# Dividing decimals

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##### Intros
###### Lessons
1. How to do long division?
How to do long division?
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##### Examples
###### Lessons
1. Divide using diagrams.
1. 3.4 ÷ 2
2. 1.8 ÷ 3
2. Divide using a number line.
1. 8 ÷ 0.5
2. 2 ÷ 0.4
3. Divide using long division.
1. 334.4 ÷ 4
4. Tim bought some of his favorite donuts at a local bakery. Each donut costs $0.50 and he spent$4.00 in total. How many donuts did he buy? Use long division to determine the answer.
###### Topic Notes
In the previous section, we learned how to multiply decimal numbers. In this section, we will learn how to divide decimal numbers. First, we will divide using diagrams. Next, we will divide using number lines. Then, we will divide using long division. When dividing using long division, the divisor must first be turned into a whole number by moving the decimal point over to the right one or more spaces. Next, the decimal point in the dividend must be moved over to the right the same number of spaces. The decimal point in the answer will line up with the decimal point in the dividend.

## Introduction to Dividing Decimals

Dividing decimals is a crucial mathematical skill that builds upon your previous knowledge of multiplying decimals. This section introduces you to the concept and techniques of decimal division, which are essential for solving complex mathematical problems. To begin, we recommend watching our introduction video, which provides a clear and concise explanation of the process. This visual guide will help you grasp the fundamental principles and steps involved in dividing decimals. The video serves as an important foundation for understanding the concept, making it easier to follow the subsequent lessons and practice exercises. By mastering the art of dividing decimals, you'll enhance your overall mathematical proficiency and problem-solving abilities. Remember, just as with multiplying decimals, precision and attention to decimal point placement are key factors in achieving accurate results when dividing decimals.

## Understanding Division with Decimals

Division with decimals can often seem daunting, but it's a fundamental mathematical skill that becomes easier with practice and understanding. Let's break down this concept using simple examples and demonstrate how we can manipulate division problems to make them more manageable.

First, let's start with a basic division problem: 10 divided by 2. This is a straightforward calculation that most people can do mentally, resulting in 5. Now, let's consider what happens when we introduce decimals into the mix.

Imagine we have 1.0 divided by 0.2. At first glance, this might seem more complicated than our initial example. However, there's a key principle we can apply here: multiplying both the dividend (the number being divided) and the divisor (the number we're dividing by) by the same value doesn't change the result of the division.

Using this principle, we can transform our decimal division problem into a whole number division. If we multiply both 1.0 and 0.2 by 10, we get:

(1.0 × 10) ÷ (0.2 × 10) = 10 ÷ 2

Now we're back to our original, simple division problem of 10 divided by 2, which we know equals 5. This demonstrates that 1.0 ÷ 0.2 also equals 5.

Let's look at another example: 0.6 ÷ 0.2. Again, we can multiply both the dividend and divisor by 10 to eliminate the decimals:

(0.6 × 10) ÷ (0.2 × 10) = 6 ÷ 2 = 3

This method of multiplying both the dividend and divisor by the same number (often 10, 100, or 1000, depending on the decimal places) is a powerful tool for converting decimals to whole numbers. It allows us to convert decimal division into whole number division, which is typically easier to calculate.

Here's another example to reinforce this concept: 2.4 ÷ 0.6. In this case, multiplying both numbers by 10 gives us:

(2.4 × 10) ÷ (0.6 × 10) = 24 ÷ 6 = 4

It's important to note that this method works because when we multiply both the top and bottom of a fraction (which is essentially what division represents) by the same number, the fraction's value remains unchanged. This principle is fundamental in mathematics and applies to both simple and complex calculations.

By mastering this technique, you can approach decimal division problems with confidence. Remember, the key steps are:

1. Identify the number you need to multiply by to convert both the dividend and divisor to whole numbers (usually a power of 10).
2. Multiply both the dividend and divisor by this number.
3. Perform the division with the resulting whole numbers.
4. The result of this whole number division is the same as the original decimal division.

Practice with various examples to become comfortable with this method. Over time, you'll find that decimal division becomes much less intimidating, and you'll be able to solve these problems with ease and accuracy.

## The Decimal Shift Method

The decimal shift method is a powerful technique that simplifies division problems involving decimals. This approach leverages our understanding of place value in division and the relationship between multiplication and division to make calculations more manageable. By shifting the decimal point in both the dividend and divisor, we can transform a complex division problem into a simpler one without changing the actual value of the quotient.

At the heart of this method is the concept of multiplying decimals by 10, 100, or 1000. This multiplication effectively shifts the decimal point to the right, eliminating decimal places and converting the problem into whole number division. The key principle to remember is that when we multiply both the top and bottom of a fraction (or division problem) by the same number, the value of the fraction remains unchanged.

Let's break down this process using the example from the video: 5.74 ÷ 0.8

Step 1: Identify the number of decimal places in the divisor (0.8 has one decimal place).

Step 2: Multiply both the dividend and divisor by 10 to shift the decimal one place to the right:

5.74 × 10 = 57.4

0.8 × 10 = 8

The new problem becomes: 57.4 ÷ 8

Step 3: If there are still decimals in the dividend, repeat the process. In this case, we'll multiply both numbers by 10 again:

57.4 × 10 = 574

8 × 10 = 80

Now we have: 574 ÷ 80

This transformation has converted our original problem with decimals into a whole number division problem, which is generally easier to solve using standard long division with decimals techniques.

The power of multiplying decimals by 10, 100, or 1000 lies in how it affects place value in division. Each time we multiply by 10, we shift the decimal point one place to the right. Multiplying by 100 shifts it two places, and by 1000 shifts it three places. This systematic movement of the decimal point is what gives the method its name - the decimal shift method.

It's important to note that while we've changed the appearance of the numbers, we haven't altered the fundamental relationship between them. The quotient of 574 ÷ 80 will be exactly the same as 5.74 ÷ 0.8. This consistency is what makes the decimal shift method so reliable and useful.

Once we've shifted the decimals and arrived at a whole number division problem, we can proceed with long division with decimals as usual. In this case, 574 ÷ 80 = 7.175. This final answer is the same result we would have gotten if we had divided 5.74 by 0.8 directly, but the process was significantly simplified by using the decimal shift method.

This technique is particularly valuable when dealing with divisors that are less than 1, as it transforms them into whole numbers, making the division process more intuitive. It's also helpful in mental math, allowing for quicker estimations and calculations without the need for complex decimal arithmetic.

In conclusion, the decimal shift method is a versatile tool in mathematics that simplifies division with decimals. By understanding and applying this technique, students can approach a wide range of division problems with greater confidence and efficiency. It reinforces the importance of place value in division in our number system and demonstrates how mathematical operations can be manipulated to our advantage without changing the underlying values. Mastering this method not only aids in solving specific problems but also enhances overall numerical fluency and problem-solving skills.

## Long Division with Decimals

Long division with decimals is a fundamental mathematical skill that builds upon basic division techniques. Let's walk through the process using the example 574 ÷ 80, demonstrating how to handle remainders and work beyond the decimal point.

Step 1: Set up the division problem. Write 80 on the left side of the division symbol and 574 under the "roof" of the division house.

Step 2: Estimate how many times 80 goes into 574. In this case, 80 goes into 574 about 7 times. Write 7 above the division symbol.

Step 3: Multiply 80 by 7, which equals 560. Write this below 574 and subtract: 574 - 560 = 14. This remainder (14) is less than our divisor (80), so we can't divide further without using decimals.

Step 4: To continue dividing, we need to add a decimal point and a zero after the remainder. Write ".0" after 14 to get 14.0. Also, add a decimal point after the 7 in your quotient.

Step 5: Now, we ask how many times 80 goes into 140. It goes 1 time with a remainder. Write 1 after the decimal point in your quotient.

Step 6: Multiply 80 by 1, which is 80. Subtract this from 140: 140 - 80 = 60.

Step 7: Again, 60 is less than 80, so we need to add another zero. Our new dividend is 600.

Step 8: 80 goes into 600 exactly 7 times. Write 7 as the next digit in your quotient.

Step 9: Multiply 80 by 7, which is 560. Subtract this from 600: 600 - 560 = 40.

Step 10: Once more, add a zero to 40 to get 400.

Step 11: 80 goes into 400 exactly 5 times. Add 5 to your quotient.

Step 12: Multiply 80 by 5, which is 400. Subtract from 400, and we get 0 as our remainder.

The final answer is 7.175. This process demonstrates several key concepts in long division with decimals:

1. Place value: We maintain the correct place value by adding zeros after the decimal point when necessary.

2. Handling remainders: When a remainder is smaller than the divisor, we add a decimal point and continue dividing.

3. Decimal point placement: The decimal point in the quotient aligns with where we add it in the dividend.

4. Continuing beyond the decimal: The process remains the same after the decimal point, adding zeros as needed.

5. Terminating vs. repeating decimals: In this case, we reached a remainder of 0, resulting in a terminating decimal. Some divisions may result in repeating decimals.

Mastering long division with decimals enhances numerical fluency and provides a foundation for more advanced mathematical concepts. It reinforces understanding of place value, improves estimation skills, and develops logical thinking in problem-solving. Practice with various examples will help solidify this essential mathematical technique.

## Terminating and Repeating Decimals

When performing division, the result can be expressed as a decimal, which falls into two main categories: terminating decimals and repeating decimals. Understanding these concepts is crucial for mastering division and working with rational numbers.

Terminating decimals are those that end after a finite number of decimal places. For example, 0.5, 0.25, and 0.375 are all terminating decimals. These occur when the division process eventually results in a remainder of zero. To identify when a division will result in a terminating decimal, consider the denominator of the fraction. If the denominator can be expressed as a product of 2 and 5 (the prime factors of 10), the division will yield a terminating decimal. For instance, 1/8 = 0.125 (terminating) because 8 = 2³.

On the other hand, repeating decimals are those where a digit or group of digits continues infinitely after the decimal point. For example, 1/3 = 0.333... (repeating), and 2/7 = 0.285714285714... (repeating). Repeating decimals occur when the division process never results in a remainder of zero. To identify when a division might lead to a repeating decimal, look at the denominator. If it contains prime factors other than 2 and 5, the result will be a repeating decimal. For instance, 1/6 = 0.166666... (repeating) because 6 = 2 × 3.

Let's examine some examples to illustrate both cases:

1. Terminating decimal: 3/8 = 0.375 (The denominator 8 = 2³, so it terminates)

2. Repeating decimal: 1/9 = 0.111111... (The denominator 9 = 3², so it repeats)

3. Terminating decimal: 17/20 = 0.85 (The denominator 20 = 2² × 5, so it terminates)

4. Repeating decimal: 5/11 = 0.454545... (The denominator 11 is prime, so it repeats)

Understanding the difference between terminating and repeating decimals is essential for various mathematical applications, including simplifying fractions, solving equations, and working with rational numbers and irrational numbers. By recognizing the patterns in division and the properties of denominators, you can quickly determine whether a division will result in a terminating or repeating decimal, saving time and improving your mathematical intuition.

## Practice Problems and Common Mistakes

To reinforce your understanding of decimal division practice problems, let's dive into a set of practice problems. We'll start with simpler exercises and gradually increase the complexity. Remember to apply the concepts we've learned and watch out for common pitfalls.

1. 12.6 ÷ 2
2. 3.5 ÷ 0.5
3. 8.4 ÷ 4
4. 1.8 ÷ 0.3
5. 5.6 ÷ 0.8

### More Complex Decimal Division Problems:

1. 23.76 ÷ 1.2
2. 0.0468 ÷ 0.06
3. 7.254 ÷ 0.03
4. 0.8645 ÷ 0.05
5. 12.3456 ÷ 0.24

As you work through these problems, be mindful of common mistakes that students often make when dividing decimals. Here are some frequent errors and how to avoid them:

### Common Mistake 1: Misplacing the Decimal Point

One of the most common errors is incorrectly placing the decimal point in the quotient. To avoid this, remember to move the decimal point in both the dividend and divisor the same number of places to the right until the divisor becomes a whole number. Then, place the decimal point directly above its position in the dividend.

### Common Mistake 2: Forgetting to Add Zeros

When the division doesn't come out evenly, students sometimes forget to add zeros in decimal division to continue the division. Always add zeros as needed to complete the division or reach the desired level of precision.

### Common Mistake 3: Ignoring Trailing Zeros

Some students overlook the importance of trailing zeros, especially when converting fractions to decimals. Remember that 0.5 is not the same as 0.50 in terms of precision. Be mindful of significant figures and the level of accuracy required in your calculations.

### Common Mistake 4: Difficulty with Divisors Less Than 1

Many students struggle when the divisor is a decimal less than 1. The key is to recognize that dividing by a number less than 1 will result in a quotient larger than the dividend. Practice problems like 1.8 ÷ 0.3 to become comfortable with this concept.

### Common Mistake 5: Rounding Errors

Improper rounding can lead to significant errors in your final answer. Always round to the specified number of decimal places and use the correct rounding rules. If rounding isn't specified, maintain the level of precision of the least precise number in the problem.

To avoid these mistakes, follow these tips:

• Double-check your decimal point placement before and after the division.
• Practice with a variety of problems, especially those involving divisors less than 1.
• Pay attention to the level of precision required and use proper rounding techniques.
• Show your work step-by-step to catch errors more easily.

Remember, mastering decimal division practice problems takes practice. Don't get discouraged if you make mistakes initially. Use them as learning opportunities to reinforce your understanding of the concepts. As you work through more problems, you'll develop a stronger intuition for decimal operations and become more confident in your abilities.

## Conclusion

### Divide using diagrams: 3.4 ÷ 2

In this guide, we will divide the decimal number 3.4 by 2 using diagrams. This method helps visualize the division process and understand the concept better.

#### Step 1: Represent the Number 3.4 Using Diagrams

First, we need to represent the number 3.4 using diagrams. We can break down 3.4 into 3 ones and 4 tenths.

We use large tiles to represent the ones and smaller tiles to represent the tenths. So, we have:

• 3 large tiles, each representing 1 (totaling 3 ones)
• 4 smaller tiles, each representing 0.1 (totaling 0.4 tenths)

#### Step 2: Divide the Ones into Two Groups

Next, we need to divide the 3 ones into two equal groups. Since 3 is an odd number, we can only evenly distribute 2 of the ones, leaving 1 one left over.

So, each group will have:

• 1 large tile (representing 1)

And we have 1 large tile left over.

#### Step 3: Convert the Remaining One into Tenths

To divide the remaining 1 one, we convert it into tenths. Since 1 one is equal to 10 tenths, we now have:

• 10 tenths from the remaining one
• 4 tenths from the original 0.4

This gives us a total of 14 tenths.

#### Step 4: Divide the Tenths into Two Groups

Now, we need to divide the 14 tenths into two equal groups. Dividing 14 by 2 gives us 7 tenths in each group.

So, each group will have:

• 7 smaller tiles (representing 0.7)

#### Step 5: Combine the Ones and Tenths in Each Group

Finally, we combine the ones and tenths in each group. Each group has:

• 1 large tile (representing 1)
• 7 smaller tiles (representing 0.7)

So, each group represents the number 1.7.

#### Conclusion

By dividing the number 3.4 into two groups using diagrams, we can see that each group represents the number 1.7. Therefore, 3.4 divided by 2 equals 1.7.

### FAQs

1. #### What is the decimal shift method in division?

The decimal shift method is a technique used to simplify decimal division problems. It involves multiplying both the dividend and divisor by the same power of 10 to eliminate decimal places, converting the problem into whole number division. This method makes the calculation easier without changing the actual value of the quotient.

2. #### How do you identify if a decimal will be terminating or repeating?

To identify if a decimal will be terminating or repeating, look at the denominator of the fraction. If the denominator can be expressed as a product of only 2 and 5 (prime factors of 10), the decimal will terminate. If the denominator contains any other prime factors, the decimal will repeat. For example, 1/8 (8 = 2³) will terminate, while 1/3 will repeat.

3. #### What are common mistakes to avoid when dividing decimals?

Common mistakes in decimal division include misplacing the decimal point, forgetting to add zeros to continue division, ignoring trailing zeros, struggling with divisors less than 1, and making rounding errors. To avoid these, always double-check decimal point placement, use estimation to verify reasonableness, and pay attention to the required level of precision.

4. #### How do you handle division when the divisor is less than 1?

When the divisor is less than 1, the quotient will be larger than the dividend. To handle this, you can use the decimal shift method to convert the divisor to a whole number. For example, to divide 1.8 by 0.3, multiply both numbers by 10 to get 18 ÷ 3, which is easier to calculate.

5. #### Why is mastering decimal division important?

Mastering decimal division is crucial for various real-life applications such as budgeting, cooking, and data analysis. It enhances numerical fluency, improves problem-solving skills, and provides a foundation for more advanced mathematical concepts. Proficiency in decimal division also helps in everyday calculations and decision-making processes involving fractions and percentages.

### Prerequisite Topics for Dividing Decimals

Understanding the process of dividing decimals requires a solid foundation in several key mathematical concepts. One of the most crucial prerequisites is multiplying decimals. This skill is essential because division is often viewed as the inverse operation of multiplication, and many division strategies involve multiplying by reciprocals.

Another fundamental concept is place value, which is critical in division of decimals. Knowing the significance of each digit's position helps in aligning decimal points correctly and understanding how to shift the decimal point when necessary. This understanding is closely tied to the negative exponent rule, which comes into play when dealing with very small decimal numbers or when converting between decimal and scientific notation.

Proficiency in adding and subtracting decimals is also crucial, as these operations are often part of the division process, especially in long division or when checking results. Additionally, the ability to convert repeating decimals to fractions can be helpful in simplifying and interpreting division results.

Understanding greatest common factors (GCF) and least common multiples can aid in simplifying division problems and finding equivalent fractions, which is often useful in decimal division. This concept extends to more advanced topics like polynomial long division, where similar principles are applied to algebraic expressions.

As students progress, they'll find that the skills learned in dividing decimals are foundational for solving problems with rational numbers in decimal form. This ability is crucial in various real-world applications and more advanced mathematical concepts.

Even seemingly unrelated topics like the rational zero theorem build upon the understanding of division and rational numbers, showcasing how fundamental decimal division is to higher-level mathematics.

By mastering these prerequisite topics, students will find themselves well-equipped to tackle the challenges of dividing decimals. Each concept provides a crucial piece of the puzzle, allowing for a deeper understanding and more efficient problem-solving in decimal division and beyond. Remember, mathematics is a cumulative subject, and a strong grasp of these foundational concepts will pave the way for success in more advanced mathematical endeavors.