Finding Equivalent Expression To 6 + (2 + 3) × 5
Introduction to Order of Operations
In mathematics, the order of operations is paramount to arriving at the correct solution. This fundamental principle dictates the sequence in which mathematical operations should be performed. Often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), this guideline ensures consistency and accuracy in mathematical calculations. Understanding and applying PEMDAS is crucial for anyone delving into mathematical problem-solving. Let's break down the PEMDAS rule to fully grasp its significance:
- Parentheses: Operations enclosed within parentheses (or brackets) are always performed first. This step takes precedence over all other operations, ensuring that expressions within parentheses are simplified before interacting with the rest of the equation.
- Exponents: Exponents, which represent repeated multiplication, are the next priority. Evaluating exponents simplifies the expression and sets the stage for subsequent operations.
- Multiplication and Division: Multiplication and division hold equal importance and are performed from left to right. This means that if both operations are present in an expression, the one appearing earlier from left to right is executed first.
- Addition and Subtraction: Similarly, addition and subtraction share the same level of precedence and are carried out from left to right. Just like multiplication and division, the order in which they appear in the expression dictates the sequence of operations.
Ignoring the order of operations can lead to drastically different and incorrect results. For instance, consider the expression 2 + 3 × 4. If we were to perform addition before multiplication, we might incorrectly calculate 5 × 4 = 20. However, adhering to PEMDAS, we first perform the multiplication: 3 × 4 = 12, followed by addition: 2 + 12 = 14. The correct answer is 14, highlighting the critical role of order of operations.
Decoding the Expression: 6 + (2 + 3) × 5
To accurately solve the expression $6 + (2 + 3) × 5$, we must meticulously follow the order of operations, encapsulated in the acronym PEMDAS. This methodical approach ensures that we arrive at the correct answer by performing each operation in its proper sequence.
Our expression is $6 + (2 + 3) × 5$, which features addition, parentheses, and multiplication. According to PEMDAS, the first step involves addressing any expressions within parentheses. In this case, we have (2 + 3), which simplifies to 5. So, the expression now becomes:
With the parentheses resolved, the next operation in line is multiplication. We see 5 × 5 in our updated expression. Performing this multiplication gives us 25. Our expression is further simplified to:
Finally, we are left with a simple addition operation. Adding 6 and 25 yields the final result:
Therefore, by strictly adhering to the order of operations, we have successfully determined that the value of the expression $6 + (2 + 3) × 5$ is 31. This methodical approach underscores the importance of PEMDAS in ensuring accurate mathematical calculations. Let's now evaluate the given expressions and find the equivalent one.
Evaluating the Expressions: A Step-by-Step Analysis
Now, let's evaluate each of the given expressions to determine which one is equal to our calculated result of 31. We will meticulously apply the order of operations (PEMDAS) to each expression, ensuring accuracy in our calculations. This systematic approach will allow us to confidently identify the expression that matches our target value.
Expression 1: $5 + 4 × (5 - 6)$
Following PEMDAS, we begin with the parentheses. Inside the parentheses, we have 5 - 6, which equals -1. The expression now becomes:
Next, we perform the multiplication: 4 × (-1) = -4. The expression is now:
Finally, we perform the addition: 5 + (-4) = 1. Thus, the value of the first expression is 1, which is not equal to 31.
Expression 2: $1 + 10 × 3$
In this expression, we have addition and multiplication. According to PEMDAS, multiplication takes precedence. So, we first multiply 10 by 3, which gives us 30. The expression becomes:
Now, we perform the addition: 1 + 30 = 31. This expression evaluates to 31, which matches our target value.
Expression 3: $(4 × 5) + 3$
Starting with the parentheses, we have 4 × 5, which equals 20. The expression simplifies to:
Performing the addition, we get 20 + 3 = 23. This expression evaluates to 23, which is not equal to 31.
Expression 4: $9 × 5 + 10$
Following PEMDAS, we perform the multiplication first: 9 × 5 = 45. The expression becomes:
Adding the numbers, we get 45 + 10 = 55. This expression evaluates to 55, which is not equal to 31.
Conclusion: Identifying the Equivalent Expression
After meticulously evaluating each of the given expressions, we have determined that only one expression yields the same result as $6 + (2 + 3) × 5$, which we calculated to be 31. The expressions were:
The only expression that equals 31 is $1 + 10 × 3$. This highlights the importance of the order of operations (PEMDAS) in ensuring accurate mathematical calculations. By following PEMDAS, we can systematically solve complex expressions and arrive at the correct answer. In this case, understanding the order in which to perform multiplication and addition was crucial in identifying the equivalent expression. This exercise reinforces the fundamental principles of arithmetic and their application in problem-solving. Mastering PEMDAS is an essential skill for anyone working with mathematical expressions, and this example serves as a practical illustration of its importance.
Therefore, the expression $1 + 10 × 3$ is the correct answer, demonstrating the power and precision of mathematical operations when applied in the correct order.
