Adding, Subtracting, Multiplying, and Dividing Whole Numbers

Learning Objectives

  • Use addition, subtraction, multiplication, and division when evaluating whole number expressions

Working with whole numbers and performing basic calculations is the backbone of all math. We’re going to assume you remember how to do single digit addition, subtraction, multiplication, and division. You will often have a calculator on hand to do these calculations, but a quick refresher will help you better understand how to work with numbers so that complex equations are less daunting.

Addition

example

Add: [latex]28+61[/latex]

Solution
To add numbers with more than one digit, it is often easier to write the numbers vertically in columns.

Write the numbers so the ones and tens digits line up vertically. [latex]\begin{array}{c}\hfill 28\\ \\ \hfill \underset{\text{____}}{+61}\end{array}[/latex]
Then add the digits in each place value.

Add the ones: [latex]8+1=9[/latex]

Add the tens: [latex]2+6=8[/latex]

[latex]\begin{array}{c}\hfill 28\\ \\ \hfill \underset{\text{____}}{+61}\\ \hfill 89\end{array}[/latex]

 

In the previous example, the sum of the ones and the sum of the tens were both less than [latex]10[/latex]. But what happens if the sum is [latex]10[/latex] or more? Let’s use our base-[latex]10[/latex] model to find out.

The graphic below shows the addition of [latex]17[/latex] and [latex]26[/latex] again.

 

An image containing two groups of items. The left group includes 1 horizontal rod with 10 blocks and 7 individual blocks 2 horizontal rods with 10 blocks each and 6 individual blocks. The label to the left of this group of items is

 

When we add the ones, [latex]7+6[/latex], we get [latex]13[/latex] ones. Because we have more than [latex]10[/latex] ones, we can exchange [latex]10[/latex] of the ones for [latex]1[/latex] ten. Now we have [latex]4[/latex] tens and [latex]3[/latex] ones. Without using the model, we show this as a small red [latex]1[/latex] above the digits in the tens place.

When the sum in a place value column is greater than [latex]9[/latex], we carry over to the next column to the left. Carrying is the same as regrouping by exchanging. For example, [latex]10[/latex] ones for [latex]1[/latex] ten or [latex]10[/latex] tens for [latex]1[/latex] hundred.

Add whole numbers

  1. Write the numbers so each place value lines up vertically.
  2. Add the digits in each place value. Work from right to left starting with the ones place. If a sum in a place value is more than [latex]9[/latex], carry to the next place value.
  3. Continue adding each place value from right to left, adding each place value and carrying if needed.

example

Add: [latex]43+69[/latex]

try it

try it

When the addends have different numbers of digits, be careful to line up the corresponding place values starting with the ones and moving toward the left.

example

Add: [latex]1,683+479[/latex].

try it

Watch the video below for another example of how to add three whole numbers by lining up place values.

Subtraction

Addition and subtraction are inverse operations. Addition undoes subtraction, and subtraction undoes addition.
We know [latex]7 - 3=4[/latex] because [latex]4+3=7[/latex]. Knowing all the addition number facts will help with subtraction. Then we can check subtraction by adding. In the examples above, our subtractions can be checked by addition.

[latex]7-3=4[/latex] because [latex]4+3=7[/latex]
[latex]13-8=5[/latex] because [latex]5+8=13[/latex]
[latex]43-26=17[/latex] because [latex]17+26=43[/latex]

To subtract numbers with more than one digit, it is usually easier to write the numbers vertically in columns just as we did for addition. Align the digits by place value, and then subtract each column starting with the ones and then working to the left.

Exercise

Subtract and then check by adding: [latex]89 - 61[/latex].

TRY IT

Subtract whole numbers

  1. Write the numbers so each place value lines up vertically.
  2. Subtract the digits in each place value. Work from right to left starting with the ones place. If the digit on top is less than the digit below, borrow as needed.
  3. Continue subtracting each place value from right to left, borrowing if needed.
  4. Check by adding.

excercise

Subtract: [latex]43 - 26[/latex].

In the example above, if we model subtracting [latex]26[/latex] from [latex]43[/latex], we would exchange [latex]1[/latex] ten for [latex]10[/latex] ones. When we do this without models, we say we borrow [latex]1[/latex] from the tens place and add [latex]10[/latex] to the ones place.

try it

Exercise

Subtract and then check by adding: [latex]207 - 64[/latex].

try it

Exercise

Subtract and then check by adding: [latex]2,162 - 479[/latex].

try it

 

Watch the video below to see another example of subtracting whole numbers by lining up place values.

Multiplication

In order to multiply without using models, you need to know all the one digit multiplication facts. Make sure you know them fluently before proceeding in this section. The table below shows the multiplication facts.

Each box shows the product of the number down the left column and the number across the top row. If you are unsure about a product, model it. It is important that you memorize any number facts you do not already know so you will be ready to multiply larger numbers.

[latex]x[/latex] [latex]0[/latex] [latex]1[/latex] [latex]2[/latex] [latex]3[/latex] [latex]4[/latex] [latex]5[/latex] [latex]6[/latex] [latex]7[/latex] [latex]8[/latex] [latex]9[/latex]
[latex]0[/latex] [latex]0[/latex] [latex]0[/latex] [latex]0[/latex] [latex]0[/latex] [latex]0[/latex] [latex]0[/latex] [latex]0[/latex] [latex]0[/latex] [latex]0[/latex] [latex]0[/latex]
[latex]1[/latex] [latex]0[/latex] [latex]1[/latex] [latex]2[/latex] [latex]3[/latex] [latex]4[/latex] [latex]5[/latex] [latex]6[/latex] [latex]7[/latex] [latex]8[/latex] [latex]9[/latex]
[latex]2[/latex] [latex]0[/latex] [latex]2[/latex] [latex]4[/latex] [latex]6[/latex] [latex]8[/latex] [latex]10[/latex] [latex]12[/latex] [latex]14[/latex] [latex]16[/latex] [latex]18[/latex]
[latex]3[/latex] [latex]0[/latex] [latex]3[/latex] [latex]6[/latex] [latex]9[/latex] [latex]12[/latex] [latex]15[/latex] [latex]18[/latex] [latex]21[/latex] [latex]24[/latex] [latex]27[/latex]
[latex]4[/latex] [latex]0[/latex] [latex]4[/latex] [latex]8[/latex] [latex]12[/latex] [latex]16[/latex] [latex]20[/latex] [latex]24[/latex] [latex]28[/latex] [latex]32[/latex] [latex]36[/latex]
[latex]5[/latex] [latex]0[/latex] [latex]5[/latex] [latex]10[/latex] [latex]15[/latex] [latex]20[/latex] [latex]25[/latex] [latex]30[/latex] [latex]35[/latex] [latex]40[/latex] [latex]45[/latex]
[latex]6[/latex] [latex]0[/latex] [latex]6[/latex] [latex]12[/latex] [latex]18[/latex] [latex]24[/latex] [latex]30[/latex] [latex]36[/latex] [latex]42[/latex] [latex]48[/latex] [latex]54[/latex]
[latex]7[/latex] [latex]0[/latex] [latex]7[/latex] [latex]14[/latex] [latex]21[/latex] [latex]28[/latex] [latex]35[/latex] [latex]42[/latex] [latex]49[/latex] [latex]56[/latex] [latex]63[/latex]
[latex]8[/latex] [latex]0[/latex] [latex]8[/latex] [latex]16[/latex] [latex]24[/latex] [latex]32[/latex] [latex]40[/latex] [latex]48[/latex] [latex]56[/latex] [latex]64[/latex] [latex]72[/latex]
[latex]9[/latex] [latex]0[/latex] [latex]9[/latex] [latex]18[/latex] [latex]27[/latex] [latex]36[/latex] [latex]45[/latex] [latex]54[/latex] [latex]63[/latex] [latex]72[/latex] [latex]81[/latex]

We know that changing the order of addition does not change the sum. We saw that [latex]8+9=17[/latex] is the same as [latex]9+8=17[/latex].

Is this also true for multiplication? Let’s look at a few pairs of factors.

[latex]4\cdot 7=28\quad 7\cdot 4=28[/latex]
[latex]9\cdot 7=63\quad 7\cdot 9=63[/latex]
[latex]8\cdot 9=72\quad 9\cdot 8=72[/latex]

When the order of the factors is reversed, the product does not change. This is called the Commutative Property of Multiplication.

Commutative Property of Multiplication

Changing the order of the factors does not change their product.

[latex]a\cdot b=b\cdot a[/latex]

example

Multiply:

[latex]8\cdot 7[/latex]
[latex]7\cdot 8[/latex]

try it

To multiply numbers with more than one digit, it is usually easier to write the numbers vertically in columns just as we did for addition and subtraction.

[latex]\begin{array}{c}\hfill 27\\ \hfill \underset{\text{___}}{\times 3}\end{array}[/latex]

We start by multiplying [latex]3[/latex] by [latex]7[/latex].

[latex]3\times 7=21[/latex]

We write the [latex]1[/latex] in the ones place of the product. We carry the [latex]2[/latex] tens by writing [latex]2[/latex] above the tens place.

No Alt Text
Then we multiply the [latex]3[/latex] by the [latex]2[/latex], and add the [latex]2[/latex] above the tens place to the product. So [latex]3\times 2=6[/latex], and [latex]6+2=8[/latex]. Write the [latex]8[/latex] in the tens place of the product.

No Alt Text
The product is [latex]81[/latex].

 

When we multiply two numbers with a different number of digits, it’s usually easier to write the smaller number on the bottom. You could write it the other way, too, but this way is easier to work with.

example

Multiply: [latex]15\cdot 4[/latex]

try it

example

Multiply: [latex]286\cdot 5[/latex]

try it

When we multiply by a number with two or more digits, we multiply by each of the digits separately, working from right to left. Each separate product of the digits is called a partial product. When we write partial products, we must make sure to line up the place values.

MultiplIcation of whole numbers

  1. Write the numbers so each place value lines up vertically.
  2. Multiply the digits in each place value.
    • Work from right to left, starting with the ones place in the bottom number.
      • Multiply the bottom number by the ones digit in the top number, then by the tens digit, and so on.
      • If a product in a place value is more than [latex]9[/latex], carry to the next place value.
      • Write the partial products, lining up the digits in the place values with the numbers above.
    • Repeat for the tens place in the bottom number, the hundreds place, and so on.
    • Insert a zero as a placeholder with each additional partial product.
  3. Add the partial products.

example

Multiply: [latex]62\left(87\right)[/latex]

try it

 

When there are three or more factors, we multiply the first two and then multiply their product by the next factor. For example:

to multiply [latex]8\cdot 3\cdot 2[/latex]
first multiply [latex]8\cdot 3[/latex] [latex]24\cdot 2[/latex]
then multiply [latex]24\cdot 2[/latex] [latex]48[/latex]

In the video below, we summarize the concepts presented on this page including the multiplication property of zero, the identity property of multiplication, and the commutative property of multiplication.m

Division

We said that addition and subtraction are inverse operations because one undoes the other. Similarly, division is the inverse operation of multiplication. We know [latex]12\div 4=3[/latex] because [latex]3\cdot 4=12[/latex]. Knowing all the multiplication number facts is very important when doing division.

We check our answer to division by multiplying the quotient by the divisor to determine if it equals the dividend. We know [latex]24\div 8=3[/latex] is correct because [latex]3\cdot 8=24[/latex].

example

Divide. Then check by multiplying.

  1. [latex]42\div 6[/latex]
  2. [latex]\frac{72}{9}[/latex]
  3. [latex]7\overline{)63}[/latex]

Solution:

1.
[latex]42\div 6[/latex]
Divide [latex]42[/latex] by [latex]6[/latex]. [latex]7[/latex]
Check by multiplying.

[latex]7\cdot 6[/latex]

[latex]42\quad\checkmark [/latex]
2.
[latex]\frac{72}{9}[/latex]
Divide [latex]72[/latex] by [latex]9[/latex]. [latex]8[/latex]
Check by multiplying.

[latex]8\cdot 9[/latex]

[latex]72\quad\checkmark [/latex]
3.
[latex]7\overline{)63}[/latex]
Divide [latex]63[/latex] by [latex]7[/latex]. [latex]9[/latex]
Check by multiplying.

[latex]9\cdot 7[/latex]

[latex]63\quad\checkmark [/latex]

try it

What is the quotient when you divide a number by itself?

[latex]\frac{15}{15}=1\text{ because }1\cdot 15=15[/latex]

Dividing any number [latex]\text{(except 0)}[/latex] by itself produces a quotient of [latex]1[/latex]. Also, any number divided by [latex]1[/latex] produces a quotient of the number. These two ideas are stated in the Division Properties of One.

Division Properties of One

Any number (except 0) divided by itself is one. [latex]a\div a=1[/latex]
Any number divided by one is the same number. [latex]a\div 1=a[/latex]

example

Divide. Then check by multiplying:

  1. [latex]11\div 11[/latex]
  2. [latex]\frac{19}{1}[/latex]

try it

Suppose we have [latex]\text{\$0}[/latex], and want to divide it among [latex]3[/latex] people. How much would each person get? Each person would get [latex]\text{\$0}[/latex]. Zero divided by any number is [latex]0[/latex].

Now suppose that we want to divide [latex]\text{\$10}[/latex] by [latex]0[/latex]. That means we would want to find a number that we multiply by [latex]0[/latex] to get [latex]10[/latex]. This cannot happen because [latex]0[/latex] times any number is [latex]0[/latex]. Division by zero is said to be undefined.

These two ideas make up the Division Properties of Zero.

Division Properties of Zero

Zero divided by any number is [latex]0[/latex]. [latex]0\div a=0[/latex]
Dividing a number by zero is undefined. [latex]a\div 0[/latex] undefined

Another way to explain why division by zero is undefined is to remember that division is really repeated subtraction. How many times can we take away [latex]0[/latex] from [latex]10?[/latex] Because subtracting [latex]0[/latex] will never change the total, we will never get an answer. So we cannot divide a number by [latex]0[/latex].

example

Divide. Check by multiplying:

  1. [latex]0\div 3[/latex]
  2. [latex]\frac{10}{0}[/latex]

try it

When the divisor or the dividend has more than one digit, it is usually easier to use the [latex]4\overline{)12}[/latex] notation. This process is called long division. Let’s work through the process by dividing [latex]78[/latex] by [latex]3[/latex].

Divide the first digit of the dividend, [latex]7[/latex], by the divisor, [latex]3[/latex].
The divisor [latex]3[/latex] can go into [latex]7[/latex] two times since [latex]2\times 3=6[/latex] . Write the [latex]2[/latex] above the [latex]7[/latex] in the quotient. CNX_BMath_Figure_01_05_043_img-02.png
Multiply the [latex]2[/latex] in the quotient by [latex]2[/latex] and write the product, [latex]6[/latex], under the[latex]7[/latex]. CNX_BMath_Figure_01_05_043_img-03.png
Subtract that product from the first digit in the dividend. Subtract [latex]7 - 6[/latex] . Write the difference, 1, under the first digit in the dividend. CNX_BMath_Figure_01_05_043_img-04.png
Bring down the next digit of the dividend. Bring down the [latex]8[/latex]. CNX_BMath_Figure_01_05_043_img-05.png
Divide [latex]18[/latex] by the divisor, [latex]3[/latex]. The divisor [latex]3[/latex] goes into [latex]18[/latex] six times. CNX_BMath_Figure_01_05_043_img-06.png
Write [latex]6[/latex] in the quotient above the [latex]8[/latex].
Multiply the [latex]6[/latex] in the quotient by the divisor and write the product, [latex]18[/latex], under the dividend. Subtract [latex]18[/latex] from [latex]18[/latex]. CNX_BMath_Figure_01_05_043_img-07.png

We would repeat the process until there are no more digits in the dividend to bring down. In this problem, there are no more digits to bring down, so the division is finished.

[latex]\text{So }78\div 3=26[/latex].

Check by multiplying the quotient times the divisor to get the dividend. Multiply [latex]26\times 3[/latex] to make sure that product equals the dividend, [latex]78[/latex].

[latex]\begin{array}{c}\hfill \stackrel{1}{2}6\\ \hfill \underset{\text{___}}{\times 3}\\ \hfill 78 \end{array}[/latex]

It does, so our answer is correct. [latex]\checkmark[/latex]

Division of whole numbers

  1. Divide the first digit of the dividend by the divisor.If the divisor is larger than the first digit of the dividend, divide the first two digits of the dividend by the divisor, and so on.
  2. Write the quotient above the dividend.
  3. Multiply the quotient by the divisor and write the product under the dividend.
  4. Subtract that product from the dividend.
  5. Bring down the next digit of the dividend.
  6. Repeat from Step 1 until there are no more digits in the dividend to bring down.
  7. Check by multiplying the quotient times the divisor.

In the video below we show another example of using long division.

example

Divide [latex]2,596\div 4[/latex]. Check by multiplying:

try it

example

Divide [latex]4,506\div 6[/latex]. Check by multiplying:

try it

Watch this video for another example of how to use long division to divide a four digit whole number by a two digit whole number.

So far all the division problems have worked out evenly. For example, if we had [latex]24[/latex] cookies and wanted to make bags of [latex]8[/latex] cookies, we would have [latex]3[/latex] bags. But what if there were [latex]28[/latex] cookies and we wanted to make bags of [latex]8?[/latex] Start with the [latex]28[/latex] cookies.

An image of 28 cookies placed at random.
Try to put the cookies in groups of eight.

An image of 28 cookies. There are 3 circles, each containing 8 cookies, leaving 3 cookies outside the circles.
There are [latex]3[/latex] groups of eight cookies, and [latex]4[/latex] cookies left over. We call the [latex]4[/latex] cookies that are left over the remainder and show it by writing R4 next to the [latex]3[/latex]. (The R stands for remainder.)

To check this division we multiply [latex]3[/latex] times [latex]8[/latex] to get [latex]24[/latex], and then add the remainder of [latex]4[/latex].

[latex]\begin{array}{c}\hfill 3\\ \hfill \underset{\text{___}}{\times 8}\\ \hfill 24\\ \hfill \underset{\text{___}}{+4}\\ \hfill 28\end{array}[/latex]

example

Divide [latex]1,439\div 4[/latex]. Check by multiplying.

try it

example

Divide and then check by multiplying: [latex]1,461\div 13[/latex].

try it

Watch the video below for another example of how to use long division to divide whole numbers when there is a remainder.

 

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