## Wednesday, June 6, 2012

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

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

1. This comment has been removed by a blog administrator.

2. Exponents, or powers, are a way of indicating that a quantity is to be multiplied by itself some number of times. In the the expression 25, 2 is called the base and 5 is called the exponent, or power. 25 is shorthand for "multiply five twos together": 25 = 2×2×2×2×2 = 32.Powers and Roots

3. Hi Eddie, how do you get a total for adding on the calculator? For example I do 1 + 2 + 3 = but I don't get the correct total. Please help! I have this same exact calculator and the manual doesn't say how to get a total. Thank you.

1. Wrk IT Girl:

Printing calculators work a bit differently from normal calculators. To add 1 + 2 + 3, press:

1 [ + ]
2 [ + ]
3 [ + ]
[*T]

In general, addition and subtraction following this key pattern:
[Clear] (if needed)
number [ + ] or [ - ]
number [ + ] or [ - ]
...
[*T]

[*T] returns the total can completes the calculation. If you want a subtotal and continue where you left off, press [ S ] instead.

Eddie

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