Use Division Notation

Suppose you have [latex]12[/latex] cookies and want to package them in bags with [latex]4[/latex] cookies in each bag. How many bags would we need?

You might put [latex]4[/latex] cookies in the first bag, [latex]4[/latex] in the second bag, and so on, until you run out of cookies. Doing it this way, you would fill [latex]3[/latex] bags.

We can write

[latex]12\div 4[/latex]

We read this as twelve divided by four and the result of latex]3[/latex] is the quotient of [latex]12[/latex] and [latex]4[/latex].  We call the number being divided, the dividend, and the number dividing it, the divisor. In this case, the dividend is [latex]12[/latex] and the divisor is [latex]4[/latex].

There are four different ways of notating division, but they all mean the same thing:  [latex]4\overline{)12}[/latex], [latex]12\div 4, 12\text{/}4, \frac{12}{4}[/latex]. In each case, the [latex]12[/latex] is the dividend and the [latex]4[/latex] is the divisor.

Dividing Integers

Division is the inverse operation of multiplication. This means that division undoes 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].

 

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.
Multiply the [latex]2[/latex] in the quotient by [latex]2[/latex] and write the product, [latex]6[/latex], under the [latex]7[/latex].
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.
Bring down the next digit of the dividend. Bring down the [latex]8[/latex].
Divide [latex]18[/latex] by the divisor, [latex]3[/latex]. The divisor [latex]3[/latex] goes into [latex]18[/latex] six times.
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].

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]

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

 

 

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

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.

Try to put the cookies in groups of eight.

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]

This is an example of the Fundamental Theorem of Arithmetic.

 

 

 

Sometimes it might not be obvious how many times the divisor goes into digits of the dividend. We will have to guess and check numbers to find the greatest number that goes into the digits without exceeding them.

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