LESSON: Operations with Whole Numbers

READ: Dividing Whole Numbers

Dividing Whole Numbers

Our final operation is division. First, let’s talk about what the word “division” actually means. To divide means to split up into groups. If multiplication means to add groups of things, then division is the opposite of multiplication.

Example

$72 \div 9 = \underline{\;\;\;\;\;\;\;\;\;\;}$

In this problem, 72 is the number being divided, it is the dividend. 9 is the number doing the dividing, it is the divisor. The answer in a division problem is called the quotient. We can complete this problem by thinking of our multiplication facts.

Sometimes, a number won’t divide evenly. When this happens, we have a remainder.

Example

$15 \div 2 =\underline{\;\;\;\;\;\;\;\;\;\;}$

Fifteen is not an even number. There will be a remainder here.

We can use an “ $r$ ” to show that there is a remainder. Our solution to $15 \div 2 =\underline{\;\;\;\;\;\;\;\;\;\;}$ is 7 r. 2 (or 7 remainder 2)

We can also divide larger numbers. We can use a division box to do this.

Example

$8 \overline{)825 \;}$

Here we have a one digit divisor, 8, and a three digit dividend, 825. We need to figure out how many 8’s there are in 825. To do this, we divide the divisor 8 into each digit of the dividend.

$& 8 \overline{)825 \;} \qquad “How \ many \ 8’s \ are \ there \ in \ 8?”\ & \qquad \qquad \ \ The \ answer \ is \ 1.$

We put the 1 on top of the division box above the 8.

$& \overset{\ 1}{8\overline{ ) 825}}\ & \underline{-8} \Bigg \downarrow\ & \quad 02$

We multiply 1 times 8 and subtract our result from the dividend. Then we can bring down the next number in the dividend. Then, we need to look at the next digit in the dividend. “How many 8’s are there in 2?” The answer is 0. We put a 0 into the answer next to the 1.

$& \overset{\ 10}{8\overline{ ) 825}}\ & \underline{-8} \;\; \Bigg \downarrow\ & \quad \ 025$

Because we couldn’t divide 8 into 2, now we can look at 25. “How many 8’s are in 25?” The answer is 3 with a remainder of 1. We can add this into our answer.

$& \overset{\ 103rl}{8\overline{ ) 825 \;}}\ & \ \underline{ -8 \ \ }\ & \ \ \ 025\ & \ \ \underline{-24}\ & \qquad 1$

We can check our work by multiplying the answer times to divisor.

$& \qquad 103\ & \ \underline {\times \quad \ \ 8 \ }\ & \qquad 824 + r \ \text{of} \ 1 = 825$

Our answer checks out.

Let’s look at an example with a two-digit divisor.

Example

$& \overset{\ 2}{12\overline{ ) 2448}} && “How \ many \ 12’s \ are \ in \ 2? \ None.”\ & \ \underline{-24} \Bigg \downarrow && “How \ many \ 12’s \ are \ in \ 24?”\ & \qquad \ 4 && \ Two\ \ & \overset{20}{12\overline{) 2448}} && “How \ many \ 12’s \ are \ in \ 4? \ None, \ so \ we \ add \ in \ a \ zero.”\ && &“How \ many \ 12’s \ are \ in \ 48?”\ && &Four\ && &There \ in \ not \ a \ remainder.\ \ &\overset{204}{12\overline{ ) 2448}}$

We check our work by multiplying $204 \times 12$ .

$& \qquad \quad 204\ & \ \underline {\times \qquad \ 12}\ & \qquad \quad 408\ & \ \underline {+ \quad \ 2040}\ & \qquad \ 2448$

Our answer checks out.

We can apply these same steps to any addition problem even if the divisor has two or three digits. We work through each value of the divisor with each value of the dividend. We can check our work by multiplying our answer times the divisor too.