Understanding and Solving Mathematical Expressions: The Case of 6÷2(12)

Mathematical expressions often appear straightforward, yet they can sometimes lead to controversy and confusion. The expression 6÷2(12) is a prime example of this. This article aims to clarify the process of solving such expressions using the correct order of operations, which are well-established conventions in mathematics. We will explore the steps taken using both BODMAS and PEMDAS—two popular acronyms used to denote the order of operations.

The BODMAS and PEMDAS Rules

Before diving into the specific example, it's essential to understand the principles behind BODMAS (Brackets, Orders, Division/Multiplication, Addition, Subtraction) and PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition, Subtraction). Both acronyms serve the same purpose: to establish a consistent order for solving complex expressions. The process is as follows:

Brackets/Parentheses: Evaluate expressions within brackets or parentheses first. Exponents: Solve any exponents or orders next. Multiplication and Division: Perform multiplication and division from left to right. Addition and Subtraction: Finally, perform addition and subtraction from left to right.

Let's see how these rules apply to the expression 6÷2(12).

Solving 6÷2(12): BODMAS and PEMDAS in Action

Consider the expression 6÷2(12).

Using BODMAS

Brackets/Parentheses: There are no specific parentheses in this expression. However, if we interpret 2(12) as a multiplication, it simplifies to 2 × 12, which equals 24. But since there's no direct bracket, we proceed to the next step. Multiplication and Division: We then perform the division from left to right. The expression can be rewritten as 6÷2 3, and then 3 × 24 72. However, if we consider the expression directly as 6÷2(12), it's ambiguous in standard notation and leads to different interpretations.

A more straightforward approach using BODMAS suggests solving it as follows:

First, we solve the bracket: 2(12) 24. Then, we perform the division: 6÷24 0.25. Finally, we multiply: 0.25 × 24 6.

However, this is not a correct interpretation according to the standard order of operations.

Using PEMDAS

Parentheses: There are no explicit parentheses in the expression 6÷2(12). However, we can treat 2(12) as a multiplication, which equals 24. The expression becomes 6÷24. Multiplication and Division: We perform the division from left to right: 6÷2 3, then 3 × 24 72. Again, this is not correct for the original expression.

A more accurate approach using PEMDAS would be:

First, solve the bracket: 2(12) 24. Then, perform the division: 6÷24 0.25. Finally, multiply: 0.25 × 24 6.

Despite these steps, the most standard and accepted interpretation is:

First, evaluate the bracket: 2(12) 24. Second, perform the division: 6÷2 3. Third, multiply: 3 × 24 72.

Therefore, the correct solution following standard rules (BODMAS/PEMDAS) is:

This confirms that the answer to 6÷2(12) is 9.

Conclusion

Mathematical expressions like 6÷2(12) can be ambiguous and lead to different interpretations based on the order of operations. Using the BODMAS and PEMDAS conventions, we have shown that the correct answer is 9. It's crucial to understand and apply these rules consistently to avoid confusion in mathematical problems. The flexibility in interpretation highlights the importance of clear notation and communication in mathematical expressions to prevent misunderstandings and ensure accurate solutions.

By adhering to the order of operations, we can solve such expressions systematically and with confidence. This article aims to provide clarity and a consistent approach for solving similar mathematical problems.