Long Division Questions With Answers

Mastering Long Division: A Comprehensive Guide with Questions and Answers

Long division, a fundamental arithmetic operation, can seem daunting at first. However, with a systematic approach and consistent practice, mastering it becomes achievable. This comprehensive guide breaks down the process of long division, providing clear explanations, step-by-step examples, and a range of practice questions with detailed answers. Understanding long division is crucial for success in higher-level mathematics, making this skill a valuable investment in your academic journey. We will cover everything from basic concepts to more advanced problems, ensuring you gain a solid understanding of this essential mathematical tool.

Understanding the Basics: Divisor, Dividend, Quotient, and Remainder

Before diving into the process, let's define the key terms involved in long division:

Dividend: The number being divided (the larger number).
Divisor: The number you are dividing by (the smaller number).
Quotient: The result of the division (the answer).
Remainder: The amount left over after the division is complete. If the division is exact, the remainder will be zero.

For example, in the expression 15 ÷ 3 = 5, 15 is the dividend, 3 is the divisor, and 5 is the quotient. There is no remainder in this case.

Step-by-Step Guide to Long Division

Long division is a process that involves several steps, best understood through practical examples. Let's work through a problem step-by-step:

Example 1: 675 ÷ 5

Set up the problem: Write the dividend (675) inside the long division symbol (⟌) and the divisor (5) outside.
```
5 ⟌ 675
```
Divide the first digit: Divide the first digit of the dividend (6) by the divisor (5). 6 divided by 5 is 1 with a remainder of 1. Write the quotient (1) above the 6.
```
   1
5 ⟌ 675
```
Multiply and subtract: Multiply the quotient (1) by the divisor (5), which equals 5. Subtract this result from the first digit of the dividend (6 - 5 = 1).
```
   1
5 ⟌ 675
   -5
    1
```
Bring down the next digit: Bring down the next digit of the dividend (7) next to the remainder (1), making it 17.
```
   1
5 ⟌ 675
   -5
    17
```
Repeat steps 2-4: Divide 17 by 5. This is 3 with a remainder of 2. Write the 3 above the 7. Multiply 3 by 5 (15) and subtract from 17 (17 - 15 = 2).
```
   13
5 ⟌ 675
   -5
    17
   -15
     2
```
Bring down the last digit: Bring down the last digit of the dividend (5) next to the remainder (2), making it 25.
```
   13
5 ⟌ 675
   -5
    17
   -15
     25
```
Final division: Divide 25 by 5, which equals 5 with no remainder. Write the 5 above the 5.
```
   135
5 ⟌ 675
   -5
    17
   -15
     25
    -25
      0
```
The quotient is 135 and the remainder is 0. Therefore, 675 ÷ 5 = 135.

Example 2: A Problem with a Remainder

Let's tackle a problem that results in a remainder: 847 ÷ 6

Setup:
```
6 ⟌ 847
```
Divide, Multiply, Subtract, Bring Down: Follow the same steps as in Example 1. You will find that 8 divided by 6 is 1 with a remainder of 2. Bringing down the 4 makes it 24. 24 divided by 6 is 4. Bringing down the 7 makes it 7. 7 divided by 6 is 1 with a remainder of 1.
```
   141
6 ⟌ 847
   -6
    24
   -24
     07
    -6
     1
```
The quotient is 141 and the remainder is 1. Therefore, 847 ÷ 6 = 141 R 1 (R denotes remainder).

Dealing with Zeroes in the Dividend

Zeroes in the dividend require careful attention. Let's look at an example: 1204 ÷ 4

Setup:
```
4 ⟌ 1204
```
Division: Following the steps, you'll find 12 divided by 4 is 3. Bringing down the zero, 0 divided by 4 is 0. Bringing down the 4, 4 divided by 4 is 1.
```
   301
4 ⟌ 1204
   -12
     00
     -0
      04
     -4
      0
```
The quotient is 301 and the remainder is 0. Therefore, 1204 ÷ 4 = 301. Notice how the zero is properly accounted for in the quotient.

Long Division with Decimals

Long division also extends to numbers with decimals. The process remains similar, but you need to handle the decimal point carefully. Let's examine: 37.5 ÷ 5

Setup:
```
5 ⟌ 37.5
```
Division: Divide as usual. 37 divided by 5 is 7 with a remainder of 2. Bring down the 5 to make it 25. 25 divided by 5 is 5.
```
   7.5
5 ⟌ 37.5
   -35
     25
    -25
      0
```
The quotient is 7.5 and the remainder is 0. Therefore, 37.5 ÷ 5 = 7.5. The decimal point in the quotient is placed directly above the decimal point in the dividend.

Long Division Practice Questions with Answers

Here are some practice questions for you to try. Remember to follow the steps outlined above:

Question 1: 936 ÷ 3

Answer 1: 312

Question 2: 1575 ÷ 15

Answer 2: 105

Question 3: 2484 ÷ 24

Answer 3: 103.5

Question 4: 7854 ÷ 12

Answer 4: 654.5

Question 5: 5678 ÷ 17

Answer 5: 334

Question 6: 34567 ÷ 23

Answer 6: 1502 R 21

Question 7: 89765 ÷ 45

Answer 7: 1994 R 35

Question 8: 123456 ÷ 67

Answer 8: 1842 R 42

Question 9: 987654321 ÷ 1111

Answer 9: 888888...

Question 10: 45.6 ÷ 12

Answer 10: 3.8

Question 11: 123.45 ÷ 5

Answer 11: 24.69

Question 12: 0.789 ÷ 3

Answer 12: 0.263

Frequently Asked Questions (FAQ)

Q: What if I get a remainder that is larger than the divisor?

A: If your remainder is larger than your divisor, you've made a mistake in your calculation. Go back and check your subtraction and division steps.

Q: How do I handle very large numbers?

A: The process remains the same, even with very large numbers. It might take longer, but the steps are consistent. Break the problem down step-by-step to avoid errors.

Q: Are there any shortcuts for long division?

A: While there aren't significant shortcuts to completely bypass the process, practicing regularly will improve your speed and accuracy. Familiarity with multiplication tables is also crucial for efficient long division.

Q: What if the divisor is a decimal?

A: If the divisor is a decimal, you need to multiply both the divisor and the dividend by a power of 10 to make the divisor a whole number. For example, if you have 12.5 ÷ 2.5, multiply both by 10 to get 125 ÷ 25.

Conclusion

Long division, while initially challenging, becomes manageable with practice and a solid understanding of the underlying principles. By consistently working through examples and utilizing the step-by-step method outlined in this guide, you can build confidence and proficiency in this essential mathematical skill. Remember to break down the problem into manageable parts, carefully check your work, and don't be afraid to practice repeatedly until you feel comfortable. Mastering long division will not only improve your mathematical abilities but also provide a strong foundation for more advanced mathematical concepts. Keep practicing, and you will see improvement!