True Number Sentence: How To Solve Division Problems

by Alex Johnson 53 views

Are you struggling with division problems involving negative numbers? Do you want to understand the rules for dividing decimals and integers? Then you've come to the right place! In this article, we will break down a common type of math problem: identifying the true number sentence from a set of options. We'll use the example question "Which number sentence is true?" and explore the concepts behind it. We'll walk through the steps to solve these problems, focusing on the rules of dividing positive and negative numbers, as well as how to handle decimals in division. By the end of this guide, you will be able to confidently tackle similar math challenges and boost your understanding of fundamental arithmetic operations. So, grab your pencil and paper, and let's dive into the world of number sentences!

Understanding the Basics of Number Sentences

Before we dive into the specific problem, let's ensure we understand the core concepts. Number sentences are mathematical statements that express a relationship between numbers using mathematical operations such as addition, subtraction, multiplication, and division. In this case, we are focusing on division. A true number sentence is one where the equation is mathematically correct. For example, 10 ÷ 2 = 5 is a true number sentence, while 10 ÷ 2 = 6 is false. The critical skill here is performing the division accurately and understanding the rules governing the signs of the numbers involved. When dividing, remember that the sign of the result depends on the signs of the numbers being divided. A positive number divided by a positive number yields a positive result. A negative number divided by a negative number also yields a positive result. However, a positive number divided by a negative number or a negative number divided by a positive number results in a negative answer. These sign rules are crucial for solving problems involving negative numbers. Additionally, it’s essential to know how to divide decimals accurately. This often involves shifting the decimal point in both the divisor and the dividend to make the divisor a whole number, and then performing long division. A solid grasp of these fundamental concepts will make identifying the true number sentence much easier and less prone to error. Let’s move on to applying these concepts to our specific problem.

Breaking Down the Problem: -39.6 ÷ (-4.8) = 8.25

Now, let's tackle the problem at hand: identifying the correct number sentence. The question asks us to determine which of the provided division problems is true. One of the options is: -39.6 ÷ (-4.8) = 8.25. To verify this, we need to perform the division and check if the result matches the stated answer. First, let's consider the signs. We are dividing a negative number (-39.6) by another negative number (-4.8). According to the rules of division, when you divide a negative by a negative, the result is positive. This aligns with the stated answer of 8.25, which is a positive number. Next, we need to perform the actual division. Dividing 39.6 by 4.8 can seem daunting, but we can simplify it by eliminating the decimals. To do this, we shift the decimal point one place to the right in both numbers. This is equivalent to multiplying both numbers by 10, which doesn’t change the result of the division. So, we transform -39.6 ÷ (-4.8) into 396 ÷ 48. Now, we can perform long division. When you divide 396 by 48, you get 8.25. This confirms that our calculation matches the answer provided in the number sentence. Therefore, based on our understanding of the sign rules and the accurate division, we can confidently say that -39.6 ÷ (-4.8) = 8.25 is indeed a true number sentence. This step-by-step verification process is key to solving these types of problems accurately.

Step-by-Step Solution: How to Verify a Number Sentence

To effectively verify a number sentence, especially one involving decimals and negative numbers, it's best to follow a systematic approach. Here's a step-by-step solution you can use:

  1. Analyze the Signs: The first step is to determine the sign of the result. Remember the rules: a positive divided by a positive is positive, a negative divided by a negative is positive, and a positive divided by a negative (or vice versa) is negative. Understanding this rule helps you quickly eliminate options that have the wrong sign. In our example, -39.6 ÷ (-4.8) involves dividing a negative by a negative, so the result should be positive. This aligns with the given answer of 8.25.

  2. Simplify the Division: If you're dealing with decimals, simplify the division by shifting the decimal point to the right in both the divisor and dividend until the divisor is a whole number. This is equivalent to multiplying both numbers by a power of 10, which doesn’t change the outcome of the division. For instance, -39.6 ÷ (-4.8) can be simplified to 396 ÷ 48 by moving the decimal one place to the right in both numbers.

  3. Perform the Division: Carry out the division using long division or a calculator. Ensure you are accurate in your calculations. When dividing 396 by 48, the result is 8.25.

  4. Compare the Result: Compare your result with the answer provided in the number sentence. If the calculated result matches the given answer, the number sentence is true. If they don't match, the number sentence is false. In our case, the calculated result of 8.25 matches the provided answer, confirming that -39.6 ÷ (-4.8) = 8.25 is a true number sentence.

  5. Double-Check Your Work: It's always a good idea to double-check your work to minimize errors. Review the signs, the decimal placement, and the division process to ensure accuracy. This systematic approach helps you solve number sentence problems effectively and confidently.

Common Mistakes to Avoid When Solving Division Problems

When solving division problems, especially those involving decimals and negative numbers, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and improve your accuracy. One frequent error is misunderstanding the sign rules. It’s crucial to remember that a negative number divided by a negative number yields a positive result, while a positive number divided by a negative number (or vice versa) results in a negative answer. Forgetting this rule can lead to incorrect answers. Another common mistake is incorrectly handling decimals. When dividing decimals, it’s essential to shift the decimal point in both the divisor and dividend to make the divisor a whole number. Failing to do this or miscounting the number of places to shift can result in inaccurate calculations. For example, in the problem -39.6 ÷ (-4.8), not shifting the decimal correctly can lead to confusion and an incorrect quotient. Another pitfall is making arithmetic errors during the division process. Division, especially long division, requires careful attention to each step. Mistakes in subtraction or bringing down the wrong digit can lead to a wrong answer. It’s always a good practice to double-check your calculations to minimize these errors. Furthermore, not simplifying the problem can make it more challenging. Look for opportunities to simplify the division, such as reducing fractions or eliminating decimals, before you start the calculation. Finally, skipping steps or trying to do too much in your head can increase the likelihood of errors. It’s better to write down each step clearly and methodically to ensure accuracy. By being mindful of these common mistakes and taking steps to avoid them, you can significantly improve your problem-solving skills and achieve more accurate results in division problems.

Practice Problems and Examples

To solidify your understanding of identifying true number sentences involving division, let's work through a few more practice problems. These examples will help you apply the steps and strategies we've discussed. Remember, the key is to analyze the signs, simplify the division if necessary, perform the division accurately, and compare the result with the given answer.

Example 1:

Which of the following number sentences is true?

A. 15.75 ÷ (-2.5) = 6.3

B. -15.75 ÷ 2.5 = -6.3

C. -15.75 ÷ (-2.5) = -6.3

D. 15.75 ÷ 2.5 = -6.3

Solution:

  • First, analyze the signs. In option B, we have a negative divided by a positive, which should result in a negative answer. In option C, we have a negative divided by a negative, which should result in a positive answer. The same logic applies to other options as well.
  • Next, perform the division for each option and compare the result with the given answer. For option B, dividing 15.75 by 2.5 gives us 6.3. Since it’s a negative divided by a positive, the result is -6.3, which matches the answer in the number sentence. Therefore, option B is the correct answer.

Example 2:

Which of the following number sentences is true?

A. -42.5 ÷ 5 = 8.5

B. 42.5 ÷ (-5) = 8.5

C. -42.5 ÷ (-5) = 8.5

D. 42.5 ÷ 5 = -8.5

Solution:

  • Analyze the signs. In option C, we have a negative divided by a negative, which should result in a positive answer. This aligns with the answer provided, 8.5.
  • Perform the division: 42.5 ÷ 5 = 8.5. Since both the calculated result and the sign match, option C is the true number sentence.

By practicing with these examples, you'll become more comfortable and confident in your ability to solve division problems involving decimals and negative numbers. Remember to take it one step at a time, double-check your work, and apply the strategies we’ve discussed.

Conclusion

Mastering division, especially with decimals and negative numbers, is a crucial skill in mathematics. By understanding the rules of signs, knowing how to handle decimals, and following a systematic approach, you can confidently solve problems that ask you to identify true number sentences. Remember to analyze the signs first, simplify the division when necessary, perform the division accurately, and always double-check your work. The examples and step-by-step solutions provided in this article should serve as a valuable resource as you continue to practice and improve. With consistent effort and a clear understanding of the fundamentals, you’ll be well-equipped to tackle any division challenge that comes your way. Keep practicing, and you’ll see your skills grow!

For further learning and practice on division and other mathematical concepts, you can visit resources like Khan Academy's Arithmetic Section. This can provide additional explanations, examples, and exercises to help you master these essential skills.