Long division is a fundamental arithmetic skill that allows you to break down complex division problems into a series of simpler steps. Whether you are dividing large numbers, decimals, or even polynomials, understanding long division is crucial. This method systematically breaks down the dividend (the number being divided) by the divisor (the number you are dividing by) to find the quotient (the result) and the remainder (if any). This guide will walk you through How To Do Long Division with clear steps and examples, making it easy to learn and master.
Understanding the Parts of Long Division
Before we dive into the process of long division, it’s important to understand the terminology. Just like any mathematical operation, long division has specific parts that play key roles in solving the problem. Let’s break down these components:
- Dividend: This is the number that you are dividing. In a long division problem, the dividend is placed inside the division symbol.
- Divisor: This is the number by which you are dividing the dividend. The divisor sits outside the division symbol, to the left of the dividend.
- Quotient: This is the result of the division. It represents how many times the divisor goes into the dividend. The quotient is written above the dividend, on top of the division symbol.
- Remainder: Sometimes, the divisor does not divide the dividend evenly. The remainder is the amount left over after the division is complete, and it’s always less than the divisor.
Let’s visualize these parts with an example:
In this example, 75 is the dividend, 4 is the divisor, 18 is the quotient, and 3 is the remainder. This visual representation helps to understand the position and role of each part in the long division process.
Step-by-Step Guide to Long Division
Long division might seem daunting at first, but it becomes straightforward once you understand the steps involved. It’s essentially a sequence of Divide, Multiply, Subtract, and Bring Down. Let’s break down the general steps for how to do long division:
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Set up the Problem: Write the dividend inside the long division symbol and the divisor to the left of it.
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Divide: Look at the first digit (or first few digits) of the dividend. Determine how many times the divisor goes into this part of the dividend. Write this number as the first digit of the quotient above the division symbol.
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Multiply: Multiply the digit you just wrote in the quotient by the divisor. Write the product directly below the part of the dividend you used in step 2.
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Subtract: Subtract the product from the part of the dividend above it. Write the difference below the line.
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Bring Down: Bring down the next digit of the dividend and write it next to the remainder from the subtraction step. This new number becomes the new dividend for the next iteration.
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Repeat: Repeat steps 2-5 until there are no more digits to bring down from the dividend.
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Remainder (if any): If there’s a number left after the last subtraction that is smaller than the divisor, this is your remainder.
These steps provide a roadmap for tackling any long division problem. To make it even clearer, let’s work through some examples.
Long Division Examples: With and Without Remainders
To solidify your understanding of how to do long division, let’s explore examples that cover different scenarios, including division with and without remainders.
Division with Remainders
Case 1: When the first digit of the dividend is greater than or equal to the divisor.
Example: Divide 435 ÷ 4
Let’s follow the steps we outlined:
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Set up:
_______ 4 | 435 -
Divide: The first digit of the dividend is 4, which is equal to the divisor 4. 4 divided by 4 is 1. Write ‘1’ as the first digit of the quotient.
1____ 4 | 435 -
Multiply: Multiply the quotient digit (1) by the divisor (4): 1 × 4 = 4. Write ‘4’ below the first digit of the dividend.
1____ 4 | 435 4 -
Subtract: Subtract 4 from 4: 4 – 4 = 0.
1____ 4 | 435 4 --- 0 -
Bring Down: Bring down the next digit of the dividend, which is 3, and place it next to the remainder 0.
1____ 4 | 435 4 --- 03 -
Repeat: Now we have ’03’, or simply 3, as the new number to divide. Since 3 is less than 4, 4 goes into 3 zero times. Write ‘0’ as the next digit in the quotient.
10___ 4 | 435 4 --- 03 0 -
Multiply: Multiply the new quotient digit (0) by the divisor (4): 0 × 4 = 0. Write ‘0’ below 3.
10___ 4 | 435 4 --- 03 0 -
Subtract: Subtract 0 from 3: 3 – 0 = 3.
10___ 4 | 435 4 --- 03 0 --- 3 -
Bring Down: Bring down the last digit of the dividend, which is 5, and place it next to the remainder 3.
10___ 4 | 435 4 --- 035 0 --- 35 -
Repeat: Now we have 35. How many times does 4 go into 35? 4 × 8 = 32, which is close to 35 without going over. Write ‘8’ as the next digit in the quotient.
108__ 4 | 435 4 --- 035 0 --- 35 -
Multiply: Multiply the new quotient digit (8) by the divisor (4): 8 × 4 = 32. Write ’32’ below 35.
108__ 4 | 435 4 --- 035 0 --- 35 32 -
Subtract: Subtract 32 from 35: 35 – 32 = 3.
108__ 4 | 435 4 --- 035 0 --- 35 32 --- 3
Since there are no more digits to bring down, and 3 is less than the divisor 4, 3 is the remainder. The quotient is 108, and the remainder is 3.
Case 2: When the first digit of the dividend is less than the divisor.
Example: Divide 735 ÷ 9
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Set up:
_______ 9 | 735 -
Divide: The first digit of the dividend is 7, which is less than the divisor 9. So, we consider the first two digits, 73. How many times does 9 go into 73? 9 × 8 = 72. Write ‘8’ as the first digit of the quotient.
8____ 9 | 735 -
Multiply: Multiply the quotient digit (8) by the divisor (9): 8 × 9 = 72. Write ’72’ below 73.
8____ 9 | 735 72 -
Subtract: Subtract 72 from 73: 73 – 72 = 1.
8____ 9 | 735 72 --- 1 -
Bring Down: Bring down the next digit of the dividend, which is 5, and place it next to the remainder 1.
8____ 9 | 735 72 --- 15 -
Repeat: Now we have 15. How many times does 9 go into 15? 9 × 1 = 9. Write ‘1’ as the next digit in the quotient.
81___ 9 | 735 72 --- 15 9 -
Multiply: Multiply the new quotient digit (1) by the divisor (9): 1 × 9 = 9. Write ‘9’ below 15.
81___ 9 | 735 72 --- 15 9 -
Subtract: Subtract 9 from 15: 15 – 9 = 6.
81___ 9 | 735 72 --- 15 9 --- 6
Since there are no more digits to bring down, and 6 is less than the divisor 9, 6 is the remainder. The quotient is 81, and the remainder is 6.
Division without Remainder
Example: Divide 900 ÷ 5
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Set up:
_______ 5 | 900 -
Divide: The first digit of the dividend is 9. How many times does 5 go into 9? 5 × 1 = 5. Write ‘1’ as the first digit of the quotient.
1____ 5 | 900 -
Multiply: Multiply the quotient digit (1) by the divisor (5): 1 × 5 = 5. Write ‘5’ below 9.
1____ 5 | 900 5 -
Subtract: Subtract 5 from 9: 9 – 5 = 4.
1____ 5 | 900 5 --- 4 -
Bring Down: Bring down the next digit of the dividend, which is 0, and place it next to the remainder 4.
1____ 5 | 900 5 --- 40 -
Repeat: Now we have 40. How many times does 5 go into 40? 5 × 8 = 40. Write ‘8’ as the next digit in the quotient.
18___ 5 | 900 5 --- 40 40 -
Multiply: Multiply the new quotient digit (8) by the divisor (5): 8 × 5 = 40. Write ’40’ below 40.
18___ 5 | 900 5 --- 40 40 -
Subtract: Subtract 40 from 40: 40 – 40 = 0.
18___ 5 | 900 5 --- 40 40 --- 0 -
Bring Down: Bring down the last digit of the dividend, which is 0.
18___ 5 | 900 5 --- 40 40 --- 00 -
Repeat: Now we have 0. How many times does 5 go into 0? 5 × 0 = 0. Write ‘0’ as the last digit in the quotient.
180__ 5 | 900 5 --- 40 40 --- 00 0 -
Subtract: Subtract 0 from 0: 0 – 0 = 0.
180__ 5 | 900 5 --- 40 40 --- 00 0 --- 0
The remainder is 0, meaning 900 is perfectly divisible by 5. The quotient is 180.
Long Division with Two-Digit Divisors
Long division by a 2 digit number follows the same principles but requires a bit more attention to multiplication and estimation. The key is to consider the first two digits of the dividend (or more if necessary) and estimate how many times the two-digit divisor goes into that number.
Example: Divide 7248 ÷ 24
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Set up:
_______ 24 | 7248 -
Divide: Consider the first two digits of the dividend, 72. How many times does 24 go into 72? 24 × 3 = 72. Write ‘3’ as the first digit of the quotient.
3____ 24 | 7248 -
Multiply: Multiply the quotient digit (3) by the divisor (24): 3 × 24 = 72. Write ’72’ below 72.
3____ 24 | 7248 72 -
Subtract: Subtract 72 from 72: 72 – 72 = 0.
3____ 24 | 7248 72 --- 0 -
Bring Down: Bring down the next digit of the dividend, which is 4.
3____ 24 | 7248 72 --- 04 -
Repeat: Now we have 4. Since 4 is less than 24, 24 goes into 4 zero times. Write ‘0’ as the next digit in the quotient.
30___ 24 | 7248 72 --- 04 0 -
Multiply: Multiply the new quotient digit (0) by the divisor (24): 0 × 24 = 0. Write ‘0’ below 4.
30___ 24 | 7248 72 --- 04 0 -
Subtract: Subtract 0 from 4: 4 – 0 = 4.
30___ 24 | 7248 72 --- 04 0 --- 4 -
Bring Down: Bring down the last digit of the dividend, which is 8.
30___ 24 | 7248 72 --- 048 0 --- 48 -
Repeat: Now we have 48. How many times does 24 go into 48? 24 × 2 = 48. Write ‘2’ as the last digit in the quotient.
302__ 24 | 7248 72 --- 048 0 --- 48 48 -
Subtract: Subtract 48 from 48: 48 – 48 = 0.
302__ 24 | 7248 72 --- 048 0 --- 48 48 --- 0
The remainder is 0. The quotient is 302. Therefore, 7248 ÷ 24 = 302.
Beyond Basic Long Division
The principles of long division extend to more complex scenarios:
Long Division of Polynomials
Long division isn’t just for numbers! You can also use it to divide polynomials. This is particularly useful in algebra when simplifying rational expressions or factoring polynomials. The process is analogous to numerical long division, but you work with terms of polynomials instead of digits. For a deeper dive, you can explore resources on Dividing Polynomials.
Long Division with Decimals
Dividing decimals using long division is also straightforward. The key is to handle the decimal point correctly. When dividing a decimal by a whole number, you place the decimal point in the quotient directly above the decimal point in the dividend. For more detailed explanations and examples, refer to resources on Dividing Decimals.
Dividing Decimals by Whole Numbers
Let’s look at an example of dividing a decimal by a whole number:
Example: Divide 36.9 ÷ 3
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Set up:
_______ 3 | 36.9 -
Divide: Divide the whole number part (36) by 3. 36 ÷ 3 = 12. Write ’12’ as the whole number part of the quotient.
12.__ 3 | 36.9 3 --- 06 6 --- 0 -
Decimal Point: When you reach the decimal point in the dividend, place the decimal point in the quotient directly above it.
12.__ 3 | 36.9 3 --- 06 6 --- 0. -
Bring Down: Bring down the digit after the decimal point, which is 9.
12.__ 3 | 36.9 3 --- 06 6 --- 0.9 -
Repeat: Divide 9 by 3. 9 ÷ 3 = 3. Write ‘3’ as the decimal part of the quotient.
12.3_ 3 | 36.9 3 --- 06 6 --- 0.9 9 -
Subtract: Subtract 9 from 9: 9 – 9 = 0.
12.3_ 3 | 36.9 3 --- 06 6 --- 0.9 9 --- 0
The remainder is 0. The quotient is 12.3. Therefore, 36.9 ÷ 3 = 12.3.
Tips and Tricks for Long Division
Mastering long division involves not just knowing the steps but also employing helpful tips and tricks to improve accuracy and efficiency:
- Remainder Check: Always remember that the remainder must be smaller than the divisor. If your remainder is larger than or equal to the divisor, you need to increase your quotient digit.
- Zero as a Placeholder: When a digit in the dividend is smaller than the divisor, and you need to bring down the next digit, make sure to place a ‘0’ in the quotient as a placeholder. This is crucial, especially when dealing with zeros within the dividend, as seen in the example of 900 ÷ 5.
- Estimation Skills: Practice estimating how many times the divisor goes into the current part of the dividend. This will speed up the process and reduce errors. For larger divisors, rounding them to the nearest ten can help with estimation.
- Verification: You can always check your answer using the division formula:
Dividend = (Divisor × Quotient) + Remainder.
If the calculation holds true, your long division is correct. In cases where the remainder is 0, you can simply check if (Divisor × Quotient) equals the Dividend. - Practice Regularly: Like any skill, proficiency in long division comes with practice. Work through various problems with different divisors and dividends, including those with remainders, two-digit divisors, and decimals.
By incorporating these tips and consistently practicing, you’ll become more confident and accurate in performing long division.
Real-World Examples of Long Division
Long division isn’t just a math exercise; it’s a practical skill used in many real-life situations. Here are a couple of examples to illustrate its relevance:
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Example 1: Distributing Trees Equally
Imagine Ron, a gardener, wants to plant 75 trees in 3 equal rows. To find out how many trees should go in each row, he uses long division: 75 ÷ 3.
As shown in the image, using long division, we find that 75 ÷ 3 = 25. Therefore, Ron should plant 25 trees in each row to distribute them equally.
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Example 2: Dividing Money Among Workers
A construction company needs to distribute $4000 equally among 25 workers for a completed project. To calculate how much each worker should receive, they use long division: 4000 ÷ 25.
Long division reveals that 4000 ÷ 25 = 160. Thus, each worker will receive $160.
These examples demonstrate how long division helps solve everyday problems involving equal distribution and division of quantities.
Practice Questions
Test your understanding of how to do long division with these practice questions:
- Divide 856 by 7. What is the quotient and remainder?
- Calculate 1938 divided by 15.
- Solve: 5292 ÷ 36. Is there a remainder?
Frequently Asked Questions (FAQs) about Long Division
Let’s address some common questions about long division to further clarify the concept:
What is Long Division in Math?
Long division is a systematic method used in mathematics to divide larger numbers by breaking down the division process into smaller, manageable steps. It’s particularly useful when the divisor is a multi-digit number, or when you need to find both the quotient and the remainder. It relies on repeated steps of division, multiplication, subtraction, and bringing down digits to progressively solve the division problem.
How to do Long Division?
To do long division, follow these basic steps:
- Divide: Divide a part of the dividend by the divisor and write the result in the quotient.
- Multiply: Multiply the quotient digit by the divisor and write the product under the dividend.
- Subtract: Subtract the product from the dividend part.
- Bring Down: Bring down the next digit of the dividend.
- Repeat: Repeat these steps until all digits of the dividend have been used.
Refer to the “Step-by-Step Guide to Long Division” section in this article for a more detailed explanation.
What are the Steps of Long Division?
The main steps of long division, often remembered by the acronym DMSBR (Divide, Multiply, Subtract, Bring Down, Repeat), are as follows:
- Divide the first part of the dividend by the divisor.
- Multiply the quotient digit by the divisor.
- Subtract the product from the dividend part.
- Bring Down the next digit of the dividend.
- Repeat the process until complete.
See the “Step-by-Step Guide to Long Division” and examples provided earlier in this article for a visual and detailed walkthrough of these steps.
How to do Long Division with 2 Digits?
Long division with 2 digits as the divisor is performed using the same steps as with single-digit divisors. The main difference is that you’re dividing by a two-digit number, which might require more estimation and multiplication. You start by considering the first two digits of the dividend (or more if necessary) and determine how many times the two-digit divisor fits into that number. For a detailed example, see the “Long Division with Two-Digit Divisors” section in this guide.
What is the Long Division of Polynomials?
In algebra, long division of polynomials is a method for dividing one polynomial by another of the same or lower degree. It’s analogous to arithmetic long division but uses polynomial terms instead of numbers. This process helps in simplifying rational functions and is a key technique in algebraic manipulation. You can find more information in resources dedicated to Dividing Polynomials.
How to do Long Division with Decimals?
Long division with decimals is very similar to regular long division. The key difference is handling the decimal point. When dividing a decimal number by a whole number, you place the decimal point in the quotient directly above the decimal point in the dividend once you reach it during the division process. For more detailed instructions and examples, refer to the “Long Division with Decimals” section and resources on Dividing Decimals.
Conclusion
Long division is an essential skill that extends beyond basic arithmetic. Mastering how to do long division provides a strong foundation for more advanced mathematical concepts, from algebra to calculus. By understanding the steps, practicing regularly, and utilizing helpful tips, you can confidently tackle division problems of any complexity. Remember, patience and practice are key to becoming proficient in long division. Keep practicing, and you’ll find yourself solving these problems with ease and accuracy!
Answer Section for Practice Questions:
Practice Question 1: Divide 856 by 7. What is the quotient and remainder?
Answer: Quotient: 122, Remainder: 2
Practice Question 2: Calculate 1938 divided by 15.
Answer: Quotient: 129, Remainder: 3
Practice Question 3: Solve: 5292 ÷ 36. Is there a remainder?
Answer: Quotient: 147, Remainder: 0 (No remainder)
