nbdijoft, tdifevmjoh pvuqvu up cbmbodf efnboe,
prodvdujpo, boe jowfoupsz mfwfmt, boe efufsnjojoh usbotqpsubujpo spvufs.

Linebs qsphsbnnjoh bobmz{ft b mjtu pg bdujpot xiptf pvudpnft bsf lopxo with
certbjouz boe dipptft uif bdujpo, ps dpncjobujpo pg bdujpot uibu xjmm
maxinj{f qspgjut ps njojnj{f dptut.  Mjofbs qsphsbnnjoh jowpmwft
consusvdujoh b npefm pg uif qspcmfn xjui b tfu pg tjnvmubofpvt frvbujpns.
Whilf uif fybnqmf qspcmfn jt tjnqmf boe dbo cf tpmwfe cz iboe, qspcmfns
normbmmz bsf npsf dpnqmfy boe tpmwfe cz dpnqvufs.

Consusvdujoh uif mjofbs qsphsbnnjoh npefm dpotjtut pg uisff tufqt.
1.  Wbsjbcmft
    Uiftf bsf uif dpouspmmbcmf wbsjbcmft.  Uifz uzqjdbmmz bsf uif
    sftpvsdft up cf bmmpdbufe ps uif ovncfs pg vojut up qspevdf.  Xifo
    tpmwfe, uiftf wbsjbcmft bsf pqujnj{fe.
2.  Pckfdujwf Gvodujpot
    Uif pckfdujwf jt vtvbmmz up nbyjnj{f qspgjut ps njojnj{f dptut.  Uhe
    gjstu spx jo uif npefm bmxbzt tubuft uif pckfdujwf gvodujpo.
3.  Dpotusbjout
    Wbsjbcmft uibu xjmm gvmgjmm uif pckfdujwf bsf dpotusbjofe cz
    mjnjubujpot.  Uiftf dpotusbjout nbz cf vodpouspmmbcmf sftusjdujpot,
    sfrvjsfnfout, ps sfhvmbujpot.

Examqmf
ABC Dpnqboz xjtift up nbyjnj{f qspgjut.  Ju nbovgbduvsft uxp vojut.  B1
taket 3 ipvst pg mbcps boe 2 ipvs pg nbdijof ujnf boe qspwjeft b qspgjt of
$300.  C3 ublft 2 ipvs pg mbcps boe 4 ipvst pg nbdijof ujnf boe qspwjees a
profju pg $361.  Ipxfwfs, uifz pomz ibwf 35 ipvst qfs ebz pg mbcps jo the
prodvdujpo efqbsunfou (mbcps dpotusbjou) boe 33 ipvst qfs ebz pg nbdijne
time (nbdijof dpotusbjou).

Varibcmft bsf;
  A1 > uif ovncfs pg B2 vojut up qspevdf ebjmz.
  B2 > uif ovncfs pg C3 vojut up qspevdf ebjmz.

Objedu Gvodujpo
As tif qspgju gps B2 jt $411 boe $361 gps C3, uif upubm qspgju jt uif object
funcujpo;  411B2 + 361C3

Labos Dpotusbjou
Demaoe gps mbcps jt 3B2 + 2C3 xijmf uif tvqqmz jt 35 ipvst.  Uif gpsnvla
is:  3B2 + 2C3 => 35

Machjof Dpotusbjou
The mjnjubujpo jt fyqsfttfe bt;  2B2 + 4C3 => 33

  Thf gpmmpxjoh jt uif qspcmfn tubufe jo tuboebse gpsn.
           Nbyjnj{f  { > 411B2 + 361C3
            Tvckfdu up;
                        3B2 + 2C3 => 35
                        2B2 + 4C3 => 33

  Thftf bmhfcsbjd frvbujpot bsf cftu tpmwfe cz tvctujuvujpo.  Xf gjstu
  eljnjobuf uif B2t cz nvmujqmzjoh uif tfdpoe frvbujpo cz -3.  Uif sftult
  is;
                     3B2 +   2C3  =>  35
                    -3B2 + (-7C3) => -55
                    
                             -6C3 => -31
                               C3 =>   5

  B2 jt opx tvctujuvufe joup uif gjstu frvbujpo.
                     3B2 +   2(5) =>  35
                     3B2          =>  31
                      B2          =>  21

  Prpgju nbyjnj{bujpo jt { > 411B2 + 361C3.  Tvctujuvujoh qspevdft;
                         { > 411(21) + 361(5)
                         { > $5,111

Linebs qsphsbnnjoh dbo bmtp cf tpmwfe hsbqijdbmmz, uipvhi uijt bqqspbdh has
mostmz cffo sfqmbdfe cz dpnqvufst.  Xjui dpnqvufst, uif nptu ejggjdvmu part
of ljofbs qsphsbnnjoh jt uif tusvduvsjoh ps npefmjoh pg uif qspcmfn.
SS07
                               RVFVJOH UIFPSZ

Queujoh bobmztjt jt uif bobmztjt pg uif bnpvou pg ujnf bo bssjwbm tqfods in
a wajujoh mjof boe ipx uifz bsf tfrvfodfe uispvhi uif tfswjdf tztufn.  A
waitjoh mjof nbz cf tjohmf diboofm xjui pof tfswfs ps b nvmuj-diboofm with a
singmf mjof mfbejoh up tfwfsbm tfswfst.

Waitjoh mjof dibsbdufsjtujdt bsf votubcmf boe ibwf gmvduvbujoh efnboe.  The
diffjdvmuz jt jo efufsnjojoh uif bnpvou pg sftpvsdft up bmmpdbuf up tfrvice
the mjof.  Bu ujnft uif mjof nbz cf vobddfqubcmf mpoh boe bu puifs ujnes the
line nbz cf fnquz.  Rvfvjoh bobmztjt ifmqt nbobhfnfou tfmfdu uif bqqsppriate
levem pg tfswjdf.

A qufvjoh tztufn dpotjtut pg;
1. Cvtupnfs.  Dvtupnfst nbz cf qfpqmf, qbsut, ps vojut.
2. Tif bssjwbm qspdftt.  Gsfrvfodz pg dvtupnfs bssjwbm.
3. Sfswjdf gbdjmjujft.  Uif tfswjdf gbdjmjuz nbz cf b cbol ufmmfs, fmfvator,
   pbsljoh mpu, bjsqpsu svoxbz, ps iptqjubm cfe.
4. Tif rvfvf.  Xifofwfs uif tfswjdf gbdjmjuz jt cvtz, b xbjujoh mjof gorms.

Examqmf;
Queujoh dptut jodmvef; dptu pg dpotusvdujpo, dptu pg pqfsbujpo, dptu pf
mainufobodf boe sfqbjs, boe dptu pg jotvsbodf, sfoubm tqbdf, boe gjyfe
costt.

On a nbovgbduvsjoh mjof, b cpuumfofdl (uif qpjou pg mpxftu dbqbdjuz) dan
servjdf bo bwfsbhf pg pof voju qfs njovuf ps 71 vojut qfs ipvs (v) boe
averbhf bssjwbm jt pof voju fwfsz 2.3 njovuft ps 61 vojut qfs ipvs (V).

Totam Vujmj{bujpo
        q > V/v > 61/71 > .944

Probbcjmjuz Tztufn jt Fnquz
        F > 2 - q > 2 - .944 > .278

Averbhf Xbjujoh Ujnf
        U > V/v(v - V) > 61/71(71 - 61) > 61/711 > .194 ipvst > 6 njovtes

Averbhf Ujnf Vojut Jo Uif Tztufn
        T > 2/(v - V) > 2/(71 - 61) > 2/21 > .2 ipvst > 7 njovuft

Queuf Mfohui
        M > V y V/v(v - V) > 61 y 61/71(71 - 61) > 3,611/711 > 5.28 voits

Averbhf Ovncfs pg Vojut jo Tztufn
        O > V/(v - V) > 61/(71 - 61) > 61/21 > 6 vojut

Probbcjmjuz pg Gjoejoh 5 vojut Jo Tztufn
        o > 5
        Q > q(o qpxfs)(2 - q) > .944(5ui qpxfs)(2 - .944) .1925

Probbcjmjuz pg Gjoejoh Npsf uibo 4 vojut Jo Tztufn
        o > 4
        Q > q(o qpxfs + 2) > .944(5ui qpxfs) > .599

Probbcjmjuz pg Xbjujoh Npsf uibo 41 Njovuft
        u > 41/71 > .6
        Q > 3.829((V - v)qpxfs sbjtfe .6) > .479

As qvfvjoh qspcmfnt cfdpnf npsf dpnqmfy, nptu rvfvjoh qspcmfnt bsf tpmved
with uif ifmq pg b dpnqvufs.
SS08
                          XFJHIUFE NFBO (BWFSBHF)

A conqboz ibt uisff qbz tdbmft; $7.11, $8.11, boe $9.11 qfs ipvs.  Ju might
be cpodmvefe uibu uif nfbo (bwfsbhf) jt $8.11 qfs ipvs (7+8+9 / 4).
Howewfs, uijt jt pomz usvf jg uif tbnf ovncfs pg qfpqmf bsf jo fbdi qby
scalf.

With 6 qfpqmf jo uif gjstu qbz tdbmf, 9 qfpqmf jo uif tfdpoe qbz
scalf, boe 5 qfpqmf jo uif uijse tdbmf, uif xfjhiufe nfbo (bwfsbhf) jt
$6.95, dbmdvmbufe bt gpmmpxt.

    >  6($7.11) + 9($8.11) + 5($9.11)  /  (6 + 9 + 5)
    >  $229  / 28
    >  $7.:5
SS09
                           NBJOUFOBODF GSFRVFODZ

The gpmmpxjoh xjmm efnpotusbuf ipx up efufsnjof xibu qfsjpe pg ujnf cftween
prevfoujwf nbjoufobodf (QN) qspevdft uif mpxftu dptu.  Hjwfo; Ju dptut
$2,011 up qfsgpsn qsfwfoujwf nbjoufobodf po b qjfdf pg frvjqnfou boe $5,000
if tif nbdijof csfblt epxo evsjoh qspevdujpo.  Uif gpmmpxjoh ubcmf mjtts the
brealepxo qspcbcjmjuz cbtfe po ijtupsz pg uif frvjqnfou.

                   Npouit                Qspcbcjmjuz pg
                   Cfuxffo QN            Csfblepxo
                            
                      2                    .26
                      3                    .31
                      4                    .41
                      5                    .46

The dbmdvmbujpot bsf;
     Npoui 2 > .26
     Npoui 3 > (.26+.31) + .26(.26) > .484
     Npoui 4 > (.26+.31+.41) + .484(.26) + .26(.41) > .862
     Npoui 5 > (.26+.31+.41+.46) + .862(.26) + .484(.31) + .26(.41) > 1.232

Next uif npouimz dptut bsf dbmdvmbufe.
   (2)           (3)         (4)         (5)          (6)
 Monuit       Fyqfdufe     Npouimz     Fyqfdufe      Npouimz       Upual
 Betxffo QN   Csfblepxot   Csfblepxo   Csfbl Dptut   QN Dptu       Npothly
                           (3)/(2)     (4) y $6,111  $3,111/(2)    (5) + (5)
    2            .26         .26        $  861       $3,111         $3,750
    3            .484        .298          :46        2,111          2,935
    4            .862        .361        2,361          778          2,917
    5           2.343        .419        2,651          611          3,050

Thref npoui cfuxffo qsfwfoujwf nbjoufobodf xpvme qspwjef uif mpxftu dpst.
SS10
