Four Function and Desktop Calculators: The [ TAX+ ] and [ TAX- ] Keys

Four Function and Desktop Calculators: The [ TAX+ ] and [ TAX- ] Keys

A good subject for a Monday (start off the work week for most of us, me included)

The TAX+ and TAX- keys up close and personal  (Casio SL-300VC)

What is needed:  a calculator with [ TAX+ ] and [ TAX- ]

Introduction

On a four function or desktop calculator with the [ TAX+ ] and [ TAX- ] keys, you can perform tax calculations and financial calculations.  The primary key purpose of these keys is for sales tax.  The procedure for these calculations are (almost) universal.  However, the procedure to set such tax varies by manufacturer.

Casio:  Hold [AC] until the display blanks and refreshes.  Enter the tax rate and press [ % ] (RATE SET). 

Texas Instruments (TI-1795SV):   Enter the tax rate, press [ rate ], [ tax+ ] (store)

Canon:  Press [AC], [ TAX+ ] (SET), enter the rate, and press [ TAX+ ]

Sharp: Press [ CE/C ], [ CE/C ] (twice), [ TAX+ ], enter the rate, press [ TAX+ ]

Printing Calculators (various):  Set tax setting switch to SET, enter the rate, set the tax setting switch back to calculator mode.  Check your manual because printing calculators may vary.

My focus will be on four function and desktop calculators.  Procedures for printing calculators will probably vary.

Note:  I did the procedures listed in today's blog entry on both the Casio SL-300VC and Casio LS-123K.  For using independent memory, check to see if you have either a combined memory recall/clear key [MRC] (or [RM/CM on the LS-123K) or two separate keys ([MR] for recall and [MC] for clear).  For illustration purposes I will use [MRC]  (once for recall, twice for clear).   Remember that you can press [AC] to clear memory. 

Left:  Canon LS-123K (Green), Right:  Casio SL-300VC (Blue)

Note:  Pressing [AC] or [MRC] [MRC] will only clear the memory register, not the tax rate.

Contents

1.  Testing What Tax Rate is Stored in Memory
2. Price After Sales Tax and the Amount of Tax
3. Adding Taxable and Non-taxable Amounts
4. Calculating Use Tax
5.  Find the Taxable Amount Given the Grand Total (Total Plus Tax)
6.  Finance: Using [ TAX+ ] to calculate Future Value
7.  Finance: Using [ TAX- ] to calculate Present Value

For all the examples today, I have the tax rate set to 9.5%.  Please see the Introduction section above on how to set the tax rate.

1.  Testing What Tax Rate is Stored in Memory

This procedure is for any calculator that does not have a recall tax feature, although it works for these calculators too.

Procedure:

100 [ TAX+ ] [ - ] 100 [ = ]

Example:  (remember I'm using the tax rate set at 9.5% for all examples on this blog entry)

100 [ TAX+ ] [ - ] 100 [ = ]
Result:  9.5

2. Price After Sales Tax and the Amount of Tax

The [ TAX+ ] key performs two operations:

*  Press [ TAX+ ] once to calculate the total plus tax
*  Press [ TAX+ ] again to calculate the tax amount

Example:

An invoice shows of a purchase of equipment totaling $250.00.  Find the total after tax and the sales tax.

250 [ TAX+ ]
Display:  273.75   (total invoice:  $273.75)

[ TAX+ ]
Display:  23.75   (sales tax:  $23.75)


3. Adding Taxable and Non-taxable Amounts

Sometimes an invoice has both taxable amount and non-taxable amounts (such as most services, installation fee, sometimes freight, some food).  We can calculate the total invoice accurately with both the TAX and memory keys.  Start by clearing out the memory. 

Procedure:

[MRC] [MRC] taxable amount [ TAX+ ] [ M+ ] n
non-taxable amount [ M+ ] 
[MRC]  (recall memory)

Example:

Suppose has the invoice has:

A scanner that subject to sales tax:  $69.99
Computer services not subject to sales tax: $34.95
Sales Tax Rate:  9.5%

[ MRC ] [ MRC ] 69.99 [ TAX+ ] [ M+ ] 34.95 [ M+ ] [MRC]

The total invoice is $111.59   (8-Digit display shows 111.58905)


4. Calculating Use Tax

In certain states, such as California, use tax on a purchase can occur.  Where you take delivery determines what sales tax rate you would pay.  If a vendor does not charge the required tax, the use tax kicks in, which constitutes the remaining amount required from the buyer.  How to pay the use tax is beyond the scope of this blog entry. 

Procedure:

taxable amount [ TAX+ ] [ TAX+ ] [ - ] sales tax charged [ = ]

Example:

A purchase of a printer from an online store has a retail price of $164.79.  Sales tax of 8% was charged on the printer in the amount of $13.18.  The purchaser lives in a tax district that has a sales tax rate of 9.5%.  What is the use tax?

164.79 [ TAX+ ] [ TAX+ ] [ - ] 13.18 [ = ]  

The use tax is $2.48  (8-Digit display shows 2.47505)


5.  Find the Taxable Amount Given the Grand Total (Total Plus Tax) 

Sometime you know only the grand total of the invoice, but you need to find what was the taxable amount and the tax applied to that amount.  That is what the key [ TAX- ] is for.

*  Press [ TAX- ] once to calculate the taxable amount
*  Press [ TAX- ] again to calculate the tax amount

This is the inverse of the [ TAX+ ] key.

Example:

During an audit, we find an invoice from an electronics retail store for a purchase of a video projector.  However, only the total invoice is readable, in the amount of $187.44.  What is the retail price and what was the sales tax charged?

187.44 [ TAX- ]
Display:  171.17809   (taxable amount:  $171.18)

[ TAX- ]
Display:  16.261917 (sales tax:  $16.26)


6.  Finance: Using [ TAX+ ] to calculate Future Value

Time for a little unorthodox use of the TAX keys to calculate simple compound interest problems.  If you have an investment and you want to know how much your account will be in n periods (usually year), you can use the [ TAX+ ] [ = ] combination.

FV = PV * (1 + r%)^n

FV = future value
PV = present value
r% = interest rate, stored as the TAX rate
n = number of periods

Procedure:

[ MRC ] [ MRC ] 
present value [ M- ]
Loop:  [ TAX+ ] [ = ]   (do this n times for n periods)
(display future value)
[ M+ ] [ MRC ] 
(display interest earned)

Example:

You deposit $1,000.00 in a moderate to aggressive investment account that pays an average of 9.5% per year.  What is the balance after 5 years?  How much interest is earned in those five years?

[ MRC] [ MRC ] 
1000 [ M- ]
[ TAX+ ] [ = ]
[ TAX+ ] [ = ]
[ TAX+ ] [ = ]
[ TAX+ ] [ = ]
[ TAX+ ] [ = ]    (loop the last two keys 5 times)
Future value:  $1,574.24  (8-Digit display: 1574.2385)

[ M+ ] [ MRC ]
Interest earned:  $574.24  (8-Digit display:  574.2385)


7.  Finance: Using [ TAX- ] to calculate Present Value

Similarly, we can use the [ TAX- ] [ = ] combination to calculate the present value of a discounted note. 

PV = FV / (1 + r%)^n

FV = future value
PV = present value
r% = interest rate, stored as the TAX rate
n = number of periods

Procedure:

[ MRC ] [ MRC ] 
future value [ M+ ]
Loop:  [ TAX- ] [ = ]   (do this n times for n periods)
(display future value)
[ M- ] [ MRC ] 
(display interest earned)

Example:

You want to save $10,000.00 in five years.  You find an account that pays 9.5% annual interest.  How much will you need to deposit today to get that $10,000.00 goal in five years?

[ MRC] [ MRC ] 
10000 [ M+ ]
[ TAX- ] [ = ]
[ TAX- ] [ = ]
[ TAX- ] [ = ]
[ TAX- ] [ = ]
[ TAX- ] [ = ]    (loop the last two keys 5 times)
Present value:  $6,352.28  (8-Digit display: 6352.2775)

[ M- ] [ MRC ]
Interest earned:  $3,647.72  (8-Digit display:  3647.723)

What the [ TAX+ ] and [ TAX- ] Keys Calculate

With the tax rate r set:

[ TAX+ ] calculates:   number in the display * ( 1 + r/100)

[ TAX- ] calculates:  number in the display / (1 + r/100 )

Happy calculating and may all your calculations, and work weeks, be successful,

Eddie


All original content copyright, © 2011-2019.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

Arithmetic, Powers, Percent Change, and Square Roots using the Printing Calculator

Printing Calcualtor Madness

Printing calcualtors, better known as adding machines and 10-key machines are calculators designed for the office. If you are a professional accountant, CEO, auditor, tax preparer, or supervisor, chances are you operate these machines at least once in a while if not every day. The Sharp EL-1750V printing calculator is pictured below.

Introduction

This blog entry will show you how to extend the operation of the printing calculator beyond the mere adding and subtracting. While a printing calculator may not be a weapon of choice for super advanced mathematics, we can use some of the features to do some cool arithmetic and algebraic problems. Amaze your office friends with these techniques!

Note: I will assume the calculator is set to Float (F) mode and not in Add mode (A, or +)

The Printing Calculator

You have a printing office calculator if you have keys such as +=, -=, *T, ♢S, and GT.

The printing calculator operates on a hybrid notation. Addition and subtraction operate by postfix notation. That is, enter the number, then press the + key or - key, and repeat. Addition and subtraction operations are terminated by a totals key, labeled *, *T, or */T. For this blog entry, I will use *T to symbolize the totals key.

Example 1: 45 + 68 + 99 - 100 = 112

Keystrokes will be in blue and Courier font.


45 +
68 +
99 +
100 - *T


Multiplication and division operate by infix notation. This is enter the first number, press either the × or the ÷ key, then enter the second number. Multiplying and dividing operations are terminated by the equals key. The equals key can be by itself (=) or coupled by a plus or minus operation (+= or -=).

Example 2: 24 × 72 ÷ 6 = 288

24 ×
72 ÷
6 =

The memory keys are your good friends on the printing calculator. Here is a basic rundown of the memory keys:

M+: Adds whatever is in the display to the memory register
M-: Subtracts whatever is in the display from the memory register
MS or M♢ : Displays and recalls the contents of the memory register
MT or M* : Recalls the contents of the memory register AND resets the memory register to 0.

For this blog entry, I will use the M ♢ and M* notation. It is often a very good idea to "clear memory" first by pressing M* before doing several of the more complex calculations.

Now that you have it, let's go on to the arithmetic calculations. Paper printing is optional but unless you have ton of tape, it is not really necessary.

Arithmetic

Couple of Golden Rules:

1. Make sure memory is clear before beginning (necessary most of the time).
2. If you remember "My Dear Aunt Sally" (multiplication and division before adding and subtracting), the rules of parenthesis, and the rules involving rational expressions, we're in business. See, algebra class pays off!
3. Do not forget to terminate adding/subtracting calculations with *T before switching to multiplying/dividing.
4. Likewise, do not forget to terminate multiplying/dividing calculations with = before switching to adding/subtracting. Before starting the adding/subtracting, press the + key once to tell the printing calculator to add the number.

Let's begin. I use the capital letters A, B, C, etc. to represent numbers (as variables).

#1. A × B + C
Keystrokes:
A × B =
+ C + *T

Example: 5 × 2 + 10 = 22
5 × 2 =
+ 12 + *T

#2. (A + B) ÷ C
Keystrokes:
A + B + *T
÷ C =

Example: (56 + 27) ÷ 3 = 27.6666666667
56 + 27 + *T
÷ 3 =

Here is where the memory keys get used. If you have to switch signs (i.e. negate a value), remember to use the plus/minus key +/- .

#3. A × B + C × D
Keystrokes:
M*
A × B = M+
C × D = M+
M*

Example 1: 4 × 1.99 + 8 × .99 = 15.88
M*
4 × 1.99 = M+
8 × .99 = M+
M*

Example 2: A bill with sales tax.
(2 × 19.95 + 10.99 + 3 × 4.99) + 8%
= (2 × 19.95 + 10.99 + 3 × 4.99) × (1 + 8 ÷ 100)
= (2 × 19.95 + 10.99 + 3 × 4.99) × 1.08
M*
2 × 19.95 = M+
10.99 M+
3 × 4.99 = M+
M* × 1.08 =

#4. A × B - C × D
Keystrokes:
M*
A × B = M+
C × D = M-
M*

Example: 3 × 4 - 2 × 3.1 = 5.8
M*
3 × 4 = M+
2 × 3.1 = M-
M*

#5. (A + B) ÷ (C + D)
The trick here is to work on the denominator first. Keystrokes:
M*
C + D + *T M+
A + B + *T
÷ M* =

Example 1: (6 + 3.9) ÷ (5 - 4.1)
M*
5 + 4.1 - *T M+
6 + 3.9 + *T
÷ M* =

Example 2: 2 × (3 + 4) ÷ (7 + 8) = .93333333333
M*
7 + 8 + *T M+
3 + 4 + *T
÷ M*
× 2 =

#6: Working With Mixed Fractions
Mixed fractions in the form of A B/C can be entered with the following keystrokes:
B ÷ C = + A + *T

Be sure to work with memory when you adding and subtracting fractions. You will get an answer in decimal format, no matter how nice your fractions are.

Example: 4 2/5 + 3/7 - 2 1/9 = 2 226/355 = 2.71746031746
M*
2 ÷ 5 = + 4 + *T M+
3 ÷ 7 = M+
1 ÷ 9 = + 2 + *T M-
M*

Cost/Sell/Margin and using those keys to find Δ%

#7. Margin
The equation for Cost/Sell/Margin calculations is:

100 × (sell - cost) ÷ sell = margin

Let A be the cost of the item and B be the selling price.
Keystrokes:
A COST
B SELL (Margin is automatically calculated)

Example: If the cost of an item is $3,200 and the selling price is $4,800, what is the margin?
3200 COST
4800 SELL

The margin is 33.3333333333%

#8. Percent Change Δ%
The equation for percent change is:

100 × (new - old) ÷ old = Δ%

Note that

100 × -1 × (old - new) ÷ old = Δ%

It looks like the Cost/Sell/Margin equation above. We can use the Cost/Sell/Margin and the change sign (+/-) to calculate Δ%.

Let A be the old number and B be the new number.
A SELL
B COST +/-

Example: Find the percent change from 17.95 to 24.65. 17.95 is the old number, 24.65 is the new number.
17.95 SELL
24.65 COST +/-

Δ% = 37.3259052924%

Reciprocals and Powers

#9. Reciprocals

The reciprocal of A is 1 ÷ A.

We can use straight division, memory, or a third technique which can be useful in more complex calculations.
A ÷ = =

Example 1. Calculate the reciprocal of 4. (4^-1)
4 ÷ ÷ =

4^-1 = 1 ÷ 4 = 2.5

Example 2. 1 ÷ (60.7 - 38.6) = (60.7 - 38.6)^-1 = .0454886877
60.7 + 38.6 - *T ÷ ÷ =

#10. Powers

To raise a number A to a power N (N is an integer), use the following technique:
1. Enter A
2. Press the × key N-1 times.
3. Press the = key.

To square A, press A × =.
To cube A, press A × × =.

For practical purposes, N should not be too high, unless you want to press × many times while keeping a mental count.

If N is negative, follow the power calculation with a reciprocal calculation. See Example 2.

Example 1: 3.6^3 = 46.656
3.6 × × =

Example 2: 5000 × 1.1^-4 = 3,415.50672768
1.1 × × × =
÷ ÷ =
× 5000 =

Example 3: (3 + 2.4)^2 - (3.1 - 2.8)^3 = 29.133
*M. (clear memory)
3 + 2.4 + *T × = M+
3.1 + 2.8 - *T × × = M-
*M

Square Root Routine

One of the most annoying things about printing calculators is the lack of a square root key (√). There is an iterative method for finding the square root using the Babylonian Method (if you want a challenge or don't feel like getting a calculator with a square root function or running Excel):

To find √S (the square root of S):

1. Start with an initial guess x0. Obviously, the better the initial guess, the faster this method goes.
2. Calculate x1 = (x0 + S ÷ x0) ÷ 2
3. As you repeat Step 2, more and more decimal places should "repeat" themselves. For each loop, set x1 as the new x0.

Keystroke wise:
To start: M* x0
Loop: M+ S ÷ M♢ = M+ M* ÷ 2 =

Example: √125 with an initial guess x0=10
M* 10

Loop #1:
M+ 125 ÷ M♢ = M+ M* ÷ 2 = (11.25)

Loop #2:
M+ 125 ÷ M♢ = M+ M* ÷ 2 = (11.1805555555)

Loop #3:
M+ 125 ÷ M♢ = M+ M* ÷ 2 = (11.1803398895)

Loop #4:
M+ 125 ÷ M♢ = M+ M* ÷ 2 = (11.1803398874)

Loop #5:
M+ 125 ÷ M♢ = M+ M* ÷ 2 = (11.1803398874). STOP

Hence √125 = 11.1803398874


Have fun impressing your co-workers and friends. It's been a blast as always! Until next time, Eddie




This blog is property of Edward Shore. © 2012