Long Division With Decimals Worksheet

Mastering Long Division with Decimals: A Comprehensive Guide with Worksheet Examples

Long division with decimals can seem daunting at first, but with a systematic approach and plenty of practice, it becomes manageable and even enjoyable. This comprehensive guide will walk you through the process step-by-step, providing clear explanations, helpful tips, and plenty of worksheet examples to solidify your understanding. We'll cover everything from the fundamental principles to tackling more complex problems, ensuring you gain confidence and mastery in this essential mathematical skill.

Understanding the Basics: A Refresher on Long Division

Before diving into decimals, let's quickly review the fundamental principles of long division. Long division is a method used to divide a large number (the dividend) by a smaller number (the divisor) to find the quotient (the result) and any remainder. Remember the basic structure:

• Dividend: The number being divided.
• Divisor: The number you are dividing by.
• Quotient: The result of the division.
• Remainder: The amount left over after the division (if any).

Let's look at a simple example without decimals: Dividing 675 by 5.

     135
5 | 675
   -5
    17
   -15
     25
    -25
      0

Here, 675 is the dividend, 5 is the divisor, and 135 is the quotient. The remainder is 0. This process involves repeatedly subtracting the divisor from the dividend until we reach a remainder smaller than the divisor.

Introducing Decimals: The Key Differences

When dealing with decimals in long division, the core process remains the same, but we need to handle the decimal point carefully. The key difference lies in the placement of the decimal point in the quotient.

Step-by-Step Guide to Long Division with Decimals

Let's break down the process into manageable steps using an example: Dividing 47.6 by 4.

Step 1: Set up the Problem

Write the problem in the standard long division format:

4 | 47.6

Step 2: Place the Decimal Point

The most crucial step! Bring the decimal point straight up into the quotient:

   .
4 | 47.6

Step 3: Divide as Usual

Now, proceed with the long division process just like you would with whole numbers. Ignore the decimal point for now and focus on dividing 4 into 47.

  1.
4 | 47.6
  -4
   07

Step 4: Continue Dividing

Continue dividing, bringing down the next digit (6) and continuing the division process:

  11.
4 | 47.6
  -4
   07
   -4
    36
   -36
    0

Step 5: Check Your Work

Multiply the quotient (11.9) by the divisor (4): 11.9 x 4 = 47.6. This confirms our answer is correct.

Dealing with Remainders and Terminating vs. Repeating Decimals

Sometimes, you might encounter a remainder that doesn't reach zero. In such cases, you might need to add zeros to the dividend and continue the division process. This could lead to either:

• Terminating Decimals: The division process stops after a finite number of steps, resulting in a decimal that ends (e.g., 0.75).
• Repeating Decimals: The division process continues indefinitely, with a repeating pattern of digits (e.g., 0.333...). In this case, you'll often indicate the repeating digits by placing a bar over them (e.g., 0.3̅).

Example with a Repeating Decimal: Dividing 1 by 3.

0.333...
3 | 1.000
  -9
   10
   -9
    10
    -9
     1

This results in 0.3̅, indicating that the digit 3 repeats infinitely.

Long Division with Decimals: Worksheet Examples

Now, let's put our knowledge into practice with some worksheet examples of varying difficulty levels.

Worksheet 1: Basic Practice

1. Divide 12.6 by 3
2. Divide 37.8 by 6
3. Divide 85.5 by 5
4. Divide 94.8 by 12
5. Divide 25.2 by 7

Worksheet 2: Intermediate Practice – Including Remainders

1. Divide 75.3 by 4
2. Divide 112.8 by 8
3. Divide 153.6 by 16
4. Divide 234.9 by 9
5. Divide 456.7 by 11

Worksheet 3: Advanced Practice – Dealing with Repeating Decimals

1. Divide 1 by 9
2. Divide 2 by 3
3. Divide 5 by 6
4. Divide 7 by 11
5. Divide 13 by 12

Answer Key (for Self-Checking):

(Remember to check your answers carefully!)

Worksheet 1:

1. 4.2
2. 6.3
3. 17.1
4. 7.9
5. 3.6

Worksheet 2:

1. 18.825
2. 14.1
3. 9.6
4. 26.1
5. 41.51818… (repeating decimal)

Worksheet 3:

1. 0.1̅
2. 0.6̅
3. 0.83̅
4. 0.63̅
5. 1.08333… (repeating decimal)

Frequently Asked Questions (FAQ)

• Q: What happens if the divisor is a decimal? A: Convert the divisor to a whole number by multiplying both the divisor and the dividend by the same power of 10. For example, if dividing by 0.25, multiply both by 100 to get 25.

• Q: How do I handle zeros in the dividend? A: Treat them just like any other digit. Bring them down as needed during the division process.

• Q: How can I improve my speed and accuracy? A: Practice regularly! Start with easier problems and gradually work your way up to more challenging ones. Use flashcards, online quizzes, and work through multiple worksheets.

Conclusion: Mastering Long Division with Decimals

Mastering long division with decimals takes time and dedicated effort. However, with a clear understanding of the steps involved, consistent practice using worksheets like those provided, and a willingness to persevere, you can confidently tackle even the most complex problems. Remember to focus on the placement of the decimal point, practice regularly, and don't be afraid to seek help when needed. With dedication and persistence, you will conquer this important mathematical skill and build a strong foundation for future mathematical studies.