Subtracting with Five Extras and Ten Extras

Short Course

This is a short course (slide show) in the basic concepts to subtract using the Five (5) Extras Ten (10) Extras principles.

The Five (5) Extras Principle for Addition

First, we need to review the five (5) extras principles for addition. This principle will be used in some subtraction calculations that will be discussed later in the chapter. As you recall, we use this principle to add the integers 1, 2, 3, 4 or 5 in a single column when the exact number of beads weren’t available, but the 5 bead was available. In this approach, we add 5 (which will be too much unless you want to add 5) but then remove the extra. For example:

• • • • We want to add 1. Instead, we add 5 and remove the extra, 4. (5 – 1).

      • We want to add 2. Instead, we add 5 and remove the extra, 3. (5 – 2)

      • We want to add 3. Instead, we add 5 and remove the extra, 2. (5 – 3)

      • We want to add 4. Instead, we add 5 and remove the extra, 1. (5 – 4)

      • We want to add 5. Nothing extra (5 – 5)

This is five (5) extras principle for addition: Add 5 and remove the extra.

When we need to add the integers 1, 2, 3, 4, or 5, but the beads we need below the bar aren’t available, but the 5 bead is available, then we can use the five (5) extras principle. To apply this principle, we will push the 5 bead to the bar (set 5) and then clear (remove) the extra beads from the bar to get the number that you originally wanted to add.

Examples

Here are a few simple examples of how this principle is applied:

• Example: 4 + 3

• Example: 423 + 144

The Five (5) Extras Principle for Subtraction

Next, we need to review the five (5) extras principle for subtraction. This principle will also be used in subtraction calculations that will be used later in the chapter.

The five (5) extras principle for subtraction is the reverse of the five (5) extras principle for addition. Instead of adding the integers 1, 2, 3, 4 or 5, we want to subtract them. In this situation, the exact number of beads we want to subtract aren’t available below the bar, but the 5 bead is set. To perform this calculation, we remove the 5 bead (which is too much unless you want to subtract 5) and add the extra.

• We want to subtract 1. Instead, we remove 5 and add the extra, 4. (-5 + 4).

• We want to subtract 2. Instead, we remove 5 and add the extra, 3. (-5 + 3)

• We want to subtract 3. Instead, we remove 5 and add the extra, 2. (-5 + 2)

• We want to subtract 4. Instead, we remove 5 and add the extra, 1. (-5 + 1)

• We want to subtract 5. Nothing extra (-5 + 0)

This is five (5) extras principle for subtraction: Remove 5 and add the extra.

When we need to subtract the integers 1, 2, 3, 4, or 5, but the beads we need below the bar aren’t available and the 5 bead is in use, then we can use the five (5) extras principle for subtraction. In this principle, we remove the 5 bead from above the bar (clear 5) and then add (set) the extra beads to the bar to get the number that you originally wanted to subtract.

Examples

Let’s see how this works in a few examples:

• Example: 5 - 3

• Example: 876 - 433

Putting it All Together

Now you are ready to apply the five (5) extras principles for addition and subtraction and the tens (10) extras principle for subtraction calculations with any size integers. To subtract any two integers or groups of integers, you will need to use one of these principles or a combination of all of these principles. Sometimes you can perform the calculation in single column. But other times, you will need two or more columns to complete the calculation. To subtract two integers, you may need to:

• Subtract a digit directly

• Add or subtract a digit using the five (5) extras principle

• Subtract a digit using the ten (10) extras principle and the five (5) extras principle for addition or subtraction using one or more columns

Before we look at some examples, let’s summarize all the steps we have learned to apply these principles to subtraction calculations.

Examples

Now, let’s look at several examples that will show you how to combine the principles you have learned. As we have done in all of the previous examples:

1. Set all the digits of the first number

2. Working left to right, subtract each digit of the second number. Continue subtracting until you completed subtracting the digit in the unit column.

3. If there are more integers, repeat step 2 for each integer until you reach the final difference.

• Example: 9 - 1 – 1 – 1 – 1 – 1 – 1 – 1 – 1 – 1 (Subtracting by 1, nine times)

• Example: 75 - 16

• Example: 64 - 27

• Example: 478 - 369

• Example: 2,911 - 737

• Example: 2,651 - 864

• Example: 5,229 - 777

• Example: 83,886 - 32,595

• Example: 79,896 - 65,887

• Example: 65,253 - 26,852

• Example: 47,595 - 39,841

• Example: 32,690 - 23,945

• Example: 684,007 - 5,308

• Example: 1,898,561 - 497,473

• Example: 969,095 - 203,618

• Example: 5,016,260 - 3,960,753

• Example: 501,949 - 489,784

Build Your Skill

Now you are ready to try some calculations on your own. Click on this link for some problems and their solutions to test your skill.