How to Do Long Division - Steps with Examples

Long division is a widely used method in mathematics for solving problems that involve dividing large numbers. It simplifies complex calculations by breaking them into smaller, manageable steps. This technique is especially helpful when mental math or simple division methods are not enough. 

Long division is not only used with whole numbers but can also be applied to decimals and algebraic expressions. In this blog, we will explain the steps to perform long division with remainders, without remainders, and without using decimals.

Read More:  Division of Fractions - Definitions, Steps & Examples

What is the Long Division Method?

The long division method is a way to divide large numbers by simplifying the process into smaller steps. Instead of solving the entire division at once, you work through it one digit at a time. At each step, you perform a basic division, subtraction, and bring down the next digit to continue the process.

This continues until there are no more digits left to work with. The method helps keep long or difficult division problems organized and easier to solve. Long division can also be thought of as repeated subtraction. You continue subtracting until you reach a remainder that’s smaller than the number you're dividing by, or until you reach zero.

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Components of the Long Division Method

The long division method involves four main components that work together during the division process. These are:

Dividend: The number that is being divided. It is usually the larger number in the problem.

Divisor: The number you are dividing by. It divides the dividend and is typically smaller than or equal to the dividend.

Quotient: The number you get after completing the division process. It represents the final answer written above the dividend.

Remainder: The amount left over after completing the division. If nothing is left, the remainder is zero. A remainder is always smaller than the divisor.

Read More:  Multiple of Fractions

Let’s understand these components with an example:

Divide 765 ÷ 11 using long division.

Dividend = 765

Divisor = 11

When you perform the long division, the result is:

Quotient = 69

Remainder = 6

So, the number 11 goes into 765 exactly 69 times, with 6 left over. That leftover part is the remainder. You can also check your answer using this formula:

(Divisor × Quotient) + Remainder = Dividend

 (5 × 87) + 2 = 435

Read More: 15 Math Tricks to Make Learning Easy

How to Do Long Division?

The long division method is a step-by-step way to divide large numbers. Instead of solving the entire problem at once, you work through it by dividing, subtracting, and bringing down digits one at a time until the entire number has been divided. 

Here are the steps to perform the long division process:

Step 1: Start from the left side of the dividend. Look at the first digit (or group of digits) that is large enough to be divided by the divisor.

Step 2: Find the closest multiple of the divisor that fits into that part of the dividend.

Step 3: Write the result (a single digit of the quotient) above the dividend.

Step 4: Multiply the divisor by the quotient digit and subtract the result from the current portion of the dividend.

Step 5: Bring down the next digit of the dividend and repeat the process.

Step 6: Continue until all digits have been used. The number left (if any) is the remainder.

Long Division Steps Without Remainder

In long division without remainder, the steps are just like regular long division. The only difference is that the division ends exactly, with no leftover value at the end.

Let’s understand this with an example:

Divide 57 by 3

Step 1: Begin with the number 57. This number is to be divided by 3 using long division.

Step 2: Consider the first digit, which is 5. 3 fits into 5 one time, since 3 × 1 = 3.  Place 1 in the quotient above the digit 5.

Step 3: Subtract 3 from 5. The result is 2. This becomes the current remainder.

Step 4: Bring down the next digit, which is 7. This forms the number 27.

Step 5: 3 fits into 27 nine times, since 3 × 9 = 27. Write 9 in the quotient next to the 1.

Step 6: Subtract 27 from 27. The result is 0. There are no more digits to bring down, and no remainder is left.

Long Division Steps With Remainder

In long division with remainder, the process follows the usual steps of divide, multiply, subtract, and bring down. However, if the final value left after all digits have been used is smaller than the divisor, it cannot be divided further and becomes the remainder. This happens when the dividend does not divide evenly by the divisor.

Let's understand the long division steps with remainder through an example:

Example: Divide 135 by 6 using long division

Step 1: Start with the first digit of the dividend, which is 1. Since 1 is less than 6, it cannot be divided. Take the first two digits together: 13.

Step 2: 6 goes into 13 two times, because 6 × 2 = 12. Write 2 in the quotient above the digit 3. Subtract 12 from 13, leaving a remainder of 1.

Step 3: Bring down the next digit of the dividend, which is 5. This makes the new number 15.

Step 4: 6 goes into 15 two times, since 6 × 2 = 12. Write 2 in the quotient. Subtract 12 from 15, which leaves a remainder of 3.

Step 5: No more digits are left to bring down. Since 3 is smaller than 6, it becomes the final remainder. 

Read More: Fractions - Definition, Types & Examples

Long Division Steps With Decimal

Long division with decimals works just like regular long division, with one key difference: when you reach the decimal point in the dividend, you place it directly in the answer (quotient) at the same position.

Let’s understand long division steps with decimals using an example below:

Divide 23.95 by 5.

Step 1: Start by dividing the whole number part. 5 goes into 23 a total of 4 times, since 5 × 4 = 20. Subtracting 20 from 23 leaves a remainder of 3. Write 4 as the first digit of the quotient.

Step 2: Now bring down the next digit, which is 9. But before continuing, notice that 9 comes after the decimal point in the original number, so you place a decimal point in the quotient now.

Step 3: Now divide 5 into 39. It fits 7 times, since 5 × 7 = 35. Subtracting 35 from 39 leaves a remainder of 4. Write 7 after the decimal in the quotient.

Step 4: Bring down the last digit, which is 5, to make 45. 5 goes into 45 exactly 9 times, with no remainder. Write 9 in the quotient.

The final answer is 4.79. So, 23.95 divided by 5 equals 4.79, with no remainder.

Read More: What is Factorization Formula?

Long Division of Polynomials

In the long division of polynomials, observe the degree of the dividend and the divisor, and write each term of the quotient such that its product with the divisor matches the leading term of the current dividend. Let's understand this in steps.

How to Do Polynomial Long Division?

Here, we will learn the steps to perform long division of a polynomial with an example: 

Let's learn to divide polynomial  4x³ + 3x² + x − 4 by x − 4 

Step 1: Arrange the polynomial terms in descending order of powers. Here, both the dividend and divisor are already in standard form:

Dividend = 4x³ + 3x² − 3x + 1

Divisor = x − 2

Step 2: Divide the first term of the dividend (4x³) by the first term of the divisor (x). 

4x³ ÷ x = 4x² → This is the first term of the quotient.

Step 3: Multiply the entire divisor by 4x²:

(x − 2) 4x² = 4x³ − 8x²

Step 4: Subtract this from the dividend: (4x³ + 3x²) − (4x³ − 8x²) = 11x² 

Bring down the next term.  Now we have 11x² - 3x

Step 5: Repeat the process. 

11x² ÷ x = 11x → Next term of the quotient.

Step 6: Multiply the divisor by 11x: 

(x − 2) × 11x = 11x² − 22x

Step 7: Subtract: (11x² - 3x) − (11x² − 22x) = 19x 

Bring down the next term.  Now we have 19x + 1

Step 8: Continue. 

19x ÷ x = 19

Step 9: Multiply: (x − 2) × 19 = 19x − 38

Step 10: Subtract: (19x + 1) − (19x − 38) = 39

The remainder is 39, and no more terms are left in the dividend.

Final Result:

Quotient = 4x² + 11x + 19

Remainder = 39

So ,(4x³ + 3x² − 3x + 1) ÷ (x − 2) = 4x² + 11x + 19 + 39⁄(x − 2)

Long Division Word Problems 

Let’s apply long division to solve real-world problems.

Problem 1: A teacher has 672 pencils and wants to divide them equally among 4 classes. How many pencils per class?

672 ÷ 4 = 168

Problem 2: A bakery has 2,310 cookies packed into 6 boxes. How many cookies in each box?

2310 ÷ 6 = 385

Also read:  Reference Angle - Formula with Examples

Long Division Practice Problems 

Now it’s your turn to practice. Try to divide the following using the steps you have learned.

1. Divide 836 ÷ 4

2. Divide 1728 ÷ 12

3. Divide the following: 999 ÷ 9

4. Divide the following: 5280 ÷ 20

5. Divide the following: 343 ÷ 7

6. Divide the polynomial: 6x³ + 5x² − 3x + 2 ÷ 2x − 1

7. Divide the decimal: 48.6 ÷ 3

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