Applying GMDAS Rule In Solving Mathematical Operations
The GMDAS rule, an acronym for Grouping, Multiplication, Division, Addition, and Subtraction, is a fundamental principle in mathematics that dictates the order in which operations must be performed in an expression. Understanding and applying this rule correctly is crucial for obtaining accurate results in mathematical calculations. In this article, we will delve into the application of the GMDAS rule by solving a series of mathematical problems, illustrating each step in detail.
Understanding the GMDAS Rule
The GMDAS rule provides a hierarchy for performing mathematical operations, ensuring consistency and accuracy in calculations. The order of operations, as dictated by GMDAS, is as follows:
- Grouping: Operations within parentheses, brackets, or other grouping symbols are performed first. This step takes precedence over all other operations.
- Multiplication and Division: Multiplication and division are performed from left to right. These operations have equal precedence, so the order in which they appear in the expression determines the sequence of execution.
- Addition and Subtraction: Addition and subtraction are performed from left to right. Similar to multiplication and division, these operations have equal precedence, and their order of appearance dictates the sequence of execution.
By adhering to the GMDAS rule, we can systematically solve complex mathematical expressions, ensuring that we arrive at the correct answer. Let's apply this rule to the problems at hand.
Solving Mathematical Problems Using GMDAS Rule
To effectively illustrate the application of the GMDAS rule, let's solve the following mathematical problems step-by-step, explaining each operation performed and the rationale behind it. By working through these examples, you'll gain a solid understanding of how to apply the GMDAS rule in various scenarios.
1. 7 x 2 - (9 + 2) =
In this problem, we encounter a combination of multiplication, subtraction, and grouping. Following the GMDAS rule, we must first address the grouping operation.
- Step 1: Grouping
- (9 + 2) = 11
- We perform the addition operation within the parentheses, resulting in 11.
- Step 2: Multiplication
- 7 x 2 = 14
- Next, we perform the multiplication operation.
- Step 3: Subtraction
- 14 - 11 = 3
- Finally, we perform the subtraction operation.
Therefore, 7 x 2 - (9 + 2) = 3
2. (6 ÷ 3) x 11 - 4 =
This problem involves division, multiplication, subtraction, and grouping. Again, we start with the grouping operation as per the GMDAS rule.
- Step 1: Grouping
- (6 ÷ 3) = 2
- We perform the division operation within the parentheses.
- Step 2: Multiplication
- 2 x 11 = 22
- Next, we perform the multiplication operation.
- Step 3: Subtraction
- 22 - 4 = 18
- Finally, we perform the subtraction operation.
Therefore, (6 ÷ 3) x 11 - 4 = 18
3. 9 x 3 + (20 - 18) =
This problem includes multiplication, addition, subtraction, and grouping. Following the GMDAS rule, we begin with the grouping operation.
- Step 1: Grouping
- (20 - 18) = 2
- We perform the subtraction operation within the parentheses.
- Step 2: Multiplication
- 9 x 3 = 27
- Next, we perform the multiplication operation.
- Step 3: Addition
- 27 + 2 = 29
- Finally, we perform the addition operation.
Therefore, 9 x 3 + (20 - 18) = 29
4. 47 - 17 + 10 x 3 =
In this problem, we have subtraction, addition, and multiplication. According to the GMDAS rule, multiplication takes precedence over addition and subtraction.
- Step 1: Multiplication
- 10 x 3 = 30
- We perform the multiplication operation first.
- Step 2: Subtraction
- 47 - 17 = 30
- Then, we perform the subtraction operation from left to right.
- Step 3: Addition
- 30 + 30 = 60
- Finally, we perform the addition operation.
Therefore, 47 - 17 + 10 x 3 = 60
5. 10 ÷ [9 - (2 x 2)] =
This problem presents a combination of division, subtraction, and nested grouping. The GMDAS rule dictates that we start with the innermost grouping.
- Step 1: Inner Grouping
- (2 x 2) = 4
- We perform the multiplication operation within the inner parentheses.
- Step 2: Outer Grouping
- [9 - 4] = 5
- Next, we perform the subtraction operation within the brackets.
- Step 3: Division
- 10 ÷ 5 = 2
- Finally, we perform the division operation.
Therefore, 10 ÷ [9 - (2 x 2)] = 2
6. 3 + 6 x 5 + 4 =
This problem involves addition and multiplication. As per the GMDAS rule, multiplication takes precedence.
- Step 1: Multiplication
- 6 x 5 = 30
- We perform the multiplication operation first.
- Step 2: Addition (Left to Right)
- 3 + 30 = 33
- We perform the addition operations from left to right.
- Step 3: Addition
- 33 + 4 = 37
- Finally, we complete the addition.
Therefore, 3 + 6 x 5 + 4 = 37
7. 26 + 11 x 2 ÷ 9 =
This problem includes addition, multiplication, and division. According to the GMDAS rule, multiplication and division are performed from left to right before addition.
- Step 1: Multiplication
- 11 x 2 = 22
- We perform the multiplication operation first.
- Step 2: Division
- 22 ÷ 9 = 2.44 (approximately)
- Next, we perform the division operation.
- Step 3: Addition
- 26 + 2.44 = 28.44 (approximately)
- Finally, we perform the addition operation.
Therefore, 26 + 11 x 2 ÷ 9 ≈ 28.44
8. 100 - 16 ÷ 12 - 8 =
This problem involves subtraction and division. As per the GMDAS rule, division takes precedence.
- Step 1: Division
- 16 ÷ 12 = 1.33 (approximately)
- We perform the division operation first.
- Step 2: Subtraction (Left to Right)
- 100 - 1.33 = 98.67 (approximately)
- We perform the subtraction operations from left to right.
- Step 3: Subtraction
- 98.67 - 8 = 90.67 (approximately)
- Finally, we complete the subtraction.
Therefore, 100 - 16 ÷ 12 - 8 ≈ 90.67
9. 8 ÷ 4 x (5 + 9) =
This problem includes division, multiplication, and grouping. The GMDAS rule dictates that we start with the grouping operation.
- Step 1: Grouping
- (5 + 9) = 14
- We perform the addition operation within the parentheses.
- Step 2: Division
- 8 ÷ 4 = 2
- Next, we perform the division operation.
- Step 3: Multiplication
- 2 x 14 = 28
- Finally, we perform the multiplication operation.
Therefore, 8 ÷ 4 x (5 + 9) = 28
10. 81 ÷ ?
This question is incomplete, lacking the full expression required to apply the GMDAS rule. To solve this, we need the complete mathematical statement. However, if the intention was to simply divide 81 by an implied number, further clarification is needed to provide an accurate solution. For instance, if the question was meant to be 81 ÷ 9, the answer would be 9. Without the complete expression, we cannot apply the GMDAS rule effectively.
Importance of GMDAS Rule
The GMDAS rule is not just a mathematical convention; it is a cornerstone of accurate mathematical calculations. Its importance stems from the following:
- Consistency: The GMDAS rule ensures that everyone arrives at the same answer when solving the same mathematical expression. Without a standardized order of operations, calculations would be subjective, leading to confusion and errors.
- Clarity: By adhering to the GMDAS rule, we can eliminate ambiguity in mathematical expressions. The rule provides a clear roadmap for solving problems, minimizing the potential for misinterpretations.
- Accuracy: The GMDAS rule guarantees the correct sequence of operations, which is crucial for obtaining accurate results. Inaccurate calculations can have significant consequences in various fields, such as engineering, finance, and science.
In conclusion, the GMDAS rule is an indispensable tool for anyone working with mathematical expressions. By mastering this rule, you can confidently solve complex problems and ensure the accuracy of your calculations. Remember to always prioritize grouping, followed by multiplication and division (from left to right), and finally, addition and subtraction (from left to right).
Conclusion
Applying the GMDAS rule is essential for accurately solving mathematical expressions involving multiple operations. By following the correct order of operations – Grouping, Multiplication and Division (from left to right), Addition and Subtraction (from left to right) – we can ensure consistency and precision in our calculations. The examples provided in this article demonstrate the practical application of the GMDAS rule, offering a step-by-step guide to solving various types of mathematical problems. Mastering this rule is a fundamental skill in mathematics, enabling us to tackle more complex equations and problems with confidence. Remember, the key to success in mathematics lies not only in understanding the concepts but also in applying them correctly, and the GMDAS rule is a critical component of this application. Whether you are a student learning the basics or a professional working with advanced calculations, the GMDAS rule is a reliable framework for achieving accurate and consistent results. Keep practicing and applying this rule, and you will find yourself becoming more proficient and confident in your mathematical abilities.
