How to Divide Decimals| Examples with Answers

Understanding how to divide decimals is a fundamental skill in mathematics, essential for solving real-world problems involving money, measurements, and data analysis. Whether you’re dividing a decimal by a whole number, another decimal, or powers of 10 (such as 10, 100, 1000), the process involves careful manipulation of the decimal point and applying principles of long division.

Wu Kong Education will with you break down the key concepts, step-by-step procedures, and examples to ensure a solid grasp of decimal division.

What is the Division of Decimals?

The division of decimals is a mathematical operation where a decimal number (the dividend) is divided by another number, which can be a whole number or a decimal (the divisor). The goal is to find the quotient, which may also be a decimal. The process is similar to dividing whole numbers, but it requires special attention to the decimal point to ensure the correct placement in the quotient.

Key Concepts:

• Decimal Number: A number with a decimal point that separates the whole number part from the fractional part (e.g., 3.14, 0.5, 12.75).
• Dividend: The number being divided (e.g., in 6.4 ÷ 2, 6.4 is the dividend).
• Divisor: The number by which the dividend is divided (e.g., 2 in the example above).
• Quotient: The result of the division.

The main challenge in decimal division is maintaining the correct place value of digits, which relies on properly positioning the decimal point in the quotient.

Dividing Decimals by Whole Numbers

Dividing a decimal by a whole number is a straightforward extension of whole-number division. The only difference is the presence of the decimal point in the dividend. Here’s how to do it step by step:

Steps for Dividing a Decimal by a Whole Number:

1. Set up the long division problem: Write the dividend under the division bar and the divisor outside, just like with whole numbers.
2. Divide as you would with whole numbers: Ignore the decimal point initially and perform the division.
3. Place the decimal point in the quotient: Once you’ve divided the whole number part, place the decimal point in the quotient directly above where it appears in the dividend. This ensures the correct place value.
4. Continue dividing: If there are remaining digits after the decimal point in the dividend, bring them down one by one and continue the division. If the dividend runs out of digits, add zeroes to the right of the decimal point to keep dividing.

Example 1: Divide 8.4 by 2

• Step 1: Set up the long division
  Write the dividend 8.4 inside the division symbol “)” and the divisor 2 outside to the left of the division symbol, that is: 2 ) 8.4
• Step 2: Divide the whole number part
  Ignore the decimal point first. Divide 8 by 2. Since 8÷2 = 4, write 4 above the division bar, aligned with the 8.
• Step 3: Place the decimal point in the quotient
  Place the decimal point in the quotient directly above the decimal point of the dividend. So we get 4.
• Step 4: Continue dividing the decimal part
  Bring down the 4 from the decimal part of the dividend. Then divide 4 by 2. Since 4÷2 = 2, write 2 next to the decimal point in the quotient. We get 4.2.
• Final answer: 4.2

Example 2: Divide 15.6 by 5

• Step 1: Set up the long division
  Write the dividend 15.6 inside the division symbol “)” and the divisor 5 outside to the left of the division symbol, that is: 5 ) 15.6
• Step 2: Divide the whole number part
  Ignore the decimal point initially. Divide 15 by 5. Since 15÷5 = 3, write 3 above the division bar, aligned with the 5 in 15. Then place the decimal point in the quotient directly above the decimal point of the dividend, getting 3.
• Step 3: Continue dividing the decimal part
  Bring down the 6 from the decimal part of the dividend. Divide 6 by 5. We have 6÷5 = 1 with a remainder of 1. Write 1 after the decimal point in the quotient.
• Step 4: Add a zero and continue dividing
  Add a zero after the remainder 1 to make it 10. Divide 10 by 5. Since 10÷5 = 2, write 2 after the 1 in the quotient. We get 3.12.
• Final answer: 3.12

Key Points:

• The decimal point in the quotient is always aligned with the dividend’s decimal point.
• Adding zeroes to the dividend (after the decimal point) allows you to continue dividing until there’s no remainder or until you reach the desired precision.

Division of a Decimal Number by Another Decimal Number

Dividing a decimal by another decimal requires an extra step: converting the divisor into a whole number to simplify the process. This is done by moving the decimal point in both the divisor and the dividend by the same number of places to the right, ensuring their ratio remains unchanged.

Steps for Dividing a Decimal by a Decimal:

1. Convert the divisor to a whole number: Count the number of decimal places in the divisor and move its decimal point to the right to eliminate the decimal. For example, if the divisor is 0.3 (1 decimal place), move the decimal point one place right to get 3.
2. Move the dividend’s decimal point the same number of places: Whatever you do to the divisor, do the same to the dividend to maintain the equality of the division problem. If the dividend has fewer decimal places, add zeroes as needed.
3. Divide the new dividend by the new whole-number divisor: Use the same long division method as when dividing by a whole number, placing the decimal point in the quotient correctly.

Example 1: Divide 7.2 by 0.6

• Divisor 0.6 has 1 decimal place. Move decimal point 1 place right: 0.6 → 6.
• Move dividend 7.2’s decimal point 1 place right: 7.2 → 72.
• Now divide 72 by 6: 12. Final answer: 12.

Example 2: Divide 4.86 by 1.2

• Divisor 1.2 has 1 decimal place. Move to get 12.
• Dividend 4.86: move 1 place right → 48.6 (add a zero if needed, but here it’s 48.6).
• Set up: 12 ) 48.6
• Divide 48 by 12: 4, place decimal point: 4.
• Bring down 6, divide 6 by 12: 0.5 (since 12×0.5=6).
• Final answer: 4.05

Key Points:

• Moving the decimal point in both numbers by the same amount ensures the division problem remains equivalent.
• Always start by converting the divisor to a whole number to simplify the long division process.

Dividing Decimals by 10, 100, and 1000

Dividing a decimal by a power of 10 (10, 100, 1000, etc.) is a special case that follows a simple pattern: the decimal point moves to the left by as many places as there are zeros in the power of 10. This is the inverse of multiplying by powers of 10, where the decimal point moves right.

Rules:

• Divide by 10: Move the decimal point one place to the left.
• Divide by 100: Move the decimal point two places to the left.
• Divide by 1000: Move the decimal point three places to the left.

Example 1: Divide 56.7 by 10

• 56.7 ÷ 10 = 5.67 (decimal point moved one place left).

Example 2: Divide 9.3 by 100

• 9.3 ÷ 100 = 0.093 (decimal point moved two places left; since there’s only one digit before the decimal, add a zero to fill the second place).

Example 3: Divide 123.456 by 1000

• 123.456 ÷ 1000 = 0.123456 (decimal point moved three places left).

Key Points:

• If there are fewer digits than places to move, add leading zeros to the left of the number (e.g., 0.5 ÷ 100 = 0.005).
• This rule simplifies calculations and is essential for unit conversions (e.g., meters to kilometers).

Dividing Decimals: Examples

Let’s explore more examples to reinforce the concepts, covering different scenarios:

Example 1: Dividing a Decimal by a Whole Number (with Remainder)

Problem: 10.5 ÷ 4

• Set up: 4 ) 10.5
• Divide 10 by 4: 2, remainder 2. Place decimal point: 2.
• Bring down 5, making 25. Divide 25 by 4: 6 (4×6=24), remainder 1.
• Add a zero: 10. Divide 10 by 4: 2 (4×2=8), remainder 2.
• Add another zero: 20. Divide 20 by 4: 5.
• Quotient: 2.625

Example 2: Dividing a Decimal by a Decimal (with Multiple Decimal Places)

Problem: 0.09 ÷ 0.03

• Divisor 0.03 has 2 decimal places. Move both decimals 2 places right: 0.03 → 3, 0.09 → 9.
• Divide 9 by 3: 3. Answer directly: 3.

Example 3: Repeating Decimal Result

Problem: 1 ÷ 3

• Set up: 3 ) 1.000…
• Divide 1 by 3: 0, place decimal: 0.
• Bring down 0: 10 ÷ 3 = 3 (3×3=9), remainder 1.
• This repeats: 10 ÷ 3 = 3, remainder 1.
• Quotient: 0.333… (a repeating decimal)

Problem: 6.48 ÷ 2.4

• Divisor 2.4 has 1 decimal place. Move both decimals 1 place right: 24 and 64.8.
• Divide 64 by 24: 2 (24×2=48), remainder 16.
• Bring down 8: 168 ÷ 24 = 7 (24×7=168).
• Exact Answer: 2.7

Frequently Asked Questions (FAQs)

How do you divide decimals step by step?

1. For Dividing by a Whole Number:
   • Set up long division, aligning the quotient’s decimal point with the dividend’s.
   • Divide as with whole numbers, bringing down digits and adding zeroes as needed.
2. For Dividing by a Decimal:
   • Convert the divisor to a whole number by moving its decimal point right.
   • Move the dividend’s decimal point the same number of places right.
   • Divide using the whole-number divisor, placing the decimal point in the quotient correctly.
3. For Powers of 10:
   • Move the dividend’s decimal point left by the number of zeros in the power of 10.

Which way do you move the decimal point when dividing by 10?

When dividing by 10, 100, or 1000, the decimal point moves to the left by 1, 2, or 3 places, respectively. For example, 75.2 ÷ 10 = 7.52 (1 place left), 75.2 ÷ 100 = 0.752 (2 places left).

How do you solve 5 divided by 100?

• Since 100 has two zeros, move the decimal point in 5 (which is 5.0) two places left: 0.05.
• Alternatively, use long division: 100 ) 5.00 → 0.05.

What is 31 over 100 as a decimal?

• “31 over 100” means 31 ÷ 100. Moving the decimal point two places left in 31 (31.0) gives 0.31.

How do you divide a decimal by 10?

• Simply move the decimal point one place to the left. For example:
  • 4.5 ÷ 10 = 0.45
  • 0.6 ÷ 10 = 0.06 (add a leading zero if needed).

Mastering Decimal Division: Key Takeaways

• Decimal Point Alignment: Always align the decimal point in the quotient with the dividend (when dividing by a whole number) or adjust both numbers when the divisor is a decimal.
• Long Division Skills: Practice long division with whole numbers first, then apply the same steps to decimals, focusing on place value.
• Power of 10 Shortcut: Remember that dividing by 10, 100, or 1000 is a quick decimal shift left, saving time on calculations.
• Repeating Decimals: Recognize when a remainder repeats, leading to a repeating quotient (e.g., 1÷3=0.333…).
• Practice Makes Perfect: Work through examples and problems to build confidence, ensuring you understand each step rather than memorizing rules.

Conclusion

By mastering these techniques, you’ll gain a good understanding of decimal division, which is crucial for advanced math topics like fractions, percentages, and algebra. Remember to take it step by step, double-check your decimal placement, and don’t hesitate to add zeroes to complete the division. With patience and practice, especially through online math classes, dividing decimals will become second nature!

This comprehensive guide covers all aspects of dividing decimals, from basic definitions to complex examples, ensuring readers can confidently apply these skills in various contexts. By following the step-by-step instructions and practicing with the problems provided, anyone can master the art of decimal division.