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Contact Information ....................................................................................................6
4
1At a Glance...
This section is designed for you if you’re already familiar with calculator operation or financial
concepts. You can use it for quick reference. The rest of the manual is filled with explanations
and examples of the concepts presented in this section.
Basics of Key Functions
Table 1-1 Basics of key functions
KeysDisplayDescription
=
] [blue]
\ [orange]
JGD|
M
\t
0.00Turns calculator on.
0.00
0.00
12_Erases last character.
0.00Clears display.
0.00Clears statistics
Displays shift
annunciator .
Displays shift
annunciator .
memory.
\N
]Oj
]OY
]OJ
]O:
\>
12 P _ Y r (message flashes, then
disappears)
BOND CLR (message flashes,
then disappears)
BR EV CLR (message flashes,
then disappears)
TVM CLR (message flashes, then
disappears)
CFLO CLR (message flashes, then
disappears)
Clears all memory.
Clears bond memory.
Clears break-even
memory.
Clears tvm registers.
Clears cash flow
memory.
Turns calculator off.
At a Glance...
1
Shift Keys
Most keys on the HP 10bII+ have three functions:
•a primary function printed in white on the key.
•a secondary function printed in orange on the bevel of the key.
•a tertiary function printed in blue above the key on the keyboard (see Figure 1).
Figure 1
As an example, the functions associated with the equals key, 4, are illustrated in the text
as follows:
•primary function (equals): 4
•secondary function (display): \5
•tertiary function (random): ]6
Boxed Key Functions
These special functions require subsequent key presses to operate. For example, the functions
associated with the clear key, M, include:
Table 1-2 Clearing functions
KeysAssociated Function
Clear display.
M
\N
Clear all memory.
]Oj
Clears bond memory.
At a Glance...2
Table 1-2 Clearing functions
KeysAssociated Function
]OY
Clears break-even memory.
]OJ
Clears TVM memory.
]O:
Clears cash flow memory.
\t
Clears statistics memory.
For more information on the calculator’s keys and basic functions, refer to chapter 2, Getting Started.
Percentages
Table 1-3 Keys for percentage calculations
KeysDescription
Percent
§
\¨
Percent change
À
¼
®
Ã
Add 15% to 17.50.
Table 1-4 Calculating the price
KeysDisplayDescription
Jj7V:1
JV§4
Cost
Price
Margin
Markup
17.50Enters number.
20.13Adds 15%.
Find the margin if the cost is 15.00 and selling price is 22.00.
At a Glance...
3
Table 1-5 Finding the margin
KeysDisplayDescription
JVÀ
15. 00E n ter s cos t.
GG¼
22.00Enters price.
®
31. 82C al c u la t e s m arg i n.
If the cost is 20.00 and the markup is 33%, what is the selling price?
Table 1-6 Calculating the price
KeysDisplayDescription
20.00Enters cost.
G:À
DDÃ
33.00Enters markup.
¼
26.60Calculates price.
For more information on percentages, refer to chapter 3, Business Percentages.
Memory Keys
Table 1-7 Memory keys
KeysDescription
Stores a constant operation.
ª
s
Stores a value in the M register (memory location).
p
Recalls a value from the M register.
m
Adds a value to the number stored in the M register.
\w
When followed by a number key, : to d, or 7 and : to d, stores a number in the
display into a numbered data storage register. There are 20 storage registers, designated 0-
19. Pre ss
\w7 followed by : through d to access registers 10-19.
v
At a Glance...4
When followed by a number key, : to d, or 7 and : to d, recalls a number from
a storage register. Press
v7 followed by : through d to access registers 10-19.
Multiply 17, 22, and 25 by 7, storing ‘× 7’ as a constant operation.
Table 1-8 Storing ‘x 7’ as a constant
KeysDisplayDescription
JjPjª
7.0 0St ore s ‘ × 7’ as a
119.00Multiplies 17 × 7.
4
154.00Multiplies 22 × 7.
GG4
175.00Multiplies 25 × 7.
GV4
Store 519 in register 2, then recall it.
Table 1-9 Storing and recalling
KeysDisplayDescription
519.00Stores 519 in register 2.
VJd\wG
0.00Clears display.
M
519.0 0Re c a lls reg i s te r 2.
vG
constant operation.
Store 1.25 into register 15, then add 3, and store the result in register 15.
Table 1-10 Storage register arithmetic
KeysDisplayDescription
J7GV
\w7V
D\w17V
1. 25In p ut s 1. 25 i nt o t he
display.
Stores 1.25 in register
15 .
3.00Adds 3 to 1.25 in
register 15 stores the
result in register 15.
0.00Clears the display.
M
v7V
4.25Recalls register 15.
For more information on number storage and storage register arithmetic, refer to chapter 4,
Number Storage and Storage Register Arithmetic.
At a Glance...
5
Time Value of Money (TVM)
Enter any four of the five values and solve for the fifth.
A negative sign in the display represents money paid out, and money received is positive.
Table 1-11 Keys for TVM calculations
KeysDescription
]OJ
Ù
\Ú
Ò
Ï
Ì
É
\¯
\Í
Clears TVM memory and the current P_YR is
displayed.
Number of payments.
Multiplies a value by the number of payments
per year and stores as N.
Interest per year.
Present value.
Payment.
Future value.
Begin or End mode.
Number of payments per year mode.
If you borrow 14,000 (PV) for 360 months (N) at 10% interest (I/YR), what is the monthly
repayment?
Set to End mode. Press
Table 1-12 Calculating the monthly payment
KeysDisplayDescription
]OJ
\¯ if BEGIN annunciator is displayed.
TVM CLR (message flashes, then
disappears)
12.00Sets payments per year.
Clears TVM memory and
displays the current P_YR.
JG\Í
DS:Ù
360.00Enters number of payments.
J:Ò
10.00Enters interest per year.
JY:::Ï
14,000.00Enters present value.
At a Glance...6
Table 1-12 Calculating the monthly payment
KeysDisplayDescription
:É
0.00Enters future value.
Ì
TVM What if...
It is not necessary to reenter TVM values for each example. Using the values you just entered,
how much can you borrow if you want a payment of 100.00?
Table 1-13 Calculating a new payment
KeysDisplayDescription
J::yÌ
Ï
...how much can you borrow at a 9.5% interest rate?
-122.86Calculates payment if paid
at end of period.
-100.00Enters new payment
amount. (Money paid out is
negative).
11,395.08Calculates amount you can
borrow.
Table 1-14 Calculating a new interest rate
KeysDisplayDescription
9.50Enters new interest rate.
d7VÒ
Ï
J:Ò
JY:::Ï
11,892.67Calculates new present
value for 100.00 payment
and 9.5% interest.
10.00Reenters original interest
rate.
14,000.00Reenters original present
value.
-122.86Calculates original payment.
Ì
For more information on TVM concepts and problems, refer to chapter 5, Picturing Financial
Problems, and chapter 6, Time Value of Money Calculations.
At a Glance...
7
Amortization
After calculating a payment using Time Value of Money (TVM), input the periods to amortize
and press \Ê. Press \Ê once for periods 1-12, and once again for payments 13-
24. Press 4 to continually cycle through the principal, interest, and balance values (indicated
by the PRIN, INT, and BAL annunciators respectively). Using the previous TVM example,
amortize a single payment and then a range of payments.
Amortize the 20th payment of the loan.
Table 1-15 Amortizing the 20th payment of the loan
KeysDisplayDescription
G:Æ
20.00Enters period to amortize.
20 – 20Displays period to amortize.
\Ê
-7.25Displays principal.
4
-115. 61D is pl ays in teres t. ( Mo ney
4
13,865.83Displays the balance
4
Amortize the 1st through 24th loan payments.
Table 1-16 Amortization example
KeysDisplayDescription
12_Enters range of periods to
JÆJG
paid out is negative).
amount.
amortize.
\Ê
1 – 12Displays range of periods
(payments).
-77.82Displays principal.
4
4
4
-1,396.5 0Di sp lays i nt ere st. (M on ey
paid out is negative).
13,922.18Displays the balance
amount.
13 – 24Displays range of periods.
\Ê
4
-85.96Displays principal.
At a Glance...8
Table 1-16 Amortization example
KeysDisplayDescription
4
-1,388.36Displays interest.
4
For more information on amortization, refer to the section titled, Amortization in chapter 6,
Time Value of Money Calculations.
Depreciation
Table 1-17 Depreciation keys
KeysDescription
Expected useful life of the asset.
13,836.22Displays the balance
amount.
Ù
Ò
Declining balance factor entered as a
percentage.
Depreciable cost of the asset at acquisition.
Ï
É
Salvage value of the asset.
]{
Straight-line depreciation.
]x
Sum-of-the-years’-digits depreciation.
]u
Declining Balance depreciation.
A metalworking machine, purchased for 10,000.00, is to be depreciated over five years. Its
salvage value is estimated at 500.00. Using the straight-line method, find the depreciation
and remaining depreciable value for each of the first two years of the machine's life.
Table 1-18 Calculating the depreciation
KeysDisplayDescription
10,000.00Inputs cost of the item.
J::::Ï
V::É
500.00Inputs the salvage value of the
item.
5.00Inputs the useful life of the asset.
VÙ
J]{
1,900.00Depreciation of the asset in year
one.
At a Glance...
9
Table 1-18 Calculating the depreciation
KeysDisplayDescription
\«
G]{
\«
For more information on depreciation, refer to chapter 7, Depreciation.
7,600.00Remaining depreciable value
after year one.
1,900.00Depreciation of the asset in year
two.
5,700.00Remaining depreciable value
after year two.
Interest Rate Conversion
To convert between nominal and effective interest rates, enter the known rate and the number
of periods per year, then solve for the unknown rate.
Ta ble 1-19 Keys fo r int er est ra te co nv ersi o n
KeysDescription
Nominal interest percent.
\Ó
\Ð
Effective interest percent.
\Í
Periods per year.
Find the annual effective interest rate of 10% nominal interest compounded monthly.
Table 1-20 Calculating the interest rate
KeysDisplayDescription
10.00Enters nominal rate.
J:\Ó
JG\Í
12.00Enters payments per year.
\Ð
For more information on interest rate conversions, refer to the section titled, Interest Rate
Conversions in chapter 6, Time Value of Money Calculations.
10.47Calculates annual effective
interest.
At a Glance...10
Cash Flows, IRR/YR, NPV, and NFV
Table 1-21 Cash flows, IRR, NPV, and NFV keys
KeysDescription
]O:
\Í
¤
number1
Æ number 2 ¤
v¤
Clears cash flow memory.
Number of periods per year (default is
12). For annual cash flows, P/YR should
be set to 1; for monthly cash flows, use
the default setting, 12 .
Cash flows, up to 45. “J” identifies the
cash flow number. When preceded by a
number, pressing
amount.
Enter a cash flow amount, followed by
¤ enters a cash flow
Æ. Enter a number for the cash flow
count followed by
flow amount and count simultaneously.
Opens editor for reviewing/editing
entered cash flows. Press
¤ to enter cash
1 or A to
\¥
\Á
\½
\½\«
scroll through the cash flows.
Number of consecutive times cash flow
“J” occurs.
Internal rate of return per year.
Net present value.
Net future value.
At a Glance...
11
If you have an initial cash outflow of 40,000, followed by monthly cash inflows of 4,700,
7,000, 7,000, and 23,000, what is the IRR/YR? What is the IRR per month?
Table 1-22 Calculating the IRR/YR and IRR per month
KeysDisplayDescription
]O:
JG\Í
Y::::y¤
Yj::¤
j:::ÆG¤
GD:::¤
v¤
CFLO CLR
(message flashes, then
disappears)
12.00Sets payments per year.
-40,000.00
(CF 0 flashes, then disappears)
4,700.00
(CF 1 flashes, then disappears)
2.00
(CFn 2 flashes, then disappears)
23,000.00
(CF 3 flashes, then disappears)
0 -40,000.00Reviews entered cash flows
Clears cash flow memory.
Enters initial outflow.
Enters first cash flow.
Enters both the cash flow
amount (7000.00) and count
(2.00) simultaneously for second
cash flow.
Enters third cash flow.
starting with the initial cash flow.
\Á
15 . 9 6C a l c u l a t e s IRR/YR.
1. 3 3C a l c u l a t e s IRR per month.
aJG4
What is the NPV and NFV if the discount rate is 10%?
Table 1-23 Calculating NPV and NFV
KeysDisplayDescription
10.0 0Ent er s I/YR.
J:Ò
622.85Calculates NPV.
\½
Press
1 to scroll through the
cash flow list to verify the cash
flow number, the amounts, and
count for each entry. Press
to exit.
M
At a Glance...12
Table 1-23 Calculating NPV and NFV
KeysDisplayDescription
\½\«
643.88Calculates NFV.
For more information on cash flows, refer to chapter 8, Cash Flow Calculations in the HP 10bII+ Financial Calculator User’s Guide.
Date and Calendar
Table 1-24 Keys used for dates and calendar functions
KeysDescription
]È
Enters dates in DD.MMYYYY or MM.DDYYYY formats. D.MY is the default.
Numbers at the far right of a calculated date indicate days of the week. 1
is for Monday; 7 is for Sunday.
Toggles between 360-and 365-day (Actual) calendars.
]Å
\Ç
\Ä
Calculates the date and day, past or future, that is a given number of days
from a given date. Based on your current setting, returned result is
calculated using either 360-day or 365-day (Actual).
Calculates the number of days between two dates. Returned result is
always calculated based on the 365-day calendar (Actual).
If the current date is February 28 2010, what is the date 52 days from now? Calculate the
date using the 365-day calendar (actual) and the M.DY settings.
If 360 is displayed, press ]Å. If D.MY is displayed, press ]È.
Table 1-25 Calculating the date
KeysDisplayDescription
G7GgG:J:
2.28Inputs the date in the
selected format.
\Ç
VG4
For more information on date and calendar functions, refer to chapter 9, Calendar Formats
and Date Calculations.
4-21-2010 3Inputs the number of days
and calculates the date
along with the day of the
week.
At a Glance...
13
Bonds
Bond calculations, primarily calculating bond price and yield, are performed by two keys,
]Ñ and ]Ô. These keys permit you to input data or return results. Pressing ]Û
only calculates a result. The other keys used in bond calculations only permit you to input the
data required for the calculations.
Table 1-26 Bond calculation keys
KeysDescription
]Oj
]Û
]Ô
]Ñ
]Î
]Ë
]È
]Å
]Â
Clears bond memory.
Calculates accrued interest only.
Yield% to maturity or yield% to call date for
given price.
Price per 100.00 face value for a given yield.
Coupon rate stored as an annual %.
Call value. Default is set for a call price per
100.00 face value. A bond at maturity has a
call value of 100% of its face value.
Date format. Toggle between day-month-year
(dd.mmyyyy) or month-day-year (mm.ddyyyy).
Day count calendar. Toggle between Actual
(365-day calendar) or 360 (30-day month/
360-day year calendar).
Bond coupon (payment). Toggle between
semiannual and annual payment schedules.
]¾
]°
What price should you pay on April 28, 2010 for a 6.75% U.S. Treasury bond maturing on
June 4, 2020, if you want a yield of 4.75%? Assume the bond is calculated on a semiannual
coupon payment on an actual/actual basis.
If SEMI is not displayed, press
If D.MY is displayed, press]Èto select M.DY format.
At a Glance...14
Settlement date. Displays the current settlement
date.
Maturity date or call date. The call date must
coincide with a coupon date. Displays the
current maturity.
]Âto select the semiannual coupon payment.
Table 1-27 Bond calculation
KeysDisplayDescription
]Oj
Y7GgG:J:
]¾
S7:YG:G:
]°
S7jV]Î
J::]Ë
Y7jV]Ô
]Ñ
BOND CLR (message
flashes, then disappears)
4-28-2010 3Inputs the settlement date
6-4-2020 4Inputs the maturity date.
6.75Inputs CPN%.
100.00Inputs call value. Optional,
4.75Inputs Yield%.
115.89Calculates the price.
Clears bond memory.
(mm.ddyyyy format).
as default is 100.
1]Û
4
For more information on bond calculations, refer to chapter 10, Bonds.
2.69Displays the current value
for accrued interest.
118.59Returns the result for total
price (value of price + value
of accrued interest). The net
price you should pay for the
bond is 11 8 . 5 9 .
At a Glance...
15
Break-even
Table 1-28 Break-even keys
KeysDescription
]OY
Clears break-even memory.
]¬
Stores the quantity of units required for a given
profit or calculates it.
For more information on break-even calculations, refer to chapter 11, Break-even.
3,200.00Calculates the current value
for the unknown item,
UNITS.
At a Glance...16
Statistical Calculations
Table 1-30 Statistics keys
KeysDescription
\t
x-data
¡
\¢
x-data
Æ y-data ¡
x-data
x-data
Æ y-data \¢
v¡
\k \«
Clear statistical registers.
Enter one-variable statistical
data.
Delete one-variable statistical
data.
Enter two-variable statistical
data.
Delete two-variable statistical
data.
Opens editor for reviewing/
editing entered statistical data.
Means of x and y.
\T \«
\h \«
\e \«
y-data
\Z \«
\W \«
x-data
]L
Mean of x weighted by y. Also
calculates b, intercept.
Sample standard deviations of
x and y.
Population standard deviations
of x and y.
Estimate of x and correlation
coefficient.
Estimate of y and slope.
Permits selection of six
regression models; linear is
default.
At a Glance...
17
Using the following data, find the means of x and y, the sample standard deviations of x and
Σ
Σ
y, and the y-intercept and the slope of the linear regression forecast line. Then, use summation
statistics to find xy.
x-data246
y-data5090160
Table 1-31 Statistics example
KeysDisplayDescription
\t
GÆV:¡
YÆd:¡
SÆJS:¡
v¡
\k
\«
0.00Clears statistics registers.
1.00Enters first x,y pair.
2.00Enters second x,y pair.
3.00Enters third x,y pair.
1 2.00Reviews entered statistical
data, starting with the initial
x-value. Press
through and verify the
entered statistical data.
Press
M to exit.
4.00Displays mean of x.
100.00Displays mean of y.
1 to scroll
\h
\«
\T\«
\W\«
]f
For more information on statistical calculations, refer to chapter 12, Statistical Calculations.
At a Glance...18
2.00Displays sample standard
deviation of x.
55.68Displays sample standard
deviation of y.
-10 .0 0D is pl ays y-intercept of
regression line.
27.50Displays slope of regression
line.
1,420.00
Displays xy, sum of the
products of x- and y-values.
Probability
Table 1-32 Probability keys
KeysDescription
]F
]o
F
]I
]o
I
]<
]9
]E
Calculates a cumulative normal probability
given a Z-value.
Calculates a Z-value given a cumulative
normal probability.
Calculates the cumulative Student’s T
probability given degrees of freedom and a
T-value.
Calculates a T-value given degrees of
freedom and the cumulative Student’s T
probability.
Calculates number of permutations of n items
taken
r at a time.
Calculates number of combinations of n taken
r at a time.
Calculates factorial of n (where -253 < n <
253).
Enter .5 as a Z-value and calculate the cumulative probability of the Z-value and the Z-value
from a given cumulative probability.
Table 1-33 Calculating the probability
KeysDisplayDescription
\5V
7V]F
0.00000Sets number display to five
digits to the right of the
decimal.
.69146Calculates the cumulative
probability of the Z-value.
.94146Adds .25.
17GV4
]oF
For more information on probability, refer to the section titled, Probability in chapter 12,
Statistical Calculations.
1.56717Calculates the Z-value from
the cumulative probability.
At a Glance...
19
Trigonometric Functions
θ
θ
θ
Table 1-34 Trigonometry keys
KeysDescription
] c, R, or C
]o
c, R, or C
]r
c, R, or C
Calculates sine, cosine, and tangent.
Calculates inverse sine, inverse cosine,
and inverse tangent.
Calculates hyperbolic sine, cosine and
tangent.
]ro
Calculates inverse hyperbolic sine,
cosine, and tangent.
c, R, or C
]3
Find Sin =.62 in degrees. If RAD is displayed, press ]3.
Table 1-35 Trigonometry example
KeysDisplayDescription
Toggles between radians and degrees
modes. Degrees is the default setting.
.62
Enters value of sine for
7SG
]oc
38.32
Calculates
.
.
At a Glance...20
Convert the results to radians using Pi.
Table 1-36 Converting to radians
KeysDisplayDescription
P\;aJg
.67Converts degrees to radians.
:4
For more information on trigonometric functions, refer to chapter 2, Getting Started.
At a Glance...
21
At a Glance...22
2Getting Started
Power On and Off
To turn on your HP 10bII+, press =. To turn the calculator off, press the orange shift key,
\, then >. To change the brightness of the display, hold down = and then
simultaneously press 1 or A.
Since the calculator has continuous memory, turning it off does not affect the information you
have stored. To conserve energy, the calculator turns itself off after five minutes of inactivity.
The calculator uses two CR2032 coin batteries. If you see the low-battery symbol () in
the display, replace the batteries. For more information, refer to the section titled, Installing Batteries in Appendix A.
Manual Conventions and Examples
In this manual, key symbols are used to indicate the key presses used in the example problems. These symbols vary in appearance according to whether they indicate the primary,
secondary, or tertiary functions required for the problem. For example, the functions associ-
ated with the equals key, 4, are illustrated in the text as follows:
•primary function (equals):4
•secondary function (display):\5
•tertiary function (random):]6
Note the symbol for the primary function of the key, in this case, =, appears on each of the
key symbols depicted above. This repetition is intended to serve as a visual aid. By looking
for the symbol of the primary function on the key, you can quickly locate the keys used for the
secondary and tertiary functions on the calculator.
Displayed text
Text that appears in the display screen of the calculator is presented in BOLD CAPITAL letters
throughout the manual.
Examples
Example problems appear throughout the manual to help illustrate concepts and demonstrate
how applications work. Unless otherwise noted, these examples are calculated with CHAIN
set as the active operating mode. To view the current mode, press v]?. The
current mode, CHAIN or ALGEBRAIC, will flash, then disappear. To change the mode, press
] followed by ?.
Getting Started
23
Basics of Key Functions
Table 2-1 Basics of key functions
KeysDisplayDescription
=
][blue]
\[orange]
JGD|
]3
M
\t
0.00Turns calculator on.
0.00
0.00
12_Erases last character.
RAD
(at the bottom of the display)
0.00Clears display.
0.00Clears statistics memory.
Displays shift annunciator .
Displays shift annunciator .
Toggles between radians and degrees.
The item before the / is the alternate; the
item after the / is the default setting.
Except for the operating mode,
annunciators in the display indicate
alternate settings are active.
\N
]Oj
]OY
]OJ
]O:
\>
12 P _Y r (message flashes, then
disappears)
BOND CLR (message flashes,
then disappears)
BR EV CLR (message flashes,
then disappears)
TVM CLR (message flashes, then
disappears)
CFLO CLR (message flashes, then
disappears)
Clears all memory.
Clears bond memory.
Clears break-even memory.
Clears tvm memory.
Clears cash flow memory.
Turns calculator off.
Getting Started24
Shift Keys
Most keys on the HP 10bII+ have three functions:
• a primary function printed in white on the key.
• a secondary function printed in orange on the bevel of the key.
• a tertiary function printed in blue above the key on the keyboard (see Figure 1).
Figure 1
When you press \ or ], a shift annunciator or is displayed to indicate that the
shifted functions are active. For example, press \ followed by 2 to multiply a number in the display by itself. To turn the shift annunciators off, press \ or ] again.
Boxed Key Functions
There are three shifted key functions on the calculator that are used to change the operation
of another key's function. These three tertiary functions,]O,]oand
]r,are bound by blue boxes to show that they operate differently. These special
functions require subsequent key presses to operate. For example, the functions associated
with the clear key,M, include:
Table 2-2 Clearing functions
KeysAssociated Function
Clear display.
M
\N
\t
]Oj
Clear all memory.
Clear statistics memory.
Clears bond memory.
Getting Started
25
Table 2-2 Clearing functions
KeysAssociated Function
]OY
Clears break-even memory.
]OJ
Clears TVM memory.
]O:
Clears cash flow memory.
Simple Arithmetic Calculations
Operating Modes
To change the operating mode, press the blue shift key, ] followed by ? to toggle
between Algebraic and Chain modes. A brief message is displayed indicating the selected
operating mode.
To view the current mode, press v]?. The current mode will flash, then disappear.
Arithmetic Operators
The following examples demonstrate using the arithmetic operators 1, A, P, and
a.
If you press more than one operator consecutively, for example 1, A, 1, P
1, all are ignored except the last one.
If you make a typing mistake while entering a number, press | to erase the incorrect digits.
Table 2-3 Example displaying calculations using arithmetic operators
KeysDisplayDescription
GY7jJ1SG7Yj4
When a calculation has been completed (by pressing
new calculation.
4), pressing a number key starts a
87.18Adds 24.71 and
62.47.
Table 2-4 Completing a calculation
KeysDisplayDescription
240.92Calculates 19 × 12.68.
JdPJG7Sg4
Getting Started26
If you press an operator key after completing a calculation, the calculation is continued.
Table 2-5 Continuing a calculation
KeysDisplayDescription
1JJV7V4
Calculations in Chain Mode
Calculations in Chain mode are interpreted in the order in which they are entered. For
example, entering the following numbers and operations as written from left to right,
356.42Completes calculation of
240.92 + 115.5.
J1GPD4, returns 9. If you press an operator key, 1,A,P, or
a, after 4, the calculation is continued using the currently displayed value.
You can do chain calculations without using 4 after each step.
Table 2-6 Chain calculations
KeysDisplayDescription
S7dPV7DVa
36.92
Pressing a displays
intermediate result (6.9 × 5.35).
40.57Completes calculation.
7dJ4
Without clearing, now calculate 4 + 9 × 3.
Table 2-7 Chain calculations
KeysDisplay
13.00Adds 4 and 9.
Y1dP
D4
39.00Completes calculation.
In Chain mode, if you wish to override the left to right order of entry, use parentheses
\q and \n to prioritize operations.
Getting Started
27
For example, to calculate 1 + (2 x 3), you may enter the problem as written from left to right,
y
x
30
8512–()
----------------------
9×
with parentheses to prioritize the multiplication operation. When entered with parentheses,
this expression returns a result of 7.
Calculations in Algebraic Mode
In Algebraic mode, multiplication and division have a higher priority than addition and
subtraction. For example, in Algebraic mode, pressing J1GPD4 returns a
result of 7.0 0 . In Chain mode, the same key presses return a result of 9.00 .
In Algebraic mode, operations between two numbers have the following priority:
•Highest priority: combinations and permutations, T probability calculations, % change, and
date calculations
•Second priority: the power function ( )
•Third priority: multiplication and division
•Forth priority: addition and subtraction.
The calculator is limited to 12 pending operations. An operation is pending when it is waiting
for the input of a number or the result of an operation of higher priority.
Using Parentheses in Calculations
Use parentheses to postpone calculating an intermediate result until you’ve entered more
numbers. You can enter up to four open parentheses in each calculation. For example,
suppose you want to calculate:
If you enter D:agVA, the calculator displays the intermediate result, 0.35.
This is because calculations without parentheses are performed from left to right as you enter
them.
To delay the division until you’ve subtracted 12 from 85, use parentheses. Closing parentheses
at the end of the expression can be omitted. For example, entering 25 ÷ (3 × (9 + 12 = is
equivalent to 25 ÷ (3 × (9 + 12)) =.
If you type in a number, for example, 53, followed by the parenthesis symbol, the calculator
considers this implicit multiplication.
Example
Table 2-8 Using parentheses in calculations
KeysDisplayDescription
D:a\qgVA
JG\n
Getting Started28
85.00No calculation yet.
73.00Calculates 85 - 12.
Table 2-8 Using parentheses in calculations
KeysDisplayDescription
P
0.41Calculates 30 ÷ 73.
d4
3.70Multiplies the result by 9.
Negative Numbers
Enter the number and press y to change the sign.
Calculate -75 ÷ 3.
Table 2-9 Changing the sign of numbers
KeysDisplayDescription
-75_Changes the sign of 75.
jVy
aD4
–25.00Calculates result.
Understanding the Display and Keyboard
Cursor
The blinking cursor ( _ ) is visible when you are entering a number.
Clearing the Calculator
Backspace
When the cursor is on, | erases the last digit you entered. Otherwise, | clears the
display and cancels the calculation.
Clear
M clears the current item on the display and replaces it with 0. If entry is in progress,
pressing M clears the current entry and replaces it with 0, but the current calculation
continues. Otherwise, M clears the display of its current contents and cancels the current
calculation.
Clear Memory
]O followed by j,Y,J,: clears a selected memory type (register). Other
memory is left intact.
Getting Started
29
Table 2-10 Clear memory keys
KeysDescription
]Oj
Clears bond memory.
]OY
Clears break-even memory.
]OJ
Clears TVM memory.
]O:
Clears cash flow memory.
\t
Clears statistics memory.
Clear All
\N all clears all memory in the calculator, with the exception of the payments per
year (P/Yr) setting. To clear all memory and reset calculator modes, press and hold down
=, then press and hold down both Ù and Ï. When you release all three, all
memory is cleared. The All Clear message is displayed.
Clearing Messages
When the HP 10bII+ is displaying an error message, |or M clears the message and
restores the original contents of the display.
Annunciators
Annunciators are symbols in the display that indicate the status of the calculator. For functions
that toggle between settings, annunciators indicate alternate settings are active. For the
defaults, no annunciators appear in the display. For example, when selecting a date format,
the default setting is month-day-year (M.DY). When day-month-year (D.MY) is active, the D.MY
in the display indicates it is the active setting. Table 2-11 lists all the annunciators that appear
in the display screen.
Getting Started30
Table 2-11 Annunciators and status
AnnunciatorStatus
,
INVInverse mode is active for trigonometric or probability functions.
RADRadians mode is active.
BEGBegin mode is active; payments are at the beginning of a
D.MYDay-month-year date format (DD.MMYYYY) is active.
360360-day calendar is active.
SEMISemi-annual coupon payment schedule (bonds) is active.
PENDAn operation is waiting for another operand.
INPUT
AMORTThe amortization annunciator is lit, together with one of the
PERThe range of periods for an amortization is displayed.
PRINThe principal of an amortization is displayed.
A shift key has been pressed. When another key is pressed, the
functions labeled in orange or blue are executed.
period.
The
Æ key has been pressed and a number stored.
Battery power is low.
following four annunciators:
INTThe interest of an amortization is displayed.
BALThe balance of an amortization is displayed.
CFLOThe cash flow annunciator is lit, together with one of the
following two annunciators:
CF The cash flow number appears briefly, then the cash flow is
shown.
NThe cash flow number appears briefly, then the number of
times the cash flow is repeated is shown.
STATThe statistics annunciator is lit, together with one of the following
two annunciators:
XThe number of the data point, n, followed by an x-value is
shown, or, if STAT is not lit, indicates that the first of two
results is displayed.
YThe number of the data point, n, followed by a y-value is
shown, or, if STAT is not lit, indicates that the second of two
results is displayed.
ERRORThe error annunciator is lit, together with one of the following
four annunciators:
TVMThere is a TVM error (such as an invalid P/Yr), or, when
ERROR is not lit, a TVM calculation returned a second result.
FULLAvailable memory for cash flows or statistics is full, or the
pending operator memory is full.
STATIncorrect data used in a statistics calculation or, when
ERROR is not lit, a statistical calculation has been performed.
Getting Started
31
Table 2-11 Annunciators and status
AnnunciatorStatus
FUNCA math error has occurred (for example, division by zero).
Input Key
The Æ key is used to separate two numbers when using two-number functions or twovariable statistics. The Æ key can also be used to enter cash flows and cash flow counts,
ordered pairs, and evaluate any pending arithmetic operations, in which case the result is the
same as pressing 4.
Swap Key
Pressing \« exchanges the following:
•The last two numbers that you entered; for instance, to change the order of division or subtraction.
•The results of functions that return two values.
The « key toggles the item in the Æ register, or swaps the top two items in the
mathematical stack. This function is used to retrieve a secondary value returned during a
calculation, as well as to swap two items during a calculation.
Statistics Keys
The statistics keys are used to access summary statistics from the statistics memory registers.
When you press ] followed by a statistics key, you can recall one of six summary statistics
with the next keystroke.
For example, press ] followed by the X key to recall the sum of the x-values entered.
Table 2-12 Statistics keys
KeysDescription
]l
]i
Sum of the squares of the xvalues.
Sum of the squares of the yvalues.
]f
Sum of the products of the x- and
y-values.
Number of data points entered.
][
Getting Started32
Table 2-12 Statistics keys
KeysDescription
]U
Sum of the y-values.
]X
Sum of the x-values.
Time Value of Money (TVM), Cash Flows, Bond, and Break-even Keys
When entering data for TVM, cash flows, bond, depreciation and break-even calculations,
results are calculated based on data entered into specific memory registers. When pressed,
the keys used for these operations:
•store data.
•enter data for a variable that is used during calculations (input only).
•calculate unknown variables based on stored data.
For more information on how these keys function, refer to the specific chapters which cover
TVM problems, cash flows, and bond and break-even calculations.
Math Functions
One-Number Functions
Math functions involving one number use the number in the display. To execute one-number
functions, with a number displayed, press the key or key combination corresponding to the
operation you wish to execute. The result is displayed. See Table 2-14 for a list of one-number
functions.
Before doing any trigonometric calculations, check whether the angle mode is set for degrees
or radians (Rad). Degrees is the default setting. The RAD annunciator in the display indicates
radians is active. Press ]3 to toggle between the settings. You will need to change
the setting if the active mode is not what your problem requires.
Table 2-13 Example displaying one number functions
KeysDisplayDescription
9.45Calcula tes squa re root.
gd7GV\B
D7Vj1G7DS\b
0.421/2.36 is calculated
first.
3.99Adds 3.57 and 1/2.36.
4
Table 2-14 lists the one-number functions of the calculator.
Getting Started
33
Table 2-14 One-number functions
KeysDescription
§
\}
\b
\B
\2
\K
\H
\E
]
Divide a number by 100.
Rounds x to the number specified by the display format.
Calculates 1/x.
Calculates the square root of x.
Calculates the square of x.
Calculates natural exponent to the power of x.
Calculates natural log.
Calculates factorial of n (where -253 < n < 253). The Gamma function is
used to calculate n! for non-integers or negative numbers.
Calculates sine, cosine, or tangent.
c, R, or C
]o
c, R, or C
]r
c, R, or C
]ro
c, R, or C
]F
]oF
Calculates inverse sine, cosine, or tangent.
Calculates hyperbolic sine, cosine, or tangent.
Calculates inverse hyperbolic sine, cosine, or tangent.
Calculates a cumulative normal probability given a Z-value.
Calculates a Z-value given a cumulative normal probability.
The random function]6, and Pi \; are special operators. They insert values
for Pi, or a random number in the range 0 < x <1, into calculations.
Getting Started34
Trigonometric and Hyperbolic Functions and Modes
Selecting Angle Format
The trigonometric angle format determines how numbers are interpreted when using
trigonometry functions. The default format for angles on the 10bII+ is degrees. To change to
radians mode, press ]3. When radians mode is active, the RAD annunciator is
displayed.
Trigonometric Functions
Table 2-15 Trigonometric functions
KeysDescription
]c
Calculates sine, written as sin.
]R
]C
]oc
]oR
]oC
Example
Perform the following trigonometric calculations. If RAD is lit in the display, press ]3.
Table 2-16 Example using various trigonometric calculations
Calculates cosine, written as
cos.
Calculates tangent, written as
tan.
Calculates inverse sine, also
written, arcsin, asin, or sin
Calculates inverse cosine, also
written, arccos, acos, or cos
Calculates inverse tangent, also
written, arctan, atan, or tan
-1
.
-1
.
-1
.
KeysDisplayDescription
0.0000Set display to four decimal places.
\5Y
JV]c
0.2588
1. 7 3 21
J1S:]C
2.7321
4
69.5127Displays inverse cosine of 0.35.
Displays sine of 15
Displays tangent of 60
Calculates 1 + tangent of 60
7DV]oR
o
.
o
.
Getting Started
o
.
35
Table 2-16 Example using various trigonometric calculations
π
π
π
A4πr
2
=
π
KeysDisplayDescription
A7SG]oR
51. 6839D isp l ay s inv er s e co s ine o f 0. 62.
4
17.8288Calculates arccos 0.35 - arccos 0.62.
\5G
17.83Return display to default format.
Pi
Pressing \; displays the value of . Although the displayed value is appears in the
current display format, the 12 digit value is actually used for calculations. is often used
during calculations in radians mode, as there are 2 radians in a circle.
Example
Find the surface area of a sphere with a radius of 4.5 centimeters. Use the formula:
Table 2-17 Example using Pi
KeysDisplayDescription
YP\;
PY7V\2
4
Hyperbolic Functions
Table 2-18 Hyperbolic and inverse hyperbolic functions
KeysDescription
]rc
]rR
]rC
]roc
3.14
20.25
254.47Calculates sphere surface area
Calculates hyperbolic sine, written as,
sinh.
Calculates hyperbolic cosine, written
as, cosh.
Calculates hyperbolic tangent, written
as, tanh.
Calculates inverse hyperbolic sine,
written as, arcsinh, asinh, or, sinh
Displays
Displays 4.5
in square centimeters.
.
2
.
-1
.
]roR
]roC
Getting Started36
Calculates inverse hyperbolic sine, also
written, arccosh, acosh, or cosh
Calculates inverse hyperbolic tangent,
also written, arctanh, atanh, or tanh
-1
.
-1
.
Example
y
x
Perform the following hyperbolic calculations.
Table 2-19 Example performing various hyperbolic calculations
KeysDisplayDescription
\5Y
0.0000Sets display to four decimal places.
J7GV]rc
1.6019Display sinh 1.25.
17Vd]rC
0.5299Displays tanh 0.59.
4
2.1318Calculates sinh 1.25 + tanh 0.59.
]roR
1.3 89 9C a lc u lat e s a c o sh 2.1318 .
\5G
1.39Returns display to default format.
Two-Number Functions
When a function requires two numbers, other than for addition, subtraction, multiplication,
division, and the power function, (), you may key in the numbers as follows: number1 Æ number 2 followed by the operation. Pressing Æ evaluates the current expression and
displays the INPUT annunciator.
In-line Functions
For calculations involving \¨, \Ä, \Ç,]9,
]<,]I, and ]oI, which require two numbers, you may also
key in the first number followed by the function keys, and then key in the second number
followed by 4 to return results. Throughout the manual, when examples are entered in this
manner without using Æ, they are referred to as in-line functions. For example, the
following keystrokes calculate the percent change between 17 and 29 using the \¨
keys as an in-line function:
Getting Started
37
Table 2-20 Example calculating percent change as an in-line function
KeysDisplayDescription
Jj\¨
17.0 0E n te rs number1, displays the PEND annunciator indicating the
calculator is awaiting
instructions.
29_Enters number 2.
Gd
4
Press M, and now calculate the same example using the Æ key to store the first number,
then key in the second number and perform the operation.
Table 2-21 Example calculating percent change using ‘INPUT’
KeysDisplayDescription
JjÆ
70.59Calculates the percent change.
17.0 0E n te rs number1, and displays
the INPUT annunciator
indicating the number has been
stored.
Gd\¨
Although the in-line function has fewer key strokes, performing this example using the Æ
key permits you to store a value and then perform other calculations following Æwithout
using parentheses.
Table 2-22 Example displaying two-number functions with chain calculation
KeysDisplayDescription
JjÆ
Gd1DD
1VYAgj
70.59Enters number 2 and calculates
the percent change.
17.0 0E n t er s number1, and displays
the INPUT annunciator.
87_Enters and performs the chain
calculation. Results are stored
and used in the next operation.
The PEND annunciator and the
blinking cursor indicate an
operation is pending as the
calculator awaits instructions.
\¨
Getting Started38
70.59Calculates the percent change
between 17 and the result of the
chain operation (29).
The Table 2-23 below lists the two-number functions of the calculator.
20
20
Table 2-23 Two-number functions
KeysDescription
1APa
\Q
\¨
]9
]<
\Ç
\Ä
]I
Addition, subtraction,
multiplication, division.
The power function.
% Change.
Combinations.
Permutations.
The date and day, past or
future, that is a given number of
days from a given date.
The number of days between
two dates.
Calculates the cumulative
Student’s t probability given
degrees of freedom and a tvalue.
Calculates a t-value given
]oI
Two-number functions may be performed in either CHAIN or ALGEBRAIC mode.
degrees of freedom and the
cumulative Student’s t
probability.
Arithmetic with One-and Two-number Functions
Math functions operate on the number in the display.
Example 1
Calculate 1/4, then calculate + 47.2 + 1.12.
Table 2-24 Calculating the expression
KeysDisplayDescription
0.25Calculates the reciprocal of 4.
Y\b
G:\B
4.47
Calculates .
Getting Started
39
Table 2-24 Calculating the expression
20
y
x
KeysDisplayDescription
1Yj7G1
51. 67
J7J\2
1. 21
4
52.88Completes the calculation.
Example 2
Calculate natural logarithm (e
Table 2-25 Calculating the logarithm value
KeysDisplayDescription
2.5
). Then calculate 790 + 4!
12.18
G7V\K
\H
2.50Calculates natural logarithm of
jd:1Y\E
24.00Calculates 4 factorial.
Calculates + 47.20.
2
2.5
.
.
Calculates 1.1
Calculates e
the result.
4
814.00Completes calculation.
Example 3
The power operator, , raises the preceding number (y-value) to the power of the following
number (x -value).
3
Calculate 125
Table 2-26 Calculating the cube root
KeysDisplayDescription
JGV\QD4
JGV\QD\b4
, then find the cube root of 125.
1/3
3
.
.
1,953,125.00
5.00Calculates the cube root of
Calculates 125
125, or 125
Getting Started40
Last Answer
When a calculation is completed by pressing 4, or a calculation is completed during
another operation, the result is stored in a memory location that contains the last calculated
result. This enables the last result of a calculation to be used during the next calculation.
To access the last calculated answer, press v4. Unlike the other stored memory
registers however, this register is automatically updated when you complete a calculation.
Example 1
Table 2-27 Using last answer
KeysDisplayDescription
VAJ7GV4
D\Qv4
4
Example 2
Table 2-28 Using last answer with ‘INPUT’
KeysDisplayDescription
V:Æ
GG1JY\¨
S:Æ
3.75Calculate 5-1.25
3.75Recall last answer.
61. 5 5
50.00Store 50 in the INPUT register.
-28.00Calculate percent change.
60.00Store 60 in the INPUT register.
Calculate 3
3.75
.
v4
36.00Recalls last calculation, 22+14.
-40.00Calculate percent change.
\¨
Display Format of Numbers
When you turn on the HP 10bII+ for the first time, numbers are displayed with two decimal
places and a period as the decimal point. The display format controls how many digits
appear in the display.
If the result of a calculation is a number containing more significant digits than can be
displayed in the current display format, the number is rounded to fit the current display setting.
Getting Started
41
Regardless of the current display format, each number is stored internally as a signed, 12-digit
number with a signed, three-digit exponent.
Specifying Displayed Decimal Places
To specify the number of displayed decimal places:
1.P r e s s \5 followed by :–d for the desired decimal setting.
2.\5 followed by 7, v, or s changes the display mode. Pressing
\7 provides the best estimate and displays as many digits as required. v is
the value for 10, and s for 11.
Table 2-29 Example displaying the number of decimal places
KeysDisplayDescription
\M
0.00Clears display.
\5D
0.000Displays three decimal places.
YV7SP
5.727
7JGVS4
\5d
5.727360000Displays nine decimal places.
\5G
5.73Restores two decimal places.
When a number is too large or too small to be displayed in DISP format, it automatically
displays in scientific notation.
Displaying the Full Precision of Numbers
To set your calculator to display numbers as precisely as possible, press \57
(trailing zeros are not displayed.) To temporarily view all 12 digits of the number in the display
(regardless of the current display format setting), press
is displayed as long as you continue holding 4. The decimal point is not shown.
Start with two decimal places
Getting Started42
\5G.
\5 and hold 4. The number
Table 2-30 Example displaying all digits
KeysDisplayDescription
Jaj4
1. 4 3D i v i d e s .
\54
142857142857Displays all 12 digits.
Scientific Notation
Scientific notation is used to represent numbers that are too large or too small to fit in the
display. For example, if you enter the number 10,000,000 x 10,000,000 =, the result is
1.0 0E 14, which means one times ten to the fourteenth power, or 1.00 with the decimal point
moved fourteen places to the right. You can enter this number by pressing J\
zJY. The E stands for exponent of ten.
Exponents can also be negative for very small numbers. The number 0.000000000004 is
displayed as 4.00E–12, which means four times ten to the negative twelfth power, or 4.0 with
the decimal point moved 12 places to the left. You can enter this number by pressing
Y\zyJG.
Interchanging the Period and Comma
To switch between the period and comma (United States and International display) used as
the decimal point and digit separator, press \8.
For example, one million can be displayed as 1,000,000.00 or 1.000.000,00.
Pressing
\8, toggles between these options.
Rounding Numbers
The calculator stores and calculates using 12-digit numbers. When 12 digit accuracy is not
desirable, use
calculation. Rounding numbers is useful when you want the actual (dollars and cents) monthly
payment.
\} to round the number to the displayed format before using it in a
Getting Started
43
Table 2-31 Example displaying rounding off numbers
KeysDisplayDescription
d7gjSVYD
9.87654321_Enters a number with more than
two nonzero decimal places.
GJ
\5G
9.88Displays two decimal places.
\54
(while you press
4).
\}
987654321000Displays all digits without the
decimal.
9.88Rounds to two decimal places
(specified by pressing
\5G).
988000000000Shows rounded, stored number.
\54
Messages
The HP 10bII+ displays messages about the status of the calculator or informs you that you
have attempted an incorrect operation. To clear a message from the display, press M or
|. For a complete list of error messages, refer to Appendix C.
Getting Started44
3Business Percentages
The Business Percentage Keys
When entering data for business percentage calculations, results are calculated based on
data entered into specific memory registers. When pressed, the keys used for these
operations:
•store data.
•enter known data for variables used during calculations.
•calculate unknown variables based on stored data.
You can use the 10bII+ to calculate simple percent, percent change, cost, price, margin, and
markup.
Percent key
The § key has two functions:
•Finding a percent
•Adding or subtracting a percent
Finding a Percent
The § key divides a number by 100 unless it is preceded by an addition or subtraction
sign.
Example
Find 25% of 200.
Table 3-1 Finding a percent
KeysDisplayDescription
200.00Enters 200.
G::P
GV§
0.25Converts 25% to a decimal.
4
50.00Multiplies 200 by 25%.
Adding or Subtracting a Percent
You can add or subtract a percent in one calculation.
Example 1
Decrease 200 by 25%.
Business Percentages
45
Table 3-2 Subtracting a percent in a calculation
KeysDisplayDescription
G::A
200.00Enters 200.
GV§
50.00Multiplies 200 by 0.25 and
subtracts 50 from 200.
150.00Completes the calculation.
4
Example 2
You borrow 1,250 from a relative, and you agree to repay the loan in a year with 7% simple
interest. How much money will you owe?
Table 3-3 Adding a percent in a calculation
KeysDisplayDescription
JGV:1j§
4
Percent Change
1,337.50Calculates loan interest, 87.50
and adds 87.50 and 1250.00 to
show the repayment amount.
Calculate the percent change between two numbers.
Example 1
Calculate the percent change between 291.7 and 316.8 using the in-line feature.
Table 3-4 Calculating the percent change
KeysDisplayDescription
291.70Enters number1.
GdJ7j\¨
DJS7g4
8.60Calculates percent change.
Example 2
Calculate the percent change between (12 × 5) and (65 + 18) using Æ.
Table 3-5 Calculating the percent change between two numbers
KeysDisplayDescription
JGPVÆ
60.00Calculates and enters number1.
Note the INPUT annunciator.
SV1Jg\¨
38.33Calculates percent change.
For more information on in-line features, refer to chapter 2, Getting Started.
Business Percentages46
Margin and Markup Calculations
The 10bII+ can calculate cost, selling price, margin, or markup.
Table 3-6 Keys for margin and markup
ApplicationKeysDescription
Margin
À, ¼, ®
Markup
À, ¼, Ã
To see any value used by the margin and markup application, press v and then the key
you wish to see. For example, to see the value stored as À, press vÀ.
Margin is markup expressed as a
percent of price.
Markup calculations are expressed
as a percent of cost.
Margin Calculations
Example
Kilowatt Electronics purchases televisions for 255. The televisions are sold for 300. What is
the margin?
Table 3-7 Calculating the margin
KeysDisplayDescription
255.00Stores cost in CST.
GVVÀ
D::¼
300.00Stores selling price in PRC.
®
15. 00Ca l cu lat es m a rg i n.
Markup on Cost Calculations
Example
The standard markup on costume jewelry at Kleiner’s Kosmetique is 60%. They just received
a shipment of chokers costing 19.00 each. What is the retail price per choker?
Table 3-8 Calculating the retail price
KeysDisplayDescription
19.0 0St or es co st.
JdÀ
S:Ã
60.00Stores markup.
Business Percentages
47
Table 3-8 Calculating the retail price
KeysDisplayDescription
¼
30.40Calculates retail price.
Using Margin and Markup Together
Example
A food cooperative buys cases of canned soup with an invoice cost of 9.60 per case. If the
co-op routinely uses a 15% markup, for what price should it sell a case of soup? What is the
margin?
Table 3-9 Calculating the margin
KeysDisplayDescription
9.60Stores i nvoic e cost.
d7SÀ
JVÃ
15. 00St or es m a rk up .
¼
11.04Calculates the price on a case of
soup.
13 . 0 4C a l c u l a t e s margin.
®
Business Percentages48
4Number Storage and Storage Register Arithmetic
Using Stored Numbers in Calculations
You can store numbers for reuse in several different ways:
•Use ª (Constant) to store a number and its operator for repetitive operations.
•Use 3 Key Memory (
keystroke.
•Use
Using Constants
Use ª to store a number and arithmetic operator for repetitive calculations. Once the
constant operation is stored, enter a number and press 4. The stored operation is
performed on the number in the display.
Example 1
Calculate 5 + 2, 6 + 2, and 7 + 2.
Table 4-1 Storing ‘+2’ as constant
KeysDisplayDescription
\w and v to store to, and recall from, the 20 numbered registers.
s, p, and m) to store, recall, and sum numbers with a single
2.00Stores + 2 as constant.
V1Gª
4
S4
j4
7.00Adds 5 + 2.
8.00Adds 6 + 2.
9.00Adds 7 + 2.
Number Storage and Storage Register Arithmetic
49
Example 2
Calculate 10 + 10%, 11 + 10%, and 25 + 10%.
Table 4-2 Storing ‘+ 10%’ as a constant
KeysDisplayDescription
J:1J:§ª
1. 0 0S t o r e s + 10% as a constant.
4
11.0 0Adds 10% to 10.
4
12.10A dds 10% t o 11.
GV4
27.50Adds 10% to 25.
Example 3
Calculate 23 and 43.
Table 4-3 Storing ‘y
KeysDisplayDescription
3
’ as a constant
3.00
G\QDª
4
8.00
Stores
Calculates 2
3
y
as constant.
3
.
Y4
64.00
Calculates 4
3
.
Number Storage and Storage Register Arithmetic50
Example 4
Calculate the percent change between 55 and 32 and store it as a constant. Then calculate
the percent change between 50 and 32, and 45 and 32.
Table 4-4 Calculating percent change
KeysDisplayDescription
VV\¨DGª
32.00Stores % change 32 as constant.
4
V:4
YV4
All of the other two-number functions on the calculator may be used with ª in the same
manner as shown in example 4. For a complete list of two-number functions, refer to the
section titled, Two -Number Functions in chapter 2.
Using the M Register
The s, p, and m keys perform memory operations on a single storage register,
called the M register. In most cases, it is unnecessary to clear the M register, since s
-41.82Calculates the % change
between 55 and 32.
-36.00Calculates the % change
between 50 and 32.
-28.89Calculates the % change
between 45 and 32.
replaces the previous contents. However, you can clear the M register by pressing :s.
To add a series of numbers to the M register, use s to store the first number and m to
add subsequent numbers. To subtract the displayed number from the number in the M register,
press y followed by m.
Table 4-5 Keys for performing memory operations
KeysDescription
Stores displayed number in the M register.
s
p
Recalls number from the M register.
m
Adds displayed number to the M register.
Number Storage and Storage Register Arithmetic
51
Example
475.6
39.15
----------------
and
560.1475.6+
39.15
---------------------------------------
Use the M register to add 17, 14.25, and 16.95. Then subtract 4.65 and recall the result.
Table 4-6 Calculating basic arithmetic operations using M register
KeysDisplayDescription
Jjs
17.00S to re s 17 in M re gi st er.
JY7GVm
14.2 5A dds 14 . 25 to M r eg i s te r.
JS7dVm
16.95Adds 16.95 to M register.
Y7SVym
-4.65Adds -4.65 to M register.
p
Using Numbered Registers
The \w and v keys access the 20 user registers, designated 0-19. The
43.55Recalls contents of the M
register.
\w key is used to copy the displayed number to a designated register. The v key
is used to copy a number from a register to the display.
To store or recall a number in two steps:
•Press \w or v. To cancel this step, press | or M.
•Press
\w followed by a number key, : to d, or 7 and : to d, to store a
number in the display into a numbered data storage register. Press
\w7 followed by
: through d to access registers 10-19.
•Press
Example
In the following example, two storage registers are used. Set the calculator for CHAIN mode
(]?) and calculate the following:
v followed by a number key, : to d, or 7 and : to d, to recall a
number from a storage register. Press
10 -19.
v7 followed by : through d to access registers
Number Storage and Storage Register Arithmetic52
Table 4-7 Calculating the expression using two storage registers
KeysDisplayDescription
YjV7S
\w7Y
aDd7JV
\wG
4
VS:7J1
v7Y
avG
4
475.60Stores 475.60 (displayed
number) in R
39.15Stores 39.15 in R
12.15Completes first calculation.
1,035.70Recalls R
NOTE: If the calculator is set for
Algebraic mode, press 4 at the
end of this step.
39.15Recalls R
26.45Completes second calculation.
.
14
.
2
.
14
.
2
With the exception of the statistics registers, you can also use \w and v for
application registers. For example, \wÒ stores the number from the display in the
Ò register. vÒ copies the contents from Ò to the display.
In most cases, it is unnecessary to clear a storage register since storing a number replaces the
previous contents. However, you can clear a single register by storing 0 in it. To clear all the
registers at once, press \N.
Doing Arithmetic Inside Registers
You can do arithmetic inside storage registers R0 through R19. The result is stored in the
register.
Table 4-8 Keys for performing arithmetic inside registers
KeysNew Number in Register
\w1 registernumber
\wA registernumber
Old contents + displayed number.
Old contents - displayed number.
\wP registernumber
Old contents × displayed number.
Number Storage and Storage Register Arithmetic
53
Table 4-8 Keys for performing arithmetic inside registers
KeysNew Number in Register
\wa registernumber
Example 1
Store 45.7 in R3, multiply by 2.5, and store the result in R3.
Table 4-9 Calculating and storing the result in the storage register
KeysDisplayDescription
Old contents ÷ displayed number.
45.70Stores 45.7 in R
YV7j
\wD
G7V
2.50Multiplies 45.7 in R
\wPD
vD
114 . 2 5D i s p l a y s R
.
3
by 2.5 and
3
stores result (114.25) in R
.
3
.
3
Example 2
Store 1.25 into register 15, then add 3, and store the result in register 15.
Table 4-10 Storage register arithmetic
KeysDisplayDescription
1. 25Inp u ts 1. 25 i n to t h e di s p lay.
J7GV
\w7V
1. 25St o re s 1.25 in R
D\w17V
3.00Adds 3 to 1.25 in R
stores the result R
0.00Clears the display.
M
v7V
4.25Recalls R
.
15
.
15
and
15
.
15
Number Storage and Storage Register Arithmetic54
5Picturing Financial Problems
How to approach a Financial Problem
The financial vocabulary of the HP 10bII+ is simplified to apply to all financial fields. For
example, your profession may use the term balance, balloonpayment, residual, maturity
value, or remainingamount to designate a value that the HP 10bII+ knows as É (future
value).
The simplified terminology of the HP 10bII+ is based on cash flow diagrams. Cash flow
diagrams are pictures of financial problems that show cash flows over time. Drawing a cash
flow diagram is the first step to solving a financial problem.
The following cash flow diagram represents investments in a mutual fund. The original
investment was 7,000.00, followed by investments of 5,000.00 and 6,000.00 at the end of
the third and sixth months. At the end of the 11th month, 5,000.00 was withdrawn. At the end
of the 16th month, 16,567.20 was withdrawn.
Figure 2 Cash flow diagram
Any cash flow example can be represented by a cash flow diagram. As you draw a cash flow
diagram, identify what is known and unknown about the transaction.
Time is represented by a horizontal line divided into regular time periods. Cash flows are
placed on the horizontal line when they occur. Where no arrows are drawn, no cash flows
occur.
Picturing Financial Problems
55
Signs of Cash Flows
In cash flow diagrams, money invested is shown as negative and money withdrawn is shown
as positive. Cash flowing out is negative, cash flowing in is positive.
For example, from the lender’s perspective, cash flows to customers for loans are represented
as negative. Likewise, when a lender receives money from customers, cash flows are
represented as positive. In contrast, from the borrower’s perspective, cash borrowed is positive
while cash paid back is negative.
Periods and Cash Flows
In addition to the sign convention (cash flowing out is negative, cash flowing in is positive) on
cash flow diagrams, there are several more considerations:
•The time line is divided into equal time intervals. The most common period is a month, but days,
quarters, and annual periods are also common. The period is normally defined in a contract and
must be known before you can begin calculating.
•To solve a financial problem with the HP 10bII+, all cash flows must occur at either the beginning or
end of a period.
•If more than one cash flow occurs at the same place on the cash flow diagram, they are added
together or netted. For example, a negative cash flow of -250.00 and a positive cash flow of 750.00
occurring at the same time on the cash flow diagram are entered as a 500.00 cash flow (750 - 250
= 500).
•A valid financial transaction must have at least one positive and one negative cash flow.
Simple and Compound Interest
Financial calculations are based on the fact that money earns interest over time. There are two
types of interest:
•Simple interest
•Compound interest
The basis for Time Value of Money and cash flow calculations is compound interest.
Simple Interest
In simple-interest contracts, interest is a percent of the original principal. The interest and
principal are due at the end of the contract. For example, say you loan 500 to a friend for a
year, and you want to be repaid with 10% simple interest. At the end of the year, your friend
owes you 550.00 (50 is 10% of 500). Simple interest calculations are done using the
key on your HP 10bII+. An example of a simple interest calculation can be found in chapter
6 under the section titled, Interest Rate Conversions.
§
Picturing Financial Problems56
Compound Interest
A compound-interest contract is like a series of simple-interest contracts that are connected.
The length of each simple-interest contract is equal to one compounding period. At the end
of each period the interest earned on each simple-interest contract is added to the principal.
For example, if you deposit 1,000.00 in a savings account that pays 6% annual interest,
compounded monthly, your earnings for the first month look like a simple-interest contract
written for 1 month at 1/2% (6% ÷ 12). At the end of the first month the balance of the account
is 1,005.00 (5 is 1/2% of 1,000).
The second month, the same process takes place on the new balance of 1,005.00. The
amount of interest paid at the end of the second month is 1/2% of 1,005.00, or 5.03. The
compounding process continues for the third, fourth, and fifth months. The intermediate results
in this illustration are rounded to dollars and cents.
Figure 3 Annual interest compounded monthly
The word compound in compound interest comes from the idea that interest previously earned
or owed is added to the principal. Thus, it can earn more interest. The financial calculation
capabilities of the HP 10bII+ are based on compound interest.
Interest Rates
When you approach a financial problem, it is important to recognize that the interest rate or
rate of return can be described in at least three different ways:
Picturing Financial Problems
57
•As a periodic rate. This is the rate that is applied to your money from period to period.
•As an annual nominal rate. This is the periodic rate multiplied by the number of periods in a year.
•As an annual effective rate. This is an annual rate that considers compounding.
In the previous example of a 1,000.00 savings account, the periodic rate is 1/2% (per month),
quoted as an annual nominal rate of 6% (1/2 × 12). This same periodic rate could be quoted
as an annual effective rate, which considers compounding. The balance after 12 months of
compounding is 1,061.68, which means the annual effective interest rate is 6.168%.
Examples of converting between nominal and annual effective rates can be found in the
section titled, Interest Rate Conversions in the next chapter.
Two Types of Financial Problems
The financial problems in this manual use compound interest unless specifically stated as
simple interest calculations. Financial problems are divided into two groups:
•TVM problems
•Cash flow problems
Recognizing a TVM Problem
If uniform cash flows occur between the first and last periods on the cash flow diagram, the
financial problem is a TVM (time value of money) problem. There are five main keys used to
solve a TVM problem.
Table 5-1 Keys for solving a TVM problem
KeysDescription
Number of periods or payments
Ù
Ò
Ï
Annual percentage interest rate (usually the annual nominal
rate)
Present value (the cash flow at the beginning of the time
line)
Periodic payment
Ì
É
You can calculate any value after entering the other four values. Cash flow diagrams for
loans, mortgages, leases, savings accounts, or any contract with regular cash flows of the
same amount are normally treated as TVM problems.
Future value (the cash flow at the end of the cash flow
diagram, in addition to any regular periodic payment).
For example, following is a cash flow diagram, from the borrower’s perspective, for a 30-year,
150,000.00 mortgage, with a payment of 1,041.40, at 7.5% annual interest, with a 10,000
balloon payment.
One of the values for PV, PMT, FV can be zero. For example, following is a cash flow diagram
(from the saver’s perspective) for a savings account with a single deposit and a single
withdrawal five years later. Interest compounds monthly. In this example, PMT is zero.
Figure 5 Cash flow diagram (Saving perspective)
Time value of money calculations are described in the next chapter titled, Time Value of
Money Calculations.
Recognizing a Cash Flow Problem
A financial problem that does not have regular, uniform payments (sometimes called uneven
cash flows) is a cash flow problem rather than a TVM problem.
Picturing Financial Problems
59
The following is a cash flow diagram for an investment in a mutual fund. This is an example
of a problem that is solved using either \½ (Net Present Value) or \Á
(Internal Rate of Return per Year).
Figure 6 Cash flow diagram (Investment in a mutual fund)
Cash flow problems are described in chapter 8 titled, Cash Flow Calculations.
Picturing Financial Problems60
6Time Value of Money Calculations
Using the TVM Application
The time value of money (TVM) application is used for compound interest calculations that
involve regular, uniform cash flows – called payments. Once the values are entered you can
vary one value at a time, without entering all the values again.
To use TVM, several prerequisites must be met:
•The amount of each payment must be the same. If the payment amounts vary, use the procedures
described in chapter 8 titled, Cash Flow Calculations.
•Payments must occur at regular intervals.
•The payment period must coincide with the interest compounding period. If it does not, convert the
interest rate using the
section titled, Interest Rate Conversions.
•There must be at least one positive and one negative cash flow.
\Ó, \Ð, and \Í keys described below in the
The TVM Keys
When entering data for TVM calculations, results are calculated based on data entered into
specific memory registers. When pressed, the keys used for these operations:
•store data.
•enter known data for variables used during calculations.
•calculate unknown variables based on stored data.
Table 6-1 Keys for performing TVM calculations
KeysStores or Calculates
Number of payments or compounding periods.
Ù
Ò
Annual nominal interest rate.
Ï
Ì
É
\Í
\Ú
Present value of future cash flows. PV is usually an initial investment or
loan amount and always occurs at the beginning of the first period.
Amount of periodic payments. All payments are equal, and none are
skipped; payments can occur at the beginning or end of each period.
Future value. FV is either a final cash flow or compounded value of a
series of previous cash flows. FV occurs at the end of the last period.
Stores the number of periods per year. The default is 12. Reset only
when you wish to change it.
Optional shortcut for storing N: number in display is multiplied by the
value in P/YR and the result is stored in N.
Time Value of Money Calculations
61
Table 6-1 Keys for performing TVM calculations
KeysStores or Calculates
\¯
Switches between Begin and End mode. In Begin mode, the BEGIN
annunciator is displayed.
Calculates an amortization table.
\Ê
To verify values, press vÙ, vÒ, vÏ, vÌ, and vÉ.
Pressing v\Ú recalls the total number of payments in years and v\Í
shows you the number of payments per year. Recalling these numbers does not change the
content of the registers.
Begin and End Modes
Before you start a TVM calculation, identify whether the first periodic payment occurs at the
beginning or end of the first period. If the first payment occurs at the end of the first period,
set your HP 10bII+ to End mode; if it occurs at the beginning of the first period, set your
calculator to Begin mode.
To switch between modes, press \¯. The BEGIN annunciator is displayed when your
calculator is in Begin mode. No annunciator is displayed when you are in End mode.
Mortgages and loans typically use End mode. Leases and savings plans typically use Begin
mode.
Loan Calculations
Example: A Car Loan
You are financing a new car with a three year loan at 10.5% annual nominal interest,
compounded monthly. The price of the car is 14,500. Your down payment is 1,500.
Part 1
What are your monthly payments at 10.5% interest? (Assume your payments start one month
after the purchase or at the end of the first period.)
Time Value of Money Calculations62
Figure 7 Cash flow diagram (Calculate PMT)
Set to End mode. Press \¯ if BEGIN annunciator is displayed.
Table 6-2 Calculating the monthly payment
KeysDisplayDescription
JG\Í
12.00Sets periods per year (optional, as
12 is the default).
36.00Stores number of periods in loan.
DPJGÙ
J:7VÒ
10.50Stores annual nominal interest rate.
JYV::A
13,000.00Stores amount borrowed.
JV::Ï
:É
Ì
Part 2
At a price of 14,500, what interest rate is necessary to lower your payment by 50.00, to
372.53?
0.00Stores the amount left to pay after 3
years.
-422.53Calculates the monthly payment.
The negative sign indicates money
paid out.
Time Value of Money Calculations
63
Table 6-3 Calculating the interest rate
KeysDisplayDescription
1V:Ì
-372.53Decreases payment from 422.53.
Ò
Part 3
If interest is 10.5%, what is the maximum you can spend on the car to lower your car payment
to 375.00?
Table 6-4 Calculating the amount
KeysDisplayDescription
2.03Calculates annual interest rate for
the reduced payment.
10.50Stores original interest rate.
J:7VÒ
DjVyÌ
-375.00Stores desired payment.
Ï
1JV::4
11,537.59Calculates amount of money to
finance.
13,037.59Adds the down payment to the
amount financed for total price
of the car.
Example: A Home Mortgage
You decide that the maximum monthly mortgage payment you can afford is 930.00. You can
make a 12,000 down payment, and annual interest rates are currently 7.5%. If you obtain a
30 year mortgage, what is the maximum purchase price you can afford?
Figure 8 Cash flow diagram (Calculate PV)
Time Value of Money Calculations64
Set to End mode. Press \¯ if BEGIN annunciator is displayed.
Table 6-5 Calculating the maximum purchase price
KeysDisplayDescription
JG\Í
12.00Sets periods per year.
D:\Ú
360.00Stores the length of the
mortgage (30 × 12).
0.00Pays mortgage off in 30 years.
:É
j7VÒ
7.50Stores interest rate.
dD:yÌ
Ï
1JG:::4
Example: A Mortgage With a Balloon Payment
You’ve obtained a 25 year, 172,500 mortgage at 8.8% annual interest. You anticipate that
you will own the house for four years and then sell it, repaying the loan with a balloon
payment. What will your balloon payment be?
-930.00Stores desired payment (money
paid out is negative).
133,006.39Calculates the loan you can
afford with a 930 payment.
145,006.39Adds 12,000 down payment
for the total purchase price.
Solve this problem using two steps:
1.Calculate the loan payment using a 25 year term.
2.Calculate the remaining balance after 4 years.
Step 1
First calculate the loan payment using a 25 year term.
Time Value of Money Calculations
65
Figure 9 Cash flow diagram (Calculate PMT)
Set to End mode. Press \¯ if BEGIN annunciator is displayed.
Table 6-6 Calculating the monthly payment
KeysDisplayDescription
JG\Í
12.00Sets periods per year.
GV\Ú
:É
300.00Stores length of mortgage (25 ×
12 = 3 00 m ont hs) .
0.00Stores loan balance after 25
years.
172,500.00Stores original loan balance.
JjGV::Ï
g7gÒ
8.80Stores annual interest rate.
Ì
-1,424.06Calculates the monthly payment.
Step 2
Since the payment is at the end of the month, the past payment and the balloon payment
occur at the same time. The final payment is the sum of PMT and FV.
Time Value of Money Calculations66
Figure 10 Cash flow diagram (Calculate FV)
The value in PMT should always be rounded to two decimal places when calculating FV or
PV to avoid small, accumulative discrepancies between non-rounded numbers and actual
(dollars and cents) payments. If the display is not set to two decimal places, press
\5G.
Table 6-7 Calculating the final amount
KeysDisplayDescription
\}Ì
YgÙ
É
1vÌ4
-1,424.06Rounds payment to two decimal
places, then stores.
48.00Stores four year term (12 × 4)
that you expect to own house.
-163,388.39Calculates loan balance after
four years.
-164 ,812. 45
Calculates the total 48
payment (PMT and FV) to pay
off the loan (money paid out is
negative).
th
Savings Calculations
Example: A Savings Account
If you deposit 2,000 in a savings account that pays 7.2% annual interest compounded
annually, and make no other deposits to the account, how long will it take for the account to
grow to 3,000?
Time Value of Money Calculations
67
Figure 11 Cash flow diagram (Calculate the number of years)
Since this account has no regular payments (PMT = 0), the payment mode (End or Begin) is
irrelevant.
Table 6-8 Calculating the number of years
KeysDisplayDescription
]OJ
0.00Clears TVM memory.
J\Í
G:::yÏ
D:::É
1. 0 0S e t s P/YR to 1 since interest is
compounded annually.
-2,000.00Stores amount paid out for the
first deposit.
3,000.00Stores the amount you wish to
accumulate.
7.20Stores annual interest rate.
j7GÒ
Ù
Since the calculated value of N is between 5 and 6, it will take six years of annual
compounding to achieve a balance of at least 3,000. Calculate the actual balance at the end
of six years.
5.83Calculates the number of years it
takes to reach 3,000.
Time Value of Money Calculations68
Table 6-9 Calculating the balance after six years
KeysDisplayDescription
SÙ
6.00Sets n to 6 years.
É
Example: An Individual Retirement Account
You opened an individual retirement account on April 14, 1995, with a deposit of 2,000.
80.00 is deducted from your paycheck and you are paid twice a month. The account pays
6.3% annual interest compounded semimonthly. How much will be in the account on
April 14, 2010?
3,035.28Calculates the amount you can
withdraw after six years.
Figure 12 Cash flow diagram (Calculate FV)
Set to End mode. Press \¯ if BEGIN annunciator is displayed.
Table 6-10 Calculating the balance amount
KeysDisplayDescription
24.00Sets number of periods per year.
GY\Í
G:::yÏ
-2,000.00Stores initial deposit.
g:yÌ
-80.00Stores regular semimonthly
deposits.
Time Value of Money Calculations
69
Table 6-10 Calculating the balance amount
KeysDisplayDescription
S7DÒ
6.30Stores interest rate.
JV\Ú
360.00Stores the number of deposits.
É
52,975.60Calculates the balance amount.
Example: An Annuity Account
You opt for an early retirement after a successful business career. You have accumulated a
savings of 400,000 that earns an average of 7% annual interest, compounded monthly. What
annuity (repetitive, uniform, withdrawal of funds) will you receive at the beginning of each
month if you wish that savings account to support you for the next 50 years?
Figure 13 Cash flow diagram (Calculate the amount)
Set to Begin mode. Press \¯ if BEGIN annunciator is not displayed.
Table 6-11 Calculating the amount at the beginning of each month
KeysDisplayDescription
12.00Sets payments per year.
JG\Í
Y:::::yÏ
jÒ
-400,000.00Stores your nest egg as an
outgoing deposit.
7.00Stores annual interest rate you
expect to earn.
600.00Stores number of withdrawals.
V:\Ú
:É
Ì
0.00Stores balance of account after
50 years.
2,392.80Calculates the amount that you
can withdraw at the beginning
of each month.
Time Value of Money Calculations70
Lease Calculations
A lease is a loan of valuable property (like real estate, automobiles, or equipment) for a
specific amount of time, in exchange for regular payments. Some leases are written as
purchase agreements, with an option to buy at the end of the lease (sometimes for as little as
1.00). The defined future value (FV) of the property at the end of a lease is sometimes called
the residual value or buy out value.
All five TVM application keys can be used in lease calculations. There are two common lease
calculations.
•Finding the lease payment necessary to achieve a specified yield.
•Finding the present value (capitalized value) of a lease.
The first payment on a lease usually occurs at the beginning of the first period. Thus, most
lease calculations use Begin mode.
Example: Calculating a Lease Payment
A customer wishes to lease a 13,500 car for three years. The lease includes an option to buy
the car for 7,500 at the end of the lease. The first monthly payment is due the day the customer
drives the car off the lot. If you want to yield 10% annually, compounded monthly, what will
the payments be? Calculate the payments from your (the dealer’s) point of view.
Figure 14 Cash flow diagram (Calculate the monthly lease payment)
Set to Begin mode. Press \¯ if BEGIN annunciator is not displayed.
Time Value of Money Calculations
71
Table 6-12 Calculating the monthly lease payment
KeysDisplayDescription
JG\Í
12.00Sets payments per year.
J:Ò
10.00Stores desired annual yield.
JDV::yÏ
-13,500.00Stores lease price.
jV::É
7,500.00Stores residual (buy out value).
DSÙ
36.00Stores length of lease, in months.
Ì
Notice that even if the customer chooses not to buy the car, the lessor still includes a cash flow
coming in at the end of the lease equal to the residual value of the car. Whether the customer
buys the car or it is sold on the open market, the lessor expects to recover 7,500.
Example: Lease With Advance Payments
Your company, Quick-Kit Pole Barns, plans to lease a forklift for the warehouse. The lease is
written for a term of four years with monthly payments of 2,400. Payments are due at the
beginning of the month with the first and last payments due at the onset of the lease. You have
an option to buy the forklift for 15,000 at the end of the leasing period.
253.99Calculates the monthly lease
payment.
If the annual interest rate is 9%, what is the capitalized value of the lease?
Figure 15 Cash flow diagram (Calculate PV of the lease)
Time Value of Money Calculations72
This solution requires four steps:
1.Calculate the present value of the 47 monthly payments: (4 × 12) - 1 = 47.
2.Add the value of the additional advance payment.
3.Find the present value of the buy option.
4.Sum the values calculated in steps 2 and 3.
Step 1
Find the present value of the monthly payments.
Set to Begin mode. Press \¯ if BEGIN annunciator is not displayed.
Table 6-13 Calculating the present value
KeysDisplayDescription
JG\Í
12.00Sets payments per year.
YjÙ
47.00Stores number of payments.
GY::yÌ
-2,400.00Stores monthly payment.
:É
0.00Stores FV for Step 1.
dÒ
9.00Stores interest rate.
Ï
Step 2
Add the additional advance payment to PV. Store the answer.
Table 6-14 Adding the advance payment
KeysDisplayDescription
1vÌy4
95,477.55Calculates the present value of
47 monthly payments.
97,877.55Adds additional advance.
payment
s
97,877.55Stores result in M register.
Step 3
Find the present value of the buy option.
Table 6-15 Calculating the present value of the last cash flow
KeysDisplayDescription
48.00Stores month when buy option
YgÙ
occurs.
Time Value of Money Calculations
73
Table 6-15 Calculating the present value of the last cash flow
KeysDisplayDescription
:Ì
0.00Stores zero payment for this step
of solution.
-15,000.00Stores value to discount.
JV:::yÉ
Ï
Step 4
Add the results of ’Step 2’ and ’Step 3’.
Table 6-16 Calculating the present value of lease
KeysDisplayDescription
108,356.77Calculates the present (capitalized)
1p4
Amortization
Amortization is the process of dividing a payment into the amount that applies to interest and
the amount that applies to principal. Payments near the beginning of a loan contribute more
interest, and less principal, than payments near the end of a loan.
10,479.21Calculates the present value of
last cash flow.
value of lease. (Rounding
discrepancies are explained on
page 67.)
Figure 16 Graph
Time Value of Money Calculations74
The AMORT key on the HP 10bII+ allows you to calculate.
•The amount applied to interest in a range of payments.
•The amount applied to principal in a range of payments.
•The loan balance after a specified number of payments are made.
The \Ê function assumes you have just calculated a payment or you have stored the
appropriate amortization values in I/YR, PV, FV, PMT, and P/YR.
Table 6-17 Keys for storing the amortization values
KeysDescription
Ò
Annual nominal interest rate.
Ï
Starting balance.
É
Ending balance.
Ì
Payment amount (rounded to the display format).
\Í
Number of payments per year.
The numbers displayed for interest, principal, and balance are rounded to the current display
setting.
To Amortize
To amortize a single payment, enter the period number and press \Ê . The HP 10bII+
displays the annunciator PER followed by the starting and ending payments that will be
amortized.
4 to see interest (INT). Press 4 again to see the principal (PRIN) and again to see
Press
the balance (BAL). Continue pressing 4 to cycle through the same values again.
To amortize a range of payments, enter starting period number
then press
and ending payments that will be amortized. Then press
interest, principal, and balance.
Press \Ê again to move to the next set of periods. This auto-increment feature saves
you the keystrokes of entering the new starting and ending periods.
\Ê. The HP 10bII+ displays the annunciator PER followed by the starting
Æ endingperiod number,
4 repeatedly to cycle through
Time Value of Money Calculations
75
If you store, recall, or perform any other calculations during amortization, pressing 4 will
no longer cycle through interest, principal, and balance. To resume amortization with the
same set of periods, press v\Ê.
Example: Amortizing a Range of Payments
Calculate the first two years of the annual amortization schedule for a 30 year, 180,000
mortgage, at 7.75% annual interest with monthly payments.
Set to End mode. Press \¯ if BEGIN annunciator is displayed.
Table 6-18 Calculating the monthly payment
KeysDisplayDescription
JG\Í
12.00Sets payments per year.
D:\Ú
360.00Stores total number of payments.
j7jVÒ
7. 75S t or es i n t er est p er y e ar.
Jg::::Ï
180,000.00Stores present value.
:É
0.00Stores future value.
Ì
-1,289.54Calculates the monthly payment.
If you already know the mortgage payment, you can enter and store it just like you store the
other four values. Next, amortize the first year.
Table 6-19 Calculating the loan balance after a year
KeysDisplayDescription
JÆJG
12_Enters starting and ending
periods.
Time Value of Money Calculations76
\Ê
4
4
4
1– 12Displays the PER and AMORT
annunciators and range.
-1,579.84Displays the PRIN annunciator
and the principal paid in the first
year.
-13,894. 67D is pl ays th e INT annunciator
and the interest paid in the first
year.
178,420 .16D is p la ys the BAL annunciator
and the loan balance after one
year.
The amount paid toward interest and principal (13,894.67 + 1,579.84 = 15,474.51) equals
the total of 12 monthly payments (12 × 1,289.54 = 15,474.51). The remaining balance equals
the initial mortgage, less the amount applied toward principal (180,000 - 1,579.84 =
178 ,42 0 .16 ) .
Amortize the second year:
Table 6-20 Calculating the remaining balance
KeysDisplayDescription
JDÆGY
13 – 24Displays PER and the next range of
periods.
\Ê
4
4
4
The amount paid toward interest and principal (13,767.79 + 1,706.69 = 15,474.51) equals the
total of 12 monthly payments (12 × 1,289.54 = 15,474.51). The remaining balance equals the
initial mortgage less the amount applied toward principal (180,000 - 1,579.84 - 1,706.69 =
176,713.49). More money is applied to principal during the second year rather than the first
year. The succeeding years continue in the same fashion.
Example: Amortizing a Single Payment
Amortize the 1st, 25th, and 54th payments of a five year car lease. The lease amount is 14,250
and the interest rate is 11.5%. Payments are monthly and begin immediately.
-1,70 6. 69D is pl ay s PRIN and the principal
paid in the second year.
-13,767.79D is pl ay s INT and the interest paid
in the second year.
176 ,713 . 49D is p l ay s BAL and the loan balance
after 24 payments.
Set to Begin mode. Press
Table 6-21 Calculating the monthly payment
KeysDisplayDescription
\¯ if BEGIN annunciator is not displayed.
12.00Sets payments per year.
JG\Í
V\Ú
60.00Stores number of payments.
JJ7VÒ
11.50Stores interest per year.
JYGV:Ï
14,250.00Stores present value.
:É
0.00Stores future value.
Time Value of Money Calculations
77
Table 6-21 Calculating the monthly payment
KeysDisplayDescription
Ì
-310.42Calculates the monthly payment.
Amortize the 1st, 25th, and 54th payments
Table 6-22 Calculating the amount
KeysDisplayDescription
1.00Enters first payment.
JÆ
1 – 1Displays PER and the amortized payment
\Ê
-310.42Displays PRIN and the first principal
4
0.00Displays INT and the interest.
4
13 ,9 3 9. 5 8D i s p l a y s BAL and the loan balance after
4
25.00Enters payment to amortize.
GVÆ
period.
payment.
one payment.
\Ê
4
4
4
VYÆ
\Ê
4
4
4
25 – 25Displays PER and the amortized payment
period.
-220.21Displays PRIN and the principal paid on
th
the 25
-90.21Displays INT and the interest paid on the
25
9,193. 28Di spl ays BAL and the balance after the
25
54.00Enters payment to amortize.
54 – 54Displays PER and the amortized payment
period.
-290.37Displays PRIN and the principal paid on
the 54
-20.05Displays INT and the interest paid on the
54
1, 801. 57D i sp l a ys BAL and the balance after the
54
payment.
th
payment.
th
payment.
th
payment.
th
payment.
th
payment.
Time Value of Money Calculations78
Interest Rate Conversions
The Interest Conversion application uses three keys: \Ó, \Ð, and
\Í. They convert between nominal and annual effective interest rates.
If you know an annual nominal interest rate and you wish to solve for the corresponding
annual effective rate:
1.Enter the nominal rate and press \Ó.
2.Enter the number of compounding periods and press \Í.
3.Calculate the effective rate by pressing \Ð.
To calculate a nominal rate from a known effective rate:
1.Enter the effective rate and press \Ð.
2.Enter the number of compounding periods and press \Í.
3.Calculate the nominal rate by pressing \Ó.
In the TVM application, \Ó and Ò share the same memory.
Interest conversions are used primarily for two types of problems:
•Comparing investments with different compounding periods.
•Solving TVM problems where the payment period and the interest period differ.
Investments With Different Compounding Periods
Example: Comparing Investments
You are considering opening a savings account in one of three banks. Which bank has the
most favorable interest rate?
First Bank6.70% annual interest, compounded quarterly
Second Bank6.65% annual interest, compounded monthly
Third Bank6.63% annual interest, compounded 360 times per year
First Bank
Table 6-23 Calculating the interest rate (First bank)
KeysDisplayDescription
6.70Stores nominal rate.
S7j\Ó
Y\Í
4.00Stores quarterly compounding
periods.
Time Value of Money Calculations
79
Table 6-23 Calculating the interest rate (First bank)
KeysDisplayDescription
\Ð
6.87Calculates the annual effective rate.
Second Bank
Table 6-24 Calculating the interest rate (Second bank)
KeysDisplayDescription
6.65Stores nominal rate.
S7SV\Ó
12.00Stores monthly compounding
JG\Í
6.86Calculates the annual effective rate.
\Ð
Third Bank
Table 6-25 Calculating the interest rate (Third bank)
KeysDisplayDescription
6.63Stores nominal rate.
S7SD\Ó
periods.
DS:\Í
360.00Stores compounding periods.
\Ð
6.85Calculates the annual effective rate.
First Bank offers a slightly better deal since 6.87 is greater than 6.86 and 6.85.
Compounding and Payment Periods Differ
The TVM application assumes that the compounding periods and the payment periods are
the same. Some loan installments or savings deposits and withdrawals do not coincide with
the bank’s compounding periods. If the payment period differs from the compounding period,
adjust the interest rate to match the payment period before solving the problem.
To adjust an interest rate when the compounding period differs from the payment period
complete the following steps:
1.Enter the nominal rate and press
in a year and press \Í. Solve for the effective rate by pressing \Ð.
\Ó. Enter the number of compounding periods
2.Enter the number of payment periods in a year and press \Í. Solve for the
adjusted nominal rate by pressing \Ó.
Time Value of Money Calculations80
Example: Monthly Payments, Daily Compounding
Starting today, you make monthly deposits of 25 to an account paying 5% interest,
compounded daily (using a 365 day year). What will the balance be in seven years?
Step1
Calculate the equivalent rate with monthly compounding.
Table 6-26 Calculating the equivalent nominal percentage rate
KeysDisplayDescription
V\Ó
5.00Stores nominal percentage rate.
DSV\Í
365.00Stores bank’s compounding
periods per year.
5.13Calculates annual effective rate.
\Ð
JG\Í
12.0 0S to re s mo nth ly p er io ds.
\Ó
Since NOM% and I/YR share the same memory, this value is ready for use in the rest of the
problem.
Step2
Calculate the future value.
Set to Begin mode. Press \¯ if BEGIN annunciator is not displayed.
5.01Calculates the equivalent
nominal percentage rate for
monthly compounding.
Table 6-27 Calculating the future value
KeysDisplayDescription
0.00Stores present value
:Ï
GVyÌ
-25.00Stores payment
j\Ú
84.00Stores total number of payments
É
2,519.61Calculates the balance after 7
years.
Time Value of Money Calculations
81
Resetting the TVM Keys
Press ]OJ to clear the TVM registers. This sets N, I/YR, PV, PMT, and FV to zero
and briefly displays TVM CLR, followed by the current value in P/Yr.
Time Value of Money Calculations82
7Depreciation
On the 10bII+, depreciation calculations are performed using the functions printed in blue on
the keyboard located under the blue bracket titled, DEPRECIATION. Depreciation calculations
are based on data entered into the Time Value of Money (TVM) keys: Ï, É, Ò, and
Ù.
Table 7-1 Depreciation keys
TVM KeyDescription
]OJ
Ù
Ï
É
]{
]x
]u
Ò
Clear TVM memory. Since the TVM and depreciation applications
share the same memory, clearing TVM resets depreciation also.
The expected useful life of the asset in years.
The depreciable cost of the asset at acquisition.
The salvage value of the asset at the end of its useful life.
Straight line is a method of calculating depreciation presuming an
asset loses a certain percentage of its value annually at an amount
evenly distributed throughout its useful life.
Sum-of-the-years' digits is an accelerated depreciation method.
In SOYD, the depreciation in year y is (Life-y +1)/SOY of the asset,
where SOY is the sum-of-the-years for the asset, or, for an asset with a
5-year life, 5+4+3+2+1=15.
Declining balance is an accelerated depreciation method that
presumes an asset will lose the majority of its value during the first few
years of its useful life.
The declining balance factor as a percentage. This is used for
declining balance method.
\«
With the calculated depreciation displayed, press
display the remaining depreciable value at the end of the given year.
\« to
The Depreciation Keys
When entering data for depreciation calculations, results are calculated based on data
entered into specific memory registers. When pressed, the keys used for these operations:
•store data.
•enter known data for variables used during calculations.
•calculate unknown variables based on stored data.
Depreciation
83
To perform a depreciation calculation:
1.Enter the original cost of the asset, using Ï.
2.Enter the salvage value of the asset, using FV. If the salvage value is zero, press
:É.
3.Enter the expected useful life of the asset (in years), followed by Ù.
4.If the declining-balance method is being used, enter the declining-balance factor (as a
percentage), followed by Ò. For example, 1-1/4 times the straight-line rate — 125
percent declining-balance — would be entered as 125.
5.Key in the number of the year for which depreciation is to be calculated followed by the
desired depreciation method:
•]{ for depreciation using the straight-line method.
•
]u for depreciation using the sum-of-the-years digits method.
•
]x for depreciation using the declining-balance method.
]{, ]u, and ]x each place the amount of depreciation in the display,
and the TVM and X annunciators are displayed. Press \« to display the remaining
depreciable value (the book value less the salvage value). After pressing \« to
display the remaining depreciable value, note the X annunciator changes to Y.
Example 1
A metalworking machine, purchased for 10,000.00, is to be depreciated over five years. Its
salvage value is estimated at 500.00. Using the straight-line method, find the depreciation
and remaining depreciable value for each of the first two years of the machine's life. See
Table 7-2.
Table 7-2 Depreciation example using SL
Keys DisplayDescription
]OJ
TVM CLR (message flashes
then disappears)
Clears TVM registers.
J::::Ï
V::É
Depreciation84
10,00 0.00En te rs 10,000.00 for the depreciable cost
of the item in the selected format.
500.00Enters 500.00 for the salvage value of
the item in the selected format.
Table 7-2 Depreciation example using SL
Keys DisplayDescription
VÙ
J]{
\«
G]{
\«
5.00Inputs 5 for the expected useful life of the
asset in the selected format.
1,900.00Enters the year for which depreciation is
to be calculated and calculates the
depreciation of the asset in year one.
TVM and X are displayed.
7,600.00Displays remaining depreciable value
after year one. X changes to Y in the
display.
1,900.00Enters the year for which depreciation is
to be calculated and calculates the
depreciation of the asset in year two.
5,700.00Displays remaining depreciable value
after year two.
Example 2
A machine was purchased for 4,000 and is to be depreciated over four years with a 1,000
salvage value. Using the sum-of-the-year's digit method, what is the depreciation during the
machine's first year and third years? What is the remaining depreciable value?
Table 7-3 Depreciation example using SOYD
KeysDisplayDescription
]OJ
Y:::Ï
TVM CLR (message flashes
then disappears)
4,000.00Enters the depreciable cost of the asset at
4.00Enters the expected useful life of the asset.
Clears TVM registers.
acquisition.
YÙ
J:::É
1,000.00Enters the salvage value.
J]x
1,200.00Calculates the depreciation for the first year.
D]x
600.00Calculates the depreciation for the third year.
\«
300.00Displays the remaining depreciable value.
Depreciation
85
Example 3
A machine was purchased for 5,000 and is to be depreciated over seven years with no
salvage value. Using the double declining balance method, what is the depreciation for the
first three years of the machine's life? What is the remaining depreciable value?
Table 7-4 Depreciation example using Declining Balance
KeysDisplayDescription
]OJ
V:::Ï
jÙ
G::Ò
:É
J]u
G]u
D]u
\«
TVM CLR (message flashes
then disappears)
5,000.00Enters the depreciable cost of the asset at
7.00Enters the expected useful life of the asset.
200.00Enters the double declining balance factor as a
0.00Enters the salvage value.
1,428.57Calculates the depreciation for the first year.
1,020.41Calculates the depreciation for the second year.
728.86Calculates the depreciation for the third year.
1,822.16Displays the remaining depreciable value.
Clears TVM registers.
acquisition.
percentage.
Resetting the TVM Keys
To clear the TVM registers and reset the TVM and depreciation functions to their default
values, press ]O, followed by J. The messages, TVM CLR and 12 P _ yr appear
briefly to indicate the TVM registers have been reset.
Depreciation86
8Cash Flow Calculations
How to Use the Cash Flow Application
The cash flow application is used to solve problems where cash flows occur over regular
intervals. Problems with regular, equal, periodic cash flows are handled more easily using the
TVM keys. To operate the cash flow system, cash flow amounts and repeat values are keyed
in either individually or together. In the following chapter, the term repeat value is used to
describe the number of times a cash flow occurs. Terms such as cash flow count, number of occurrences, or cash flow group are also used to describe the repeat value.
If a new cash flow is entered, the calculator auto-increments the current cash flow count by 1.
A value of 1 is automatically entered for a repeat value. To enter a repeat value for the current
cash flow entry, enter a value using \¥. To enter the cash flow and a repeat value
together, enter the cash flow value followed by Æ, then enter the repeat value followed by
¤.
In general, use the following steps for cash flow calculations on the HP 10bII+:
1.Organize your cash flows on paper. A cash flow diagram is useful.
2.Clear the cash flow memory.
3.Enter the number of periods per year.
4.Enter the amount of the initial investment (CF
The CF0 value may have a repeated value. To enter the cash flow amount and repeat
value simultaneously, enter a cash flow amount, followed byÆ, then enter a number
for the repeat value followed by
5.Unless the cash flow and repeat value have already been entered as described in step 4
using Æ and ¤, as an alternative, enter the repeat value using \¥.
6.Repeat steps 4 and 5 for each cash flow and repeat value.
7.To calculate net present value and net future value, you must first enter a value for the
annual interest rate and press
¤.
Ò; then press \½. With NPV calculated, press
) using ¤ to enter the cash flow value.
0
\«to display Net Future Value.
8.To calculate IRR, press \Á.
Cash Flow Calculations
87
Table 8-1 Cash Flow Keys
KeyDescription
]O:
Clears cash flow memory.
\Í
number1
number1
number 2 ¤
number 2
¤
Æ
\¥
Number of periods per year (default is 12). For annual cash flows, P/YR should
be set to 1; for monthly cash flows, use the default setting, 12.
Cash flows, up to 45. J identifies the cash flow number. When preceded by a
number, pressing
Enter a cash flow amount, followed by Æ. Enter a number for the repeat
value followed by
simultaneously.
An alternative for entering repeat value for cash flow J .
¤ enters a cash flow amount.
¤. This enters cash flow amount and repeat value
v¤Opens editor for reviewing or editing entered cash flows. Press 1 or
A to scroll through the cash flow data.
\Á
Internal rate of return per year.
\½
Net present value.
\½\«
Net future value.
]X
With cash flow editor open, displays total of cash flows.
]U
With cash flow editor open, displays total number of cash flows.
Clearing the Cash Flow Memory
It is always a good idea to clear the cash flow memory before beginning. To clear cash flows,
use ]O:. A brief message appears, CFLO CLR, to indicate the cash flow memory
has been reset.
On the 10bII+, there is always space reserved for up to 15 cash flows. In addition, up to 30
additional cash flows may be stored in memory shared with the statistics memory, as shown
in Figure 1 below.
Cash Flow Calculations88
Figure 1
As illustrated in Figure 1, if no more than 15 data points are stored in the statistics memory,
you may store up to 45 cash flows with the shared memory space.
If more than 15 data points are stored in the statistics memory, the total memory available for
storing cash flows is reduced. For example, in Figure 2, there are 25 data points stored, and
the amount of available shared memory has therefore decreased by 10 slots.
Figure 2
If data storage in the calculator memory resembles Figure 2, and you have a cash flow
calculation requiring more than 35 data points, clearing unneeded statistical information will
free up more space for information. When available memory is reached (see Figure 3), the
FULL annunciator indicates there is not enough space to continue saving data. If you attempt
to enter another cash flow at this point, the ERROR annunciator is displayed. In this case, no
additional cash flow data can be entered until some data in the statistics memory is removed
and the shared memory is once again available.
Figure 3
Example 1: A Short Term Investment
The following cash flow diagram represents an investment in stock over three months.
Purchases were made at the beginning of each month, and the stock was sold at the end of
the third month. Calculate the annual internal rate of return and the monthly rate of return.
Cash Flow Calculations
89
Calculating Internal Rate of Return
1.P r e s s ]O:, and store the desired number of periods per year in P/YR.
2.Enter the cash flows using Æ and ¤.
3.Press \Á.
Figure 4 Cash flow diagram (Investments in stock)
Table 8-2 Example 1: a short term investment
KeysDisplayDescription
]O:
CFLO CLR
(message flashes, then
disappears)
12.00Set payments per year.
Clears cash flow memory.
JG\Í
V:::y¤
G:::y¤
-5,000.00
(CF 0 flashes, then
disappears)
-2,000.00
(CF 1 flashes, then
disappears)
Enters initial cash flow. Note the
CFLO and CF annunciators.
Enters first cash flow. Note the
CFLO and CF annunciators.
Cash Flow Calculations90
Table 8-2 Example 1: a short term investment
KeysDisplayDescription
Y:::y¤
JJjSV7Gd¤
-4,000.00
(CF 2 flashes, then
disappears)
11 , 7 6 5 . 2 9
(CF 3 flashes, then
disappears)
38.98Calculates annual nominal yield.
Enters second cash flow. Note
the CFLO and CF annunciators.
Enters third cash flow. Note the
CFLO and CF annunciators.
\Á
aJG
3.25Monthly yield.
NPV and IRR/YR: Discounting Cash Flows
Chapter 5 titled, Picturing Financial Problems demonstrates the use of cash flow diagrams to
clarify financial problems. This section describes discounted cash flows. The NPV, NFV and IRR/YR functions are frequently referred to as discounted cashflowfunctions.
When a cash flow is discounted, you calculate its present value. When multiple cash flows
are discounted, you calculate the present values and add them together.
The net present value (NPV) function finds the present value of a series of cash flows. The
annual nominal interest rate must be known to calculate NPV.
The net future value (NFV) function finds the value of the cash flows at the time of the last cash
flow, discounting the earlier cash flows by the value set for the annual nominal interest rate.
The internal rate of return (IRR/YR) function calculates the annual nominal interest rate that is
required to give a net present value of zero.
The utility of these two financial tools becomes clear after working a few examples. The
next two sections describe organizing and entering your cash flows. Examples of NPV, NFV, and IRR/YR calculations follow.
Organizing Cash Flows
The cash flow series is organized into an initial cash flow (CF0) and succeeding cash flow groups (up to 44 cash flows). CF0 occurs at the beginning of the first period. A cash flow
group consists of a cash flow amount and the number of times it repeats.
For example, in the following cash flow diagram, the initial cash flow is -11,000. The next
group of cash flows consists of six flows of zero each, followed by a group of three 1,000
cash flows. The final group consists of one 10,000 cash flow.
Cash Flow Calculations
91
Figure 5 Initial cash flow and cash flow groups
Whenever you enter a series of cash flows, it is important to account for every period on the
cash flow diagram, even periods with cash flows of zero.
Example
Enter the cash flows from the preceding diagram and calculate the IRR/YR. Assume there are
12 periods per year.
Table 8-3 Example calculating IRR and effective interest rate
KeysDisplayDescription
]O:
JG\Í
JJ:::y¤
:¤
S\¥
Cash Flow Calculations92
CFLO CLR
(message flashes, then
disappears)
12.00Set payments per year.
-11,000.00
(CF 0 flashes, then
disappears)
0.00
(CF 1 flashes, then
disappears)
6.00
(CFn 1 flashes, then
disappears)
Clears cash flow memory.
Enters initial cash flow. Displays
cash flow group number and
amount. Note the CFLO and CF
annunciators.
Enters first cash flow group
amount. Note the CF
annunciator.
Enters number of repetitions.
Note the CFLO and N
annunciators.
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