Long Division Of Decimals Worksheet

Mastering Long Division of Decimals: A Comprehensive Guide with Worksheets

Long division of 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, practical examples, and downloadable worksheets to solidify your understanding. We'll explore the underlying principles, tackle common challenges, and equip you with the confidence to conquer any decimal division problem. This guide covers everything from the basics to more advanced scenarios, making it ideal for students of all levels.

Understanding the Fundamentals: Decimals and Division

Before diving into the complexities of long division with decimals, let's refresh our understanding of decimals and the division process itself.

• Decimals: Decimals represent numbers that are not whole numbers. They are fractions expressed in a base-10 system, using a decimal point to separate the whole number part from the fractional part. For example, 3.14 represents three and fourteen hundredths (3 + 14/100).

• Division: Division is essentially the process of splitting a number (the dividend) into equal parts, determined by another number (the divisor). The result is called the quotient. For example, 12 ÷ 3 = 4 means that 12 can be split into four groups of 3.

Long Division of Decimals: A Step-by-Step Approach

The key to mastering long division of decimals lies in a methodical approach. Here's a breakdown of the steps, illustrated with examples:

Step 1: Set up the Problem

Write the division problem in the standard long division format. The dividend (the number being divided) goes inside the long division symbol, and the divisor (the number you're dividing by) goes outside.

Example 1: Divide 15.75 by 5

     _____
5 | 15.75

Step 2: Handle the Decimal Point

The most crucial step is dealing with the decimal point. To simplify the process, move the decimal point in the divisor to the right until it becomes a whole number. Then, move the decimal point in the dividend the same number of places to the right. If the dividend doesn't have enough digits, add zeros as placeholders.

Example 1 (continued):

Since the divisor (5) is already a whole number, we don't need to move any decimal points.

     _____
5 | 15.75

Example 2: Divide 3.78 by 0.06

Here, we move the decimal point in the divisor (0.06) two places to the right to get 6. We also move the decimal point in the dividend (3.78) two places to the right to get 378. This gives us:

    _____
6 | 378.

Step 3: Perform Long Division

Now, perform the long division as you would with whole numbers. Divide, multiply, subtract, and bring down the next digit, repeating the process until you reach the end. Remember to place the decimal point in the quotient directly above the decimal point in the dividend (after adjusting in Step 2).

Example 1 (continued):

    3.15
5 | 15.75
   15
    07
    05
    025
    025
    000

Therefore, 15.75 ÷ 5 = 3.15

Example 2 (continued):

   63
6 | 378
   36
    18
    18
     0

Therefore, 3.78 ÷ 0.06 = 63

Step 4: Add Zeros if Necessary

Sometimes, the division doesn't result in a whole number. You might need to add zeros to the dividend to continue the division process and obtain a more precise quotient. You can continue adding zeros as needed to reach the desired level of accuracy or until you get a repeating decimal.

Example 3: Divide 7.2 by 5

   1.44
5 | 7.20
   5
   22
   20
    20
    20
     0

Therefore, 7.2 ÷ 5 = 1.44

Step 5: Check Your Answer

Always verify your answer by multiplying the quotient by the divisor. The result should equal the dividend.

Example 1 (Verification): 3.15 x 5 = 15.75 (Correct!)

Dealing with Remainders and Repeating Decimals

Sometimes, long division doesn't result in a clean whole number quotient. This leads to either remainders or repeating decimals.

• Remainders: If you have a remainder, you can express it as a fraction by placing the remainder over the divisor. For example, if you have a remainder of 2 after dividing by 7, you would write the remainder as 2/7.

• Repeating Decimals: These occur when a digit or group of digits repeats indefinitely. For example, 1 ÷ 3 = 0.333... (the 3 repeats infinitely). These are often represented with a bar over the repeating digit(s), like this: 0.3̅.

Advanced Scenarios and Applications

Let's explore some more complex scenarios you might encounter:

• Dividing by a Decimal Greater Than 1: The process remains the same, ensuring you adjust the decimal point in both dividend and divisor before performing the long division.

• Dividing by a Decimal Less Than 1: Here, you will need to move the decimal point to the right in both numbers, adding zeros to the dividend as needed.

• Dividing Large Decimals: The principle remains the same even with larger numbers; meticulous execution of each step is critical. Break down the problem into manageable parts if it seems overwhelming.

• Real-World Applications: Long division of decimals is crucial in various real-life scenarios, including calculating unit prices, converting measurements, sharing amounts proportionally, and determining average values.

Long Division of Decimals Worksheets

To reinforce your understanding and develop proficiency, consistent practice is vital. Here are some examples of problems you can practice with:

Worksheet 1: Basic Practice

1. 25.5 ÷ 5
2. 18.9 ÷ 3
3. 42.6 ÷ 6
4. 75.2 ÷ 8
5. 10.5 ÷ 1.5
6. 27.3 ÷ 0.3
7. 5.67 ÷ 0.9
8. 36.48 ÷ 1.2
9. 0.456 ÷ 0.006
10. 105.6 ÷ 0.24

Worksheet 2: Advanced Practice (Includes Remainders and Repeating Decimals)

1. 17.8 ÷ 7
2. 3.14 ÷ 9
3. 45.6 ÷ 11
4. 6.7 ÷ 0.13
5. 100.2 ÷ 0.005
6. 1 ÷ 0.11
7. 2 ÷ 0.03
8. 15 ÷ 0.006
9. 100 ÷ 0.125
10. 1.23456 ÷ 0.0001

Worksheet 3: Real-world Applications

1. A box of 12 oranges weighs 4.8kg. What is the weight of one orange?
2. Sarah drove 273.6 miles in 6 hours. What was her average speed?
3. John bought 2.5 liters of juice for $10. What is the cost per liter?
4. 5 friends split a pizza that cost $22.50. How much does each friend pay?
5. A piece of fabric is 15.75 meters long. If it is cut into 0.25-meter pieces, how many pieces will there be?

Remember to show your work and check your answers!

Frequently Asked Questions (FAQ)

• Q: What if I get a remainder in decimal long division?

A: You can express the remainder as a fraction (remainder/divisor) or continue adding zeros to the dividend to get a more precise decimal answer.

• Q: How can I improve my speed in long division of decimals?

A: Practice regularly, break down complex problems into smaller steps, and focus on accuracy over speed initially. Speed will naturally improve with practice.

• Q: Are there any online tools or calculators to help with long division of decimals?

A: While various online calculators can perform the calculation, working through the problems manually is crucial for understanding the underlying process. Using calculators for verification is beneficial.

• Q: What are some common mistakes to avoid?

A: Common mistakes include incorrect placement of the decimal point, errors in subtraction, and not properly managing zeros when extending the dividend. Careful attention to detail will greatly minimize errors.

Conclusion

Long division of decimals, while initially challenging, becomes a manageable skill with consistent practice and a methodical approach. By understanding the step-by-step process, handling decimal points effectively, and practicing regularly using worksheets, you'll build the confidence to solve any decimal division problem accurately and efficiently. Remember to always check your answers and don't hesitate to seek clarification if needed. With dedication and perseverance, mastering this essential mathematical skill is entirely within your reach. Remember to utilize the provided worksheets and try different problem sets to solidify your understanding and build your confidence. Good luck!