Mastering Long Division: A Step-by-Step Guide

Long division is a fundamental arithmetic skill that builds a strong foundation for more advanced mathematical concepts. While calculators can quickly provide the answer, understanding the process of long division is crucial for developing number sense, problem-solving abilities, and a deeper appreciation for how numbers work. This comprehensive guide will walk you through the steps of long division, providing clear explanations and examples to help you master this essential skill.

## What is Long Division?

Long division is a method used to divide large numbers into smaller, more manageable parts. It’s especially useful when you’re dividing a number (the dividend) by a number that doesn’t easily divide into it evenly (the divisor). The goal of long division is to find the quotient (the result of the division) and the remainder (any amount left over).

## The Parts of a Long Division Problem

Before we dive into the steps, let’s define the different parts of a long division problem:

* **Dividend:** The number being divided (the larger number inside the division bracket).
* **Divisor:** The number you are dividing by (the number outside the division bracket).
* **Quotient:** The result of the division (the number written above the division bracket).
* **Remainder:** The amount left over after the division is complete (if the divisor doesn’t divide evenly into the dividend).

## The Steps of Long Division: D-M-S-B-R

Long division follows a consistent set of steps, which can be easily remembered using the acronym **D-M-S-B-R**:

* **D**ivide
* **M**ultiply
* **S**ubtract
* **B**ring Down
* **R**epeat (or Remainder)

Let’s break down each step with examples:

### Step 1: Divide

* **Determine if the divisor can go into the first digit (or digits) of the dividend.** Look at the first digit of the dividend. Can the divisor be divided into that digit? If not, consider the first two digits of the dividend.
* **Write the quotient above the division bracket.** How many times does the divisor go into the part of the dividend you’re considering? Write that number (the quotient) above the division bracket, directly above the last digit of the part of the dividend you used.

**Example 1:** Divide 72 by 3.

* We write the problem as: 3 | 72
* Can 3 go into 7? Yes, it can.
* How many times does 3 go into 7? It goes in 2 times (2 x 3 = 6).
* Write the ‘2’ above the ‘7’ in the division bracket.

2
3 | 72

**Example 2:** Divide 156 by 12.

* We write the problem as: 12 | 156
* Can 12 go into 1? No, it can’t. So we consider the first two digits, 15.
* How many times does 12 go into 15? It goes in 1 time (1 x 12 = 12).
* Write the ‘1’ above the ‘5’ in the division bracket.

1
12 | 156

### Step 2: Multiply

* **Multiply the quotient (the number you just wrote above the division bracket) by the divisor.**
* **Write the result of the multiplication below the part of the dividend you considered in the previous step.**

**Example 1 (Continuing):** Divide 72 by 3.

* We wrote ‘2’ above the ‘7’.
* Multiply the quotient (2) by the divisor (3): 2 x 3 = 6.
* Write the ‘6’ below the ‘7’.

2
3 | 72
6

**Example 2 (Continuing):** Divide 156 by 12.

* We wrote ‘1’ above the ‘5’.
* Multiply the quotient (1) by the divisor (12): 1 x 12 = 12.
* Write the ’12’ below the ’15’.

1
12 | 156
12

### Step 3: Subtract

* **Subtract the number you wrote down in the previous step from the part of the dividend you considered.**
* **Write the difference below the line.**

**Example 1 (Continuing):** Divide 72 by 3.

* We wrote ‘6’ below the ‘7’.
* Subtract 6 from 7: 7 – 6 = 1.
* Write the ‘1’ below the ‘6’.

2
3 | 72
6
—
1

**Example 2 (Continuing):** Divide 156 by 12.

* We wrote ’12’ below the ’15’.
* Subtract 12 from 15: 15 – 12 = 3.
* Write the ‘3’ below the ’12’.

1
12 | 156
12
—
3

### Step 4: Bring Down

* **Bring down the next digit of the dividend next to the difference you just calculated.**
* **This new number becomes the new part of the dividend you’ll work with.**

**Example 1 (Continuing):** Divide 72 by 3.

* The next digit in the dividend is ‘2’.
* Bring down the ‘2’ next to the ‘1’.
* We now have ’12’.

2
3 | 72
6
—
12

**Example 2 (Continuing):** Divide 156 by 12.

* The next digit in the dividend is ‘6’.
* Bring down the ‘6’ next to the ‘3’.
* We now have ’36’.

1
12 | 156
12
—
36

### Step 5: Repeat (or Remainder)

* **Repeat steps 1-4 (Divide, Multiply, Subtract, Bring Down) using the new number you created in the ‘Bring Down’ step.**
* **Continue repeating these steps until you have brought down all the digits of the dividend.**
* **If the divisor doesn’t divide evenly into the final number, the amount left over is the remainder.**

**Example 1 (Continuing):** Divide 72 by 3.

* **Divide:** How many times does 3 go into 12? It goes in 4 times.
* Write the ‘4’ above the ‘2’ in the division bracket.

24
3 | 72
6
—
12

* **Multiply:** Multiply the quotient (4) by the divisor (3): 4 x 3 = 12.
* Write the ’12’ below the ’12’.

24
3 | 72
6
—
12
12

* **Subtract:** Subtract 12 from 12: 12 – 12 = 0.
* Write the ‘0’ below the ’12’.

24
3 | 72
6
—
12
12
—
0

* Since there are no more digits to bring down and the result of the subtraction is 0, the division is complete.
* **The quotient is 24, and the remainder is 0.**

**Example 2 (Continuing):** Divide 156 by 12.

* **Divide:** How many times does 12 go into 36? It goes in 3 times.
* Write the ‘3’ above the ‘6’ in the division bracket.

13
12 | 156
12
—
36

* **Multiply:** Multiply the quotient (3) by the divisor (12): 3 x 12 = 36.
* Write the ’36’ below the ’36’.

13
12 | 156
12
—
36
36

* **Subtract:** Subtract 36 from 36: 36 – 36 = 0.
* Write the ‘0’ below the ’36’.

13
12 | 156
12
—
36
36
—
0

* Since there are no more digits to bring down and the result of the subtraction is 0, the division is complete.
* **The quotient is 13, and the remainder is 0.**

## Long Division with Remainders

Sometimes, the divisor won’t divide evenly into the dividend, leaving a remainder. The remainder is the amount left over after you’ve divided as far as you can.

**Example 3:** Divide 85 by 4.

* We write the problem as: 4 | 85

* **Divide:** How many times does 4 go into 8? It goes in 2 times.
* Write the ‘2’ above the ‘8’.

2
4 | 85

* **Multiply:** Multiply the quotient (2) by the divisor (4): 2 x 4 = 8.
* Write the ‘8’ below the ‘8’.

2
4 | 85
8

* **Subtract:** Subtract 8 from 8: 8 – 8 = 0.
* Write the ‘0’ below the ‘8’.

2
4 | 85
8
—
0

* **Bring Down:** Bring down the ‘5’ next to the ‘0’.

2
4 | 85
8
—
05

* **Divide:** How many times does 4 go into 5? It goes in 1 time.
* Write the ‘1’ above the ‘5’.

21
4 | 85
8
—
05

* **Multiply:** Multiply the quotient (1) by the divisor (4): 1 x 4 = 4.
* Write the ‘4’ below the ‘5’.

21
4 | 85
8
—
05
4

* **Subtract:** Subtract 4 from 5: 5 – 4 = 1.
* Write the ‘1’ below the ‘4’.

21
4 | 85
8
—
05
4
—
1

* Since there are no more digits to bring down, the division is complete.
* **The quotient is 21, and the remainder is 1.** We write this as 21 R 1.

## Long Division with Larger Numbers

The same D-M-S-B-R steps apply to larger numbers. You just might need to repeat them more times.

**Example 4:** Divide 3456 by 16.

* We write the problem as: 16 | 3456

* **Divide:** How many times does 16 go into 3? It doesn’t. How many times does 16 go into 34? It goes in 2 times.
* Write the ‘2’ above the ‘4’.

2
16 | 3456

* **Multiply:** Multiply the quotient (2) by the divisor (16): 2 x 16 = 32.
* Write the ’32’ below the ’34’.

2
16 | 3456
32

* **Subtract:** Subtract 32 from 34: 34 – 32 = 2.
* Write the ‘2’ below the ’32’.

2
16 | 3456
32
—
2

* **Bring Down:** Bring down the ‘5’ next to the ‘2’.

2
16 | 3456
32
—
25

* **Divide:** How many times does 16 go into 25? It goes in 1 time.
* Write the ‘1’ above the ‘5’.

21
16 | 3456
32
—
25

* **Multiply:** Multiply the quotient (1) by the divisor (16): 1 x 16 = 16.
* Write the ’16’ below the ’25’.

21
16 | 3456
32
—
25
16

* **Subtract:** Subtract 16 from 25: 25 – 16 = 9.
* Write the ‘9’ below the ’16’.

21
16 | 3456
32
—
25
16
—
9

* **Bring Down:** Bring down the ‘6’ next to the ‘9’.

21
16 | 3456
32
—
25
16
—
96

* **Divide:** How many times does 16 go into 96? It goes in 6 times.
* Write the ‘6’ above the ‘6’.

216
16 | 3456
32
—
25
16
—
96

* **Multiply:** Multiply the quotient (6) by the divisor (16): 6 x 16 = 96.
* Write the ’96’ below the ’96’.

216
16 | 3456
32
—
25
16
—
96
96

* **Subtract:** Subtract 96 from 96: 96 – 96 = 0.
* Write the ‘0’ below the ’96’.

216
16 | 3456
32
—
25
16
—
96
96
—
0

* Since there are no more digits to bring down and the result of the subtraction is 0, the division is complete.
* **The quotient is 216, and the remainder is 0.**

## Tips for Success in Long Division

* **Practice Regularly:** The more you practice, the more comfortable you’ll become with the steps.
* **Know Your Multiplication Facts:** Strong multiplication skills are essential for long division. Review your times tables if needed.
* **Be Neat and Organized:** Keep your numbers aligned and your work organized to avoid errors. Use graph paper if that helps you with alignment.
* **Double-Check Your Work:** After completing a long division problem, multiply the quotient by the divisor and add the remainder. The result should equal the dividend. For example, in 85 / 4, we got 21 R 1. (21 * 4) + 1 = 84 + 1 = 85. This confirms our answer is correct.
* **Break Down Problems:** If you’re struggling with a particular step, break it down into smaller parts.
* **Use Estimation:** Before you start, estimate the answer to give you a sense of what to expect. For example, in 3456/16, you might estimate that 3200/16 = 200, so the answer should be around 200.
* **Understand the ‘Why’:** Don’t just memorize the steps; understand why they work. This will help you remember them and apply them in different situations.

## Common Mistakes to Avoid

* **Misaligned Numbers:** Ensure that your numbers are properly aligned to avoid subtraction errors.
* **Incorrect Multiplication:** Double-check your multiplication facts to avoid errors in the ‘Multiply’ step.
* **Skipping Steps:** Don’t skip any steps, even if they seem obvious. Skipping steps can lead to errors.
* **Forgetting to Bring Down:** Always remember to bring down the next digit of the dividend in the ‘Bring Down’ step.
* **Misinterpreting the Remainder:** Remember that the remainder is the amount left over after you’ve divided as far as you can.

## Long Division with Decimals

Long division can also be used to divide numbers that result in decimal quotients. If you reach a point where you’ve brought down all the digits of the dividend and still have a remainder, you can add a decimal point to the end of the dividend and add zeros after the decimal point. Then, continue the long division process as before, bringing down the zeros.

**Example 5:** Divide 25 by 2.

* We write the problem as: 2 | 25

* Following the initial steps of long division, we get to:

12
2 | 25
2
—
05
4
—
1

* We’ve brought down all the digits, but we have a remainder of 1. Add a decimal point and a zero to the dividend (25 becomes 25.0).

12
2 | 25.0
2
—
05
4
—
10

* Bring down the ‘0’.

12
2 | 25.0
2
—
05
4
—
10

* **Divide:** How many times does 2 go into 10? It goes in 5 times.
* Write the ‘5’ after the decimal point in the quotient.

12.5
2 | 25.0
2
—
05
4
—
10

* **Multiply:** Multiply the quotient (5) by the divisor (2): 5 x 2 = 10.
* Write the ’10’ below the ’10’.

12.5
2 | 25.0
2
—
05
4
—
10
10

* **Subtract:** Subtract 10 from 10: 10 – 10 = 0.
* Write the ‘0’ below the ’10’.

12.5
2 | 25.0
2
—
05
4
—
10
10
—
0

* The quotient is 12.5, and the remainder is 0.

## Long Division with Multi-Digit Divisors

The process is the same, even if the divisor has multiple digits. The key is to focus on manageable chunks of the dividend. Estimation is even more important here.

**Example 6:** Divide 1728 by 36.

* We write the problem as: 36 | 1728

* **Divide:** How many times does 36 go into 1? It doesn’t. How many times does 36 go into 17? It doesn’t. How many times does 36 go into 172? This is where estimation helps. We might think, “36 is close to 40, and 40 goes into 160 four times.” So let’s try 4.
* Write the ‘4’ above the ‘2’.

4
36 | 1728

* **Multiply:** Multiply the quotient (4) by the divisor (36): 4 x 36 = 144.
* Write the ‘144’ below the ‘172’.

4
36 | 1728
144

* **Subtract:** Subtract 144 from 172: 172 – 144 = 28.
* Write the ’28’ below the ‘144’.

4
36 | 1728
144
—
28

* **Bring Down:** Bring down the ‘8’ next to the ’28’.

4
36 | 1728
144
—
288

* **Divide:** How many times does 36 go into 288? Since we know 4 x 36 = 144, then 8 x 36 might work.
* Write the ‘8’ above the ‘8’.

48
36 | 1728
144
—
288

* **Multiply:** Multiply the quotient (8) by the divisor (36): 8 x 36 = 288.
* Write the ‘288’ below the ‘288’.

48
36 | 1728
144
—
288
288

* **Subtract:** Subtract 288 from 288: 288 – 288 = 0.
* Write the ‘0’ below the ‘288’.

48
36 | 1728
144
—
288
288
—
0

* The quotient is 48, and the remainder is 0.

## Practice Problems

To solidify your understanding, try solving these practice problems:

1. 96 ÷ 8
2. 147 ÷ 7
3. 256 ÷ 4
4. 385 ÷ 5
5. 576 ÷ 12
6. 1024 ÷ 16
7. 73 ÷ 3
8. 158 ÷ 5
9. 283 ÷ 6
10. 491 ÷ 9
11. 1357 ÷ 23
12. 2784 ÷ 32
13. 4914 ÷ 54
14. 6879 ÷ 71
15. 9216 ÷ 96

## Conclusion

Long division might seem daunting at first, but with practice and a clear understanding of the steps, you can master this important skill. Remember the D-M-S-B-R mnemonic, stay organized, and don’t be afraid to break down problems into smaller parts. By mastering long division, you’ll build a solid foundation for more advanced mathematical concepts and develop valuable problem-solving skills.