Calculator Tricks - Part 2

Fractions

Welcome to Part 2 of the Calculator Tricks series. We are using a simple ("four banger" calculator) to tackle common mathematical problems. If you missed Part 1,
you can check it right here by clicking on this link.

Simple Fractions

Simple calculators do not have the ability to display numbers as fractions, just their decimal equivalents.

Here are the decimal equivalent of some fractions that are handy to remember:

1/8 = 0.125
1/4 = 0.25
1/3 ≈ 0.3333333
3/8 = 0.375
1/2 = 0.5
5/8 = 0.625
2/3 ≈ 0.6666667
3/4 = 0.75
7/8 = 0.875

Remember the order you press the keys is critical, since the simple calculator operates in Chain Mode.

This blog assumes that you are working with an 8-digit calculator.

1. Adding and Subtracting Fractions

Adding and subtracting fractions will require the use of memory. Remember to always clear memory before beginning a calculation.

---------------
Adding Fractions: A/B + C/D

Keystrokes:
MC (clear memory because we will need it)
A ÷ B = M+
C ÷ D = M+
MR
---------------
Subtracting Fractions: A/B - C/D

Keystrokes:
MC
A ÷ B = M+
C ÷ D = M-
MR
---------------
Example 1: 4/11 - 3/99 = 1/3 ≈ 0.3333333

The decimal equivalent (0.3333333 on a 8-digit calculator) is what we are after in this example.

Keystrokes:
MC (Yes, I can't emphasize clearing the memory at the start enough!)
4 ÷ 11 = M+
3 ÷ 99 = M-
MR (Display: 0.3333333 M)
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Example 2: 1/7 + 3/8 - 4/9 = 37/504 ≈ 0.073412698

We can use this technique to add and subtract more than two fractions.

Keystrokes:
MC
1 ÷ 7 = M+ (adding 0.1428571)
3 ÷ 8 = M+ (adding 0.375)
4 ÷ 9 = M- (subtracting 0.4444444)
MR (Display: 0.0734127 M)

Depending on the calculator the last digit may be rounded or not. This result is correct to eight decimal places.

2. Multiplying and Dividing Fractions

Let's multiply the fractions A/B × C/D. Using some algebra, we can simplify this expression.

A/B × C/D
= (A × C)/(B × D)
= (A × C)/B × 1/D
= (A × C) ÷ B ÷ D

---------------
Multiplying Fractions: A/B × C/D

Keystrokes: A × C ÷ B ÷ D =

Multiply the numerators, divide the denominators.
---------------
Example: 4/7 × 2/3 = 8/21 ≈ 0.3809524

Keystrokes: 4 × 2 ÷ 7 ÷ 3 =
---------------

Let's divide the fraction A/B by C/D. (Calculate A/B ÷ C/D) To divide number by a fraction, multiply it by the fraction's reciprocal.

A/B ÷ C/D
= A/B × D/C
= (A × D)/(B × C)
= (A × D)/B × 1/C
= (A × D) ÷ B ÷ C

----------------
Dividing Fractions: A/B ÷ C/D

Keystrokes: A × D ÷ B ÷ C =
---------------
Example: 7/9 ÷ 2/5 = 7/9 × 5/2 = 35/18 ≈ 1.9444444

Keystrokes: 7 × 5 ÷ 9 ÷ 2 =
----------------

Mixed Fractions

A strategy is to to covert mixed fractions to simple (improper fractions) first.

Adding Mixed Fractions

For the mixed fraction A B/C is converted into the simple fraction (A × C + B)/C. Adding two mixed fractions will require the use of the memory register.

---------------
Adding Mixed Fractions:

A B/C + D E/F = (A × C + B)/C + (D × F + E)/F

Keystrokes: MC
A × C + B ÷ C = M+
D × F + E ÷ F = M+
MR
---------------
Example: 8 1/9 + 7 3/5 = 15 32/45 ≈ 15.711111

Keystrokes: MC
8 × 9 + 1 ÷ 9 = M+
7 × 5 + 3 ÷ 5 = M+
MR
---------------

Multiplying Mixed Fractions

Unlike multiplying simple fractions, multiplying mixed fractions will require the use of memory. We can get a form for an algorithm by simplifying.

A B/C × D E/F
= (A × C + B)/C × (D × F + E)/F
= ((A × C + B) × (D × F + E))/(B × C)
= ((A × C + B) × (D × F + E)) ÷ B ÷ C

---------------
Multiplying Mixed Fractions

A B/C × D E/F = ((A × C + B) × (D × F + E)) ÷ C ÷ F

Keystrokes:
MC
A × C + B M+
D × F + E =
× MR =
÷ B ÷ F
---------------
Example 1: 4 1/2 × 8 1/3 = ((4 × 2 + 1) × (8 × 3 + 1)) ÷ 2 ÷ 3 = 37.5

Keystrokes:
MC (Start with the memory cleared!)
4 × 2 + 1 = M+ (Store 4 × 1 + 2 = 9 in memory)
8 × 3 + 1 = (Display: 25 M)
× MR = (numerator is done, Display: 225)
÷ 2 ÷ 3 = (no divide by the denominators)

The display should be 37.5 (with the memory indicator). This is a complex algorithm, so it will take some practice. Let's do another example.
---------------
Example 2: 5 5/9 × 16 2/3 = ((5 × 9 + 5) × (16 × 3 + 2)) ÷ 9 ÷ 3 ≈ 92.59259

Keystrokes:
MC (Start with the memory cleared!)
5 × 9 + 5 = M+ (Store 4 × 1 + 2 = 50 in memory)
16 × 3 + 2 = (Display: 50 M)
× MR = (numerator is done, Display: 2500 M)
÷ 9 ÷ 3 = (no divide by the denominators)

The display should have 92.59259.
---------------

In the next example we will add and multiply mixed fractions. Be willing to write immediate results on a note pad with complex problems.

---------------
Example 3:
4 1/8 + 3 3/7 × 6 2/3 = 1511/56 ≈ 26.982143

The Order of Operations says we must multiply the mixed fractions first. However, we only have one memory register, and we don't have the ability to "switch" whatever is in the display with memory. Tackling this problem requires a plan. Here is one way:

1. Change 4 1/8 into a fraction and write it down.
2. Multiply the mixed fractions 3 3/7 × 6 2/3.
3. Add the resulting decimal equivalent of 4 1/8 to the result obtained from step 2.

Step 1 Keystrokes:
4 × 8 + 1 ÷ 8 =
Note the result, which is 4.125. Don't store this number in memory, but on a notepad.

Step 2 Keystrokes:
MC
3 × 7 + 3 = M+ (Display: 24 M)
6 × 3 + 2 = (Display: 20 M)
× MR = (Display: 480 M)
÷ 7 ÷ 3 = (Display: 22.857142 M)

Step 3 Keystrokes:
Add 4.125 obtained from step 1 to 22.857142 obtained from step 2.
+ 4.125 = (Display: 26.982142 M)

We have arrived at our result.

The lesson here is to plan your calculation.

Another strategy to calculate 4 1/8 + 3 3/7 × 6 2/3 is to change all fractions to their decimal equivalents first, noting the decimal equivalents on paper. Let's see how this works out:

1. Change each mixed fraction to their decimal equivalent.
2. Complete the calculation.

Step 1 Keystrokes:
4 × 8 + 1 ÷ 8 = (4 1/8 = 4.125)
3 × 7 + 3 ÷ 7 = (3 3/7 ≈ 3.4285714)
6 × 3 + 2 ÷ 3 = (6 2/3 ≈ 6.6666666)

Now we have 4.125 + 3.4285714 × 6.6666666

Step 2 Keystrokes:
3.4285714 × 6.6666666 + 4.125 =
Result: 26.982142


The next time we meet, we will work with square numbers, reciprocals, and some geometry.

Eddie


This blog is property of Edward Shore, 2012.