,W LV QRW QHFHVVDU\ WR UHHQWHU 790 YDOXHV IRU HDFK H[DPSOH 8VLQJ WKH YDOXHV \RX MXVW HQWHUHG SDJH KRZ PXFK FDQ \RX ERUURZ LI \RX ZDQW D SD\PHQW RI "

100.00

11,395.08
(QWHUV QHZ SD\PHQW DPRXQW 0RQH\ SDLG RXW LV QHJDWLYH &DOFXODWHV DPRXQW \RX FDQ ERUURZ LQWHUHVW UDWH" (QWHUV QHZ LQWHUHVW UDWH &DOFXODWHV QHZ SUHVHQW YDOXH IRU SD\PHQW DQG LQWHUHVW 5HHQWHUV RULJLQDO LQWHUHVW UDWH 5HHQWHUV RULJLQDO SUHVHQW YDOXH &DOFXODWHV RULJLQDO SD\PHQW
+RZ PXFK FDQ \RX ERUURZ DW D

9.50 11,892.67

F " P

10.00 14,000.00 122.86

at_a_glance.fm Page 16 Sunday, June 25, 2000 2:21 PM

AmortizationAt a Glance

$IWHU FDOFXODWLQJ D SD\PHQW XVLQJ 7LPH 9DOXH RI 0RQH\ 790 HQWHU WKH SHULRGV WR DPRUWL]H DQG SUHVV 7KHQ SUHVV WR FRQWLQXDOO\ F\FOH WKURXJK WKH SULQFLSDO LQWHUHVW DQG EDODQFH YDOXHV LQGLFDWHG E\ WKH PRIN INT DQG BAL DQQXQFLDWRUV UHVSHFWLYHO\
8VLQJ WKH SUHYLRXV 790 H[DPSOH SDJH DPRUWL]H D VLQJOH SD\PHQW DQG WKHQ D UDQJH RI SD\PHQWV $PRUWL]H WKH WK SD\PHQW RI WKH ORDQ
$PRUWL]H WKH VW WKURXJK WK ORDQ SD\PHQWV

12_ 77.82 1,396.50

H B 1 H B 1

20.20 7.25 115.61

13,865.83
(QWHUV SHULRG WR DPRUWL]H 'LVSOD\V SHULRG WR DPRUWL]H 'LVSOD\V SULQFLSDO 'LVSOD\V LQWHUHVW 0RQH\ SDLG RXW LV QHJDWLYH 'LVSOD\V EDODQFH

13,922.18

(QWHUV UDQJH RI SHULRGV WR DPRUWL]H 'LVSOD\V UDQJH RI SHULRGV SD\PHQWV 'LVSOD\V SULQFLSDO 'LVSOD\V LQWHUHVW 0RQH\ SDLG RXW LV QHJDWLYH 'LVSOD\V EDODQFH
at_a_glance.fm Page 17 Sunday, June 25, 2000 2:21 PM
Interest Rate ConversionAt a Glance
7R FRQYHUW EHWZHHQ QRPLQDO DQG HIIHFWLYH LQWHUHVW UDWHV HQWHU WKH NQRZQ UDWH DQG WKH QXPEHU RI SHULRGV SHU \HDU WKHQ VROYH IRU WKH XQNQRZQ UDWH

B B B B B B

1RPLQDO LQWHUHVW SHUFHQW (IIHFWLYH LQWHUHVW SHUFHQW 3HULRGV SHU \HDU QRPLQDO LQWHUHVW
)LQG WKH DQQXDO HIIHFWLYH LQWHUHVW UDWH RI FRPSRXQGHG PRQWKO\

10.00 12.00 10.47

(QWHUV QRPLQDO UDWH (QWHUV SD\PHQWV SHU \HDU &DOFXODWHV DQQXDO HIIHFWLYH LQWHUHVW
at_a_glance.fm Page 18 Sunday, June 25, 2000 2:21 PM
IRR/YR and NPVAt a Glance

B D Bg BW Bx

1XPEHU RI SHULRGV SHU \HDU GHIDXOW LV &DVK IORZV XS WR M LGHQWLILHV WKH FDVK IORZ QXPEHU 1XPEHU RI FRQVHFXWLYH WLPHV FDVK IORZ M RFFXUV ,QWHUQDO UDWH RI UHWXUQ SHU \HDU 1HW SUHVHQW YDOXH
,I \RX KDYH DQ LQLWLDO FDVK RXWIORZ RI IROORZHG E\ PRQWKO\ FDVK LQIORZV RI DQG ZKDW LV WKH ,55<5" :KDW LV WKH ,55 SHU PRQWK"
B8B "@D "%D %D Bg !D BW 1 d F Bx
0.00 12.00 40,000.00 4,700.00 7,000.00 2.00

23,000.00 15.96 1.33

&OHDUV DOO PHPRU\ 6HWV SD\PHQWV SHU \HDU (QWHUV LQLWLDO RXWIORZ (QWHUV ILUVW FDVK IORZ (QWHUV VHFRQG FDVK IORZ (QWHUV QXPEHU RI FRQVHF XWLYH WLPHV FDVK IORZ RFFXUV (QWHUV WKLUG FDVK IORZ &DOFXODWHV ,55<5 &DOFXODWHV ,55 SHU PRQWK
:KDW LV WKH 139 LI WKH GLVFRXQW UDWH LV "

10.00 622.85

(QWHUV ,<5 &DOFXODWHV 139
fm Page 19 Sunday, June 25, 2000 2:21 PM

StatisticsAt a Glance

\YDOXH [YDOXH
e Bd e H Bd H B6Bi Bi BXBi BwBi BQBi BR BRBi
QXPEHU QXPEHU QXPEHU QXPEHU QXPEHU QXPEHU
&OHDU VWDWLVWLFDO UHJLVWHUV (QWHU RQHYDULDEOH VWDWLVWLFDO GDWD 'HOHWH RQHYDULDEOH VWDWLVWLFDO GDWD (QWHU WZRYDULDEOH VWDWLVWLFDO GDWD 'HOHWH WZRYDULDEOH VWDWLVWLFDO GDWD 0HDQV RI [ DQG \ 0HDQ RI [ ZHLJKWHG E\ \ 6DPSOH VWDQGDUG GHYLDWLRQV RI [ DQG \ 3RSXODWLRQ VWDQGDUG GHYLDWLRQV RI [ DQG \ (VWLPDWH RI [ DQG FRUUHODWLRQ FRHIILFLHQW (VWLPDWH RI \ \LQWHUFHSW DQG VORSH
8VLQJ WKH IROORZLQJ GDWD ILQG WKH PHDQV RI [ DQG \ WKH VDPSOH VWDQGDUG GHYLDWLRQV RI [ DQG \ DQG WKH \LQWHUFHSW DQG WKH VORSH RI WKH OLQHDU UHJUHVVLRQ IRUHFDVW OLQH 7KHQ XVH VXPPDWLRQ VWDWLVWLFV WR ILQG Q DQG [\

x-data y-data

at_a_glance.fm Page 20 Sunday, June 25, 2000 2:21 PM
Bj H#e "H'e $H$e B6 Bi BX Bi BR Bi A" A'
0.00 1.00 2.00 3.00 4.00 100.00 2.00
&OHDUV VWDWLVWLFV UHJLVWHUV (QWHUV ILUVW [\ SDLU (QWHUV VHFRQG [\ SDLU (QWHUV WKLUG [\ SDLU 'LVSOD\V PHDQ RI [ 'LVSOD\V PHDQ RI \ 'LVSOD\V VDPSOH VWDQ GDUG GHYLDWLRQ RI [ 'LVSOD\V VDPSOH VWDQ GDUG GHYLDWLRQ RI \ 'LVSOD\V \LQWHUFHSW RI UHJUHVVLRQ OLQH SUH GLFWHG y YDOXH IRU [ 'LVSOD\V VORSH RI UHJUHVVLRQ OLQH 'LVSOD\V Q QXPEHU RI GDWD SRLQWV HQWHUHG 'LVSOD\V [\ VXP RI WKH SURGXFWV RI [ DQG \YDOXHV

1,420.00

fm Page 21 Sunday, June 25, 2000 2:21 PM

Keyboard Map

Time value of money (page 53) Separate two numbers (page 28) Percent (page 33) Constant (page 37) Store and recall (page 40) Change sign (page 24) Statistics key (page 27) Shift key (page 27) Clear display, cancel operation (page 25) 10. Clear all memory (page 25) 11. On (page 87) 12. Statistical functions (page 87)
1. 2. 3. 4. 5. 6. 7. 8. 9.
13. n through xy: statistical summation registers (page 88) 14. 3-key memory (page 39) 15. Backspace (page 25) 16. Accumulate statistical data (page 86) 17. Cash flows (page 75) 18. Business functions: margin, markup, cost, price (page 35) 19. Interest conversion (page 72) 20. Amortization (page 67) 21. Annunciator lines (page 26)

Clearing Memory

Description
&OHDUV DOO PHPRU\ 'RHV QRW UHVHW PRGHV &OHDUV VWDWLVWLFDO PHPRU\
HqrCQ 7DDhrirshrrrhhtr $#7rtvhq@qhtr $$ hqurqvyhs hhtr !(
7R FOHDU DOO PHPRU\ DQG UHVHW FDOFXODWRU PRGHV SUHVV DQG KROG GRZQ WKHQ SUHVV DQG KROG GRZQ ERWK DQG :KHQ \RX UHOHDVH DOO WKUHH DOO PHPRU\ LV FOHDUHG 7KH All Clear PHVVDJH LV GLVSOD\HG
10B.book Page 26 Wednesday, June 21, 2000 5:19 PM

Annunciators

$QQXQFLDWRUV DUH V\PEROV LQ WKH GLVSOD\ WKDW LQGLFDWH WKH VWDWXV RI WKH FDOFXODWRU

Annunciator

PEND BEGIN INPUT AMOR T BAL INT PRIN PER C - FLOW CF N ERROR TVM
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

Status

10B.book Page 27 Wednesday, June 21, 2000 5:19 PM

FULL STAT

Status (Continued)

FUNC STAT

0RUH WKDQ FDVK IORZV KDYH EHHQ HQWHUHG RU PRUH WKDQ XQVROYHG EUDFNHWV XVHG ,QFRUUHFW GDWD XVHG LQ D VWDWLVWLFV FDOFXODWLRQ RU ZKHQ ERROR LV QRW OLW D VWDWLVWLFDO FDOFXODWLRQ KDV EHHQ SHUIRUPHG $ PDWK HUURU KDV RFFXUUHG IRU H[DPSOH GLYLVLRQ E\ ]HUR $ VWDWLVWLFDO FDOFXODWLRQ KDV EHHQ SHUIRUPHG

Shift Key

0RVW RI WKH +3 %,, NH\V KDYH D VHFRQG RU VKLIWHG IXQFWLRQ SULQWHG LQ RUDQJH RQ WKH NH\ 7KH RUDQJH VKLIW NH\ LV XVHG WR DFFHVV WKHVH IXQFWLRQV

QXPEHU LQ WKH GLVSOD\ E\ LWVHOI
B )RU H[DPSOH SUHVV B IROORZHG E\ h DOVR VKRZQ DV Bh WR PXOWLSO\ D A :KHQ \RX SUHVV A WKH VWDWLVWLFV DQQXQFLDWRU A
7KH VWDWLVWLFV NH\ FRORUHG PDXYH LV XVHG WR DFFHVV VXPPDU\ VWDWLVWLFV IURP WKH VWDWLVWLFV PHPRU\ UHJLVWHUV GLVSOD\HG 7KLV LQGLFDWHV WKDW \RX FDQ UHFDOO RQH RI VL[ VXPPDU\ VWDWLVWLFV ZLWK WKH QH[W NH\VWURNH VHH SDJH 7R WXUQ WKH STATS DQQXQFLDWRU RII SUHVV DJDLQ

STATS

:KHQ \RX SUHVV WKH VKLIW DQQXQFLDWRU SHIFT LV GLVSOD\HG WR LQGLFDWH WKDW WKH VKLIWHG IXQFWLRQV DUH DFWLYH 7R WXUQ WKH SHIFT DQQXQFLDWRU RII SUHVV DJDLQ

Statistics Key

)RU H[DPSOH SUHVV YDOXHV HQWHUHG
A IROORZHG E\ Y WR UHFDOO WKH VXP RI WKH [
10B.book Page 28 Wednesday, June 21, 2000 5:19 PM

INPUT Key

7KH NH\ LV XVHG WR VHSDUDWH WZR QXPEHUV ZKHQ XVLQJ WZRQXPEHU IXQFWLRQV RU WZRYDULDEOH VWDWLVWLFV 7KH NH\ FDQ DOVR EH XVHG WR HYDOXDWH DQ\ SHQGLQJ DULWKPHWLF RSHUDWLRQV LQ ZKLFK FDVH WKH UHVXOW LV WKH VDPH DV SUHVVLQJ

SWAP Key

3UHVVLQJ
7KH ODVW WZR QXPEHUV WKDW \RX HQWHUHG IRU LQVWDQFH WR FKDQJH WKH RUGHU RI GLYLVLRQ RU VXEWUDFWLRQ 7KH UHVXOWV RI IXQFWLRQV WKDW UHWXUQ WZR YDOXHV 7KH [ DQG \YDOXHV ZKHQ XVLQJ VWDWLVWLFV
Bi H[FKDQJHV WKH IROORZLQJ

Math Functions

One-Number Functions. 0DWK IXQFWLRQV LQYROYLQJ RQH
QXPEHU XVH WKH QXPEHU LQ WKH GLVSOD\

&' #B !#% ! $B 1

9.45 0.42 3.99
&DOFXODWHV VTXDUH URRW LV FDOFXODWHG ILUVW $GGV DQG
.book Page 29 Wednesday, June 21, 2000 5:19 PM
Two-Number Functions. :KHQ D IXQFWLRQ UHTXLUHV WZR QXPEHUV WKH QXPEHUV DUH HQWHUHG OLNH WKLV QXPEHU QXPEHU IROORZHG E\ WKH RSHUDWLRQ 3UHVVLQJ HYDOXDWHV WKH FXUUHQW H[SUHVVLRQ DQG GLVSOD\V WKH INPUT DQQXQFLDWRU )RU H[DPSOH WKH IROORZLQJ NH\VWURNHV FDOFXODWH WKH SHUFHQW FKDQJH EHWZHHQ DQG

%H ' BT

17.00 29_ 70.59
(QWHUV QXPEHU GLVSOD\V WKH INPUT DQQXQFLDWRU (QWHUV QXPEHU &DOFXODWHV WKH SHUFHQW FKDQJH

'%H !$&BT

291.70 8.60
(QWHUV Q &DOFXODWHV SHUFHQW FKDQJH
Example. &DOFXODWH WKH SHUFHQW FKDQJH EHWZHHQ DQG

4#H $#&BT

60.00 38.33
&DOFXODWHV DQG HQWHUV Q &DOFXODWHV SHUFHQW FKDQJH
10B.book Page 35 Wednesday, June 21, 2000 5:19 PM
Margin and Markup Calculations
7KH +3 %,, FDQ FDOFXODWH FRVW VHOOLQJ SULFH PDUJLQ RU PDUNXS

Application

0DUJLQ 0DUNXS

U , , U , , )

0DUJLQ LV PDUNXS H[SUHVVHG DV D SHUFHQW RI SULFH 0DUNXS FDOFXODWLRQV DUH H[SUHVVHG DV D SHUFHQW RI FRVW
7R VHH DQ\ YDOXH XVHG E\ WKH 0DUJLQ DQG 0DUNXS DSSOLFDWLRQ SUHVV DQG WKHQ WKH NH\ \RX ZLVK WR VHH )RU H[DPSOH WR VHH WKH YDOXH VWRUHG DV SUHVV 0DUJLQ DQG 0DUNXS VKDUH WKH VDPH VWRUDJH UHJLVWHU )RU H[DPSOH LI \RX VWRUH LQ WKHQ SUHVV \RX ZLOO VHH GLVSOD\HG

Margin Calculations

Example.LORZDWW (OHFWURQLFV SXUFKDVHV WHOHYLVLRQV IRU 7KH
WHOHYLVLRQV DUH VROG IRU :KDW LV WKH PDUJLQ"

##U !

255.00 300.00 15.00
6WRUHV FRVW LQ &67 6WRUHV VHOOLQJ SULFH LQ 35& &DOFXODWHV PDUJLQ
Markup on Cost Calculations
Example. 7KH VWDQGDUG PDUNXS RQ FRVWXPH MHZHOU\ DW.OHLQHUV
.RVPHWLTXH LV 7KH\ MXVW UHFHLYHG D VKLSPHQW RI FKRNHUV FRVWLQJ HDFK :KDW LV WKH UHWDLO SULFH SHU FKRNHU"

'U $)

19.00 60.00 30.40
6WRUHV FRVW 6WRUHV PDUNXS &DOFXODWHV UHWDLO SULFH
10B.book Page 36 Wednesday, June 21, 2000 5:19 PM
Using Margin and Markup Together
Example.$ IRRG FRRSHUDWLYH EX\V FDVHV RI FDQQHG VRXS ZLWK DQ LQYRLFH FRVW RI SHU FDVH ,I WKH FRRS URXWLQHO\ XVHV D PDUNXS IRU ZKDW SULFH VKRXOG LW VHOO D FDVH RI VRXS" :KDW LV WKH PDUJLQ" Keys:

'$U #)

9.60 15.00 11.04 13.04
6WRUHV LQYRLFH FRVW 6WRUHV PDUNXS &DOFXODWHV WKH SULFH RQ D FDVH RI VRXS &DOFXODWHV PDUJLQ
10B.book Page 37 Wednesday, June 21, 2000 5:19 PM
Number Storage and Arithmetic
Using Stored Numbers in Calculations
<RX FDQ VWRUH QXPEHUV IRU UHXVH LQ VHYHUDO GLIIHUHQW ZD\V n 8VH &RQVWDQW WR VWRUH D QXPEHU DQG LWV RSHUDWRU IRU UHSHWLWLYH RSHUDWLRQV n 8VH .H\ 0HPRU\ DQG WR VWRUH UHFDOO DQG VXP QXPEHUV ZLWK D VLQJOH NH\VWURNH n 8VH DQG WR VWRUH WR DQG UHFDOO IURP WKH QXPEHUHG UHJLVWHUV

B3 ( E

Using Constants
8VH WR VWRUH D QXPEHU DQG DULWKPHWLF RSHUDWRU IRU UHSHWLWLYH FDOFXODWLRQV 2QFH WKH FRQVWDQW RSHUDWLRQ LV VWRUHG HQWHU D QXPEHU DQG SUHVV 7KH VWRUHG RSHUDWLRQ LV SHUIRUPHG RQ WKH QXPEHU LQ WKH GLVSOD\
3: Number Storage and Arithmetic

FV = 7,500.00 PMT = ? 1 2
Money received by lessor is positive
I/YR = 10% N = 36 P/YR = 12 PV = -13,500.00 Begin Mode
Money paid out by lessor is negative

B F !#@ %#G !$ P

13,500.00 7,500.00 36.00 253.99
6HWV SD\PHQWV SHU \HDU 6WRUHV GHVLUHG DQQXDO \LHOG 6WRUHV OHDVH SULFH 6WRUHV UHVLGXDO EX\ RXW YDOXH 6WRUHV OHQJWK RI OHDVH LQ PRQWKV &DOFXODWHV PRQWKO\ OHDVH SD\PHQW
1RWLFH WKDW HYHQ LI WKH FXVWRPHU FKRRVHV QRW WR EX\ WKH FDU WKH OHVVRU VWLOO LQFOXGHV D FDVK IORZ FRPLQJ LQ DW WKH HQG RI WKH OHDVH HTXDO WR WKH UHVLGXDO YDOXH RI WKH FDU :KHWKHU WKH FXVWRPHU EX\V WKH FDU RU LW LV VROG RQ WKH RSHQ PDUNHW WKH OHVVRU H[SHFWV WR UHFRYHU
10B.book Page 65 Wednesday, June 21, 2000 5:19 PM
Example: Lease With Advance Payments. <RXU FRPSDQ\ 4XLFN.LW 3ROH %DUQV SODQV WR OHDVH D IRUNOLIW IRU WKH ZDUHKRXVH 7KH OHDVH LV ZULWWHQ IRU D WHUP RI \HDUV ZLWK PRQWKO\ SD\PHQWV RI 3D\PHQWV DUH GXH DW WKH EHJLQQLQJ RI WKH PRQWK ZLWK WKH ILUVW DQG ODVW SD\PHQWV GXH DW WKH RQVHW RI WKH OHDVH <RX KDYH DQ RSWLRQ WR EX\ WKH IRUNOLIW IRU DW WKH HQG RI WKH OHDVLQJ SHULRG
,I WKH DQQXDO LQWHUHVW UDWH LV ZKDW LV WKH FDSLWDOL]HG YDOXH RI WKH OHDVH"
Begin Mode PV = ? I/YR = 9% N = 48 P/YR = 4
PMT = -2,400.00 (48th payment due up front) FV = -15,000.00
7KLV VROXWLRQ UHTXLUHV IRXU VWHSV &DOFXODWH WKH SUHVHQW YDOXH RI WKH PRQWKO\ SD\PHQWV $GG WKH YDOXH RI WKH DGGLWLRQDO DGYDQFH SD\PHQW )LQG WKH SUHVHQW YDOXH RI WKH EX\ RSWLRQ 6XP WKH YDOXHV FDOFXODWHG LQ VWHSV DQG
Step 1 )LQG WKH SUHVHQW YDOXH RI WKH PRQWKO\ SD\PHQWV
10B.book Page 66 Wednesday, June 21, 2000 5:19 PM

B "% "@P G 'F

12.00 47.00 2,400.00 0.00 9.00 95,477.55
6HWV SD\PHQWV SHU \HDU 6WRUHV QXPEHU RI SD\PHQWV 6WRUHV PRQWKO\ SD\PHQW 6WRUHV )9 IRU VWHS 6WRUHV LQWHUHVW UDWH &DOFXODWHV SUHVHQW YDOXH RI PRQWKO\ SD\PHQWV
Step 2. $GG WKH DGGLWLRQDO DGYDQFH SD\PHQW WR 39 6WRUH WKH DQVZHU

97,877.55 97,877.55

$GGV DGGLWLRQDO DGYDQFH SD\PHQW 6WRUHV UHVXOW LQ 0 UHJLVWHU
Step 3. )LQG WKH SUHVHQW YDOXH RI WKH EX\ RSWLRQ

48.00 0.00

"& P #@G 1

15,000.00 10,479.21

6WRUHV PRQWK ZKHQ EX\ RSWLRQ RFFXUV 6WRUHV ]HUR SD\PHQW IRU WKLV VWHS RI VROXWLRQ 6WRUHV YDOXH WR GLVFRXQW &DOFXODWHV SUHVHQW YDOXH RI ODVW FDVK IORZ
Step 4. $GG WKH UHVXOWV RI VWHSV DQG Display:

108,356.77

&DOFXODWHV SUHVHQW FDSLWDOL]HG YDOXH RI OHDVH 5RXQGLQJ GLVFUHSDQFLHV DUH H[SODLQHG RQ SDJH

7R HQWHU GDWD IRU FDOFXODWLQJ WKH ZHLJKWHG PHDQ HQWHU HDFK GDWD YDOXH DV [ DQG LWV FRUUHVSRQGLQJ ZHLJKW DV \
If statistical data causes the value of a register to exceed 9.99999999999 10499, the HP 10BII displays a temporary overflow warning (OFLO).
10B.book Page 87 Wednesday, June 21, 2000 5:19 PM
Correcting Statistical Data
,QFRUUHFW HQWULHV FDQ EH GHOHWHG XVLQJ ,I HLWKHU YDOXH RI DQ [\ SDLU LV LQFRUUHFW \RX PXVW GHOHWH DQG UHHQWHU ERWK YDOXHV
Correcting One-Variable Data
7R GHOHWH DQG UHHQWHU VWDWLVWLFDO GDWD .H\ LQ WKH [YDOXH WR EH GHOHWHG 3UHVV WR GHOHWH WKH YDOXH 7KH QYDOXH LV GHFUHDVHG E\ RQH (QWHU WKH FRUUHFW YDOXH XVLQJ
Correcting Two-Variable Data
7R GHOHWH DQG UHHQWHU [\ SDLUV RI VWDWLVWLFDO GDWD .H\ LQ WKH [YDOXH SUHVV DQG WKHQ NH\ LQ WKH \YDOXH 3UHVV WR GHOHWH WKH YDOXHV 7KH QYDOXH LV GHFUHDVHG E\ RQH (QWHU WKH FRUUHFW [\ SDLU XVLQJ DQG
Summary of Statistical Calculations
6RPH IXQFWLRQV UHWXUQ WZR YDOXHV 7KH STAT DQQXQFLDWRU LQGLFDWHV WKDW WZR YDOXHV KDYH EHHQ UHWXUQHG 3UHVV WR VHH WKH KLGGHQ YDOXH

Bi to Display

$ULWKPHWLF PHDQ 0HDQ DYHUDJH RI WKH DYHUDJH RI WKH [YDOXHV \YDOXHV LI \RX HQWHUHG \GDWD 0HDQ RI WKH [YDOXHV ZHLJKWHG E\ WKH \YDOXHV 6DPSOH VWDQGDUG GHYLDWLRQ RI WKH [ YDOXHV 6DPSOH VWDQGDUG GHYLDWLRQ RI WKH \YDOXHV LI \RX HQWHUHG \GDWD
.book Page 88 Wednesday, June 21, 2000 5:19 PM
3RSXODWLRQ VWDQGDUG GHYLDWLRQ RI WKH \YDOXHV LI \RX HQWHUHG \GDWD

BQ [YDOXH BR B R

\YDOXH
3RSXODWLRQ VWDQGDUG GHYLDWLRQ RI WKH [ YDOXHV
(VWLPDWH RI [ IRU D JLYHQ &RUUHODWLRQ FRHIILFLHQW YDOXH RI \ (VWLPDWH RI \ IRU D JLYHQ 6ORSH P RI FDOFXODWHG YDOXH RI [ OLQH \LQWHUFHSW E RI WKH FDOFXODWHG OLQH 6ORSH P RI WKH FDOFXODWHG OLQH
UurhyrhqhqqrvhvhruhurqhhvhhyvtshyhtrpyrrrsqhhUur yhvhqhqqrvhvhruhurqhhpvrurrvryhv
Uurpryhvprssvpvrvhirvurhtr utu uhrhrupyryurqhhsvur phypyhrqyvr6hyrs vqvphrhrsrpvvrpryhvhq vqvphrhrsrprthvr pryhv6hyrpyrrvqvphruryvrvhsv

AC AY Aa A Ag Af

1XPEHU RI GDWD SRLQWV HQWHUHG 6XP RI WKH [YDOXHV 6XP RI WKH \YDOXHV 6XP RI WKH VTXDUHV RI WKH [YDOXHV 6XP RI WKH VTXDUHV RI WKH \YDOXHV 6XP RI WKH SURGXFWV RI WKH [ DQG \YDOXHV
Mean, Standard Deviations, and Summation Statistics

#B #@P "#% &@ G

180.00 250.00

'LYLGHV QRPLQDO UDWH E\ 5DLVHV H WR SRZHU &DOFXODWHV DQQXDO HIIHFWLYH UDWH 6WRUHV HIIHFWLYH UDWH 6HWV SD\PHQWV SHU \HDU &DOFXODWHV DQQXDO QRPLQDO UDWH IRU D PRQWKO\ SD\PHQW SHULRG
DQQXQFLDWRU LV GLVSOD\HG 6WRUHV QXPEHU RI PRQWKV 6WRUHV UHJXODU SD\PHQW 6WRUHV FXUUHQW EDODQFH DV D QHJDWLYH YDOXH OLNH DQ LQLWLDO LQYHVWPHQW &DOFXODWHV DFFRXQW EDODQFH DIWHU \HDUV RI SD\PHQWV ZLWK LQWHUHVW FRPSRXQGHG FRQWLQXRXVO\

4,572.80

297,640.27
Yield of a Discounted (or Premium) Mortgage
7KH DQQXDO \LHOG RI D PRUWJDJH ERXJKW DW D GLVFRXQW RU SUHPLXP FDQ EH FDOFXODWHG JLYHQ WKH RULJLQDO PRUWJDJH DPRXQW 39 LQWHUHVW UDWH ,<5 SHULRGLF SD\PHQW 307 EDOORRQ SD\PHQW DPRXQW )9 DQG WKH SULFH SDLG IRU WKH PRUWJDJH QHZ 39 5HPHPEHU WKH FDVK IORZ VLJQ FRQYHQWLRQ PRQH\ SDLG RXW LV QHJDWLYH PRQH\ UHFHLYHG LV SRVLWLYH
8: Additional Examples 99
10B.book Page 100 Wednesday, June 21, 2000 5:19 PM
Example. $Q LQYHVWRU ZLVKHV WR SXUFKDVH D PRUWJDJH WDNHQ
RXW DW IRU \HDUV 6LQFH WKH PRUWJDJH ZDV LVVXHG PRQWKO\ SD\PHQWV KDYH EHHQ PDGH 7KH ORDQ LV WR EH SDLG LQ IXOO D EDOORRQ SD\PHQW DW WKH HQG RI LWV ILIWK \HDU :KDW LV WKH \LHOG WR WKH SXUFKDVHU LI WKH SULFH RI WKH PRUWJDJH LV "
Step 1. &DOFXODWH 307 0DNH VXUH )9

12.00 9.00 240.00

DQQXQFLDWRU LV GLVSOD\HG

B 'F B @ G P

100,000.00 0.00 899.73
6HWV SD\PHQWV SHU \HDU 6WRUHV LQWHUHVW UDWH 6WRUHV QXPEHU RI PRQWKV 6WRUHV RULJLQDO DPRXQW RI PRUWJDJH (QWHUV DPRXQW OHIW WR SD\ DIWHU \HDUV &DOFXODWHV UHJXODU SD\PHQW
Step 2. (QWHU WKH QHZ YDOXH IRU 1 LQGLFDWLQJ ZKHQ WKH EDOORRQ RFFXUV WKHQ ILQG )9 WKH DPRXQW RI WKH EDOORRQ

899.73

60.00 88,706.74
5RXQGV SD\PHQW WR WZR GHFLPDO SODFHV IRU DFFXUDF\ 6WRUHV QXPEHU RI SD\PHQWV XQWLO EDOORRQ &DOFXODWHV EDOORRQ SD\PHQW DGG WR ILQDO SD\PHQW
100 8: Additional Examples
10B.book Page 101 Wednesday, June 21, 2000 5:19 PM
Step 3. (QWHU DFWXDO FXUUHQW YDOXHV IRU 1 DQG 39 WKHQ ILQG WKH QHZ
,<5 IRU WKH GLVFRXQWHG PRUWJDJH ZLWK EDOORRQ

(" %'@ F

8: Additional Examples 107
.book Page 108 Wednesday, June 21, 2000 5:19 PM

Savings

Saving for College Costs
6XSSRVH \RX VWDUW VDYLQJ QRZ WR DFFRPPRGDWH D IXWXUH VHULHV RI FDVK RXWIORZV $Q H[DPSOH RI WKLV LV VDYLQJ PRQH\ IRU FROOHJH 7R GHWHUPLQH KRZ PXFK \RX QHHG WR VDYH HDFK SHULRG \RX PXVW NQRZ ZKHQ \RXOO QHHG WKH PRQH\ KRZ PXFK \RXOO QHHG DQG DW ZKDW LQWHUHVW UDWH \RX FDQ LQYHVW \RXU GHSRVLWV
Example. <RXU ROGHVW GDXJKWHU ZLOO DWWHQG FROOHJH LQ \HDUV DQG \RX
DUH VWDUWLQJ D IXQG IRU KHU HGXFDWLRQ 6KH ZLOO QHHG DW WKH EHJLQQLQJ RI HDFK \HDU IRU IRXU \HDUV 7KH IXQG HDUQV DQQXDO LQWHUHVW FRPSRXQGHG PRQWKO\ DQG \RX SODQ WR PDNH PRQWKO\ GHSRVLWV VWDUWLQJ DW WKH HQG RI WKH FXUUHQW PRQWK 7KH GHSRVLWV FHDVH ZKHQ VKH EHJLQV FROOHJH +RZ PXFK GR \RX QHHG WR GHSRVLW HDFK PRQWK" 7KLV SUREOHP LV VROYHG LQ WZR VWHSV )LUVW FDOFXODWH WKH DPRXQW \RXOO QHHG ZKHQ VKH VWDUWV FROOHJH 6WDUW ZLWK DQ LQWHUHVW UDWH FRQYHUVLRQ EHFDXVH RI WKH PRQWKO\ FRPSRXQGLQJ

$15,000 I/YR = 9%

Year 1

Year 2

Year 3

Year 4

'B B B
DQQXQFLDWRU LV QRW GLVSOD\HG 6WRUHV DQQXDO QRPLQDO UDWH 6WRUHV QXPEHU RI FRPSRXQGLQJ SHULRGV XVHG ZLWK WKLV QRPLQDO UDWH &DOFXODWHV DQQXDO HIIHFWLYH UDWH
108 8: Additional Examples
10B.book Page 109 Wednesday, June 21, 2000 5:19 PM
:KHQ FRPSRXQGLQJ RFFXUV RQO\ RQFH SHU \HDU WKH HIIHFWLYH UDWH DQG WKH QRPLQDO UDWH DUH WKH VDPH

B #P " G

1.00 15,000.00 4.00 0.00
6WRUHV HIIHFWLYH UDWH DV DQQXDO UDWH
DQQXQFLDWRU LV QRW GLVSOD\HG 6HWV SD\PHQW SHU \HDU 6WRUHV DQQXDO ZLWKGUDZDO 6WRUHV QXPEHU RI ZLWKGUDZDOV 6WRUHV EDODQFH DW HQG RI IRXU \HDUV &DOFXODWHV DPRXQW UHTXLUHG ZKHQ \RXU GDXJKWHU VWDUWV FROOHJH

52,713.28

7KHQ XVH WKDW 39 DV WKH )9 RQ WKH IROORZLQJ FDVK IORZ GLDJUDP DQG FDOFXODWH WKH 307

I/YR = 9% 1 PMT = ? 4

FV from previous calculation. 144
8: Additional Examples 109
book Page 110 Wednesday, June 21, 2000 5:19 PM

@G B "" 'F P

0.00 12.00 144.00 9.00 204.54
DQQXQFLDWRU LV GLVSOD\HG 6WRUHV DPRXQW \RX QHHG 6WRUHV DPRXQW \RX DUH VWDUWLQJ ZLWK 6HWV SD\PHQWV SHU \HDU 6WRUHV QXPEHU RI GHSRVLWV 6WRUHV LQWHUHVW UDWH &DOFXODWHV PRQWKO\ GHSRVLW UHTXLUHG
Gains That Go Untaxed Until Withdrawal
<RX FDQ XVH WKH 790 DSSOLFDWLRQ WR FDOFXODWH WKH IXWXUH YDOXH RI D WD[ IUHH RU WD[GHIHUUHG DFFRXQW &XUUHQW WD[ ODZV DQG \RXU LQFRPH GHWHUPLQH ZKHWKHU ERWK LQWHUHVW DQG SULQFLSDO DUH WD[IUHH <RX FDQ VROYH IRU HLWKHU FDVH 7KH SXUFKDVLQJ SRZHU RI WKDW IXWXUH YDOXH GHSHQGV XSRQ WKH LQIODWLRQ UDWH DQG WKH GXUDWLRQ RI WKH DFFRXQW

Resetting the calculator

7XUQ WKH FDOFXODWRU RYHU DQG UHPRYH WKH EDWWHU\ FRYHU ,QVHUW WKH HQG RI D SDSHU FOLS LQWR WKH VPDOO URXQG KROH ORFDWHG EHWZHHQ WKH EDWWHULHV ,QVHUW WKH FOLS DV IDU DV LW ZLOO JR +ROG IRU RQH VHFRQG DQG WKHQ WKHQ UHPRYH WKH FOLS 3UHVV ,I WKH FDOFXODWRU LV VWLOO QRW UHVSRQGLQJ HUDVH WKH PHPRU\ VHH EHORZ DQG UHSHDW VWHSV WR DERYH RQH PRUH WLPH
Erasing the calculators memory
+ROG GRZQ WKH NH\ +ROG GRZQ WKH DQG WKHQ WKH NH\ 5HOHDVH DOO WKUHH NH\V 0HPRU\ LV FOHDUHG DQG All Clear VKRXOG EH GLVSOD\HG
n The calculator doesnt respond to keystrokes (nothing
happens when you press the keys):
5HVHW WKH FDOFXODWRU VHH DERYH DQG LI QHFHVVDU\ HUDVH WKH PHPRU\ VHH DERYH 7KH All Clear PHVVDJH VKRXOG QRZ EH GLVSOD\HG ,I WKLV LV QRW WKH FDVH WKH FDOFXODWRU UHTXLUHV D VHUYLFH
A: Assistance, Batteries, and Service 121
10B.book Page 122 Wednesday, June 21, 2000 5:19 PM
n The calculator responds to keystrokes but you suspect that it

is malfunctioning:

,W LV OLNHO\ WKDW \RXYH PDGH D PLVWDNH LQ RSHUDWLQJ WKH FDOFXODWRU 7U\ UHUHDGLQJ SRUWLRQV RI WKH PDQXDO DQG FKHFN $QVZHUV WR &RPPRQ 4XHVWLRQV RQ SDJH &RQWDFW WKH &DOFXODWRU 6XSSRUW GHSDUWPHQW 7KH DGGUHVV DQG SKRQH QXPEHU DUH OLVWHG RQ WKH LQVLGH EDFN FRYHU
Limited One-Year Warranty

What Is Covered

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

What Is Not Covered

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

A: Assistance, Batteries, and Service 125
Disposal of Waste Equipment by Users in Private Household in the European Union
This symbol on the product or on its packaging indicates that this product must not be disposed of with your other household waste. Instead, it is your responsibility to dispose of your waste equipment by handing it over to a designated collection point for the recycling of waste electrical and electronic equipment. The separate collection and recycling of your waste equipment at the time of disposal will help to conserve natural resources and ensure that it is recycled in a manner that protects human health and the environment. For more information about where you can drop off your waste equipment for recycling, please contact your local city office, your household waste disposal service or the shop where you purchased the product.
End-user terms and conditions
HP 10B II Calculator Warranty period: 12 months 1. HP warrants to you, the end-user customer, that HP hardware, accessories and supplies will be free from defects in materials and workmanship after the date of purchase, for the period specified above. If HP receives notice of such defects during the warranty period, HP will, at its option, either repair or replace products which prove to be defective. Replacement products may be either new or like-new. 2. HP warrants to you that HP software will not fail to execute its programming instructions after the date of purchase, for the period specified above, due to defects in material and workmanship when properly installed and used. If HP receives notice of such defects
126 A: Assistance, Batteries, and Service
during the warranty period, HP will replace software media which does not execute its programming instructions due to such defects. 3. HP does not warrant that the operation of HP products will be uninterrupted or error free. If HP is unable, within a reasonable time, to repair or replace any product to a condition as warranted, you will be entitled to a refund of the purchase price upon prompt return of the product. 4. HP products may contain remanufactured parts equivalent to new in performance or may have been subject to incidental use. 5. Warranty does not apply to defects resulting from (a) improper or inadequate maintenance or calibration, (b) software, interfacing, parts or supplies not supplied by HP, (c) unauthorized modification or misuse, (d) operation outside of the published environmental specifications for the product, or (e) improper site preparation or maintenance. 6. Hewlett-Packard makes no other express warranty or condition whether written or oral. To the extent allowed by local law, any implied warranty or condition of merchantability, satisfactory quality, or fitness for a particular purpose is limited to the duration of the express warranty set forth above. Some countries, states or provinces do not allow limitations on the duration of an implied warranty, so the above limitation or exclusion might not apply to you. This warranty gives you specific legal rights and you might also have other rights that vary from country to country, state to state, or province to province. 7. To the extent allowed by local law, the remedies in this warranty statement are your sole and exclusive remedies. Except as indicated above, in no event will Hewlett-Packard or its suppliers be liable for loss of data or for direct, special, incidental, consequential (including lost profit or data), or other damage, whether based in contract, tort, or otherwise. Some countries, States or provinces do not allow the exclusion or limitation of incidental or consequential damages, so the above limitation or exclusion may not apply to you. 8. For consumer transactions in Australia and New Zealand: the warranty terms contained in this statement, except to the extent lawfully permitted, do not exclude, restrict or modify and are in addition to the mandatory statutory rights applicable to the sale of this product to you.

A: Assistance, Batteries, and Service 127
10B.book Page 128 Wednesday, June 21, 2000 5:19 PM
appendix_B.fm Page 129 Thursday, June 22, 2000 12:33 PM

More About Calculations

IRR/YR Calculations
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
Possible Outcomes of Calculating IRR/YR
7KHVH DUH WKH SRVVLEOH RXWFRPHV RI DQ ,55<5 FDOFXODWLRQ n Case 1. 7KH FDOFXODWRU GLVSOD\V D SRVLWLYH DQVZHU 7KLV LV WKH RQO\ SRVLWLYH DQVZHU +RZHYHU RQH RU PRUH QHJDWLYH DQVZHUV PD\ H[LVW n Case 2. 7KH FDOFXODWRU ILQGV D QHJDWLYH DQVZHU EXW D VLQJOH SRVLWLYH DQVZHU DOVR H[LVWV ,W GLVSOD\V Pos Irr Also 7R VHH WKH QHJDWLYH DQVZHU SUHVV WR FOHDU WKH PHVVDJH 7R VHDUFK IRU WKH SRVLWLYH DQVZHU \RX PXVW LQSXW D JXHVV 5HIHU WR (QWHULQJ D *XHVV IRU ,55<5 EHORZ 7KHUH PLJKW DOVR EH DGGLWLRQDO QHJDWLYH DQVZHUV n Case 3. 7KH FDOFXODWRU GLVSOD\V D QHJDWLYH DQVZHU DQG QR PHVVDJH 7KLV LV WKH RQO\ DQVZHU n Case 4. 7KH FDOFXODWRU GLVSOD\V WKH PHVVDJH Not Found 7KLV LQGLFDWHV WKDW WKH FDOFXODWLRQ LV YHU\ FRPSOH[ ,W PLJKW LQYROYH PRUH WKDQ RQH SRVLWLYH RU QHJDWLYH DQVZHU RU WKHUH PD\ EH QR VROXWLRQ 7R FRQWLQXH WKH FDOFXODWLRQ \RX PXVW VWRUH D JXHVV VHH EHORZ
B: More About Calculations 129
appendix_B.fm Page 130 Thursday, June 22, 2000 12:33 PM
n Case 5. 7KH FDOFXODWRU GLVSOD\V No Solution 7KHUH LV QR DQVZHU
7KLV VLWXDWLRQ PLJKW EH WKH UHVXOW RI DQ HUURU VXFK DV D PLVWDNH LQ NH\LQJ LQ WKH FDVK IORZV $ FRPPRQ PLVWDNH WKDW UHVXOWV LQ WKLV PHVVDJH LV SXWWLQJ WKH ZURQJ VLJQ RQ D FDVK IORZ $ YDOLG FDVKIORZ VHULHV IRU DQ ,55<5 FDOFXODWLRQ PXVW KDYH DW OHDVW RQH SRVLWLYH DQG RQH QHJDWLYH FDVK IORZ

B: More About Calculations 131
.fm Page 132 Thursday, June 22, 2000 12:33 PM

Equations

35& &267 0$5 = 35& 35& &= &267
Time Value of Money (TVM)
3D\PHQW 0RGH )DFWRU 6 IRU (QG PRGH IRU %HJLQ PRGH I/YR L = P/YR
L L S 0 = PV + + --------------- PMT ------------------------------------ L ------

N L + FV 1 + -------

,L DFFXPXODWHG LQWHUHVW DFFXPXODWHG SULQFLSDO SHULRGLF LQWHUHVW UDWH
%$/ LV LQLWLDOO\ 39 URXQGHG WR WKH FXUUHQW GLVSOD\ VHWWLQJ 307 LV LQLWLDOO\ 307 URXQGHG WR WKH FXUUHQW GLVSOD\ VHWWLQJ , <5 i = ----------------------------3 <5
132 B: More About Calculations
appendix_B.fm Page 133 Thursday, June 22, 2000 12:33 PM
)RU HDFK SD\PHQW DPRUWL]HG ,17
,%$/QHZ ,17QHZ 351QHZ %$/ L ,17
LV URXQGHG WR WKH FXUUHQW GLVSOD\ VHWWLQJ ,17 IRU SHULRG LQ %HJLQ PRGH ,17

ZLWK VLJQ RI ,17

%$/ROG 351 ,17ROG ,17 351ROG 351
P YR 120 = 1 + ----------------------------- 3 <5
B: More About Calculations 133
.fm Page 134 Thursday, June 22, 2000 12:33 PM

Cash-Flow Calculations

L M &)M QM N 1M SHULRGLF LQWHUHVW UDWH WKH JURXS QXPEHU RI WKH FDVK IORZ DPRXQW RI WKH FDVK IORZ IRU JURXS M QXPEHU RI WLPHV WKH FDVK IORZ RFFXUV IRU JURXS M WKH JURXS QXPEHU RI WKH ODVW JURXS RI FDVK IORZV

1l<j

WRWDO QXPEHU RI FDVK IORZV SULRU WR JURXS M
j L 1 1 + ------- L - nj NPV = CF 0 + CF j ------------------------------------ 1 + ------- L ------j=1

:KHQ 139 UHWXUQ

WKH VROXWLRQ IRU L
LV WKH SHULRGLF LQWHUQDO UDWH RI
134 B: More About Calculations
.fm Page 135 Thursday, June 22, 2000 12:33 PM

Statistics

[\ [ y [ = ------- y = ------ [ Z = --------Q Q \ ( [) [ -------------Q -----------------------------Q ( y) y ------------Q ----------------------------Q
( [) [ -------------Q ------------------------------ y = Q
( y) y ------------Q ----------------------------Q
x y xy -------------n r = --------------------------------------------------------------------------( x) ( y) x 2 ------------- y 2 ------------- n n x y xy -------------n m = ------------------------------2 ( x) 2 x ------------n yb b = y mx x = ---------- y = mx + b m

New! HP 39gs Graphing Calculator Classroom Kit
Article Next Announcing the HP 39gs Graphing Calculator Classroom Kit for Middle Grades. Easy to use and powerful, HP Calculators for grades 6-8 are designed to perform above expectations for math and science students on all levels. These reliable calculators are equipped with easy-to-use problem solving tools, flexible connectivity & customizing options, plus award-winning HP support. Our HP 39gs Graphing Calculator Classroom Kit is a unique solution that provides teachers with the tools and training to engage and motivate middle school students in mathematics. We believe that successful integration of technology into the mathematics classroom is enhanced when it is undertaken as a group or team effort, a belief that is reinforced by independent studies, so we designed the Classroom Kit to help middle school teachers establish a mathematics leadership team on campus. The Classroom Kit features a three-day professional development workshop and supporting curricula to help ensure the successful integration of the HP 39gs Graphing Calculator into math classes. The HP 39gs Graphing Calculator is the heart of the HP Graphing Calculator Classroom Kit, and this all-inone kit includes calculators, carrying case, classroom-ready teaching materials, and a 3-day teacher workshop. The HP 39gs Graphing Calculator was designed by teachers for classroom use and to align with NCTM (National Council of Teachers of Mathematics) Standards. Importantly, and uniquely, the HP 39gs has dedicated keys for symbolic, graphic and numeric views. The Classroom Kit allows teachers to save time in their digital classroom. Using our kit and included calculator emulator software, the teacher can project a virtual calculator during the lesson so students can follow along easily. Plus, teachers can use HP Aplets, either that they've written or downloaded from numerous free resources online. Teachers can set up powerful examples beforehand, saved with common names to ease retention, and then send them to students wirelessly! By utilizing this powerful feature, and given the ability to collect or create these aplets beforehand, teachers save precious class time, and are secure in the knowledge that students will quickly get on the same page, ready to learn. See the Graphing Calculator Classroom Kit for more information.
Feature Calculator of the Month: HP 10bII Financial Calculator
Previous Article Next The HP 10bII Financial Calculator is the smart choice for business, finance and accounting needs, for professionals and students alike. With over 100 time-saving business functions you can calculate loan payments, interest rates and conversions, standard deviation, TVM, cash flows and more. Algebraic notation makes it easy to learn and use. With so many features, at a great price, the HP 10bII adds up to a wise investment. The HP 10bII is permitted for use on the CFP Certification Exam [1]. Fun facts about the HP 10bII Financial Calculator: 2
Born: Introduced December 1st, 2001 Replaced the HP 10b, which was introduced January 3rd, 1989. HP also introduced the HP 20s and HP 21s on the same date. The HP 10BII has a memory called the M register that is usable directly from the basic keyboard. There are three keys associated with the M register. The M key will store the number presently displayed into the M register. Note that this will overwrite any value previously stored in the M register. The RM key will recall the number presently in the M register to the display where it can be used. The M+ key will add the number presently in the display to the number already in the M register. If pressed repeatedly, it will add the number in the display to the number in the M register each time. This is often used to keep up with a running total. For example, to solve ( 2 + 3 ) ( ) using the M register, press 5 - 2 = M 2 + 3 RM = To quickly find the total of 12, 44, 17 and 36, press 12 M 44 M+ 17 M+ 36 M+ RM It is HP's lowest cost financial model, well suited for finance students who do not need the advanced features of other HP models, but who want HP reliability. It provides a simple interface, all the important operations needed in common financial calculations but avoids the complications of programming, scientific functions and RPN.

Click here for more information about the HP 10bII Financial Calculator. [1] CFP is a registered trademark of the Certified Financial Planner Board of Standards, Inc.

RPN Tip #4

Previous Article Next Introduction (by Wlodek Mier-Jedrzejowicz, Richard J. Nelson, & Jake Schwartz) Hewlett-Packard has always made calculators with a user interface, the method used to solve problems, most suitable for the general class of problems to be solved. The four basic types of calculator user interface are: Arithmetic, ATH; Algebraic, ALG; Reverse Polish Notation, RPN; and Command Line Interface, CLI. RPN is, over all, the most effective and efficient way to solve a very large class of problems, and HP is well known for its RPN machines. RPN is a Really Productive Notation RPN is different, and to most people it is unfamiliar. It is easy to learn and it is the most efficient way to solve a large class of problems because you treat each part of the problem in exactly the same way. The rules, compared to Algebraic, ALG, are fewer and simpler. The purpose of the Newsletter series of RPN Tips is to describe RPN techniques so new and experienced users may improve their calculator skill. The User Interface The two most popular general purpose user interfaces for midrange calculators are RPN and ALG. The user interface is the means by which the calculator user presses keys to solve problems. The user interface is the "rules" that the user (and the machine) must follow in order to give the machine the numbers (data) and especially the operators +, -, , & so the machine solves the problem correctly. Suppose you had the problem of adding one and two and multiplying the result by three. The problem would be mathematically written as (1 + 2) 3 = 9. In order for the machine to know that the problem is not 1 + (2 3) = 7, both the user and the calculator must know the rules of what the problem is, and how it is to be solved. Using parentheses and inserting the operators between the data is the problem form taught 3
in school and most people are familiar with it. The use of parentheses (and brackets and curly braces) is usually taught in a beginning algebra class. The problem, 1+23=, and pressing the keys as shown, is often used to identify the user interface. If the answer is 7 it is ALG. If the answer is 9 it is ATH. It is interesting to note that without parentheses you cant solve the problem using RPN. This is another example of the rules. The calculator, however, cant think like we can so it must be presented with the data and the operators in a very precise order. This precise order is where the rules are very important. The simple example problem above requires the machine to understand the proper sequence of performing the operators. The parentheses group the various portions of the problem and they show how the portions are related. In addition to knowing the correct order of the data and the operators, the machine must also be able to store intermediate results so that they may be correctly integrated with the current calculation. In this case the addition of one and two is performed first and the result is then multiplied by three. This method of solving the problem, e.g. (1 + 2) 3, and representing and solving problems of this type is called Algebraic, ALG, notation. From a mathematical logic perspective this is called infix notation. In order to present the problem to the calculator the three numbers must be entered, the two operators (+ and ) must be entered, and the order in which calculation is to be made must be conveyed. In addition to this, the machine must know that a previously calculated result is to be used with another number and operator. The parentheses group the various operations and the general rule is to solve the problem from the inside out by performing the operations in the inner most parentheses first. This is true no matter what calculator you are using. There is another notation called postfix notation. Using postfix notation the problem above may be represented as + 3. Solving the problem in postfix notation is what the machine does internally regardless if the user interface is ALG or RPN. Infix is 1 + 2 3; Postfix is + 3. RPN uses postfix notation as the logical interface. RPN does not require an equal key. RPN does not use parentheses. The machine must, however, know when the user presses the 1 and the 2 key that there are two numbers of interest, 1 and 2, and not the single number 12. In other words any number keyed into the calculator must be terminated (completed). RPN machines use the ENTER key to terminate numeric inputs. In terms of the keys pressed to solve our example problem the notation would be: 1 ENTER 2 + 3. The desired answer, 9, appears in the display immediately when the multiply key, , is pressed. The intermediate result, of 1 + 2, also appears when the add operator is pressed and 3 appears in the display. Equal (or a key) is not used for the RPN user interface. The ability to integrate parts of a calculator problem by storing intermediate results is a very important aspect of calculator problem solving rules that must be learned by the user. For the ALG interface the use of parentheses and the hierarchy of operations are the most important rules. The use of parentheses is the primary means of storing intermediate results. RPN does not have a list of hierarchy rules. ALG may six or more, in some cases as many as 11, depending on how many operators the machine has. RPN uses an automatic memory stack, see below, to store intermediate results. Neither ALG nor RPN is able to solve all problems exactly as they are written on paper. A more accurate form of using mixed data and operators utilizes a user interface that is predominantly found on the higher end calculators, the Graphing Calculators, and it is called Command Line Interface, CLI. All graphics calculators (and computers) use CLI. Arithmetic, ATH, is the user interface found on adding machines. It is the simplest and it doesnt have any rules except that it does the operators such as +, -, , & when the keys are pressed.

The RPN Stack The ENTER key, , and a group of four data storage registers, called an automatic RPN stack, is what makes RPN very powerful and simple to use. The stack may be visualized as a stack of shoe boxes as shown in figure 1. Each stack shoe box, register, is vertically related to each other. The contents of the X register are shown in the display. The original RPN machine, the HP-35A, utilized a single line display. More recent machines that have a multiline display may also show additional stack registers in the order of Y to T. Register Name T Z Y X Stack Register
Fig. 1 Automatic four high stack. Basic RPN Aside from the general rule we learned in school that is used for all moderately complex problem solving perform the operations by solving the parentheses from the inside out the basic RPN rule is: if an operation is indicated, do it. The example given above: (1+2) x 3 =? is 1 ENTER 2 + 3. Pressing the ENTER key performs three operations. (1) Pressing ENTER terminates the number so the calculator knows that the complete number has been keyed into the calculator. (2) Pressing ENTER also makes a copy of the X register data by also storing it on the stack in the Y register. The contents of the registers above are pushed up i.e. the contents of the Y resister are copied into the Z register. The contents of the Z register are copied into the T register and the contents of the top register, T, are lost. (3) Pressing ENTER also prepares the machine for accepting additional data or a keyboard operation. RPN Stack Operators (see Table 1) Using and controlling the RPN stack utilizes the five RPN operators. RPN utilizes postfix notation and just important is that RPN implies the automatic stack as shown in figure one above. Postfix notation, the use of a stack, and five stack operators is what defines RPN as a user interface. The five basic RPN Stack operators are: 1. ENTER,, is the most important RPN operator. See the description under Basic RPN above. ENTER is never a shifted operator on the RPN keyboard. 2. is the second most important RPN operator. exchanges the contents of the X and Y registers. is never a shifted operator on the RPN keyboard.

3. Roll down, R : The third most important RPN operator. Roll down rotates the (shoe box) stack downward. The contents of X are copied into T, T into Z, X into Y, and Y into X. Roll down is a primary operator on all RPN models except the HP-34C, 37E, and 38E/C. 4. LAST X is the very important error correction operator. LAST X recalls the value of the X Register prior to the most recent operation performed. LAST X is a shifted operator on all RPN models except the six early models that do not have it the HP-35A, 21, 22, 37E, 70A, and 80A. 5. Roll up, R , rotates the stack upward. The contents of X are copied into Y, Y into Z, Z into T, and T into X. R is a shifted operator and it is not found on 24 of the 43 RPN models (58%). The roll down, R , operator allows a quick verification of the stack contents by pressing the primary stack roll key four times in succession. 5
RPN Stack Diagrams In the days before RPN calculators were programmable and the basic RPN calculator was the optimum choice for complex problem solving, it was useful to make a stack diagram of the keystrokes used to repeatedly solve very complex problems. This was especially useful for iterative solutions and engineers carried 3 x 5 cards in their shirt pockets with these keystroke solutions written on them. To illustrate a stack diagram lets show the stack to make a calculation involving three values, each used twice. The best approach is one that utilizes the stack in such a way that the values are keyed only once to avoid errors. An example problem is shown below. Where: A =4 B =3 C = -2
The stack registers are identified at the left. The tilde symbol indicates any (dont care) value. The press row indicates the inputs that are keyed to solve the problem. These are the keystrokes that are recorded to solve the problem. The stack diagram is primarily used for illustration and analysis. The values involved in the example are shown in both symbolic and numerical form (where practical) so the user may solve the problem by pressing the inputs as indicated and see the stack values. The solution steps are numbered above the stack diagram. Step 9 T Z Y ~ ~ ~ ~ ~ B=3 B=3 ~ B=3 B=3 B=18 T Z Y X press C=-2 C=-2 B=3 (A+B)(A-C)=C=-2 C=-2 B=3 R 42 C=-2 B=3 C=-42 C=-2 B-C=42 B-C=5 C=-2 C=-42 B-C=5 R C=-2 C=-3 B=3 B=3 B=3 A=4 A=4 B=3 B=3 B=3 A+B=7 + 5 B=3 B=3 A+B=7 A=4 LASTX B=3 A+B=7 A=4 C=-2 C=-7 B=3 B=3 A+B=7 A-C=B=3 A+B=7 A-C=6 C=-2 LASTX 8 C=-2 B=3 A+B=7 A-C=6 R

B=3 B=3 B=3 B=3

X B=3 press B=3 Step 17
The reader is encouraged to solve the problem for practice. The stack diagram is useful for the following reasons. (1) A stack diagram illustrates how the stack works. (2) Solving complex problems repeatedly is faster if an optimum keystroke solution is known. What is the value of the above expression if: A = 1.41421357348, B = -1.73205091868, and 6
C = 2.23606808860? Solve the problem without looking at the stack diagram. See the correct answer at the end. Did you get this answer on your first try without using the stack diagram? (3) A stack diagram shows the values needed for future use in order to determine the best solution sequence especially in preparation for reason 4. (4) The optimum solution for a program that may solve the problem multiple times and program solution speed is an issue. The stack diagram is a powerful optimization tool. RPN Stack Operator Table Table 1 below shows all RPN models to date (April 2008) grouped into their series in approximate introduction order. It also shows the classic RPN stack operators with a weight value assigned to each of them. The weight value is used to compare the various models to each other for their RPNness in terms of the RPN keyboard and the included operators. The colors provide a quick overview of the model and the operator use on that model, i.e. if it is used, and if it is a shifted operator. The ideal RPN machine would have an index of 9. Obviously there are many keyboard tradeoffs that must be made for the intended class of problem the model has to solve. Table 1 HP RPN Calculator Stack Operators And Keystroke Index y = full value, s or ALPHA = half value, n = no value 1 Ideal Index = 9 Ks LASTx Index n 7 s 7.5 s 7.5 * = on top row when s 6 undefined * = on top row when s 6 undefined n 7 n 7 * = key marked SAVE n 7 s 7.5 s 7.5 n 7 s 7.5 s 7.5 s 7.5 s 7.5 s 7 n 6 s 7.5 s 8 s 7.5 s 8 s 7.5 s 8 s 8 s 7.5

Weight

3 ENTER y y y y y y y* y y y y y y y y y y y y y y y y y y
2 R y y y s* s* y y y y y y y y y y s s s y y y y y y y
2 X<>Y y y y s* s* y y y y y y y y y y y y y y y y y y y y
1 R n n n s s n n n n n n n n n n s n n ALPHA n s n s s n
35A 45A 55A Classic Series 65A 67A 70A 80A 29C 31E 32E 33E/C 34C 37E 38E/C 41C/CV/CX 10C 11C 12C 15C 16C 12Cp

Voyager Series

Spice/Spike Series

Woodstoc k

Pioneer Series

95C 97 19C 32S 32SII 42S

y y y y y y y y y* y* y* y y* y*
y y y y y y y y y y y y y y
y y y s n n s ALPHA n n s s n n
s s s s s s s s s s s s s s

8.5 8.5 8.7.5 7.8

Topcat Series
17BII 19BII 33S 35S 17BII+ Gold 03 17BII+ Silver 07 Total: 43 machines

* = SMALL key marked 7.5 INPUT 7.5 * = key marked INPUT 8 * = SMALL ENTER key 8 7.5 * = key marked INPUT 7.5 * = key marked INPUT Table by Jake Schwartz 1/08
Color legend: Green indicates primary operator. Red indicates stack operator not on the machine. Yellow indicates shifted operator. RPN And ALG All of the four basic user interfaces mentioned above (ATH, ALG, RPN, and CLI) have been used in HP calculators e.g. the 10 model number has been used five times since 1972. One was ATH, two were ALG, and two were RPN. All graphing machines are CLI. Each interface has its advantages and disadvantages. Over all, RPN is the most effective (requires less thinking, which is hard work, is faster, and usually requires fewer keystrokes) for a very large class of problems. ALG often seems easier because your problem is often expressed in an algebraic form. If you understand RPN and you dont use your calculator frequently you will always be able to remember the rules of how to solve problems because the rules are very few and they are obvious. There are two primary reasons that RPN is more effective compared to ALG. The first is the basic structure of RPN. The second reason is that you are able to see the problem unfold as work your way through it. All too often ALG keeps parts of the problem hidden until the final EQUAL. RPN users do not need to clear anything (the stack, or the X register) before starting a problem. In fact, the vast majority of RPN machines do not have a clear stack operator. You just start keying in the data. ALG users, however, know that if they dont start with a clean slate so to speak, they will get wrong answers and may have to do the problem all over again which usually means doing it over twice for insurance that they have the correct answer. ALG users soon learn to clear their machines before starting and if you watch an ALG user solve a problem you will see them pressing clear multiple times, just to be sure. In a similar way you will notice an inexperienced RPN user press zero and ENTER three times to clear the stack, just to be sure. The user interface of choice is often the one we first learned. Taking the time to master RPN, however, will save you endless key pressing hours. If, however, you are an HP calculator user it you may select the either ALG or RPN on many models. The HP-42S Owners Manual provides a very nice list of the advantages of RPN.

Advantages of RPN Remember: This method of entering numbers, called Reverse Polish Notation (RPN), is unambiguous and therefore does not need parentheses. It has the following advantages: You never work with more than two numbers at a time. Pressing a function key immediately executes that function so there i s n o n e e d f o r a n = key. Intermediate results appear as they are calculated, so you can check each step as you go. Intermediate results are automatically stored. They reappear as they are needed for the calculationthe last result stored is the first to come back out. You can calculate in the same order as you would with pencil and p a p e r. If you make a mistake during a complicated calculation, you don't have to start over (Correcting mistakes is covered in chapter 2,) Calculations with other types of data (such as complex numbers and matrices) follow the same rules. Calculations in programs follow the same steps as when you execute them manually. Keyboard or Program Solution Many RPN calculators are programmable. A program is simply a memorized list of keys pressed to solve a problem. In addition to the normal operators provided on the keyboard a programmable model provides a means of naming, starting, and stopping, a program. Another feature of a program is a format for inputting the data and running the program to get an answer. Once a program is proved to solve a complex problem (debugged) it may be run whenever the problem needs to be solved. This may be long after you have forgotten how to solve the problem. Another very important aspect of solving your problem using a program is that the program may test values and make decisions based on these tests. This feature is what makes running a program so powerful. Still another advantage of a program is executing the same series of keystrokes multiple times. This is called looping. Looping is an important feature of a programmable calculator. The straightforward logic and structure of RPN makes keystroke programming easy to learn and use. In some instances the program may make many calculations in a loop and speed becomes an important consideration. RPN saves keystrokes and speeds up programs. Conclusion No matter what midrange calculator you use, all problems are not solvable as you write them on paper. It takes a high end graphing calculator with its Command Line Interface to be able to do this. You may find examples where one of the simpler machines will solve the problem correctly, but as you increase the complexity of the problem you will have to know more of the user interface rules. The Arithmetic, ATH, user interface has no built in logic. This leaves ALG and RPN. Of these two, RPN has fewer rules and is always consistent in the process in which you solve problems. Like any well thought out system there are many ways in which a problem may be solved. The HP Solve newsletter series of RPN Tips provides techniques to better understand and improve your RPN skills. Answer to three variable, A,B,C problem: Click here to learn more about RPN.

Lets make music with math!
Previous Article Next It may come as a surprise to some that music and mathematics are intimately connected. Both are consequences of human creativity, both involve intricate patterns, both can be at once beautiful and daunting, and both can be appreciated more deeply through conscientious study. Pleasant harmonious patterns of music reveal themselves in exquisite numerical patterns of mathematics. This activity focuses on some of the connections between mathematics and music. Exercise 1: Start the MUSICMATH aplet on the HP39gs. You are presented with two columns of numbers in a table. The numbers in column C1 are used to designate different musical notes. The numbers in column C2 are the frequencies. The chart below is provided for those with some musical background, and relates the numbers in C1 to the names of different musical notes. The frequencies are repeated here, rounded to the nearest hundredth, for your convenience. Number 13 Note A4 A# B4 C5 C# D5 D# E5 F5 F# G5 G# A5 Frequency (hz) 440 466.16 493.88 523.25 554.37 587.33 622.25 659.26 698.46 739.99 783.99 830.61 880
The 39gs has a built-in speaker, and a command to play any note for any length of time. To play the first note in the table (A4) for 2 seconds, go to the HOME screen and enter the command BEEP 440;2 as shown in the screen below.
Find another student with a 39gs. Have your partner play the last note (A5 or 880 hz) at the same time you play the first note (A4 or 440 hz). Hold the calculators close together, so the notes blend. Try playing two E's together by executing BEEP 660;2 on one calculator, and BEEP 1320;2 on another at the same time. (Notice that 660 is pretty close to the frequency for E5).
The harmonies you just heard results from playing the same note in two different octaves (an octave consists of one complete set of twelve notes). These are the simplest type of harmonies. 1. What relationship do you notice between the frequencies of these harmonious notes (440 with 880; 660 with 1320)? Create a second list of frequencies one octave above those in column C2 by executing the command shown in the screen shot below.
You can play the entire scale by using the column name with the BEEP command. Try it!
Playing octaves on two calculators Then try playing the scale in C2 while your partner plays the scale one octave higher, in C3, as shown above on the right. Try to start the scales simultaneously. Now try playing 440 (A4) and 660 (E5) together. 2. Do they sound harmonious to you? (If the speakers on the 39gs are not adequate to hear the harmonies, try the first web resource, or a piano!). 3. Can you see any connection between the frequencies 440 hz and 660 hz? Legend has it that Pythagoras (the same guy from the Pythagorean Theorem) was one of the first to recognize the harmonious tone that results from playing notes such as these together. 4. Try playing an entire scale where the frequencies are in the ratio 3:2 by multiplying C2 by 1.5 and storing the result in C3. Use two calculators to play the resulting frequencies along with the originals. Then repeat after multiplying C2 by a nastier ratio like 17/13. What do you find? Exercise 2: What's So Special About Harmonious Notes In Exercise 1, you saw (or rather heard!) that certain musical notes sound nice when played together. What attributes do these notes have that produces the pleasing sound? Let's look at ratios. 1. What is the ratio of the frequency of A5 to the frequency of A4? 2. What is the ratio of the frequency of E5 to A4? 11

Let's look for notes whose frequencies are in small integer ratios. The 39gs can help in the hunt! First, we'll round the frequencies we have to integers. Then we'll multiply the rounded frequencies by small integers. Finally, we'll search the table of frequencies for matches. Execute the commands shown in the screen shots:
3. Look around in the table for frequencies that are approximately the same. Then try playing them together and listen for the harmony. Record your results, and compare with other groups. Harmonies occur when the notes are in "nice" integer ratios, such as 2:1 or 3:2. Notes in the ratio 2:1 are called an octave. Those in the ratio 3:2 are called a fifth. Ratios of 4:3 are called a fourth. C and F produce a fourth, since 4 523.25 = 2093 , while 3 698.46 = 2095.38. Pretty close. Try playing notes in the ratio 1:3 (such as A3, 220, and E5, 660) at the same time. Notes in the ratios 1:2, 1:3, 1:4, 1:5, etc are called harmonics. You may encounter the harmonic series in Precalculus or Calculus. It is the sum
+ + + +. , where the ellipsis means the sum goes on forever. 4
Sometimes, three notes played together produce a pleasant sound. Find another partner and play A, C#, and E together. This chord is called a major triad. 4. Can you spot the ratio of the frequencies? Hint: Round the frequency of C# to 550. Exercise 3: A Musical Model Take another look at the scale on the first page. Notice as we go from 440 hz (A4) to the next higher A, 880 hz (A5), we've doubled the frequency. We saw in Exercise 1 that pairs of notes that are in the ratio 1:2 produce a certain pleasant harmony. But how do the notes progress within one scale? What choices could there be? One approach would be to divide the scale so that the arithmetic difference between adjacent notes was a constant. For example, 880 hz 440 hz = 440 hz. Divide this into 12 equal steps. 1. If the scale were made this way, what would the "step" between each note be? 2. Fill in the table for the notes from 440 hz to 880 hz according to this "arithmetic" scale. Does this arithmetic scale match the scale from the table given in Exercise 1. Number 12 Note A4 A# B4 C5 C# D5 D# E5 F5 F# Frequency (hz) 440

G5 G# A5

Since harmonies occur when the ratio of notes are small whole numbers, it makes more sense to arrange the scale so that the ratios of adjacent notes is the same. Let's calculate these ratios using the values in the MUSICMATH aplet. In the 39gs, the first value in column C2 is referred to as C2(1). Calculate the ratio C2(2)/C2(1), as shown.

3. What is the ratio? After you've calculated this ratio, press the up arrow key, highlight the expression C2(2)/C2(1), and press the soft COPY key. Then edit (using the DEL key and inserting replacement digits) to produce the ratio C2(3)/C2(2). Repeat for C2(4)/C2(3). 4. What are C2(3)/C2(2) and C2(4)/C2(3) ? Clearly, the ratios of adjacent notes are the same through the scale. If we call this ratio r, then all we do to get the next note in the scale is to multiply the previous note by r. That is, with A4 = 440 hz, A# is 440r. Store the ratio you calculated into the calculator variable R, as shown. Just press the STO> soft key on the HOME screen right after you calculated one of the ratios, then press ALPHA R.
Now to get the next note above A4, calculate 440 R as shown. Again, use ALPHA R to type the letter R:
The next note is always the previous note multiplied by R. To get B4, we could enter 440 R R. (The COPY feature will help you out!).
This quickly becomes tedious. Mathematicians use exponents as a shorthand for repeated multiplications like this. For example 440 R R = 440 R , 440 R R R = 440 R , and so on. On the 39gs, exponents
are entered using the key labeled XY. Use this key to evaluate 440 R and 440 R.
440 R11 = ______________________ and 440 R12 = _____________________.
6. So, when R is multiplied by itself 12 times, what do you get? That is, what is R12 ? We call the number R the twelfth root of 2 and write it like this: R = 12 2. This number, like irrational. Overview This exploration into the connection between music and mathematics begins with listening to octaves, which are notes with frequencies in the ratio of 2:1. Most, but not all, westerners sense the harmonies from these octaves. Also in exercise 1 is a 3:2 harmony. This exercise is largely qualitative. Exercise 2 extends and quantifies the observations from Exercise 1. By making a large table of frequencies that are small integer multiples of each other, and searching for values that are close, students find more chords that produce harmonious tones. Exercise 2 also includes a triad (cord with 3 notes). In Exercise 3, students explore the relationship between frequencies within one scale. In modern western music, the scale is based on "equal temperament", where the ratio between adjacent notes is exactly the same through the scale. Since there are 12 divisions within the scale, and the next octave has frequencies twice those in the previous, each adjacent frequency pair have a ratio of

2 and , is

The last observations in Exercise 3 lead to a mathematical model for an equal temperament scale, and this is explored in Extension 1. The extension should be used with students who have some experience graphing equations. Students graph the frequencies of the notes, enter the model, and see that the model fits the frequencies. They also create a scale based on equal differences between adjacent notes, and se that the graph of these leads to a linear model (though no attempt is made to create the symbolic form of the model). Note that Exercise 3 stands by itself, and could be presented without doing Exercise 1 or 2 at all. Exercises 1 and 2 should be done together. Extension 1 fits with Exercise 3. Extension 2 fits with Exercises 1 and 2. Extension 1: Enter the notes created by the arithmetic scale in Exercise 3 into column C4 of the MUSICMATH aplet. Then press SYMB. Define two scatter plots as shown below.
Press VIEWS and select Autoscale. 1. Describe the difference in the shapes of the two graphs.
2. Evaluate the expression 12 when X = 1 and when X = 2. Recall that the first value (i.e. when X = 1) in our column C2 of notes is 440, and that the second is about 466.16.
Back in the SYMBolic view, enter 12 for S1FIT. Go to the PLOT, press the soft MENU key, then press the soft FIT key. You should see that S1FIT hits all the points on the S1 scatter plot! Extension 2: Research the musical term dissonance, and find out what it has to do with this activity. Extension 3: Find music for a song and distribute a measure to each student. Have them put the notes into a list, and organize themselves to play back the song. Teacher Notes The following TEKS (and others) can be found in this activity set: 6.1C, 6.1F, 6.2C, 6.3A, 6.5, 6.11A, 6.11D, 6.13, 7.2B, 7.2D, 7.14A, 7.15, 8.1C, 8.2D, 8.4, 8.14A, 8.15A, 8.16A Here are the TEKS organized by Exercise and Extension: Table 5: TEKS Covered, by Activity Activity TEKS Covered Exercise 1 6.1C, 6.1F, 6.2C, 6.11A, 6.11D, 7.2B, 8.2D, 8.14A Exercise 2 6.1C, 6.1F, 6.2C, 6.11A, 6.11D, 7.2B, 8.14A, 8.16A Exercise 3 6.3A, 6.13, 7.2B, 7.2D, 8.1C, 8.2D, 8.14A Extension 1 6.3A, 6.5, 7.14A, 7.15A, 7.15B, 8.4, 8.14A, 8.15A Extension 2 6.11A, 8.14A Web Resources Select and play combinations of notes and see their wave forms superimposed on this musical scale. Read the Wikipedia entry on Consonance and Dissonance. See the selection of questions and answers about music and math from Dr Math.

Answers Exercise 1 1. The frequencies are in the ratio 1:2 2. Most students should answer yes. To some, the harmony is not apparent. 3. There's no guarantee students will see what we want here. The relevant connection is the 2:3 ratio. 4. The scales in the ratio 3:2 sound harmonious to most people; the scales in the ratio 17:13 sound less pleasant. Exercise 2 15
1. 2:1 2. 3:2 3. One possibility is C and F, in the ratio 4:3. A and D are also in the ratio 4:3. In general, notes that are seven rows apart in the table are in the ratio 3:2. Notes five rows apart are in the ratio 4:3. 4. 4:5:6 Exercise 3 1. 440/12 = 36.67 2. Number 13 3. 1.059 4. Both are 1.059 5. 830.61 and 880, which are the frequencies of G# and A5 6. 2 Note A4 A# B4 C5 C# D5 D# E5 F5 F# G5 G# A5 Frequency (hz) 440 476.67 513.586.67 623.696.67 733.806.67 843.33 880
Extension 1 1. S1 is curved; S2 is linear 2. 440 and 466.15, the first two notes in our scale. Extension 2 According to Wikipedia, dissonance refers to harmonies which create more complex acoustical interactions (called 'beats'). Consonant harmonies are made up of tones that complement and increase each other's resonance. Generally speaking, dissonant intervals involve frequencies with ratios that are larger integers. (Or, at least, that's what this non-music theorist believes!) However, this notion seems to be culturally dependent.
HPs most powerful financial calculator is now sleeker than ever
Previous Article Next The new HP 17bII+, HPs most powerful financial calculator, makes a great gift for Mothers Day this May. The attractive, sleek design makes it very desirable for professionals in real estate, finance, accounting, economics and business. It is the only HP calculator with menu keys which is easy to use and allows you to customize and program your keys. The HP 17bII+ is easy to use and approved for many courses and exams. The calculator is approved by the CFP and MFA and is extremely popular among finance and business professionals with over 250 functions. With this powerful financial calculator, you can quickly calculate loan payments, interest rates and conversions, standard deviation, percent, TVM, NPV, IRR, cash flows, bonds and more. It even has special unique features such as RPN and Algebraic data entry, clock, appointments, calendar, HP solve application, menu prompts and messages. Now you can solve all your financial calculations and more with one calculator. HP provides FREE calculator training specific to our calculator models. Check out the links below: CLICK HERE FOR FREE HP 17bII+ CBT TRAINING CLICK HERE FOR FREE HP 17bII+ INTRO TO FINANCE VIDEO TRAINING CLICK HERE FOR FREE HP 17bII+ LEARNING MODULES