Expressions Equivalent To 6 + (2 + 3) × 5
Introduction
In this article, we will delve into the process of solving mathematical expressions using the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). We aim to identify which of the provided expressions is equivalent to the given expression: 6 + (2 + 3) × 5. Understanding the correct sequence of operations is crucial in mathematics to arrive at the accurate result. This article will break down each expression step by step, ensuring a clear and comprehensive understanding.
Evaluating the Original Expression: 6 + (2 + 3) × 5
To begin, let’s evaluate the original expression, 6 + (2 + 3) × 5, following the order of operations (PEMDAS/BODMAS). This will serve as our benchmark against which we will compare the other expressions. The order of operations dictates that we first address any expressions within parentheses, then exponents, followed by multiplication and division (from left to right), and finally addition and subtraction (from left to right). This methodical approach ensures that we solve the expression accurately.
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Parentheses: Start by solving the operation inside the parentheses: (2 + 3) = 5
Now the expression becomes:
6 + 5 × 5
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Multiplication: Next, perform the multiplication operation:
5 × 5 = 25
The expression is now:
6 + 25
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Addition: Finally, perform the addition:
6 + 25 = 31
Therefore, the value of the original expression 6 + (2 + 3) × 5 is 31. This result is critical as we will compare it with the values of the other expressions to determine which one is equivalent. The step-by-step breakdown helps to minimize errors and ensures clarity in our calculations.
Evaluating the First Expression: 5 + 4 × (5 - 6)
Now, let's evaluate the first expression, 5 + 4 × (5 - 6), following the same order of operations to ensure accuracy. This expression involves parentheses, multiplication, and addition, providing a good example of how PEMDAS guides the solution process. Breaking it down step by step helps to avoid common mistakes and ensures a clear understanding of each operation's role.
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Parentheses: Start by solving the operation inside the parentheses:
(5 - 6) = -1
The expression becomes:
5 + 4 × (-1)
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Multiplication: Next, perform the multiplication operation:
4 × (-1) = -4
The expression is now:
5 + (-4)
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Addition: Finally, perform the addition:
5 + (-4) = 1
Thus, the value of the expression 5 + 4 × (5 - 6) is 1. Comparing this result to our original value of 31, we can conclude that this expression is not equivalent. The step-by-step evaluation highlights the importance of following the correct order of operations to achieve the correct result. Each step is crucial in ensuring the final answer is accurate.
Evaluating the Second Expression: 9 × 5 + 10
Next, we evaluate the second expression, 9 × 5 + 10, by adhering to the order of operations. This expression includes both multiplication and addition, providing a straightforward example of how PEMDAS dictates the sequence of calculations. By breaking down the expression into manageable steps, we can clearly see how each operation contributes to the final result, ensuring accuracy and comprehension.
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Multiplication: First, perform the multiplication operation:
9 × 5 = 45
The expression becomes:
45 + 10
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Addition: Now, perform the addition:
45 + 10 = 55
The value of the expression 9 × 5 + 10 is 55. Comparing this to our original result of 31, it is clear that this expression is not equivalent. This step-by-step evaluation underscores the importance of adhering to the order of operations to prevent errors and arrive at the correct solution. Each calculation is performed deliberately, ensuring clarity and precision in the final answer.
Evaluating the Third Expression: 1 + 10 × 3
Now, let's evaluate the third expression, 1 + 10 × 3, using the order of operations to ensure we arrive at the correct value. This expression includes both addition and multiplication, offering a practical example of how PEMDAS guides us through the solution. Breaking down the expression step by step makes it easier to understand each operation’s impact on the final result.
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Multiplication: Begin by performing the multiplication operation:
10 × 3 = 30
The expression becomes:
1 + 30
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Addition: Next, perform the addition operation:
1 + 30 = 31
The value of the expression 1 + 10 × 3 is 31. When we compare this result to the value of the original expression, which is also 31, we can see that this expression is indeed equivalent. This step-by-step evaluation demonstrates how following the order of operations leads to the correct result. Each step is clearly defined, ensuring accuracy and ease of understanding.
Evaluating the Fourth Expression: (4 × 5) + 3
Finally, we will evaluate the fourth expression, (4 × 5) + 3, by strictly following the order of operations. This expression involves parentheses, multiplication, and addition, providing a comprehensive example of how PEMDAS helps in solving mathematical expressions. Breaking down the expression into steps allows us to clearly see how each operation contributes to the final result, ensuring both accuracy and understanding.
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Parentheses: Start by solving the operation inside the parentheses:
(4 × 5) = 20
The expression becomes:
20 + 3
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Addition: Now, perform the addition operation:
20 + 3 = 23
Thus, the value of the expression (4 × 5) + 3 is 23. Comparing this result to our original value of 31, it is evident that this expression is not equivalent. The step-by-step evaluation highlights the importance of adhering to the order of operations to avoid errors and ensure the correct solution. Each calculation is meticulously performed, promoting clarity and precision in the final answer.
Conclusion
In conclusion, after evaluating all the given expressions, we have determined that the expression 1 + 10 × 3 is the only one equivalent to the original expression 6 + (2 + 3) × 5. Both expressions evaluate to 31. This exercise underscores the critical importance of adhering to the order of operations (PEMDAS/BODMAS) in mathematics. Consistent application of this order ensures accurate calculations and reliable results. By breaking down each expression step by step, we can clearly see how the order of operations influences the final outcome.
The step-by-step approach not only helps in solving individual expressions but also enhances our overall understanding of mathematical principles. This knowledge is invaluable in various fields, from basic arithmetic to more complex mathematical problems. Mastering the order of operations is a fundamental skill that builds a strong foundation for further mathematical studies and applications. The ability to accurately evaluate expressions is essential for problem-solving and critical thinking in mathematics and beyond.
