Mastering Math Operations A Step By Step Guide To Arithmetic Problems

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Hey guys! Let's dive into the exciting world of math operations! In this guide, we'll tackle a series of arithmetic problems, breaking them down step by step to ensure you not only get the right answers but also understand the underlying principles. Whether you're brushing up on your skills or learning these concepts for the first time, this article is here to help. Get ready to sharpen your pencils and your minds!

Problem Set Overview

We're going to work through ten different math problems, each involving a mix of addition, subtraction, multiplication, and division. These problems are designed to test your understanding of the order of operations (PEMDAS/BODMAS) and how to work with negative numbers. Don't worry if it seems daunting at first; we'll take it one problem at a time.

Key Concepts to Remember

Before we jump into the problems, let's quickly recap some key concepts:

  • Order of Operations (PEMDAS/BODMAS): This is your golden rule! It tells you the order in which to perform operations: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
  • Negative Numbers: Remember that multiplying or dividing two negative numbers results in a positive number. Multiplying or dividing a positive and a negative number results in a negative number. Adding a negative number is the same as subtracting, and subtracting a negative number is the same as adding.

With these concepts in mind, let's get started!

Problem 1: 4 - 8 × (-5)

In this first problem, let's solve the equation 4 - 8 × (-5), we have subtraction and multiplication. According to the order of operations, multiplication comes first. So, we multiply 8 by -5, which gives us -40. Now our equation looks like this: 4 - (-40). Remember that subtracting a negative number is the same as adding its positive counterpart. So, 4 - (-40) becomes 4 + 40. Adding these together, we get 44. Therefore, the solution to the equation 4 - 8 × (-5) is 44. This highlights the importance of following the correct order of operations to arrive at the correct answer. Understanding these fundamental rules helps in tackling more complex mathematical problems with confidence.

  • Step 1: Multiplication 8 × (-5) = -40
  • Step 2: Subtraction (becomes addition) 4 - (-40) = 4 + 40
  • Step 3: Final Answer 4 + 40 = 44

Problem 2: 7 + (-3) × 9

Moving on to the second problem, let's delve into solving the equation 7 + (-3) × 9. Here, we encounter both addition and multiplication. As per the order of operations, multiplication takes precedence. Thus, we begin by multiplying -3 by 9, which yields -27. Our equation now transforms into 7 + (-27). Adding a negative number is equivalent to subtraction, so 7 + (-27) becomes 7 - 27. Performing this subtraction, we arrive at -20. Hence, the solution to the equation 7 + (-3) × 9 is -20. This problem further emphasizes the necessity of adhering to the correct order of operations to ensure accuracy in calculations. Mastery of these basic arithmetic principles forms the bedrock for tackling more intricate mathematical challenges.

  • Step 1: Multiplication (-3) × 9 = -27
  • Step 2: Addition 7 + (-27) = 7 - 27
  • Step 3: Final Answer 7 - 27 = -20

Problem 3: 9 - (-6) × (-7)

Now, let's tackle the third problem: 9 - (-6) × (-7). This problem involves subtraction and multiplication, with the added twist of dealing with negative numbers. Following the order of operations, we first address the multiplication. We multiply -6 by -7. A negative number multiplied by a negative number gives a positive result, so -6 × -7 equals 42. Our equation now looks like this: 9 - 42. Next, we perform the subtraction: 9 - 42. This gives us -33. Therefore, the solution to the equation 9 - (-6) × (-7) is -33. This problem underscores the importance of remembering the rules for multiplying negative numbers and applying the correct order of operations. Handling such intricacies is crucial for building a solid foundation in arithmetic.

  • Step 1: Multiplication (-6) × (-7) = 42
  • Step 2: Subtraction 9 - 42
  • Step 3: Final Answer 9 - 42 = -33

Problem 4: (-2 × 3) + (-5)

For the fourth problem, let's explore the solution to (-2 × 3) + (-5). This equation features multiplication within parentheses and addition. According to the order of operations, we first tackle the operation inside the parentheses. We multiply -2 by 3, which results in -6. Our equation now reads: -6 + (-5). Adding a negative number is equivalent to subtraction, so -6 + (-5) becomes -6 - 5. Performing this subtraction, we get -11. Consequently, the solution to the equation (-2 × 3) + (-5) is -11. This problem reinforces the significance of addressing operations within parentheses before proceeding with other operations. It also emphasizes the handling of negative numbers in both multiplication and addition scenarios.

  • Step 1: Parentheses (Multiplication) (-2 × 3) = -6
  • Step 2: Addition -6 + (-5) = -6 - 5
  • Step 3: Final Answer -6 - 5 = -11

Problem 5: (-15) × (-11) + 75

In the fifth problem, let's solve (-15) × (-11) + 75. We have multiplication and addition here. Following the order of operations, we perform the multiplication first. Multiplying -15 by -11 gives us 165 because a negative times a negative is a positive. Now, our equation looks like this: 165 + 75. We then add 165 and 75, which equals 240. Therefore, the solution to the equation (-15) × (-11) + 75 is 240. This problem reinforces the rules of multiplying negative numbers and the importance of the correct order of operations in achieving accurate results. These are crucial skills for anyone looking to build their mathematical proficiency.

  • Step 1: Multiplication (-15) × (-11) = 165
  • Step 2: Addition 165 + 75
  • Step 3: Final Answer 165 + 75 = 240

Problem 6: (-21) × 17 - (-12)

Moving on to the sixth problem, let's solve the equation (-21) × 17 - (-12). This problem includes multiplication and subtraction, with a negative number being subtracted. According to the order of operations, we handle multiplication first. Multiplying -21 by 17 results in -357. Our equation now transforms to -357 - (-12). Subtracting a negative number is the same as adding its positive counterpart, so -357 - (-12) becomes -357 + 12. Adding these together, we get -345. Thus, the solution to the equation (-21) × 17 - (-12) is -345. This problem highlights the importance of correctly applying the rules for operations with negative numbers, especially when subtraction is involved. It's a good practice to double-check these steps to avoid common errors.

  • Step 1: Multiplication (-21) × 17 = -357
  • Step 2: Subtraction (becomes addition) -357 - (-12) = -357 + 12
  • Step 3: Final Answer -357 + 12 = -345

Problem 7: 25 × (-19) + 7

For the seventh problem, let's tackle 25 × (-19) + 7. Here, we have multiplication and addition. Following the order of operations, we multiply first. Multiplying 25 by -19, we get -475. Now, our equation looks like -475 + 7. Adding 7 to -475 gives us -468. So, the solution to 25 × (-19) + 7 is -468. This problem reinforces the importance of the order of operations and the rules for multiplying positive and negative numbers. Ensuring a firm grasp on these concepts is key to succeeding in arithmetic and beyond.

  • Step 1: Multiplication 25 × (-19) = -475
  • Step 2: Addition -475 + 7
  • Step 3: Final Answer -475 + 7 = -468

Problem 8: 28 + (-29) × 9

Now, let's solve the eighth problem: 28 + (-29) × 9. This problem combines addition and multiplication. Following the order of operations, we must perform the multiplication first. Multiplying -29 by 9, we obtain -261. Our equation now becomes 28 + (-261). Adding a negative number is the same as subtracting its positive counterpart, so 28 + (-261) becomes 28 - 261. Subtracting 261 from 28 yields -233. Therefore, the solution to the equation 28 + (-29) × 9 is -233. This problem reiterates the importance of adhering to the correct order of operations and effectively managing negative numbers to avoid errors. Consistent practice with these principles is essential for mathematical proficiency.

  • Step 1: Multiplication (-29) × 9 = -261
  • Step 2: Addition 28 + (-261) = 28 - 261
  • Step 3: Final Answer 28 - 261 = -233

Problem 9: -44 ÷ 4 + (-39)

In the ninth problem, let's solve -44 ÷ 4 + (-39). This equation involves division and addition. According to the order of operations, division comes before addition. So, we divide -44 by 4, which gives us -11. Now, our equation looks like this: -11 + (-39). Adding a negative number is the same as subtracting, so -11 + (-39) becomes -11 - 39. Performing this subtraction, we get -50. Therefore, the solution to -44 ÷ 4 + (-39) is -50. This problem underscores the necessity of following the correct order of operations and accurately handling negative numbers in arithmetic calculations. A clear understanding of these principles is crucial for solving more complex mathematical problems.

  • Step 1: Division -44 ÷ 4 = -11
  • Step 2: Addition -11 + (-39) = -11 - 39
  • Step 3: Final Answer -11 - 39 = -50

Problem 10: 76 + 56 ÷ (-8)

Finally, for our tenth problem, let's tackle the equation 76 + 56 ÷ (-8). This problem features both addition and division. Following the order of operations (PEMDAS/BODMAS), division takes precedence. So, we start by dividing 56 by -8, which results in -7. Our equation now looks like this: 76 + (-7). Adding a negative number is the same as subtracting, so 76 + (-7) becomes 76 - 7. Performing this subtraction, we arrive at 69. Therefore, the solution to the equation 76 + 56 ÷ (-8) is 69. This final problem reinforces the importance of adhering to the order of operations and handling operations with negative numbers accurately. These skills are essential for building a solid foundation in arithmetic and progressing to more advanced mathematical concepts.

  • Step 1: Division 56 ÷ (-8) = -7
  • Step 2: Addition 76 + (-7) = 76 - 7
  • Step 3: Final Answer 76 - 7 = 69

Conclusion

Alright guys, we've made it through all ten problems! We’ve successfully navigated a range of arithmetic challenges, emphasizing the crucial role of the order of operations (PEMDAS/BODMAS) and the rules for working with negative numbers. Each problem provided a valuable opportunity to reinforce these fundamental concepts. Remember, the key to mastering math is consistent practice and a solid understanding of the basics. Keep practicing, and you'll be tackling even more complex problems in no time! You've got this!