Calculating The Value Of The Expression 48 ÷ 6 + (5 × 67) × 13 + 6

Jul 10, 2025 by ADMIN 67 views

Calculate the Value of the Expression: 48 ÷ 6 + (5 × 67) × 13 + 6

In the realm of mathematics, evaluating expressions accurately is a fundamental skill. This article delves into the step-by-step process of calculating the value of the expression 48 ÷ 6 + (5 × 67) × 13 + 6. We'll break down the expression, applying the order of operations (PEMDAS/BODMAS) to arrive at the correct solution. Understanding the order of operations is crucial for solving mathematical expressions efficiently and correctly.

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we dive into the calculation, let's briefly review the order of operations, often remembered by the acronyms PEMDAS or BODMAS:

Parentheses / Brackets
Exponents / Orders
Multiplication and Division (from left to right)
Addition and Subtraction (from left to right)

This order dictates the sequence in which mathematical operations should be performed to ensure a consistent and accurate result. Failing to adhere to this order can lead to incorrect answers.

Step-by-Step Calculation

Now, let's apply the order of operations to the expression 48 ÷ 6 + (5 × 67) × 13 + 6:

1. Parentheses/Brackets

We begin by evaluating the expression within the parentheses:

5 × 67 = 335

So, the expression now becomes:

48 ÷ 6 + 335 × 13 + 6

2. Multiplication and Division (from left to right)

Next, we perform multiplication and division from left to right. First, we have the division:

48 ÷ 6 = 8

Then, we perform the multiplication:

335 × 13 = 4355

Now, the expression looks like this:

8 + 4355 + 6

3. Addition and Subtraction (from left to right)

Finally, we perform addition and subtraction from left to right:

8 + 4355 = 4363

4363 + 6 = 4369

Therefore, the value of the expression 48 ÷ 6 + (5 × 67) × 13 + 6 is 4369.

Detailed Breakdown of Each Step

To further solidify your understanding, let's delve into each step with more detail:

Step 1: Parentheses First (5 × 67)

The initial step in solving this mathematical expression involves tackling the operation enclosed within the parentheses. According to the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), the content within parentheses must be addressed before any other operation. In our expression, 48 ÷ 6 + (5 × 67) × 13 + 6, the parentheses contain the multiplication operation 5 × 67. Performing this calculation is crucial as it sets the stage for the subsequent steps. By focusing on this enclosed operation first, we ensure that the rest of the expression is evaluated using the correct intermediate result. The correct multiplication of 5 and 67 gives us 335. This number will then be used in the following multiplication operation as we continue to unravel the expression. Ignoring the parentheses at this stage would lead to a completely different and incorrect answer. Therefore, adhering strictly to the order of operations, we prioritize the multiplication within the parentheses, laying the groundwork for the accurate evaluation of the entire expression.

Step 2: Division and Multiplication (48 ÷ 6 and 335 × 13)

After resolving the parentheses, the next crucial phase in calculating the value of the expression 48 ÷ 6 + (5 × 67) × 13 + 6 involves handling division and multiplication. The order of operations dictates that these two operations should be performed from left to right. This means we first tackle the division operation: 48 ÷ 6. Dividing 48 by 6 yields 8, a critical intermediate result. Following this, we proceed to the next multiplication operation: 335 × 13. Multiplying 335 by 13 results in 4355. It's imperative to note that division and multiplication hold equal precedence in the order of operations, and thus, their sequence is determined by their position from left to right within the expression. If we were to deviate from this left-to-right approach, we would undoubtedly arrive at an erroneous final answer. Accurately performing these division and multiplication steps is not just about getting the numbers right; it's about respecting the fundamental rules of mathematical precedence. These intermediate results, 8 and 4355, will then feed into the final stage of the calculation, where addition will lead us to the ultimate solution.

Step 3: Addition (8 + 4355 + 6)

The concluding step in calculating the value of the mathematical expression 48 ÷ 6 + (5 × 67) × 13 + 6 is the addition phase. Having addressed the parentheses, division, and multiplication, we now turn our attention to the remaining operations, which are additions. The expression has been simplified to 8 + 4355 + 6. According to the order of operations, addition and subtraction are performed from left to right, similar to multiplication and division. In this case, we have only addition operations, so we proceed sequentially. First, we add 8 and 4355, which results in 4363. This intermediate sum then becomes the first addend in the next addition operation. We then add 6 to 4363. This final addition is critical, as it combines all the previous results into the ultimate solution. Correctly performing this addition is the last piece of the puzzle, ensuring that the entire expression is accurately evaluated. The sum of 4363 and 6 is 4369, which is the final value of the original expression. This carefully executed addition, following all prior steps, provides the definitive answer to our mathematical problem.

Common Mistakes to Avoid

When evaluating mathematical expressions, it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

Ignoring the Order of Operations: This is the most frequent mistake. Always follow PEMDAS/BODMAS to ensure accuracy.
Incorrectly Performing Arithmetic Operations: Double-check your calculations, especially during multiplication and division.
Skipping Steps: Break down the expression into smaller steps to minimize errors.
Misinterpreting Parentheses: Pay close attention to parentheses, as they indicate the order of operations.

By avoiding these common mistakes, you can improve your accuracy and confidence in solving mathematical expressions.

Alternative Methods for Calculation

While following the order of operations is the standard approach, there might be alternative methods or strategies you could employ, although they are less relevant for this specific expression. One such method involves using a calculator or computational tool, which automates the process and reduces the risk of manual calculation errors. However, it's still crucial to input the expression correctly and understand the order of operations to interpret the result. Another strategy involves rearranging the terms using the commutative and associative properties of addition and multiplication, but this is more applicable when dealing with longer expressions or when simplifying algebraic equations. For instance, if the expression had multiple addition operations, you could group numbers that are easier to add together. However, for our expression 48 ÷ 6 + (5 × 67) × 13 + 6, the standard order of operations approach is the most straightforward and efficient way to arrive at the correct answer. Alternative methods are generally more valuable in more complex scenarios, whereas the step-by-step breakdown following PEMDAS/BODMAS provides clarity and accuracy in this case.

Conclusion

In conclusion, evaluating the expression 48 ÷ 6 + (5 × 67) × 13 + 6 requires a systematic approach that adheres to the order of operations. By following the PEMDAS/BODMAS rule, we first addressed the parentheses, then performed multiplication and division from left to right, and finally, carried out addition. This step-by-step process led us to the correct answer of 4369. Understanding and applying the order of operations is essential for solving mathematical expressions accurately and efficiently. Remember to break down complex expressions into smaller, manageable steps and double-check your calculations to avoid common mistakes. Whether you're a student learning the basics or someone looking to brush up on their math skills, mastering the order of operations is a crucial skill for success in mathematics and related fields. Practice and careful attention to detail will help you confidently tackle a wide range of mathematical problems.