Lesson 2

Whole Number Subtraction

Introduction

Here you’ll learn how to subtract whole numbers using vertically arranged subtraction problems.

Vocabulary

Here are the vocabulary words that are found in this Concept.

Subtrahend the number being subtracted. IE minuend - subtrahend = difference

Minuend the number being subtracted from.

Difference  the answer to a subtraction problem

Regroup when you need to borrow from the next column in subtraction and reduce the corresponding minuend digit. Used with the subtractive method.

Borrow  when you need to borrow a digital from the next column. Used with the additive method.

Payback  when you need to return a borrowed digit to the next column's subtrahend.Used with the additive method.

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INFORMATION

A general concept overvew statement or word problem exercise can be placed in the concept introduction section.

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Guidance

You have a lot of experience with subtraction of whole numbers. Let's just think about what subtraction is for a short time.

Subtraction is the opposite of addition. 

What does that mean exactly? It means that if you can add two numbers and get the sum of the numbers, then you can subtract one of those numbers from that total and end up with the difference and the difference will be the other starting number.

In other words, Subtraction is the opposite of addition. When you add two numbers you get a sum, when you subtract two numbers, you get the difference. Let's look at that in a little more detail.

6 + 9 = 14

 That is easy. So if we subtract 9 from 14 we have the following:

14 − 9 = 6

In the same way we can subtract the 6 from 14 and get this:

14 − 6 = 9

So while 6 + 9 and 9 + 6 give the same result - 14, subtraction lets us get two values and that is very useful.

So now we need to get down to learning how to put the problem into vertical form just as we did with long-addition.

1 4
− 9
0 6

You can easily do this simple subtraction using mental arithmetic. But what happens if you had more digits? Look at this:

12456 − 237 = ?

That is not so easy understand and solve so we arrange the numbers vertically being very careful to arrange them in their place values.

What does this look like in a place value table? Here it is. All we have to do is include the minus sign to make this a subtraction problem.

Ten Thousands Thousands Hundreds Tens Units
TTh Th H T U
1 2 4 5 6
    2 3 7
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INFORMATION

Subtraction Teaching Methods

There are two international methods of teaching long-subtraction, arguable one of the more difficult arithmetic concepts to learn.

The regroup or the subtractive method is common in the U.S. and many other places. The  additive method is still used in some countries. Both are presented here as Guided Practice options to show that the required teaching method can easily be accomodated for any locale.

This module is middle-school maths so it is assumed the basic concepts of addition and subtraction have already been taught and reasonably understood, and that the concept is being extended to more complicated numerical expressions.

By design the IGP:Digital Publisher construction, ALL-IN Interactive Learning Library and AZARDI:Content Fulfilment delivery system support all languages and localization needs for education publishers in any country and any language.

Guided Tutorial 1. Subtractive Method

Let's do it. You can see it step by step in this tutorial. Remember to make sure you understand each step and slide back and forwards if you need to.

 Step 1. Place values

  TTh Th H T U
  1 7 5 5 6
  −   5 3 7
           

We are going to Subtract 537 from 17,556.

The minuend is 17,556. It has a Ten-thousands place value. The minuend must go on the top.

The subtrahend is 537 and is the smaller number. It has a Hundreds place value. The subtrahend must go on the bottom.

See how the Units (U), Tens (T) and Hundreds (H) are lined up vertically in their correct column.

 Step 2. The vocabulary

           
  1 2 4 5 6
  − 1 5 3 7
           
 

We are now ready to start the subtraction.

We have set the sum up vertically.

Remember the top number is the minuend. 

The bottom smaller number is the subtrahend.

We are subtracting the subtrahend from the minuend. The result will be the difference.

 Step 3. Subtract the Units

        4 16
  1 2 4 5/ 6/
  −   2 4 7
          9

First we are going to subtract the Units column. Subtracting  7 from 6 is not possible.

We borrow 1 from the Tens to make the minuend 16. 7 can now be subtracted from 16.

The answer 9 is placed in the Units answer line.

Now we regroup the Tens column by subtracting the 1 that was borrowed.  5 minus 1 is 4.

Strike out the 5 and write 4 above.

 Step 4 Subtract the Tens

        4 16
  1 2 4 5/ 6/
  −   2 3 7
        1 9

Next we subtract in the Tens column.

We regrouped the Tens so we are now subtracting 3 from 4.

The answer is 1. We place this into the difference line.

 Step 5. Subtract the Hundreds

          1
  1 2 4 5 6
  −   2 3 7
      2 1 9

We now subtract in the Hundreds column.

Here we are subtracting 2 from 4.

Because the minuend was bigger than the subtrahend we don't have to borrow in this column.

The result is 2 and that is added to the difference line.

 

 Step 6. Subtract the Thousands

          1
  1 2 4 5 6
  −   2 3 7
  1 2 2 1 9

We have now run out of numbers in the bottom (subtrahend) line. They are effectively zero.

For clarity, 2 minus nothing is 2.

The 2 in the Thousands column can be placed directly into the difference line in the Thousands place value column.

 Step 7. Subtract the Ten-thousands

          1
  1 2 4 5 6
  −   2 3 7
  1 2 2 1 9

We have now run out of numbers in the bottom (subtrahend) line. They are effectively zero.

For clarity, 1 minus nothing is 1.

The 1 in the Ten-thousands column can be placed directly into the difference line in the Ten-thousands place value column.

 Step 8. The Answer

          1
  1 7 5 5 6
  −   2 4 7
  1 2 2 1 9

Here is the final answer 12,219.

The difference of 12,345 minus 237 is 12,219. 

Remember the terms for subtraction.

We subtracted the subtrahend 237 from the minuend 12,456 to get the difference 12,219.

= END OF TUTORIAL =

◄ Slide for more ►

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Guided Tutorial 2. Additive Method

The additive method of subtraction involves borrowing a digit from the next column when required and then paying it back.

It is borrowed into the subtrahend and adds 10 to the number. It is paid back into the minuend and added to the minuend before the subtraction takes place.

That means there is a borrow row at the top and a payback row under the subtrahend.

Let’s put that into action and write it vertically into our arithmetic subtraction. You can see this in action in this tutorial. Go steadily and make sure you understand each step.

 Step 1. Place values

  TTh Th H T U
  1 2 4 5 6
  − 1 5 3 7
           
           

We are going to Subtract 537 from 17,556.

The minuend is 17,556. It has a Ten-thousands place value. The minuend must go on the top.

The subtrahend is 537 and is the smaller number. It has a Hundreds place value. The subtrahend must go on the bottom.

See how the Units (U), Tens (T) and Hundreds (H)  are all each lined up vertically in their correct column.

 Step 2. Subtraction Vocabulary

           
  1 2 4 5 6
  − 1 5 3 7
           
           
 

We are now ready to start the subtraction.

We have set the sum up vertically.

Remember the top number is the minuend. 

The bottom smaller number is the subtrahend.

We are subtracting the subtrahend from the minuend. The result will be the difference.

 Step 3. Subtract the Units

          1
  1 2 4 5 6
  − 1 5 3 7
           
          9

First we are going to subtract the Units column.

Subtracting 7 from 6 is not possible.

We borrow 1 from the Tens to make the minuend 16. 7 can now be subtracted from 16.

The answer 9 is placed in the Units position on the difference line.

 Step 4. Subtract the Tens

          1
  1 2 4 5 6
  − 1 5 3 7
        1  
        1 9

Next we subtract in the Tens column.

We borrowed a 1 from  the Tens column to use in the Units column.

We payback the 1 by writing it under/next to the subtrahend.

We add the payback 1 to the subtrahend 3. That is 4. Now we subtract 4 from the 5 in the minuend.

The result for the Tens column is 1.

 Step 5. Subtract the Hundreds

      1   1
  1 2 4 5 6
  − 1 5 3 7
        1  
      9 1 9

We now subtract in the Hundreds column.

Subtracting 5 from 4 is not possible.

Again we must borrow 1 from the Thousands column.

14 - 5 = 9

We write 9 in the Hundreds difference line.

 Step 6. Subtract the Thousands

      1   1
  1 2 4 5 6
  − 1 5 3 7
    1   1  
  1 0 9 1 9

Next we subtract in the Thousands column.

We borrowed a 1 from  the Thousands column to use in the Hundreds column.

We payback the 1 by writing it under/next to the subtrahend.

We add the payback 1 to the subtrahend 1. That is 2. Now we subtract 2 from the 2 in the minuend.

The Thousands column is 0.

 Step 7. Subtract the Ten-thousands

      1   1
  1 2 4 5 6
  − 1 5 3 7
    1   1  
  1 0 9 1 9

Next we subtract in the Ten-Thousands column.

We have now run out of numbers in the bottom (subtrahend) line. They are zero.

The 1 in the Ten-Thousands column can be placed directly into the difference line in the correct place position.

 Step 8. The Answer

      1   1
  1 2 4 5 6
  − 1 5 3 7
    1   1  
  1 0 9 1 9

Here is the final answer 10,919.

The difference for 12,456 minus 1,537 is 10,919. 

Remember the terms for subtraction.

We subtracted the subtrahend 1,537 from the minuend 12,456 to get the difference 10,919.

= END OF TUTORIAL =

◄ Slide for more ►

Video Review

These videos will help you with your whole number subtraction if you have Internet access.

Interactive Practice

Practice your subtraction here

Interactive Test

Test your subtraction here

Take the time to complete all the Exercises and use the Practice and Test Interactive tools to make sure subtraction becomes your friend. Learn to enjoy the numbers. They are fun and these exercises let you do it all at your own speed anytime you want.

  

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