Lesson Objectives

- Demonstrate an understanding of place value
- Demonstrate an understanding of subtraction
- Demonstrate an understanding of multiplication
- Demonstrate an understanding of single-digit division
- Demonstrate an understanding of division with remainders
- Learn how to divide multi-digit whole numbers using long division

## How to Divide Multi-Digit Whole Numbers using Long Division

In our last lesson, we learned the basic properties of the division operation. At this point, you should be fairly comfortable with dividing single-digit whole numbers. What happens when we want to divide 2-digit, 3-digit, or large multi-digit whole numbers? To divide multi-digit whole numbers, we normally use a process known as long division. Long division is a way to simplify our multi-digit division problem into a series of manageable division, multiplication, and subtraction operations. First and foremost, we will draw the long division symbol. This can be drawn as a right parenthesis with an attached horizontal line (the horizontal line is known as a vinculum). Additionally, we may see a straight line instead of a right parenthesis: To set up our long division problem, we place the dividend under the long division symbol. The divisor will be placed to the left of the long division symbol. As an example, suppose we want to divide 216 by 9 using long division.

216 ÷ 9:

216 - dividend

9 - divisor We place our dividend (216) under the long division symbol. The divisor (9) goes to the left of the long division symbol. Now that we have set up the problem, we are ready to start the long division operation. Many times in math we will use an acronym to remember a sequence of events or steps. When we encounter long division we will usually run into:

DMSBR

On its own, DMSBR doesn't really do much for the memory. We can relate this acronym to members of the family unit. This may help us to remember the division steps:

DMSBR:

Following the DMSBR long division acronym, our first step is to divide. Instead of trying to determine ? x 9 = 216 from the start, we will attack the dividend one digit at a time. We will try to divide the leftmost digit of our dividend by our divisor. We would hear this asked as:

How many times does 9 go into 2? In other words, what is 2 ÷ 9 or how many groups of 9 can be made from 2: 9 doesn't go into 2, because 2 isn't large enough to create even 1 group of 9. When this happens we can expand our selection to include the next digit of the dividend. Now we would ask:

How many times does 9 go into 21?

In other words, what is 21 ÷ 9? We will think about this question using a related multiplication statement:

9 x ? = 21

9 x 1 = 9

9 x 2 = 18

9 x 3 = 27

We can see that there isn't a whole number that multiplies 9 and produces 21 as a result. We will select the number that produces a product that is as close to 21 without going over. In this case, we see that 9 x 2 = 18. Here 18 is as close as we can get to 21 without going over. This means we will use 2 as our answer. This also means we will have a remainder for our division problem. Since 21 - 18 = 3, this will be our remainder or leftover amount from the division operation:

21 ÷ 9 = 2 R3

We want to only think about the quotient part and not the remainder or leftover amount. We would say 9 goes into 21 twice (2 times) since we can at most make two groups of 9 (9 x 2 = 18) out of 21. We write our answer directly over the dividend above the horizontal line. We want to line up our answer from each step according to the place value of the current part of our problem. Here we worked with 21 ÷ 9, so we place our answer with respect to the number 21. We will write the answer (2) above the ones' place for the number 21 (directly over the 1 in the number 216):

correct: incorrect: DMSBR:

Once we have completed the division, we are ready to move onto the next step, which is to multiply. We take our answer from the division problem (2) and multiply by the divisor 9. The result (18) will be written under the dividend from the last problem (21).

2 x 9 = 18

Write the result under 21: DMSBR:

Once we have completed the multiplication and lined up the result correctly, we are ready to move on to our next step, which is to subtract. We subtract the dividend we were working with (21) minus the result from the multiplication (18), we write the result (3) below and maintain place value.

21 - 18 = 3 DMSBR:

Once we have completed our subtraction and lined up the result correctly, we are ready to move on to our next step, which is to bring down. The term "bring down" tells us to bring down the next digit of the dividend (6): DMSBR:

Once we have completed our "bring down" step, we repeat the process of: division, multiplication, subtraction, and bring down. This will continue until there are no more digits in the dividend to bring down. At that point, the long division process is over. We may or may not have a remainder, depending on the problem. Let's start our process over with division. Our new dividend is 36, the number formed as a result of the subtraction and the bring down steps combined. The divisor (9) remains the same throughout the problem.

DMSBR:

We want to divide 36 by 9:

36 ÷ 9 = 4

Write the result (4) next to the 2 in the answer. Each digit flows one to the right after the initial. DMSBR:

Once we have completed the division, we are ready to multiply. We take our answer (4) and multiply by the divisor (9). The result (36) will be written under the dividend from the last problem (36).

4 x 9 = 36 DMSBR:

We subtract the dividend we were working with (36) minus the result from the multiplication (36), we write the result (0) below and maintain place value.

36 - 36 = 0 DMSBR:

Once we have completed the subtraction, we move into the "bring down step". We now have no more digits in the dividend (216) to bring down. When this occurs, we are finished with our long division operation. Since the result of the last subtraction operation (36 - 36 = 0) was zero, we don't have a remainder:

We can report our answer as 24, the number directly over the dividend.

Let's take a look at a few more examples:

Example 1: Find each quotient:

465 ÷ 15

Let's set up our long division problem:

465 - dividend

15 - divisor DMSBR:

Divide:

Our first step is to divide. We will first ask:

How many times will 15 go into 4? 15 doesn't go into 4, so let's expand our selection. We will now ask:

How many times will 15 go into 46?

In other words what is 46 ÷ 15? 15 x ? = 46

15 x 1 = 15

15 x 2 = 30

15 x 3 = 45

15 x 4 = 60

We can see that 15 x 3 = 45, this result (45) is as close to 46 as we can get without going over. We will use 3 as our answer and have a remainder of 1 (46 - 45 = 1)

46 ÷ 15 = 3 R1

We can ignore any remainders at this point. We write the answer 3 above the 6 in the number 465. DMSBR:

Multiply:

Multiply 3 x 15, the result (45) is placed below 46.

3 x 15 = 45 DMSBR:

Subtract:

Subtract 46 - 45, the result (1) is placed directly below.

46 - 45 = 1 DMSBR:

Bring Down:

Bring down the next digit of the dividend (5) DMSBR:

Repeat or Remainder:

We will repeat the process. We will be working with 15 as the dividend.

DMSBR:

We ask the division question:

How many times will 15 go into 15?

15 ÷ 15 = 1

We write the answer (1) next to the 3 in our answer. DMSBR:

Multiply:

Multiply 1 x 15, the result (15) is placed below 15.

1 x 15 = 15 DMSBR:

Subtract:

Subtract 15 - 15, the result (0) is placed directly below.

15 - 15 = 0 DMSBR:

Bring Down:

There are no more digits to bring down, and the result of our last subtraction (15 - 15 = 0) was 0:

This tells us our answer is 31 with no remainder.

465 ÷ 15 = 31

Let's take a look at an example with a remainder.

Example 2: Find each quotient:

971 ÷ 29

Let's set up our long division problem:

971 - dividend

29 - divisor DMSBR:

Divide:

Our first step is to divide. We will first ask:

How many times will 29 go into 9? 29 doesn't go into 9, so let's expand our selection. We will now ask:

How many times will 29 go into 97?

In other words what is 97 ÷ 29? 29 x ? = 97

29 x 1 = 29

29 x 2 = 58

29 x 3 = 87

29 x 4 = 116

We can see that 29 x 3 = 87, this result (87) is as close as to 97 as we can get without going over. We will use 3 as our answer and have a remainder of 10 (97 - 87 = 10)

97 ÷ 29 = 3 R10

We can ignore any remainders at this point. We write the answer 3 above the 7 in the number 971. DMSBR:

Multiply:

Multiply 3 x 29, the result (87) is placed below 97.

3 x 29 = 87 DMSBR:

Subtract:

Subtract 97 - 87, the result (10) is placed directly below.

97 - 87 = 10 DMSBR:

Bring Down:

Bring down the next digit of the dividend (1) DMSBR:

Repeat or Remainder:

We will repeat the process. We will be working with 101 as the dividend.

DMSBR:

We ask the division question:

How many times will 29 go into 101?

We know from earlier that:

29 x 3 = 87 and 29 x 4 = 116

116 is too large, so we will use 3 for the answer.

101 ÷ 29 = 3 R14

Remember to ignore any remainders at this point.

We write the answer (3) next to the 3 in our answer. DMSBR:

Multiply:

Multiply 3 x 29, the result (87) is placed below 101.

3 x 29 = 87 DMSBR:

Subtract:

Subtract 101 - 87, the result (14) is placed directly below.

101 - 87 = 14 DMSBR:

Bring Down:

There are no more digits to bring down, and the result of our last subtraction (101 - 87 = 14) was 14:

This tells us our answer is 33 with a remainder of 14. We can write "R" followed by the remainder (14) to the right of our answer. 971 ÷ 29 = 33 R14

216 ÷ 9:

216 - dividend

9 - divisor We place our dividend (216) under the long division symbol. The divisor (9) goes to the left of the long division symbol. Now that we have set up the problem, we are ready to start the long division operation. Many times in math we will use an acronym to remember a sequence of events or steps. When we encounter long division we will usually run into:

DMSBR

On its own, DMSBR doesn't really do much for the memory. We can relate this acronym to members of the family unit. This may help us to remember the division steps:

- D - Dad
- M - Mom
- S - Sister
- B - Brother
- R - Rover (dog)

- D - Divide
- M - Multiply
- S - Subtract
- B - Bring Down
- Repeat the steps or Remainder
- Not every problem ends with a remainder. We are finished with the problem when there are no more digits in the dividend to bring down.

DMSBR:

Following the DMSBR long division acronym, our first step is to divide. Instead of trying to determine ? x 9 = 216 from the start, we will attack the dividend one digit at a time. We will try to divide the leftmost digit of our dividend by our divisor. We would hear this asked as:

How many times does 9 go into 2? In other words, what is 2 ÷ 9 or how many groups of 9 can be made from 2: 9 doesn't go into 2, because 2 isn't large enough to create even 1 group of 9. When this happens we can expand our selection to include the next digit of the dividend. Now we would ask:

How many times does 9 go into 21?

In other words, what is 21 ÷ 9? We will think about this question using a related multiplication statement:

9 x ? = 21

9 x 1 = 9

9 x 2 = 18

9 x 3 = 27

We can see that there isn't a whole number that multiplies 9 and produces 21 as a result. We will select the number that produces a product that is as close to 21 without going over. In this case, we see that 9 x 2 = 18. Here 18 is as close as we can get to 21 without going over. This means we will use 2 as our answer. This also means we will have a remainder for our division problem. Since 21 - 18 = 3, this will be our remainder or leftover amount from the division operation:

21 ÷ 9 = 2 R3

We want to only think about the quotient part and not the remainder or leftover amount. We would say 9 goes into 21 twice (2 times) since we can at most make two groups of 9 (9 x 2 = 18) out of 21. We write our answer directly over the dividend above the horizontal line. We want to line up our answer from each step according to the place value of the current part of our problem. Here we worked with 21 ÷ 9, so we place our answer with respect to the number 21. We will write the answer (2) above the ones' place for the number 21 (directly over the 1 in the number 216):

correct: incorrect: DMSBR:

Once we have completed the division, we are ready to move onto the next step, which is to multiply. We take our answer from the division problem (2) and multiply by the divisor 9. The result (18) will be written under the dividend from the last problem (21).

2 x 9 = 18

Write the result under 21: DMSBR:

Once we have completed the multiplication and lined up the result correctly, we are ready to move on to our next step, which is to subtract. We subtract the dividend we were working with (21) minus the result from the multiplication (18), we write the result (3) below and maintain place value.

21 - 18 = 3 DMSBR:

Once we have completed our subtraction and lined up the result correctly, we are ready to move on to our next step, which is to bring down. The term "bring down" tells us to bring down the next digit of the dividend (6): DMSBR:

Once we have completed our "bring down" step, we repeat the process of: division, multiplication, subtraction, and bring down. This will continue until there are no more digits in the dividend to bring down. At that point, the long division process is over. We may or may not have a remainder, depending on the problem. Let's start our process over with division. Our new dividend is 36, the number formed as a result of the subtraction and the bring down steps combined. The divisor (9) remains the same throughout the problem.

DMSBR:

We want to divide 36 by 9:

36 ÷ 9 = 4

Write the result (4) next to the 2 in the answer. Each digit flows one to the right after the initial. DMSBR:

Once we have completed the division, we are ready to multiply. We take our answer (4) and multiply by the divisor (9). The result (36) will be written under the dividend from the last problem (36).

4 x 9 = 36 DMSBR:

We subtract the dividend we were working with (36) minus the result from the multiplication (36), we write the result (0) below and maintain place value.

36 - 36 = 0 DMSBR:

Once we have completed the subtraction, we move into the "bring down step". We now have no more digits in the dividend (216) to bring down. When this occurs, we are finished with our long division operation. Since the result of the last subtraction operation (36 - 36 = 0) was zero, we don't have a remainder:

We can report our answer as 24, the number directly over the dividend.

Let's take a look at a few more examples:

Example 1: Find each quotient:

465 ÷ 15

Let's set up our long division problem:

465 - dividend

15 - divisor DMSBR:

Divide:

Our first step is to divide. We will first ask:

How many times will 15 go into 4? 15 doesn't go into 4, so let's expand our selection. We will now ask:

How many times will 15 go into 46?

In other words what is 46 ÷ 15? 15 x ? = 46

15 x 1 = 15

15 x 2 = 30

15 x 3 = 45

15 x 4 = 60

We can see that 15 x 3 = 45, this result (45) is as close to 46 as we can get without going over. We will use 3 as our answer and have a remainder of 1 (46 - 45 = 1)

46 ÷ 15 = 3 R1

We can ignore any remainders at this point. We write the answer 3 above the 6 in the number 465. DMSBR:

Multiply:

Multiply 3 x 15, the result (45) is placed below 46.

3 x 15 = 45 DMSBR:

Subtract:

Subtract 46 - 45, the result (1) is placed directly below.

46 - 45 = 1 DMSBR:

Bring Down:

Bring down the next digit of the dividend (5) DMSBR:

Repeat or Remainder:

We will repeat the process. We will be working with 15 as the dividend.

DMSBR:

We ask the division question:

How many times will 15 go into 15?

15 ÷ 15 = 1

We write the answer (1) next to the 3 in our answer. DMSBR:

Multiply:

Multiply 1 x 15, the result (15) is placed below 15.

1 x 15 = 15 DMSBR:

Subtract:

Subtract 15 - 15, the result (0) is placed directly below.

15 - 15 = 0 DMSBR:

Bring Down:

There are no more digits to bring down, and the result of our last subtraction (15 - 15 = 0) was 0:

This tells us our answer is 31 with no remainder.

465 ÷ 15 = 31

Let's take a look at an example with a remainder.

Example 2: Find each quotient:

971 ÷ 29

Let's set up our long division problem:

971 - dividend

29 - divisor DMSBR:

Divide:

Our first step is to divide. We will first ask:

How many times will 29 go into 9? 29 doesn't go into 9, so let's expand our selection. We will now ask:

How many times will 29 go into 97?

In other words what is 97 ÷ 29? 29 x ? = 97

29 x 1 = 29

29 x 2 = 58

29 x 3 = 87

29 x 4 = 116

We can see that 29 x 3 = 87, this result (87) is as close as to 97 as we can get without going over. We will use 3 as our answer and have a remainder of 10 (97 - 87 = 10)

97 ÷ 29 = 3 R10

We can ignore any remainders at this point. We write the answer 3 above the 7 in the number 971. DMSBR:

Multiply:

Multiply 3 x 29, the result (87) is placed below 97.

3 x 29 = 87 DMSBR:

Subtract:

Subtract 97 - 87, the result (10) is placed directly below.

97 - 87 = 10 DMSBR:

Bring Down:

Bring down the next digit of the dividend (1) DMSBR:

Repeat or Remainder:

We will repeat the process. We will be working with 101 as the dividend.

DMSBR:

We ask the division question:

How many times will 29 go into 101?

We know from earlier that:

29 x 3 = 87 and 29 x 4 = 116

116 is too large, so we will use 3 for the answer.

101 ÷ 29 = 3 R14

Remember to ignore any remainders at this point.

We write the answer (3) next to the 3 in our answer. DMSBR:

Multiply:

Multiply 3 x 29, the result (87) is placed below 101.

3 x 29 = 87 DMSBR:

Subtract:

Subtract 101 - 87, the result (14) is placed directly below.

101 - 87 = 14 DMSBR:

Bring Down:

There are no more digits to bring down, and the result of our last subtraction (101 - 87 = 14) was 14:

This tells us our answer is 33 with a remainder of 14. We can write "R" followed by the remainder (14) to the right of our answer. 971 ÷ 29 = 33 R14

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